Redshift z

coldcreation · March 30, 2011

Since 'global curvature' is not mathematically addressed in either of Fischer's papers, my first query still remains on the table: what is the mathematical justification for a redshift caused by a non-zero 'global curvature'?

Nonsense. The entire paper Homogeneous cosmological solutions of the Einstein equation (Ernst Fischer, 2009), treats mathematically the topic of global curvature.

...attention is now focused on Fisher. His model also fails because of that aforementioned irresolvable problem dealt with in a previous post.

The "irresolvable problem" in your thought experiment, was dealt with and resolved above (Post #748).

However there is more criticism that can be directed at Fischer. In his paper "Homogeneous cosmological solutions of the Einstein equation" he displays a perspective of GR that is somewhat naive; his 'grasp' of the subject is questionable.

It's always good to question someone's perspective, although here we disagree. I was under the impression that Ernst Fischer's grasp or GR was rather robust.

In looking at your criticism so far, it seems as if you have not yet grasped some of the essentials of general relativity: the covariance of the basic equations under coordinate transformations and the tensorial properties of their ingredients.

Below, in quotes, you will find Ernst Fischer's response (via personal communication) to your queries.

He fails to mention that if the g₀₀ is only a function of distance then a time coordinate can be chosen to ensure that g₀₀ = -1 , thus simplifying the equations considerably without any loss of generality because the original case, where g₀₀ is a function of distance, can be determined by transforming back to the original time coordinate.

"Covariant transformation of the field equations from a g₀₀, which depends on distance, to g₀₀ = -1 leads to an additional term in the field equations. If such transformations were possible without further changes, one could as well eliminate the time dependence from expanding space solutions by a simple rescaling of the radial coordinate. Allowing dependence of the time scale on distance adds an additional degree of freedom to possible solutions of the field equations. The motivation of this addition is to maintain Lorentz ivariance not only locally but also in extended systems (more on this later)."

At the start of his section "4 Properties of static solutions" he makes the statement: [...]
without presenting any justification or reference for it. This formula has the same structure as the formula that relates the distance across the surface of a sphere, between two points on the sphere, to the length of the perpendicular dropped from one of the points to the radius connecting the other point to the centre of the sphere. Thus Fischer's use of x_i herein is as a Schwarzschild radial coordinate r ( 2(pi)r is the "proper circumference" of a circle whose points have a radial coordinate r in a Schwarzschid spherical coordinate system centered on the centre of the circle; the actual length of the radius of such a circle in a constant curvature space is a*actan[r /sqrt{a²-r²)] ). There are other local orthonormal coordinate systems for which the proper distance is a different function of the coordinates. Fischer seems to be aware of only the system he employed.

"I think that this formula needs no further justification. It simply states the relation of the arc length between two points with coordinates x=0 and x=xi on the periphery of a circle with radius a. It follows immediately from elementary geometry. What enters into the field equations is the curvature, that means the second derivative at x=0. This is a constant value, related only to the radius of the circle, independent on the choice of the coordinate system, which has been used in the calculations."

In the first paragraph of his section "5 Energy considerations" he has the statement: [...]
which displays a misunderstanding of the physics involved. The pressure can only come from some matter/radiation field because it is only there by virtue of it being a component of the energy momentum tensor. Thus his model necessitates an all pervading matter/radiation field that has a negative pressure. So his model has nothing to contribute to the understanding of this universe.

"Einstein introduced a negative pressure term, the cosmological constant, in his static solution to establish a balance with gravitational attraction, as he considered a solution of the field equations with g₀₀ = -1. With a solution, where the time scale depends on distance such a term is not required. The solution automatically contains a term, which has a similar form as Einstein's cosmological term, but which results not directly from the matter distribution but from the curvature, which, of course, is related to the matter field. The 2/a² term in the field equations can really be interpreted as an “all pervading field with negative pressure”, if we regard curvature as the geometrical equivalent to the gravitational potential and identify the 2/a² term with the tensor of potential energy.

The main problem [...] is to accept a fact, which is not easy to understand by intuition for many people, that time is a locally defined quantity and that the time scale can change with distance. In special relativity we have learned from many observations that the time scale depends on the relative motion of the reference systems. In the case of accelerated systems, be it geodesic motion in expanding space with its continuous acceleration in the direction of motion or in curved space, where acceleration is perpendicular to the direction of motion, we try to go back to an absolute time scale, defining things like global time [or cosmic time] and an age of the universe. If we take Lorentz invariance seriously, we must regard accelerated motion as a sequence of infinitesimal Lorentz transformations with the correponding changes of length and time scale, leading by integration in a homogeneous medium to a change of the time scale proportional to distance." (Ernst Fischer)

cruel2Bkind · March 30, 2011

Sci, the webpage I referred to may well have been Errors in Tired Light Cosmology

As photons are asymmetrical, as noted since they cannot all pass through the crystal lattice of a polarizing substance, their length may be acted on unevenly by gravitation, causing a differential velocity gradient.

Can the concept of asymmetry be meaningfully applied to photons? Can polarization of EMR, either as wave or as photon, be validly interpreted as an asymmetry?

Wikipedia has a reasonable exposition of polarization of electromagnetic radiation:

The diagrams on the last four pages show a representation of the way the magnitude and direction of the electric field change along the propagation path of the electromagnetic radiation. They are not in any way to be interpreted as indicating 'shape'.

The thought experiment argument is not substantiated, nor is the conclusion derived. The situation of A and B is completely symmetric. Therefore the constant k=b1/a1 must be equal to 1 [1]. Your arguments are based on the erroneous proposition k>1. All the further calculations are nonsense.

[1]

I do not dispute that k = 1 but I arrive at the conclusion that k = 1 by logical argument whereas you maintain two contradictory statements at the same time; viz:

mutually stationary observers (within a space of constant global curvature) observe each other to be red-shifted,
because the situation of A and B is completely symmetric, post #739's factor k = b1/a1 = 1, i.e. b1 = a1

(Given the definitions of a1 & b1 as given in post #739, "b1 = a1" is equivalent to the statement that "the duration of the interval, as measured by B, between receiving the first and the second signals from A is the same as the duration of the interval, as measured by A between sending the two signals to B. I.E. neither sees any dilation in the other's time" or to put it another way "neither sees any redshift in the radiation from the other".)

A further point to make according to general relativity, there is no unique method by which vectors at points separated by great distances can be compared in a curved spacetime, i.e., a definition of curved spacetime is the inability to compare vectors at different points. Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift [2]. The interpretation of cosmological redshift z remains open.

[2]

If the proper distance between a pair of observers is increasing then it

is

possible to determine the Doppler component of any redshift they may observe in signals they send to each other. A speeding motorist would be lying if he were to claim that he is only apparently traveling at speed because 'space is expanding' or that the Doppler radar measurement is a result of him being in a deeper gravitational well than the radar transceiver.

Generally, in a curved spacetime manifold, the observed shift in frequency of a photon can be interpreted as a kinematic effect or a gravitational frequency shift (or even both together, superimposed), depending on the choice of coordinates.

If the proper distance to the source of the photon is constant then no choice of components can justify a Doppler interpretation.

In the case of the cosmological gravitational redshift interpretation, observer A and observer B (separated by a large distance) see each other's signal as if emitted at a lower elevation in a gravity field. Events at great distances appear to take longer than in the frame of the observer (clocks run slower when further removed). Observer A detects the phenomenon of time dilation and redshift from signal emitted by observer B. Observer B sees frequencies shifted and time dilation in the signal emitted by A. The relationship between the two stationary observers in entirely symmetric.

Referring to the thought experiment described in post #739, if observers A and B see each other as red-shifted then they also see each others' time intervals as being dilated when compared to appropriately corresponding intervals in their own time frames.

Thus when B receives two temporally separated signals from A, the length of the interval between receiving the signals is, in B's time frame, longer than the interval B perceives A to have experienced between sending the signals. So from B's perspective b1 > a1 (ie. k = b1/a1 > 1).

In exactly the same fashion, because they have a completely symmetric relationship, when A receives two temporally separated signals from B, the length of the interval between receiving the signals is, in A's time frame, longer than the interval A perceives B to have experienced between sending the signals. From A's perspective a2 > b1 (ie. k = a1/b1 > 1). The remainder of the mathematical explanation in post #739 logically follows.

The unavoidable conclusion is that if it is true that k > 1 then the durations of intervals between successive pairs of signals increases in geometric progression, which is easily seen to be untenable. Therefore it is not true that k > 1. In a similar way it can be shown that it is not true that k < 1. I.E. it must be the case that k = 1. But if k = 1 then the duration of the interval in B's time frame, between sending/receiving the signals, is the same size as the duration of the interval in A's time frame between receiving/sending the signals. Therefore neither of them see the other as experiencing time dilation, so neither sees the other as red-shifted.

This is an example of a type of argument called a 'reductio ad absurdum' argument, characterized by a demonstration that an assumption logically implies its own contradiction. Such an assumption can only be false. In the case at hand the assumption was that mutual observations of redshift are possible for mutually stationary observers. So the logical conclusion is that it is impossible for mutually stationary observers to observe each other as red-shifted.

Any shift in frequency can be described as gravitational or Doppler. There is no "fact" about the cause of redshift z. The conclusion chosen is a function of the coordinate system or calculation method. (See here for example).

In your reference

"The kinematic origin of the cosmological redshift" by E.F.Bunn and D.W.Hogg (2009)

the authors state "the most natural interpretation of the redshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts." They are not advocates of any constant-volume-constant-global-curvature-space cosmology; nowhere in their paper is there any claim that the 'proper distance' between earth and a far distant red-shifted galaxy can be regarded as constant. Neither can that hypothesis be consistently incorporated into their exposition.

In another way, curvature manifests itself increasingly with distance, i.e., it becomes more and more apparent, the further one looks (even though curvature is constant).

Thus redshift increases with distance, along with the associated time dilation factor.

In one of the constant-volume hyper-spherical universe models, light is not increasingly redshifted with distance. Thus your 'thus' is invalid.

The deviation from the inverse square law of gravitation was mentioned earlier not to explain the cause of redshift or time delay.[3] These result from the constant Gaussian curvature itself. Such a deviation of the inverse-square force law would explain why the gravitational potential (a la Newton) would not diverge toward infinity in an homogeneous universe, thus opening the route to static solutions. A perturbed gravitational inverse-square law modifies the Poisson equation, which affects the growth of over-dense regions and thus affects the large-scale structures (e.g., superclusters) here claimed to be gravitationally bound systems, without the need of dark matter (CDM) or dark energy (DE) [4]. This is a deviation from Newtonian gravity at large scales.

[3]

Why then did you raise such an irrelevancy in a discussion of the validity of the claim of a cosmological redshift in a constant-volume universe?

What are your references for the statements in the rest of your paragraph.

[4]

Rejection of 'dark energy' (ie. a non-zero Cosmological Constant) by those who favour a constant-volume universe is rather puzzling. The field equations can be derived from the Principle of Least Action using the techniques of the Calculus of Variations. In that technique, if there is a constraint, such as the specification of constancy of that hyper-volume of the region of space-time over which the 'action' is to be minimized, then that constraint is incorporated into the 'action' by way of multiplying it by an unspecified constant (called a 'Lagrangian multiplier' or an 'undetermined multiplier' in the variational calculus) and adding the result to the action before employing the minimization technique. The end result is the version of the field equations with the added term wherein the Cosmological Constant is just the aforementioned Lagrangian multiplier; its incorporation is thus a necessary part of the derivation of the field equations in the situation where the space is constant-volume. This was one of the reasons that mainstream Cosmology was so loathe to involve the Cosmological Constant for most of the last century.

(Wikipedia has reasonable pages on the Calculus of Variations, Euler-Lagrange Equations and Lagrangian multipliers.)

Certainly, such a deviation of the inverse square law of gravitation would affect the propagation of photons. What would manifest itself is nonlinearity. Redshift z and time dilation increase with distance from the reference frame of the observer regardless of the deviation. What changes is that the regime is now nonlinear. There is no longer a one to one relation between distance and redshift [5]. High-z objects (e.g., SNe Ia) will appear further than their standard redshift-distance would indicate, and light curve rises times will appear slower or longer than would in an otherwise linear regime.

[5]

If "(t)here is no longer a one to one relation between distance and redshift" what is the relationship? Is it 'one to many', 'many to one' or 'many to many'. A relationship between two variables can be non-linear and one to one. Such a relationship is called a function.

coldcreation · March 30, 2011

I do not dispute that k = 1 but I arrive at the conclusion that k = 1 by logical argument whereas you maintain two contradictory statements at the same time; viz:
mutually stationary observers (within a space of constant global curvature) observe each other to be red-shifted,

because the situation of A and B is completely symmetric, post #739's factor k = b1/a1 = 1, i.e. b1 = a1
(Given the definitions of a1 & b1 as given in post #739, "b1 = a1" is equivalent to the statement that "the duration of the interval, as measured by B, between receiving the first and the second signals from A is the same as the duration of the interval, as measured by A between sending the two signals to B. I.E. neither sees any dilation in the other's time" or to put it another way "neither sees any redshift in the radiation from the other".)

As long as the spacetime manifold in which observers A and B are located is of constant positive Gaussian curvature, both will observe redshift and time dilation.

If the proper distance between a pair of observers is increasing then it is possible to determine the Doppler component of any redshift they may observe in signals they send to each other. A speeding motorist would be lying if he were to claim that he is only apparently traveling at speed because 'space is expanding' or that the Doppler radar measurement is a result of him being in a deeper gravitational well than the radar transceiver.

This is a fallacy known as irrelevant conclusion: you are diverting attention away from a fact in dispute rather than addressing it directly.

We are not talking about things we know are moving radially (e.g., cars). Galaxies are not seen to move. The interpretation of radial motion comes from redshift, which can be interpreted as either a Doppler effect, expansion or stretching of space, or a curved spacetime effect. That was the point of the link above.

If the proper distance to the source of the photon is constant then no choice of components can justify a Doppler interpretation.

Again, in the case of distant galaxies, judging by redshfit z, there is no way to know for a fact wether the distance to the source is constant or not. It depends on the model.

Referring to the thought experiment described in post #739, if observers A and B see each other as red-shifted then they also see each others' time intervals as being dilated when compared to appropriately corresponding intervals in their own time frames.

Granted.

Thus when B receives two temporally separated signals from A, the length of the interval between receiving the signals is, in B's time frame, longer than the interval B perceives A to have experienced between sending the signals. So from B's perspective b1 > a1 (ie. k = b1/a1 > 1).
In exactly the same fashion, because they have a completely symmetric relationship, when A receives two temporally separated signals from B, the length of the interval between receiving the signals is, in A's time frame, longer than the interval A perceives B to have experienced between sending the signals. From A's perspective a2 > b1 (ie. k = a1/b1 > 1). The remainder of the mathematical explanation in post #739 logically follows.

You are affirming the consequent: drawing a conclusion from a premises that does not support the conclusion.

Since the situation is perfectly symmetric they both observe redshift and time dilation.

The unavoidable conclusion is that if it is true that k > 1 then the durations of intervals between successive pairs of signals increases in geometric progression, which is easily seen to be untenable. Therefore it is not true that k > 1. In a similar way it can be shown that it is not true that k < 1. I.E. it must be the case that k = 1. But if k = 1 then the duration of the interval in B's time frame, between sending/receiving the signals, is the same size as the duration of the interval in A's time frame between receiving/sending the signals. Therefore neither of them see the other as experiencing time dilation, so neither sees the other as red-shifted.

The "unavoidable conclusion" is in fact entirely avoidable, since, again, the situation is symmetric.

In a curved spacetime manifold, where k = 1, the duration of the interval sent by A will be longer than is B's time frame, and visa versa. Both of them see the other as experiencing time dilation (as if immersed at a lower altitude in a gravitational potential well), so both sees the other's signal as red-shifted.

This is an example of a type of argument called a 'reductio ad absurdum' argument, characterized by a demonstration that an assumption logically implies its own contradiction. Such an assumption can only be false. In the case at hand the assumption was that mutual observations of redshift are possible for mutually stationary observers. So the logical conclusion is that it is impossible for mutually stationary observers to observe each other as red-shifted.[/indent]

Nonsense. It is the conclusion of the thought experiment that is absurd, since it is based on an erroneous pemise.

In your reference "The kinematic origin of the cosmological redshift" by E.F.Bunn and D.W.Hogg (2009) the authors state "the most natural interpretation of the redshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts." They are not advocates of any constant-volume-constant-global-curvature-space cosmology; nowhere in their paper is there any claim that the 'proper distance' between earth and a far distant red-shifted galaxy can be regarded as constant. Neither can that hypothesis be consistently incorporated into their exposition.

You missed entirely the point:

"We show that, in any spacetime, an observed frequency shift can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon's path.

[...]our focus is on the question of interpretation: given that a photon does not arrive at the observer conveniently labeled Doppler shift, gravitational shift, or stretching of space...

In general, in any curved spacetime, the observed frequency shift in a photon can be interpreted as either a kinematic effect (a Doppler shift) or as a gravitational shift. The two interpretations arise from different choices of coordinates, or equivalently from imagining different families of observers along the photon's path. [...]

We can construct two such families of observers: one in which each of the shifts is a Doppler shift, and one in which each is a gravitational shift. By reference to these families, we can interpret the observed shift as the accumulation of either many small Doppler shifts or many small gravitational shifts. [...]

The Doppler and gravitational families provide two of many different ways of interpreting the observed shift. In any given situation, we can argue about which (if either) of the two families is natural to consider.[...]

To construct the gravitational family of observers, we demand that each member be at rest relative to her neighbor at the moment the photon passes by, so that there are no Doppler shifts. Initially, it may seem impossible in general to satisfy this condition simultaneously with the condition that the first and last observers be at rest relative to the emitter and absorber, but in fact it is always possible to do so. [...]

One of the key ideas of general relativity is the importance of distinguishing between coordinate-independent and coordinate-dependent statements. Another is the idea that spacetime is always locally indistinguishable from Minkowski spacetime. [...]

The common belief that the cosmological redshift can "only" be explained in terms of the stretching of space is based on confating the properties of a specifc coordinate system with properties of space itself. This is precisely the opposite of the correct frame of mind in which to understand relativity.

(

Source

)

In the context of the nonexpanding Universe, the gravitational interpretation corresponds to a family of stationary observers and hence seems to be the more natural one.

In one of the constant-volume hyper-spherical universe models, light is not increasingly redshifted with distance. Thus your 'thus' is invalid.

Which "one of the constant-volume hyper-spherical universe models" are you referring to? The flat universe with k = 0? If so you conclusion "thus your thus is invalid", is invalid.

Why then did you raise such an irrelevancy in a discussion of the validity of the claim of a cosmological redshift in a constant-volume universe?
What are your references for the statements in the rest of your paragraph.

That was already explained above.

For the rest, see: The Cosmological Woes of Newtonian Gravitation Theory - John D. Norton

CC

cruel2Bkind · April 10, 2011

Because Ernst Fischer's ideas have been quoted in support of an alternative explanation of the cosmological redshift, a close analysis of his article "Homogeneous cosmological solutions of the Einstein equation" published in the peer-reviewed journal "Astrophysics and Space Science" is warranted. This critique concentrates on the sections in the article that deal with the formulation of the field equations for the space-time that he proposes. His explication leaves much to be desired and his calculations are ambiguous and dubious. His proposal is easily shown to be inapplicable to reality.

3 Homogeneous solutions

... The metric tensor of such a homogeneous system will be fully described by relations between the temporal and spatial distances, but will not contain any absolute points of space or time. The spacetime distance element can be written in the form

ds² = f(τ,σ)dτ² + h(τ,σ)dσ²
(6)

where the variable τ measures the temporal distance and σ measures the spatial distance, which can be expressed locally by some orthonormal basis dσ² = dx₁² + dx₂² + dx₃². ...

The symbol "σ" is used in two different ways in equation 6. Since "h(τ,σ)" is a component of the metric tensor, "σ" must be either a function of the coordinates, or a member of a coordinate system. However if equation 6 is to be applicable to a 4D space-time then the "dσ²" therein cannot refer to the square of a single coordinate infinitesimal but must be understood as a shorthand notation, as stated at the end of the sentence. So "σ" is a function of the coordinates of the system employed, whereas "dσ" is given meaning by the relation "dσ² = dx₁² + dx₂² + dx₃²". Nevertheless, as a function of the coordinates, "dσ" already has a well defined meaning as the exact differential:

dσ = σ

_,1

dx

₁

+ σ

_,1

dx

₂

+ σ

_,1

dx

₃

where "σ_,i" means the partial derivative of σ(x₁,x₂,x₃) with respect to the variable x_i. So in this context

dσ² = Σ

_i=1

ⁱ⁼³

(σ

_,i

)²dx

_i

² + 2 Σ

_i!=j

σ

_,i

σ

_,j

dx

_i

dx

_j

,

which is completely at odds with the previous meaning of "dσ²". If the two uses of "σ" were not different then each of the σ_,i would have to equal both 1 and 0 concurrently. Imprecision and ambiguity in the use of symbols is not the hallmark of good mathematics.

4 Properties of static solutions

While solving (3--5) is very laborious in the general case, homogeneous solutions are quite easy to obtain. With no point and no direction in space being preferred, it is sufficient to solve the equations for the neighbourhood of one point and the distance in one direction.[1] In a homogeneously curved space of curvature radius a, the relation between the distance s_i and the coordinate x_i near x_i = 0 in a local orthonormal coordinate system is given by [2]

s_i = aarctan(x_i/sqrt[a² - x_i²)
(10)

The derivative, needed to determine the metric and the Ricci tensor, is

ds_i = a/sqrt[a² - x_i²)dx [3]
(11)
As the functions f and h depend only on σ , this value of the derivative is valid in every direction, yielding

dσ² = (a²/[a² - σ²])(dx₁² + dx₂² + dx₃²)[4] or h(σ) = a²/[a² - σ²]
(12)

Solving "the equations for the neighbourhood of one point and the distance in one direction" does not mean that the other directions in the neighbourhood of that point should be ignored. Determining the correct form of the field equations necessitates consideration of those other directions.
"In a homogeneously curved space of curvature radius a, the relation between the distance s_i and the coordinate x_i near x_i = 0 in a local orthonormal coordinate system" defines the type of coordinate system employed. Different types of orthonormal coordinate systems have different relations between the distance of a point from the origin and the coordinate values of that point. It is easy to transform from the system which Fischer uses to another system where the relation is linear.
Does equation 11 contain a typographic omission or is it another example of the author's imprecision?
If Fischer's previous defining but unnumbered equation for "dσ²" is used in conjunction with this equation 12 then:
dσ² = (a²/[a² - σ²])dσ²

and a rather ridiculous result ensues, viz: σ = 0. The proper equation should be :
ds² = f(σ)dτ² + (a²/[a² - σ²])(dx
₁
² + dx
₂
² + dx
₃
²)

which is in the class of "isotropic coordinates". It can be transformed to isotropic spherical coordinates (y₁,y₂,y₃) = (r,θ,φ), where r is a radial coordinate (not to be confused with the actual radius which is the integral of (a²/[a² - σ²])dr from the orign to a point at r), θ is the azimuthal angle and φ is the equatorial angle. The line-element is then
ds² = f(σ)dτ² + (a²/[a² - σ²])(dy
₁
² + y
₁
²dy
₂
² + y
₁
²
sin
²(y
₂
)dy
₃
²).

In light of Fischer's earlier words, this expression suggests that the answer to the over riding question, of the funtional dependence of σ on the coordinates, that Fischer fails to directly acknowledge, is that σ = r. I.E.:

ds² = f(y
₁
)dτ² + (a²/[a² - y
₁
²])(dy
₁
² + y
₁
²dy
₂
² + y
₁
²
sin
²(y
₂
)dy
₃
²).

The assumption that σ is the isotropic radial coordinate enables the transformation to another system of spherical coordinates where the new radial coordinate is the actual radial distance to the origin. The other two coordinates are still the azimuthal and equatorial angles. However simple inspection of the line-element for this new set of coordinates reveals that the space is not positively curved. Its negative curvature is revealed by the ratio of the circumference to the radius of circles centered on the origin. As the radius increases so to does the ratio; when the radius = a, the ratio is infinite.

This leads to the well known result that the first derivatives of the spatial components of the metric tensor and thus the corresponding Γ^k_ij vanish at σ = 0, as must be expected from the isotropy condition.[1] In the case of the static Einstein universe (f(σ) = -1), (4) leads to R_ij = (2/a²)δ_ij for the spatial components (i, j = 1, 2, 3) and R₀₀ = 0. In the general case, where f depends on σ , the components of the LeviCivita connection, containing the derivatives of f , do not vanish. We have
Γ⁰_i0 = Γ⁰_0i = (1/2)f'/f ,
Γⁱ₀₀ = - (1/2)f'/h ,
Γ⁰₀₀ = Γ^j_i0 = Γ^j_0i = 0 [2]
(13)

Some of the values of Γ^k_ij are zero at the origin because space-time is locally flat and the metric is diagonal but isotropy does not necessitate that all such spatial connection coefficients (components of the LeviCivita connection) are identically zero. Fischer should have included expressions for Γ^a_ab (no summation) and Γ^a_bb as these are not identically equal to zero. His inclusion of 3f'h'/4h² in his expression for the R₀₀ in equation 14 shows awareness of the need to differentiate h, so his exclusion of all spatial connections shows inconsistency.
To obtain these connection coefficients, the aforementioned functional dependence must be known, otherwise the partial derivatives of the metric with respect to the coordinates cannot be stated. It does not suffice to just equate the partial derivatives with the normal derivative of the metric components with respect to σ, because to do so is tantamount to demanding a linear relationship between σ and the coordinates. Such a relationship is not isotropic; it does not have full rotational symmetry. In the current situation presented by Fischer, where that functional dependence is not specified, the only correct course of action is to incorporate the partial derivatives of σ with respect to the coordinates. The result follows (with (1)neither a,b or c being zero, (2)a, b and c are all different, (3)the Einstein summation convention not invoked, i.e. no summation over the same index occurring in both raised and lowered positions):
Γ
⁰
_0a
= +(f'/2f)σ
_,a

Γ
^a
₀₀
= -(f'/2h)σ
_,a

Γ
^a
_aa
= +(h'/2h)σ
_,a

Γ
^a
_ab
= +(h'/2h)σ
_,b

Γ
^a
_bb
= -(h'/2h)σ
_,a

In the neighbourhood of σ = 0 the components of the Ricci tensor (4) are

R₀₀ = 3f'²/4fh - 9f'h'/4fh - 3f"/2h + 3f'h'/4h² , [1]
(14)

R_ii = f'²/4f² - f'h'/4fh - f"/2f + 3h'²/2h² - 2h"/h (i = 1, 2, 3) [2]
(15)

Within Fischer's own mistaken scheme, this equation contains two errors: (1) the term 9f'h'/4fh should not be present, (2) the numeric denominator of 3f'h'/4h² should be 2 and not 4. Using only the non-zero connection coefficients given in equation 13, his equation for R₀₀ should have been:
R
₀₀
= 3f'²/4fh - 3f"/2h + 3f'h'/2h² .

Unless he did actually not disregard the non-vanishing spatial connection coefficients: in which case, using also:
Γ
^a
_aa
= Γ
^a
_ab
= - Γ
^a
_bb
= (h'/2h) ,

his equation should have been:
R
₀₀
= 3f'²/4fh - 3f"/2h - 3f'h'/4h² .

These equations can be obtained by using the general expression for R₀₀:

R
₀₀
= Γ
ⁱ
_00,i
-Γ
ⁱ
_0i,0
+ Γ
ⁱ
₀₀
Γ
^m
_mi
- Γ
^m
_n0
Γ
ⁿ
_m0
,

expanding out all summation terms, eliminating all terms containing connection coefficients that are identically zero, eliminating all terms that involve differentiation with respect to the time coordinate and substituting in the values given for the connection coefficients.

Using the same procedure with the above expressions for the connection coefficients that do contain the partial derivatives of σ, yields:
R
₀₀
= [(f'²)/(4fh)-(f"/2h)-(f'h')/(4h²)][(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] - [f'/2h][σ
_,1,1
+σ
_,2,2
+σ
_,3,3
]

Comparison with Fischer's version reveals that he does indeed regard the aforementioned functional dependence as linear.
The inclusion of the terms f'h'/4fh , 3h'²/2h² and 2h"/h shows that Fischer did actually use the Γ^a_aa , Γ^a_ab and Γ^a_bb .
For the case involving the σ_,a the expressions for the spatial components of the Ricci tensor are:
R
₁₁
=

[h'²/4h²][4(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] -

[h"/2h][2(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] +

[f'h'/4fh][(σ
_,1
)²-(σ
_,2
)²-(σ
_,3
)²] +

[(f'²/4f²)-(f"/2f)](σ
_,1
)² - [(h'/2h)+(f'/2f)][σ
_,1,1
]-

[h'/2h][σ
_,1,1
+σ
_,2,2
+σ
_,3,3
]

R₂₂ and R₃₃ can be obtained by cyclically permuting the indexes of R₁₁.
This equation is included as further evidence that Fisher regards the functional dependence of σ on the coordinates as linear. Substitution of the value 1 for each σ_,a and the value 0 for each σ_,a,a yields Fishers expression for R_ii.

All the offdiagonal elements are zero. Of course, at σ = 0, the spatial derivative h' is zero also in this case. The components of the Riccitensor thus are reduced to

R₀₀ = 3f'²/4fh - 3f"/2h ,
(16)

R_ii = - f"/2f + f'²/4f² - 2h"/h (i = 1, 2, 3)
(17)

and the Ricciscalar to

R = (1/f)R₀₀ + (3/h)R_ii =- 3f"/2fh + 3f'²/2hf² - 6h"/h²
(18)

Premature simplification of equations is always bad form, and especially so in the case of determining non-linear differential equations.

Using the correction to Fischer's R₀₀ and his unsimplified R₁₁

R

₀₀

= 3f'²/4fh - 3f"/2h - 3f'h'/4h²

R

_ii

= f'²/4f² - f'h'/4fh - f"/2f + 3h'²/2h² - 2h"/h ,

the full expression for the curvature scalar is:

R = (1/f)R

₀₀

+ (3/h)R

_ii

R = (1/f)[3f'²/4fh - 3f"/2h - 3f'h'/4h²] + (3/h)[f'²/4f² - f'h'/4fh - f"/2f + 3h'²/2h² - 2h"/h]

R = 3f'²/2f²h - 3f"/fh - 3f'h'/2fh² + 9h'²/2hh² - 6h"/h²

With the condition h(0) = 1, f (0) =-1 and h"(0) = 2/a² , the field equations (8) are simplified to
G₀₀ =R₀₀ - (f/2)R = 6/a² = κε ,
(19)

G_ii =R_ii - (h/2)R = -2Y(σ) - 2/a² = κp
(20)

with
Y(σ) = f'²/4f² - f"/2f
(21)

The full calculation for the 00 component of the gravitational tensor gives:

G

₀₀

= R

₀₀

- (f/2)R

= 3f'²/4fh - 3f"/2h - 3f'h'/4h² - (f/2)[3f'²/2f²h - 3f"/fh - 3f'h'/2fh² + 9h'²/2hh² - 6h"/h²]

= 3fh"/hh - 9fh'²/4hh²

= (3f/4h)[(4h"/h)-3(h'/h)²]

The expression for h can now be used to arrive at an easily analyzed equation. There is no need to indulge in any naive simplifications relating to the values of f, h. h' or h" at the origin.

h = a²/[a² - σ²],

h' = 2a²σ/[(a² - σ²)²],

h" = [2a²/(a² - σ²)](a² + 3σ²)/[(a² - σ²)²]

(h'/h)² = 4σ²/[(a² - σ²)²],

h"/h = 2(a² + 3σ²)/[(a² - σ²)²]

It is now easily shown that

G

₀₀

= (3f/a²)(2a²+3σ²)/(a²-σ²) = κε

However f is negative and (1/a²)(2a²+3σ²)/(a²-σ²) must be positive. Therefore G₀₀ must be negative!

I.E. the energy density ε of matter/radiation must be negative in such a space-time.

Thus Fischer's proposal contributes nothing to the determination of the structure of this universe.

One is left wondering about the standard of the peer review process at "Astrophysics and Space Science".

coldcreation · April 13, 2011

.

Posted below is Ernst Fischer's response to cruel2Bkind's critique of

Homogeneous cosmological solutions of the Einstein equation

Astrophys Space Sci (2010) 325: 69–74, DOI 10.1007/s10509-009-0159-7 (SpringerLink):

Because Ernst Fischer's ideas have been quoted in support of an alternative explanation of the cosmological redshift, a close analysis of his article "Homogeneous cosmological solutions of the Einstein equation" published in the peer-reviewed journal "Astrophysics and Space Science" is warranted. This critique concentrates on the sections in the article that deal with the formulation of the field equations for the space-time that he proposes. His explication leaves much to be desired and his calculations are ambiguous and dubious. His proposal is easily shown to be inapplicable to reality.

"First I have to thank Cruel2Bkind for his critical and detailed review of my paper. It really contains a few formal inconsistencies, which I had overlooked. But these formal errors do not change the physical content and its application to realistic models of the universe. I do not see where the calculations are ambiguous or dubious and Cruel2Bkind was apparently able to follow and came to the same results (apart from the mentioned errors)."

The symbol "σ" is used in two different ways in equation 6. Since "h(τ,σ)" is a component of the metric tensor, "σ" must be either a function of the coordinates, or a member of a coordinate system. However if equation 6 is to be applicable to a 4D space-time then the "dσ²" therein cannot refer to the square of a single coordinate infinitesimal but must be understood as a shorthand notation, as stated at the end of the sentence. So "σ" is a function of the coordinates of the system employed, whereas "dσ" is given meaning by the relation "dσ² = dx₁² + dx₂² + dx₃²". Nevertheless, as a function of the coordinates, "dσ" already has a well defined meaning as the exact differential:
dσ = σ
_,1
dx
₁
+ σ
_,1
dx
₂
+ σ
_,1
dx
₃
where "σ_,i" means the partial derivative of σ(x₁,x₂,x₃) with respect to the variable x_i. So in this context
dσ² = Σ
_i=1
ⁱ⁼³
(σ
_,i
)²dx
_i
² + 2 Σ
_i!=j
σ
_,i
σ
_,j
dx
_i
dx
_j
,
which is completely at odds with the previous meaning of "dσ²". If the two uses of "σ" were not different then each of the σ_,i would have to equal both 1 and 0 concurrently. Imprecision and ambiguity in the use of symbols is not the hallmark of good mathematics.

"In homogeneous space one can always find an orthonormal basis so that locally the non-diagonal terms of the metric tensor vanish. In this case the spatial structure can be expressed by some local functional, which relates some point to the neigbouring point at distance dσ. "dσ²" is used as a shorthand for the square of this infinitesimal distance element. One should keep in mind that differential geometry is a strictly local description and that the elements of the metric tensor are locally defined functionals and not functions, which can be extended arbitrarily into space. The use of "σ" in the definition in the elements of the metric tensor is only to express the local functionality."

Solving "the equations for the neighbourhood of one point and the distance in one direction" does not mean that the other directions in the neighbourhood of that point should be ignored. Determining the correct form of the field equations necessitates consideration of those other directions.

"In a homogeneously curved space of curvature radius a, the relation between the distance s_i and the coordinate x_i near x_i = 0 in a local orthonormal coordinate system" defines the type of coordinate system employed. Different types of orthonormal coordinate systems have different relations between the distance of a point from the origin and the coordinate values of that point. It is easy to transform from the system which Fischer uses to another system where the relation is linear.

Does equation 11 contain a typographic omission or is it another example of the author's imprecision?

If Fischer's previous defining but unnumbered equation for "dσ²" is used in conjunction with this equation 12 then:
dσ² = (a²/[a² - σ²])dσ²

and a rather ridiculous result ensues, viz: σ = 0. The proper equation should be :
ds² = f(σ)dτ² + (a²/[a² - σ²])(dx
₁
² + dx
₂
² + dx
₃
²)

which is in the class of "isotropic coordinates". It can be transformed to isotropic spherical coordinates (y₁,y₂,y₃) = (r,θ,φ), where r is a radial coordinate (not to be confused with the actual radius which is the integral of (a²/[a² - σ²])dr from the orign to a point at r), θ is the azimuthal angle and φ is the equatorial angle. The line-element is then
ds² = f(σ)dτ² + (a²/[a² - σ²])(dy
₁
² + y
₁
²dy
₂
² + y
₁
²
sin
²(y
₂
)dy
₃
²).

In light of Fischer's earlier words, this expression suggests that the answer to the over riding question, of the funtional dependence of σ on the coordinates, that Fischer fails to directly acknowledge, is that σ = r. I.E.:

ds² = f(y
₁
)dτ² + (a²/[a² - y
₁
²])(dy
₁
² + y
₁
²dy
₂
² + y
₁
²
sin
²(y
₂
)dy
₃
²).

The assumption that σ is the isotropic radial coordinate enables the transformation to another system of spherical coordinates where the new radial coordinate is the actual radial distance to the origin. The other two coordinates are still the azimuthal and equatorial angles. However simple inspection of the line-element for this new set of coordinates reveals that the space is not positively curved. Its negative curvature is revealed by the ratio of the circumference to the radius of circles centered on the origin. As the radius increases so to does the ratio; when the radius = a, the ratio is infinite.

"Cruel2Bkind is right that eq.(11) contains a typing error. The last term shoud not be “x” but “x_i”. There is no need to regard other coordinate systems to determine the functional dependence of h(σ) and its derivatives, as we are interested only on the local values of the functional. They depend only on the radius of curvature and are equal in any direction. The quantity σ has nothing to do with the coordinate “r” in a radially symmetric geometry, where the proprties of the metric depend on the radial distance from a gravitating source. In homogeneous space the only parameter is the global radius of curvature. The “ridiculous” equation dσ² = (a²/[a² - σ²])dσ² is correct only at σ=0. More than that is not expected."

Some of the values of Γ^k_ij are zero at the origin because space-time is locally flat and the metric is diagonal but isotropy does not necessitate that all such spatial connection coefficients (components of the LeviCivita connection) are identically zero. Fischer should have included expressions for Γ^a_ab (no summation) and Γ^a_bb as these are not identically equal to zero. His inclusion of 3f'h'/4h² in his expression for the R₀₀ in equation 14 shows awareness of the need to differentiate h, so his exclusion of all spatial connections shows inconsistency.

To obtain these connection coefficients, the aforementioned functional dependence must be known, otherwise the partial derivatives of the metric with respect to the coordinates cannot be stated. It does not suffice to just equate the partial derivatives with the normal derivative of the metric components with respect to σ, because to do so is tantamount to demanding a linear relationship between σ and the coordinates. Such a relationship is not isotropic; it does not have full rotational symmetry. In the current situation presented by Fischer, where that functional dependence is not specified, the only correct course of action is to incorporate the partial derivatives of σ with respect to the coordinates. The result follows (with (1)neither a,b or c being zero, (2)a, b and c are all different, (3)the Einstein summation convention not invoked, i.e. no summation over the same index occurring in both raised and lowered positions):
Γ
⁰
_0a
= +(f'/2f)σ
_,a

Γ
^a
₀₀
= -(f'/2h)σ
_,a

Γ
^a
_aa
= +(h'/2h)σ
_,a

Γ
^a
_ab
= +(h'/2h)σ
_,b

Γ
^a
_bb
= -(h'/2h)σ
_,a

"Here again Cruel2Bkind apparently confounds homogeneous solutions with rotational symmetric solutions like the Schwarzschild metric. Homogeneous solutions require that changes in some direction x_i are equal to changes in direction –x_i which can be fulfilled only by dh/dx_i=dh/d(-x_i)=0. The advantage to take a suitable coordinate system, in our case local Cartesian coordinates, makes calculations easier, as derivatives of the metric tensor elements in every spatial direction are of the same form.

It is correct that the purely spatial connection coefficients are not zero and have the form given by Cruel2Bkind. But I have never claimed that they should be zero. Explicitly mentioned in my paper are only the Γs, which contain the time dependence, as these are set to zero in Einstein’s static solution. But only the derivative of these spatial connection coefficients contributes to the homogeneous solution, as the Γ themselves contain h´, which is zero at σ = 0."

Within Fischer's own mistaken scheme, this equation contains two errors: (1) the term 9f'h'/4fh should not be present, (2) the numeric denominator of 3f'h'/4h² should be 2 and not 4. Using only the non-zero connection coefficients given in equation 13, his equation for R₀₀ should have been:
R
₀₀
= 3f'²/4fh - 3f"/2h + 3f'h'/2h² .

Unless he did actually not disregard the non-vanishing spatial connection coefficients: in which case, using also:

Γ
^a
_aa
= Γ
^a
_ab
= - Γ
^a
_bb
= (h'/2h) ,

his equation should have been:
R
₀₀
= 3f'²/4fh - 3f"/2h - 3f'h'/4h² .

These equations can be obtained by using the general expression for R₀₀:

R
₀₀
= Γ
ⁱ
_00,i
-Γ
ⁱ
_0i,0
+ Γ
ⁱ
₀₀
Γ
^m
_mi
- Γ
^m
_n0
Γ
ⁿ
_m0
,

expanding out all summation terms, eliminating all terms containing connection coefficients that are identically zero, eliminating all terms that involve differentiation with respect to the time coordinate and substituting in the values given for the connection coefficients.

Using the same procedure with the above expressions for the connection coefficients that do contain the partial derivatives of σ, yields:
R
₀₀
= [(f'²)/(4fh)-(f"/2h)-(f'h')/(4h²)][(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] - [f'/2h][σ
_,1,1
+σ
_,2,2
+σ
_,3,3
]

Comparison with Fischer's version reveals that he does indeed regard the aforementioned functional dependence as linear.

The inclusion of the terms f'h'/4fh , 3h'²/2h² and 2h"/h shows that Fischer did actually use the Γ^a_aa , Γ^a_ab and Γ^a_bb .
For the case involving the σ_,a the expressions for the spatial components of the Ricci tensor are:
R
₁₁
=

[h'²/4h²][4(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] -

[h"/2h][2(σ
_,1
)²+(σ
_,2
)²+(σ
_,3
)²] +

[f'h'/4fh][(σ
_,1
)²-(σ
_,2
)²-(σ
_,3
)²] +

[(f'²/4f²)-(f"/2f)](σ
_,1
)² - [(h'/2h)+(f'/2f)][σ
_,1,1
]-

[h'/2h][σ
_,1,1
+σ
_,2,2
+σ
_,3,3
]

R₂₂ and R₃₃ can be obtained by cyclically permuting the indexes of R₁₁.
This equation is included as further evidence that Fisher regards the functional dependence of σ on the coordinates as linear. Substitution of the value 1 for each σ_,a and the value 0 for each σ_,a,a yields Fishers expression for R_ii.

"I have not controlled whether there is some error in my or in Cruel2Bkind’s calculations. But as long as the differences occur only in the terms containing h', the errors cancel out anyway, as h'(0) is zero in a homogeneous solution."

Premature simplification of equations is always bad form, and especially so in the case of determining non-linear differential equations.
Using the correction to Fischer's R₀₀ and his unsimplified R₁₁
R
₀₀
= 3f'²/4fh - 3f"/2h - 3f'h'/4h²
R
_ii
= f'²/4f² - f'h'/4fh - f"/2f + 3h'²/2h² - 2h"/h ,

the full expression for the curvature scalar is:
R = (1/f)R
₀₀
+ (3/h)R
_ii
R = (1/f)[3f'²/4fh - 3f"/2h - 3f'h'/4h²] + (3/h)[f'²/4f² - f'h'/4fh - f"/2f + 3h'²/2h² - 2h"/h]
R = 3f'²/2f²h - 3f"/fh - 3f'h'/2fh² + 9h'²/2hh² - 6h"/h²

"Setting h'=0 in these equations leads just to the formulas given in my paper. But there is an error in equations (19) and (20), which has confused Cruel2Bkind. The sign of the terms 6/a² resp. 2/a² has to be changed. I first had introduced ϰ positive, but then changed the sign to be compatible with Einstein’s original metric. But with this change I have introduced a sign error."

The full calculation for the 00 component of the gravitational tensor gives:
G
₀₀

= R
₀₀
- (f/2)R
= 3f'²/4fh - 3f"/2h - 3f'h'/4h² - (f/2)[3f'²/2f²h - 3f"/fh - 3f'h'/2fh² + 9h'²/2hh² - 6h"/h²]
= 3fh"/hh - 9fh'²/4hh²
= (3f/4h)[(4h"/h)-3(h'/h)²]
The expression for h can now be used to arrive at an easily analyzed equation. There is no need to indulge in any naive simplifications relating to the values of f, h. h' or h" at the origin.
h = a²/[a² - σ²],
h' = 2a²σ/[(a² - σ²)²],
h" = [2a²/(a² - σ²)](a² + 3σ²)/[(a² - σ²)²]
(h'/h)² = 4σ²/[(a² - σ²)²],
h"/h = 2(a² + 3σ²)/[(a² - σ²)²]

It is now easily shown that
G
₀₀
= (3f/a²)(2a²+3σ²)/(a²-σ²) = κε

However f is negative and (1/a²)(2a²+3σ²)/(a²-σ²) must be positive. Therefore G₀₀ must be negative!
I.E. the energy density ε of matter/radiation must be negative in such a space-time.
Thus Fischer's proposal contributes nothing to the determination of the structure of this universe.

One is left wondering about the standard of the peer review process at "Astrophysics and Space Science".

"With the correct sign in eqs.(19) and (20) the energy density is positive, as it should be. Setting f' and f" to zero the static solution of Einstein is recovered with its negative pressure. But allowing the dependence of time on distance solves the problem of red shift and of the cosmological constant in a universe without expansion. It is an essential contribution to the determination of the structure of this universe.

I regret that the peer review process at "Astrophysics and Space Science" has not detected the errors in my paper, but I should take more care to control the final version of papers myself."

[Ernst Fischer]

EDIT: This response by Ernst Fischer is from a private email communication. It is reproduced here at Scienceforums.com with his permission.

CC

cruel2Bkind · April 21, 2011

Continuing a discussion with Ernst Fischer via coldcreation about his paper Homogeneous cosmological solutions of the Einstein equation. (Many thanks to coldcreation for facilitating this)

"In homogeneous space one can always find an orthonormal basis so that locally the non-diagonal terms of the metric tensor vanish. In this case the spatial structure can be expressed by some local functional, which relates some point to the neighbouring point at distance dσ. "dσ²" is used as a shorthand for the square of this infinitesimal distance element. One should keep in mind that differential geometry is a strictly local description and that the elements of the metric tensor are locally defined functionals and not functions, which can be extended arbitrarily into space. The use of "σ" in the definition in the elements of the metric tensor is only to express the local functionality."

Where Ernst Fischer uses the word "functional" as a noun, he is actually misusing the word. It has a specific meaning in mathematics which is not applicable to his use. It is quite correct to say that components of the metric tensor are functions of the coordinates. If it is possible to construct a global coordinate system that can be used to label each and every point in a unique way then the components of the metric tensor can be expressed as globally applicable functions of the coordinates. If it is impossible to construct a global coordinate system then the space needs to be partitioned into regions each of which has its own coordinate system. In each region the metric components are functions, of the region's coordinates, that are defined as being applicable only to that region.

"Cruel2Bkind is right that eq.(11) contains a typing error. The last term shoud not be “x” but “xi”. There is no need to regard other coordinate systems to determine the functional dependence of h(σ) and its derivatives, as we are interested only on the local values of the functional. They depend only on the radius of curvature and are equal in any direction. The quantity σ has nothing to do with the coordinate “r” in a radially symmetric geometry, where the proprties of the metric depend on the radial distance from a gravitating source. In homogeneous space the only parameter is the global radius of curvature. The “ridiculous” equation dσ² = (a²/[a² - σ²])dσ² is correct only at σ=0. More than that is not expected."

Ernst Fischer entirely misses the point of my criticism. He writes as though there is one unique expression for the spatial components of the metric tensor and thereby seems to fail to recognize that the form of those components depends entirely on the sort of coordinates used.

He also seems to labour under the misapprehension that a radially symmetric geometry is only to be used in the case of a centrally located gravitational source. The Friedmann–Lemaître–Robertson–Walker metric is a radially symmetric system applied globally to a space-time containing an homogeneous distribution of gravitational sources.

He may reject identifying [imath]\sigma[/imath] with [imath]r[/imath] but he gives no other definition of the relationship between his [imath]\sigma[/imath] and his coordinates. His metric is given as a function of [imath]\sigma[/imath], if he wants to derive connection coefficients, which are defined in terms of partial derivatives of the metric tensor components with respect to the coordinates, then he must supply a proper definition of the way [imath]\sigma[/imath] relates to the coordinates. Merely talking of [imath]\sigma[/imath] as a generic distance from the origin and giving the connection coefficients as expressions involving total derivatives of the metric components with respect to [imath]\sigma[/imath] is meaningless.

He also misses my point in describing the equation [imath]\small{d\sigma^{2}=\frac{a^{2}}{a^{2}-\sigma^{2}}d\sigma^{2}}[/imath] as ridiculous. Not only does it imply that [imath]\sigma=0[/imath] but also, by that very implication, it implies that [imath]d\sigma=0[/imath], thereby rendering any further development pointless.

"Here again Cruel2Bkind apparently confounds homogeneous solutions with rotational symmetric solutions like the Schwarzschild metric. Homogeneous solutions require that changes in some direction xi are equal to changes in direction -xi which can be fulfilled only by dh/dxi=dh/d(-xi)=0. The advantage to take a suitable coordinate system, in our case local Cartesian coordinates, makes calculations easier, as derivatives of the metric tensor elements in every spatial direction are of the same form.

It is correct that the purely spatial connection coefficients are not zero and have the form given by Cruel2Bkind. But I have never claimed that they should be zero. Explicitly mentioned in my paper are only the Γs, which contain the time dependence, as these are set to zero in Einstein’s static solution. But only the derivative of these spatial connection coefficients contributes to the homogeneous solution, as the Γ themselves contain h´, which is zero at σ = 0."

By claiming that a reader "apparently confounds homogeneous solutions with rotational symmetric solutions like the Schwarzschild metric" Ernst Fischer seemingly displays ignorance of where considerations of 'rotational symmetry' can be applied, confounding 'rotational symmetry' with systems containing only a centrally located gravitational source. If isotropy pertains at a point,i.e. all directions have identical properties, then space has rotational symmetry at that point.

But he also once again fails to realize that he cannot establish derivatives of the metric tensor elements with respect to the coordinates when those metric elements are only given as functions of some ill-defined quantity labeled [imath]\sigma[/imath] . His [imath]h'[/imath] is a derivative with respect to [imath]\sigma[/imath], it is not a derivative with respect to any coordinate.

For a diagonal metric [imath]\Gamma_{0a}^{0}=\frac{1}{2}g^{00}g_{00,a}[/imath]. If these are equated to [imath]\frac{1}{2f}\frac{df}{d\sigma}[/imath] then either [imath]\sigma=x_{1}+x_{2}+x_{3}[/imath] or [imath]\sigma[/imath] is just a place holder for [imath]x_{1},x_{2},x_{3}[/imath] depending on which is used in the differentiations.

In the former case, [imath]h(\sigma)= \frac{a^{2}}{a^{2}-\sigma^{2}}=\frac{a^{2}}{a^{2}-(x_{1}+x_{2}+x_{3})^{2}}[/imath] which is unsatisfactory.

In the latter case:

[imath]h(\sigma)=h(x_{1})= \frac{a^{2}}{a^{2}-x_{1}^{2}}[/imath] when finding [imath]\Gamma_{a1}^{a}[/imath] & [imath]\Gamma_{aa}^{1}[/imath],

[imath]h(\sigma)=h(x_{2})= \frac{a^{2}}{a^{2}-x_{2}^{2}}[/imath] when finding [imath]\Gamma_{a2}^{a}[/imath] & [imath]\Gamma_{aa}^{2}[/imath] and

[imath]h(\sigma)=h(x_{3})= \frac{a^{2}}{a^{2}-x_{3}^{2}}[/imath] when finding [imath]\Gamma_{a3}^{a}[/imath] & [imath]\Gamma_{aa}^{3}[/imath], which is also unsatisfactory. Trying to hide such unsatisfactory situations under the cover of [imath]h'(0)=0[/imath] is just naively mistaken.

"I have not controlled whether there is some error in my or in Cruel2Bkind’s calculations. But as long as the differences occur only in the terms containing h', the errors cancel out anyway, as h'(0) is zero in a homogeneous solution."

1) Ernst Fischer seems oblivious to the fact that by evaluating his equations at the origin he ends up with equations that only express relationships between values. Such relationships cannot be used to form differential equations.

His original equation for [imath]R_{00}[/imath] is:

[math]R_{00}= \frac{3f'^{2}}{4fh}-\frac{9f'h'}{4fh}-\frac{3f''}{2h}+\frac{3f'h'}{4h^{2}}[/math]

which for illustrative purposes will be rewritten as:

[math]R_{00}(\sigma)= \frac{3f'(\sigma)^{2}}{4f(\sigma)h(\sigma)}-\frac{9f'(\sigma)h'(\sigma)}{4f(\sigma)h(\sigma)}-\frac{3f''(\sigma)}{2h(\sigma)}+\frac{3f'(\sigma)h'(\sigma)}{4h(\sigma)^{2}}[/math]

Now by setting [imath]\sigma=0[/imath] and [imath]h'(0)=0[/imath] he arrives at a relationship between values, which he writes as his equation 16:

[math]R_{00}= \frac{3f'^{2}}{4fh}-\frac{3f''}{2h}[/math]

but what it actually means can better be illustrated by writing it properly as:

[math]R_{00}(0)= \frac{3f'(0)^{2}}{4f(0)h(0)}-\frac{3f''(0)}{2h(0)}[/math]

In similar vein his equation 16 for [imath]R_{ii}[/imath] and his equation 17 for R can be better written as:

[math]R_{ii}(0)= -\frac{f''(0)}{2f(0)}+\frac{f'(0)^{2}}{4f(0)^{2}}-\frac{2h''(0)}{h(0)^{2}}\; (i=1,2,3)[/math]

[math]R(0)= -\frac{3f''(0)}{2f(0)h(0)}+\frac{3f'(0)^{2}}{2h(0)f(0)^{2}}-\frac{6h''(0)}{h(0)^{2}}[/math]

Consequently his equations 20 & 21 for [imath]G_{ii}[/imath] & [imath]Y(\sigma)[/imath], which he gives as

[math]G_{ii}=R_{ii}-\frac{1}{2}R=-2Y(\sigma)-\frac{2}{a^{2}}=\kappa p[/math]

[math]Y(\sigma)=\frac{f'^{2}}{4f^{2}}-\frac{f''}{2f}[/math]

should be written as :

[math]G_{ii}(0)=R_{ii}(0)-\frac{1}{2}R(0)=-2Y(0)-\frac{2}{a^{2}}=\kappa p[/math]

[math]Y(0)=\frac{f'(0)^{2}}{4f(0)^{2}}-\frac{f''(0)}{2f(0)}[/math]

I.E the equation

[math]-2(\frac{f'^{2}}{4f^{2}}-\frac{f''}{2f})- \frac{2}{a^{2}}=\kappa p[/math]

actually means

[math]-2(\frac{f'(0)^{2}}{4f(0)^{2}}-\frac{f''(0)}{2f(0)})-\frac{2}{a^{2}}=\kappa p[/math]

It is not a differential equation. It should not be treated as a differential equation. Attempting to solve it as a DE is mistaken, as it is no more than just a relationship between values. The equations that result from using his unsimplified [imath]R_{00}[/imath] and [imath]R_{ii}[/imath] are the actual DEs that are pertinent to his endeavours.

2) Ernst Fischer's original equation for [imath]R_{00}[/imath] is:

[math]R_{00}= \frac{3f'^{2}}{4fh}-\frac{9f'h'}{4fh}-\frac{3f''}{2h}+\frac{3f'h'}{4h^{2}}[/math]

A definition for the Ricci tensor is:

[math]R_{ij}=\Gamma_{ij,k}^{k}-\Gamma_{ik,jb}^{k}+\Gamma_{ij}^{k}\Gamma_{km}^{m}-\Gamma_{ik}^{m}\Gamma_{mj}^{k}[/math]

where the comma notation is used for partial differentiation, and the Einstein summation convention applies.

Replacing the connection coefficients by their values:

[imath]\Gamma_{0a}^{0}=+\frac{f'}{2f}\; \; \; \Gamma_{00}^{a}=-\frac{f'}{2h}\; \; \; \Gamma_{aa}^{a}=+\frac{h'}{2h}\; \; \; \Gamma_{ab}^{b}=+\frac{h'}{2h}\; \; \; \Gamma_{bb}^{a}=-\frac{h'}{2h}[/imath] ,

(in which a is not equal to b, neither a nor b is 0 and Einstein summation does not apply) in the above expression for the Ricci tensor, the expressions for [imath]R_{00}[/imath] and [imath]R_{ii}[/imath] are found to be

[math]R_{00}= \frac{3f'^{2}}{4fh}-\frac{3f''}{2h}-\frac{3f'h'}{4h^{2}}[/math]

[math]R_{ii}= \frac{f'^{2}}{4f^{2}}-\frac{f'h'}{4fh}-\frac{f''}{2f}+\frac{3h'^{2}}{2h^{2}}-\frac{2h''}{h^{2}}\; (i=1,2,3)[/math]

So Ernst Fischer's original equation for [imath]R_{00}[/imath] is WRONG ! Simple inspection shows that the term 9f'h'/4fh should not be there, careful simplification shows that the numerical coefficient of the term containing f'h'/h² .is -3/4 and not +3/4 as Ernst Fischer states. He must take responsibility for such errors, trying to dismiss them by saying "but as long as the differences occur only in the terms containing h', the errors cancel out anyway, as h'(0) is zero in a homogeneous solution" is just not good enough!. Such an erroneous approach to the maths leads only to a relationship between values, not to the correct differential equation.

"Setting h'=0 in these equations leads just to the formulas given in my paper. But there is an error in equations (19) and (20), which has confused Cruel2Bkind. The sign of the terms 6/a² resp. 2/a² has to be changed. I first had introduced ? positive, but then changed the sign to be compatible with Einstein’s original metric. But with this change I have introduced a sign error."

For a perfect fluid, which is a good model for the material/radiation in an homogeneous isotropic universe, the stress-energy tensor is:

[math]T_{\lambda \mu}=\left({\varepsilon +p}\right)u_\lambda u_\mu +pg_{\lambda \mu}[/math]

The four-velocity of the fluid is [imath]\left({u^{0},u^{1},u^{2},u^{3}}\right)=\left({u^{0},0,0,0}\right)[/imath], so [imath]u^{\mu}g_{\mu \nu}u^{\nu}=-1[/imath] becomes [imath]u^{0}g_{00}u^{0}=-1[/imath]. Since [imath]u_{\mu}=g_{\mu \nu}u^{\nu}[/imath], the covariant components of the four-velocity are [imath]\left({u_{0},u_{1},u_{2},u_{3}}\right)=\left({g_{00}u^{0},0,0,0}\right)[/imath]. Also [imath]u_{0}u_{0}=g_{00}u^{0}g_{00}u^{0}=- g_{00}[/imath], so

[math]T_{00}=\left({\varepsilon +p}\right)u_0 u_0 +pg_{00}=\left({\varepsilon +p}\right)\left({- g_{00}}\right)+pg_{00}=-\varepsilon g_{00}[/math]

[math]T_{ii}=\left({\varepsilon +p}\right)u_i u_i +pg_{ii}=pg_{ii}[/math]

The field equations read [imath]R_{ij}-\frac{1}{2}R=\kappa T_{ij}[/imath] where [imath]\kappa=\frac{8\pi G}{c^{4}}[/imath] and since [imath]G[/imath], the gravitational constant, is positive so [imath]\kappa[/imath] is also positive. [imath]T_{00}=\varepsilon[/imath] is always positive.

Using the correct expression that Ernst Fischer should have used for [imath]R_{00}[/imath], using the expression for [imath]R_{ii}[/imath] that he did obtain, using the expression for [imath]h[/imath], i.e. [imath]h(\sigma)= \frac{a^{2}}{a^{2}-\sigma^{2}}[/imath], and using [imath]f=g_{00}[/imath], the time-time field equation becomes

[math]\frac{3f\left[ {2a^2 + 3\sigma ^2}\right]}{a^2\left[{a^2 - \sigma ^2}\right]}=-\kappa \varepsilon f[/math]

Therefore

[math]\varepsilon=- \frac{3\left[ {2a^2 + 3\sigma ^2}\right]}{\kappa a^2\left[{a^2 - \sigma ^2}\right]}[/math]

I.e. [imath]\varepsilon[/imath] is negative, approaching negative infinity as [imath]\sigma[/imath] approaches [imath]a[/imath], and homogeneity of the energy density cannot apply.

Ernst Fischer ignored the off-diagonal components of the Ricci tensor presumably because, by the field equations, these equate to zero. However whilst the [imath]R_{0i}[/imath] are identically zero, the [imath]R_{ij}[/imath] for [imath] \left\{{i,j}\right\}\subset\left\{{1,2,3}\right\}[/imath] are not. Using the values for [imath]\Gamma_{\lambda}^{\mu \nu}[/imath] given above:

[math]R_{ij} = \frac{3{h'}^2}{4h^2}-\frac{h''}{2h}+\frac{{f'}^2}{4f^2}-\frac{f''}{2f}+\frac{f'h'}{2fh}=0[/math]

Since this is equal to zero, there results an equation that Ernst Fischer completely overlooked:

[math]\frac{f''}{f}-\frac{{f'}^2}{2f^2}+\frac{h''}{h}=\frac{3{h'}^2}{2h^2}+\frac{f'h'}{fh}[/math]

Using the correct expression that Ernst Fischer should have used for [imath]R_{00}[/imath], using the expression for [imath]R_{ii}[/imath] that he did obtain, and using the value for [imath]T_{ii}=pg_{ii}=ph[/imath], the spatial field equations, without any premature simplifications, are:

[math]G_{ii}=\frac{f''}{f}-\frac{{f'}^2}{2f^2}+\frac{h''}{h}+\frac{f'h'}{2fh}-\frac{3{h'}^2}{4h^2}=\kappa ph[/math]

Incorporating the 'overlooked' equation gives:

[math]\frac{3h'^2}{4h^2}+\frac{3f'h'}{2fh}=\kappa ph[/math]

Since [imath]h'(0)=0[/imath] then [imath]p(0)=0[/imath] and if that situation pertains elsewhere at points other than [imath]\sigma=0[/imath] the equation simplifies to:

[math]\frac{h'}{2h}+\frac{f'}{f}=0[/math]

As [imath]f(0)=-1[/imath] the general solution is:

[math]f=\frac{-\sqrt{a^2-\sigma ^2}}{a}[/math]

A result is that [imath]f(a)=0[/imath]. I.e there is an event horizon at [imath]\sigma =a[/imath]. A little thought by those who are mathematically adept reveals that an event horizon for one observer is an event horizon for all other observers at points not on the horizon. Also the mathematically adept reader can easily appreciate that in Ernst Fischer's space-time, if an observer at [imath]\sigma=0[/imath] sees the light from another relatively stationary observer as red shifted then that other observer sees the light from the first as blue shifted.

In conclusion, not only is there a gross lack of proper definition of [imath]\sigma[/imath], but also there is a naive oversimplification of equations that, when treated properly, reveal a complete lack of applicability to our universe because the required energy density must be negative and approach negative infinity as [imath]\sigma \to a[/imath], the pressure must be zero and there is an event horizon at [imath]\sigma =a[/imath].

cruel2Bkind · April 22, 2011

As long as the spacetime manifold in which observers A and B are located is of constant positive Gaussian curvature, both will observe redshift and time dilation.

You repeatedly claim this to be true despite a mathematical argument to the contrary. Either supply mathematical proof of your claim or prove that the disproof is in error. Merely reiterating a statement does not establish its veracity.

This is a fallacy known as irrelevant conclusion: you are diverting attention away from a fact in dispute rather than addressing it directly.[1]

We are not talking about things we know are moving radially (e.g., cars). Galaxies are not seen to move.[2] The interpretation of radial motion comes from redshift, which can be interpreted as either a Doppler effect, expansion or stretching of space, or a curved spacetime effect. That was the point of the link above.

This is a case of making a general claim, viz: "Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift " and shifting the goal post when a counter example is given. Instead of accusing another of fallacious reasoning you should have the courage to admit that you made an invalid general claim and did no use sufficient precision in your original statement so as to avoid any ambiguity.
Humanity's observation time of galaxies is very short and galaxies have not as yet been seen by humanity to move. However this situation does not mean that it is in principle impossible to distinguish between cosmic expansion (as per the rubber sheet analogy), relative velocity or gravitation, as the cause of redshift. Given sufficient observation time, the cause of red-shift could be determined.

.

Again, in the case of distant galaxies, judging by redshfit z, there is no way to know for a fact wether the distance to the source is constant or not. It depends on the model.

I restate (with amendment): If the proper distance to the source of the photon is constant then no choice of components coordinates can justify a Doppler interpretation. If a galaxy is receding then after sufficient length of time its angular extent in the sky will be smaller (if space is flat or negatively curved). If a series of signals could be sent to a galaxy, and be immediately returned on reception by observers there, then the galaxy's relative radial velocity could be determined.

You are affirming the consequent: drawing a conclusion from a premises that does not support the conclusion.[1]

Since the situation is perfectly symmetric they both observe redshift and time dilation.[2]

Affirming the consequent is reasoning in the form:
- If P, then Q.
- Q.
- Therefore, P.

In accusing me of committing this fallacy you are displaying an inability to discern the nature of a logical argument. The argument I presented has the formal structure of:

If P, then Q
Not Q
Therefore, Not P.

[*] This is a case of Begging the question

The "unavoidable conclusion" is in fact entirely avoidable, since, again, the situation is symmetric.

In a curved spacetime manifold, where k = 1, the duration of the interval sent by A will be longer than is B's time frame, and visa versa. Both of them see the other as experiencing time dilation (as if immersed at a lower altitude in a gravitational potential well), so both sees the other's signal as red-shifted.

Again this is a case of Begging the question

Repetition of a claim does not establish its veracity.

Nonsense. It is the conclusion of the thought experiment that is absurd, since it is based on an erroneous pemise.

In a previous post you retorted with:

"The thought experiment argument is not substantiated, nor is the conclusion derived. The situation of A and B is completely symmetric. ... All the further calculations are nonsense."

Mere derision of a mathematical argument does not establish its invalidity. However, such derision can be interpreted as indicative of mathematical ineptitude.

If a mathematical argument cannot be comprehended or mathematically proven incorrect then it should be repeatedly reread until either it can be comprehended or a mathematical disproof can be established.

You missed entirely the point:
"We show that, in any spacetime, an observed frequency shift can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon's path. ...
(other quotes form Bunn and Hogg)"[1]

In the context of the nonexpanding Universe, the gravitational interpretation corresponds to a family of stationary observers and hence seems to be the more natural one.[2]

In the post to which I was responding, you claimed "(a)ny shift in frequency can be described as gravitational or Doppler", giving the paper by Bunn & Hogg as an example.

Since you advocate the belief that cosmic redshift has a curved space cause I was only informing you that Bunn & Hogg cannot be cited in support of that belief. Also you appear to be under the impression that Bunn and Hogg make an incontravertible argument. They do not! The above quote from their paper is highly debatable. The curvature or flatness of space-time is not a result of the choice of coordinates. In a flat space-time where a frequency shift can be determined to be a purely Doppler shift by using signal exchange to establish that the invariant proper distance between source and sink is constantly changing, any choice of a set of increasingly accelerating observers in an attempt to justify a 'gravitational-shift' interpretation of a frequency shift, does not actually establish the correctness of the gravitational-shift interpretation. Such an interpretation is really no more than a figment of the coordinate systems used for each of the set of observers; it has no reality.
Before any value judgment can be made in regard to a gravitational interpretation of frequency shift in a non-expanding Universe, the existence of such a frequency shift must be established.

Which "one of the constant-volume hyper-spherical universe models" are you referring to? The flat universe with k = 0? If so you conclusion "thus your thus is invalid", is invalid.

I was referring to a model that Ernst Fisher also referred to in his paper. My presumption that you were also familiar with the model was apparently mistaken.

coldcreation · April 27, 2011

Continuing a discussion with Ernst Fischer via coldcreation about his paper Homogeneous cosmological solutions of the Einstein equation. (Many thanks to coldcreation for facilitating this)

What follows is Ernst Fischer's response to the queries raised by cruel2Bkind regarding Homogeneous cosmological solutions of the Einstein equation (Fischer 2009).

Where Ernst Fischer uses the word "functional" as a noun, he is actually misusing the word. It has a specific meaning in mathematics which is not applicable to his use. It is quite correct to say that components of the metric tensor are functions of the coordinates. If it is possible to construct a global coordinate system that can be used to label each and every point in a unique way then the components of the metric tensor can be expressed as globally applicable functions of the coordinates. If it is impossible to construct a global coordinate system then the space needs to be partitioned into regions each of which has its own coordinate system. In each region the metric components are functions, of the region's coordinates, that are defined as being applicable only to that region.

I fear that Cruel2Bkind has not really grasped the essentials of differential geometry. Generally speaking it describes the relations between points of a manifold by the local spatial connections between neighbouring points, expressed by the curvature tensor. Only in special cases it is possible to introduce global coordinates (if there exists a Killing vector field, so that the Lie derivative along this vector field vanishes). To get an idea, how differential geometry works, Cruel2Bkind should try to describe the geometry of a spherical surface by a global geometry, using only coordinates within this surface, or still more simple, to describe a circle only using a coordinate along the circumference.

The complete spatial part of the formulas used in my paper are absolutely nothing new, but can be found in similar form in every textbook on differential geometry to describe a surface of constant curvature. The only thing that is new is the extension to the space-time manifold of general relativity, but now under the assumption of global Lorentz invariance.

Ernst Fischer entirely misses the point of my criticism. He writes as though there is one unique expression for the spatial components of the metric tensor and thereby seems to fail to recognize that the form of those components depends entirely on the sort of coordinates used.

He also seems to labour under the misapprehension that a radially symmetric geometry is only to be used in the case of a centrally located gravitational source. The Friedmannï¿½Lemaï¿½treï¿½Robertsonï¿½Walker metric is a radially symmetric system applied globally to a space-time containing an homogeneous distribution of gravitational sources.

Homogeneity includes radial symmetry, but it is more. It is radial symmetry about every point of a manifold. My impression was that Cruel2Bkind tried to reduce the definition of distance in homogeneous space to the radial distance in spherically symmetric systems.

He may reject identifying [imath]\sigma[/imath] with [imath]r[/imath] but he gives no other definition of the relationship between his [imath]\sigma[/imath] and his coordinates. His metric is given as a function of [imath]\sigma[/imath], if he wants to derive connection coefficients, which are defined in terms of partial derivatives of the metric tensor components with respect to the coordinates, then he must supply a proper definition of the way [imath]\sigma[/imath] relates to the coordinates. Merely talking of [imath]\sigma[/imath] as a generic distance from the origin and giving the connection coefficients as expressions involving total derivatives of the metric components with respect to [imath]\sigma[/imath] is meaningless.

He also misses my point in describing the equation [imath]\small{d\sigma^{2}=\frac{a^{2}}{a^{2}-\sigma^{2}}d\sigma^{2}}[/imath] as ridiculous. Not only does it imply that [imath]\sigma=0[/imath] but also, by that very implication, it implies that [imath]d\sigma=0[/imath], thereby rendering any further development pointless.

I do not see, what should be problematic in the definition of a distance between points or in the definition of an infinitesimal distance element in curved space.

These are quantities, which do not depend on the choice of a special system of coordinates. It may be more or less laborious to calculate distances between points, depending on the coordinate system, which you use, but the distance will not change.

By claiming that a reader "apparently confounds homogeneous solutions with rotational symmetric solutions like the Schwarzschild metric" Ernst Fischer seemingly displays ignorance of where considerations of 'rotational symmetry' can be applied, confounding 'rotational symmetry' with systems containing only a centrally located gravitational source. If isotropy pertains at a point,i.e. all directions have identical properties, then space has rotational symmetry at that point.

But he also once again fails to realize that he cannot establish derivatives of the metric tensor elements with respect to the coordinates when those metric elements are only given as functions of some ill-defined quantity labeled [imath]\sigma[/imath] . His [imath]h'[/imath] is a derivative with respect to [imath]\sigma[/imath], it is not a derivative with respect to any coordinate.
For a diagonal metric [imath]\Gamma_{0a}^{0}=\frac{1}{2}g^{00}g_{00,a}[/imath]. If these are equated to [imath]\frac{1}{2f}\frac{df}{d\sigma}[/imath] then either [imath]\sigma=x_{1}+x_{2}+x_{3}[/imath] or [imath]\sigma[/imath] is just a place holder for [imath]x_{1},x_{2},x_{3}[/imath] depending on which is used in the differentiations.
In the former case, [imath]h(\sigma)= \frac{a^{2}}{a^{2}-\sigma^{2}}=\frac{a^{2}}{a^{2}-(x_{1}+x_{2}+x_{3})^{2}}[/imath] which is unsatisfactory.

[...]

Trying to hide such unsatisfactory situations under the cover of [imath]h'(0)=0[/imath] is just naively mistaken.

Again Cruel2Bkind makes the mistake of extending the local description in the form of differential properties to an extended volume, and thus arrives at inconsistencies. This naïve interpretation of differential geometry appears to be the main problem in all his discussions. But as he has discovered in the text below, the equations, which he derives, are not differential equations, but relations between values, between the values of some functions f and h and their derivatives at some fixed (but arbitrary) point. Relations over finite distances can be obtained only by integration along some path, regarding the values of the functions f and h and their derivatives as commoving information.

1) Ernst Fischer seems oblivious to the fact that by evaluating his equations at the origin he ends up with equations that only express relationships between values. Such relationships cannot be used to form differential equations.
His original equation for [imath]R_{00}[/imath] is:

[math]R_{00}= \frac{3f'^{2}}{4fh}-\frac{9f'h'}{4fh}-\frac{3f''}{2h}+\frac{3f'h'}{4h^{2}}[/math]

[...]

It is not a differential equation. It should not be treated as a differential equation. Attempting to solve it as a DE is mistaken, as it is no more than just a relationship between values. The equations that result from using his unsimplified [imath]R_{00}[/imath] and [imath]R_{ii}[/imath] are the actual DEs that are pertinent to his endeavours.

2) Ernst Fischer's original equation for [imath]R_{00}[/imath] is:

[...]

So Ernst Fischer's original equation for [imath]R_{00}[/imath] is WRONG ! Simple inspection shows that the term 9f'h'/4fh should not be there, careful simplification shows that the numerical coefficient of the term containing f'h'/hï¿½ .is -3/4 and not +3/4 as Ernst Fischer states. He must take responsibility for such errors, trying to dismiss them by saying "but as long as the differences occur only in the terms containing h', the errors cancel out anyway, as h'(0) is zero in a homogeneous solution" is just not good enough!. Such an erroneous approach to the maths leads only to a relationship between values, not to the correct differential equation.

Here Cruel2Bkind tries to extend the solution at σ=0 to extended spatial sections in contradiction to the rules of differential geometry. Integration of the field equations in curved space to obtain geodesic lines requires a continuous shift of the coordinate system, so that the direction of change remains tangential to the actual direction of motion.

For a perfect fluid, which is a good model for the material/radiation in an homogeneous isotropic universe, the stress-energy tensor is:

[math]T_{\lambda \mu}=\left({\varepsilon +p}\right)u_\lambda u_\mu +pg_{\lambda \mu}[/math]

[...]

I.e. [imath]\varepsilon[/imath] is negative, approaching negative infinity as [imath]\sigma[/imath] approaches [imath]a[/imath], and homogeneity of the energy density cannot apply.

Ernst Fischer ignored the off-diagonal components of the Ricci tensor presumably because, by the field equations, these equate to zero. However whilst the [imath]R_{0i}[/imath] are identically zero, the [imath]R_{ij}[/imath] for [imath] \left\{{i,j}\right\}\subset\left\{{1,2,3}\right\}[/imath] are not. Using the values for [imath]\Gamma_{\lambda}^{\mu \nu}[/imath] given above:

[...]

A result is that [imath]f(a)=0[/imath]. I.e there is an event horizon at [imath]\sigma =a[/imath]. A little thought by those who are mathematically adept reveals that an event horizon for one observer is an event horizon for all other observers at points not on the horizon. Also the mathematically adept reader can easily appreciate that in Ernst Fischer's space-time, if an observer at [imath]\sigma=0[/imath] sees the light from another relatively stationary observer as red shifted then that other observer sees the light from the first as blue shifted.

In conclusion, not only is there a gross lack of proper definition of [imath]\sigma[/imath], but also there is a naive oversimplification of equations that, when treated properly, reveal a complete lack of applicability to our universe because the required energy density must be negative and approach negative infinity as [imath]\sigma \to a[/imath], the pressure must be zero and there is an event horizon at [imath]\sigma =a[/imath].

Again here the application to extended space-time of the local definition of curvature with its differential connections between neighboring points leads to nonsensical results like the existence of horizons and singularities. The consequences, which Cruel2Bkind draws from his wrong interpretation of the field equations, are not valid. The solution is homogeneous with equal properties at every point of the manifold.

The assumption that the energy density must be negative is wrong. It may be caused by the sign error in my eq.(19). But apart from the fact that the time scale changes with distance, the solution of the field equations discussed in my paper is identical with Einstein’s static universe, where the coefficient κ is chosen just in the way to obtain the correct positive energy density in the Newtonian limit. It is a valid solution, which is in agreement with the observed properties of space-time, not only with respect to red shift, but which also is stable without requiring something like dark energy.

[Ernst Fischer, from an email communication with Coldcreation dated April 26, 2011]

coldcreation · May 7, 2011

As long as the spacetime manifold in which observers A and B are located is of constant positive Gaussian curvature, both will observe redshift and time dilation.

You repeatedly claim this to be true despite a mathematical argument to the contrary. Either supply mathematical proof of your claim or prove that the disproof is in error. Merely reiterating a statement does not establish its veracity.

Certainly, in a flat Newtonian, Euclidean or Minkowski universe cosmological redshift and time dilation do not occur (unless the universe is expanding for whatever reason).

Most certainly too, the geometry of curved spacetime (such as that of an Einstein universe) affects the electromagnetic spectrum. It would be ludicrous to imagine a general relativistic universe (of constant positive or negative Gaussian curvature) where the propagation of light remains unaffected from the point of view of an observer as she peers into the deep cosmos...

Einstein's general postulate of relativity (GR) did away with flat spacetime, and in doing so (simultaneously) introduced the notion (dare I say empirical fact) that light is affected by the geometry of space, i.e., spacetime curvature alters spectral lines (leading to redshift and time dilation).

[1] This is a case of making a general claim, viz: "Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift " and shifting the goal post when a counter example is given. Instead of accusing another of fallacious reasoning you should have the courage to admit that you made an invalid general claim and did no use sufficient precision in your original statement so as to avoid any ambiguity.

[2] Humanity's observation time of galaxies is very short and galaxies have not as yet been seen by humanity to move. However this situation does not mean that it is in principle impossible to distinguish between cosmic expansion (as per the rubber sheet analogy), relative velocity or gravitation, as the cause of redshift. Given sufficient observation time, the cause of red-shift could be determined.
.

[1] GR makes the general claim. I merely support it.

[2] Sufficient observation time needed to distinguish between the two effects (e.g., to actually 'see' galaxies move radially via an increase in redshift) would be on the order of millions of years. So yes, it is in principle possible, but in all practical purposes impossible.

I restate (with amendment): If the proper distance to the source of the photon is constant then no choice of components coordinates can justify a Doppler interpretation. If a galaxy is receding then after sufficient length of time its angular extent in the sky will be smaller (if space is flat or negatively curved). If a series of signals could be sent to a galaxy, and be immediately returned on reception by observers there, then the galaxy's relative radial velocity could be determined.

Too many if's.

Everyone agrees that it should, could or would be possible in principle to differentiate between a Doppler effect and a gravitational effect, but I don't think anyone wants to wait millions of years for the answer. :)

EDIT> Judging from redshift alone there is no way to differentiate between cosmological expansion (Doppler effect) and cosmological gravitational redshift (due to the curvature of space). There are however ways to make the correct interpretation, without having to wait millions of years to see the redshift of any given galaxy increase over time. As mentioned above, the Hubble's program (1926-1934) set out to find space curvature from the galaxy count: the Gauss protocol (Hubble, Tolman, 1935, ApJ 82, 302). There are other ways as well, related to angular diameter distance, luminosity distance, etc.

In a previous post you retorted with:
"The thought experiment argument is not substantiated, nor is the conclusion derived. The situation of A and B is completely symmetric. ... All the further calculations are nonsense."
Mere derision of a mathematical argument does not establish its invalidity. However, such derision can be interpreted as indicative of mathematical ineptitude.
If a mathematical argument cannot be comprehended or mathematically proven incorrect then it should be repeatedly reread until either it can be comprehended or a mathematical disproof can be established.

Your arguments were based on the erroneous proposition k>1. It was erroneous because the situation was not symmetric. In other words, you began you series of calculations with a false premise, an incorrect proposition.

Since your premise (proposition, or assumption) was not correct, the conclusion drawn was in error. Though your argument may have been logically valid, your conclusion was demonstrably wrong, because its first premise was false. A logical analysis might not reveal the error in your argument if that analysis accepted as true the argument's premise.

The truth of your premises must be first be established otherwise what follows can be interpreted as pure nonsense.

[1] In the post to which I was responding, you claimed "(a)ny shift in frequency can be described as gravitational or Doppler", giving the paper by Bunn & Hogg as an example.

Since you advocate the belief that cosmic redshift has a curved space cause I was only informing you that Bunn & Hogg cannot be cited in support of that belief. [...]

[2] Before any value judgment can be made in regard to a gravitational interpretation of frequency shift in a non-expanding Universe, the existence of such a frequency shift must be established.

[1] I don't believe that "cosmic redshift has a curved space cause" any more than I believe that cosmic redshift has a Doppler cause.

It has been merely pointed out that, empirically, the two interpretations are viable to certain extents (see the OP).

[2] Just as it is possible to extrapolate the Doppler effect as the cause of cosmological redshift, the gravitational redshift can be extrapolated as a cause of cosmological redshift. The existence of such a frequency shift has been established and is embodied within Einstein's general theory of relativity.

In another way, curvature manifests itself increasingly with distance, i.e., it becomes more and more apparent, the further one looks (even though curvature is constant). Thus redshift increases with distance, along with the associated time dilation factor.

In one of the constant-volume hyper-spherical universe models, light is not increasingly redshifted with distance. Thus your 'thus' is invalid.

Which "one of the constant-volume hyper-spherical universe models" are you referring to? The flat universe with k = 0? If so you conclusion "thus your thus is invalid", is invalid.

I was referring to a model that Ernst Fisher also referred to in his paper. My presumption that you were also familiar with the model was apparently mistaken.

The only static model where light is not increasingly redshifted with distance is the flat, Euclidean/Newtonian/Minkowski model.

Thus your thus is invalid is invalid. :)

CC

cruel2Bkind · May 13, 2011

I fear that Cruel2Bkind has not really grasped the essentials of differential geometry. Generally speaking it describes the relations between points of a manifold by the local spatial connections between neighbouring points, expressed by the curvature tensor. Only in special cases it is possible to introduce global coordinates (if there exists a Killing vector field, so that the Lie derivative along this vector field vanishes). To get an idea, how differential geometry works, Cruel2Bkind should try to describe the geometry of a spherical surface by a global geometry, using only coordinates within this surface, or still more simple, to describe a circle only using a coordinate along the circumference.

The complete spatial part of the formulas used in my paper are absolutely nothing new, but can be found in similar form in every textbook on differential geometry to describe a surface of constant curvature. The only thing that is new is the extension to the space-time manifold of general relativity, but now under the assumption of global Lorentz invariance.

I regard E. Fischer as not having the requisite expertise to make valid assessments of anyone else's competency in, or comprehension of, differential geometry. Fischer would do well to concern himself with his own level of comprehension of differential geometry, and basic differential calculus, before he passes value judgments on the abilities of others.

Although the spatial parts of the formulae used in his paper may be similar to those that appear in text books on differential geometry they are fundamentally different to the spatial line-element for a space of constant positive curvature, in ways in which he is apparently unable to appreciate.

The spatial part of the metric for a hypersherical space-time of radius [imath]a[/imath] in a system of spherical coordinates, where [imath]0 \le x^1 \le a[/imath] is a radial coordinate (but not a true radial length, as it is defined by dividing the circumference of a circle centered on the origin by [imath]2\pi[/imath] ), [imath]0 \le x^2 \le \pi[/imath] is an azimuthal angle, [imath]0 \le x^3 \le 2\pi[/imath] is an equatorial angle, is given by:

[math]\left(1\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = \frac{{a^2 {\left({dx^1}\right)}^2}}{{a^2 - {\left({x^1}\right)}^2}} + {\left({x^1}\right)}^2 {\left({dx^2}\right)}^2 + {\left({x^1}\right)}^2 {\sin}^2 \left({x^2}\right){\left({dx^3}\right)}^2[/math]

By a transformation [imath]\left({x^1 ,x^2 ,x^3}\right) \to \left({y^1 ,y^2 ,y^3}\right) = \left({a \arcsin \left({\frac{{x^1}}{a}}\right),x^2 ,x^3}\right)[/imath] , where [imath]0 \le y^1 \le a[/imath] is a true radial length, the line-element becomes:

[math]\left(2\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = {\left({dy^1}\right)}^2 + a^2 {\sin}^2 \left({\frac{{y^1}}{a}}\right){\left({dy^2}\right)}^2 + a^2 {\sin}^2 \left({\frac{{y^1}}{a}}\right){\sin}^2 \left({y^2}\right){\left({dy^3}\right)}^2[/math]

Transformation to a cartesian coordinate system [imath]\left({z^1,z^2,z^3}\right)[/imath] [imath]=[/imath] [imath]\left({x^1\sin\left({x^2}\right)\cos\left({x^3}\right),x^1\sin\left({x^2}\right)\sin\left({x^3}\right),x^1\cos\left({x^2}\right)}\right)[/imath] , gives:

[math]\left(3\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = \frac{{\left({z^1 dz^1 + z^2 dz^2 + z^3 dz^3}\right)}^2}{{a^2 - \left[{{\left({z^1}\right)}^2 + {\left({z^2}\right)}^2 + {\left({z^3}\right)}^2}\right]}} + {\left({dz^1}\right)}^2 + {\left({dz^2}\right)}^2 + {\left({dz^3}\right)}^2[/math]

A transformation to any other cartesian system [imath]\left({w^1,w^2,w^3}\right) = \left({A_k^1 z^k,A_k^2 z^k,A_k^3 z^k}\right)[/imath] sharing the same origin (wherein the [imath]A_k^j[/imath] are the elements of an orthogonal matrix), yields an expression for the line-element that has exactly the same structure but with [imath]w^1,w^2,w^3[/imath] instead of [imath]z^1,z^2,z^3[/imath] .

Along any of the coordinate axes [imath]z^i[/imath] where [imath]z^j=0[/imath] and [imath]dz^j=0[/imath] for all [imath]j \ne i[/imath] the line element takes on a simpler form:

[math]\left(4\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = \frac{{a^2}}{{a^2 - {\left({z^i}\right)}^2}}{\left({dz^i}\right)}^2[/math]

This result can be expressed in a form that applies to any line-element lying along a ray that goes through the origin. If the line-element is located at a distance [imath]r[/imath] from the origin then:

[math]\left(5\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = \frac{{a^2}}{{a^2 - r^2}}{\left({dr}\right)}^2[/math]

Since [imath]{\left({dr}\right)}^2 ={\left({dw^1}\right)}^2 +{\left({dw^2}\right)}^2 +{\left({dw^3}\right)}^2[/imath] for any cartesian coordinate system centered on the same origin, this expression can be given as:

[math]\left(6\right) \; \; \; \; \; \; \; \; {\left({d\varsigma}\right)}^2 = \frac{{a^2}}{{a^2 - r^2}}\left[{{\left({dw^1}\right)}^2 +{\left({dw^2}\right)}^2 +{\left({dw^3}\right)}^2}\right][/math]

This is the same expression as in Fischer's equation (12), only the symbols used to represent variables are different. (As explained in a previous post and repeated below, in Fischer's equations the [imath]\sigma[/imath] in [imath]d\sigma[/imath] has a different meaning to the [imath]\sigma[/imath] in the term [imath]\frac{{a^2}}{{a^2 - \sigma^2}}[/imath] ). However, by the very nature of the restrictions that enabled this simplification, it cannot be used as an expression for the general line-element of the hypersurface of the hypersphere because it excludes transverse-to-radial coordinate infinitesimals and thereby omits the transverse-to-radial contributions to the line-element. Not only does such a restricted formulation present the problem of lack of invariance of the form of the "restricted metric" under transformations to coordinate systems whose origins lie on neighbouring points, but it also presents problems for the definition of the Riemann tensor whether that be given by way of transport around a closed contour or by way of consideration of geodesic deviation. Differential geometry is formulated in terms of neighbourhoods of points and not in terms of points in isolation from their surroundings. Consideration of neighbourhoods is not the same as "extension to extended space". Appreciation of the difference is necessary for a proper comprehension of differential geometry.

The simplified expression cannot define a general line-element in a space of constant positive curvature, and neither can Fischer's equation (12). The status of proper definer of the spatial part of the line-element in a space of constant positive curvature belongs to the three expressions (1), (2) and (3) for the line-element given above.

Homogeneity includes radial symmetry, but it is more. It is radial symmetry about every point of a manifold. My impression was that Cruel2Bkind tried to reduce the definition of distance in homogeneous space to the radial distance in spherically symmetric systems.

A space is homogeneous if a translation has no effect on equations of motion. A distribution of mass/energy is homogeneous if it has the same value at every point.Within either use of the word 'isotropy' is not a necessary part of 'homogeneity'. A space can be homogeneous and non-isotropic. Homogeneity is not "radial symmetry" at every point. A space may have the property that at every point there is a preferred direction. If that direction is the same for every point in the space then the space may be homogeneous, but it does not have the property of being spherically symmetric about every point. Correct use of the language is essential to a proper understanding of general relativity.

I do not see, what should be problematic in the definition of a distance between points or in the definition of an infinitesimal distance element in curved space.

These are quantities, which do not depend on the choice of a special system of coordinates. It may be more or less laborious to calculate distances between points, depending on the coordinate system, which you use, but the distance will not change.

There is nothing inherently "problematic in the definition of a distance between points or in the definition of an infinitesimal distance element in curved space" when that distance is well-defined. However problems do arise when that distance is not well-defined. The concept of something needing to be 'well-defined' in order to be meaningful is a basic concept in mathematics.

I note with interest that Fischer fails to respond to my reasons for regarding [imath]\small{d\sigma^{2}=\frac{a^{2}}{a^{2}-\sigma^{2}}d\sigma^{2}}[/imath] as ridiculous.

The following argument comes from simple algebra, mastery of which is a prerequisite for understanding differential calculus. Comprehension of differential calculus is also a prerequisite for understanding differential geometry:

Given Fischer's equation " [imath]{\left({d\sigma}\right)}^2 ={\left({dx_1}\right)}^2+{\left({dx_2}\right)}^2+{\left({dx_3}\right)}^2[/imath] " and his equation (12) " [imath]{\left({d\sigma}\right)}^2=\frac{{a^2}}{{a^2-\sigma ^2}}\left({{\left({dx_1}\right)}^2+{\left({dx_2}\right)}^2+{\left({dx_3}\right)}^2}\right)[/imath] ", then [imath]{\left({d\sigma}\right)}^2 = \frac{{a^2}}{{a^2 - \sigma ^2}}{\left({d\sigma}\right)}^2[/imath].

If [imath]d\sigma \ne 0[/imath] then [imath]1 = \frac{{a^2}}{{a^2 - \sigma ^2}}[/imath] in which case [imath]\sigma = 0[/imath]. Since [imath]\sigma[/imath] is constant, [imath]d\sigma[/imath] must be zero, in contradiction to the assumption that [imath]d\sigma \ne 0[/imath].

Therefore either [imath]d\sigma = 0[/imath], or the [imath]\sigma[/imath] in " [imath]d\sigma[/imath] " is not the same [imath]\sigma[/imath] as that in " [imath]\frac{{a^2}}{{a^2 - \sigma ^2}}[/imath] ", or the [imath]\sigma[/imath] in "[imath]{\left({d\sigma}\right)}^2={\left({dx_1}\right)}^2+{\left({dx_2}\right)}^2+{\left({dx_3}\right)}^2[/imath] " means something completely different to the [imath]\sigma[/imath] in Fischer's equation (12).

In the first two cases, [imath]\sigma[/imath] necessarily equals zero which means that the space must have only one point and the concepts of connection coefficients and curvature tensors are inapplicable. The last two cases provide evidence for a lack of comprehension of the need for symbols to be unambiguous

Failure to appreciate such an argument can be regarded as indicative of a lack of comprehension of the required mathematics.

Again Cruel2Bkind makes the mistake of extending the local description in the form of differential properties to an extended volume [1], and thus arrives at inconsistencies. This naïve interpretation of differential geometry appears to be the main problem in all his discussions. But as he has discovered in the text below, the equations, which he derives [2], are not differential equations, but relations between values, between the values of some functions f and h and their derivatives at some fixed (but arbitrary) point. Relations over finite distances can be obtained only by integration along some path, regarding the values of the functions f and h and their derivatives as commoving information [3].

[1] This is a grossly inappropriate response. What I wrote has absolutely nothing to do with "extending the local description in the form of differential properties to an extended volume". Such an interpretation of what was written raises serious questions about the level of comprehension of the mathematics that must be mastered in order to understand differential geometry. Belittling a critic's comprehension of a topic does not constitute a valid response to a criticism that reveals a flaw in a mathematical presentation.

The flaw is made obvious by the following argument. It requires an appropriate response.

Given [imath]h = h\left( \sigma \right)[/imath] then [imath]\frac{{\partial h}}{{\partial x^1}} = \frac{{dh}}{{d\sigma}}\frac{{\partial \sigma}}{{\partial x^1}}[/imath] and [imath]\frac{{\partial h}}{{\partial x^2}} = \frac{{dh}}{{d\sigma}}\frac{{\partial \sigma}}{{\partial x^2}}[/imath] and [imath]\frac{{\partial h}}{{\partial x^3}} = \frac{{dh}}{{d\sigma}}\frac{{\partial \sigma}}{{\partial x^3}}[/imath].

From Fischer's equations for the Ricci components it must be the case that [imath]\frac{{\partial h}}{{\partial x^1}} = \frac{{\partial h}}{{\partial x^2}} = \frac{{\partial h}}{{\partial x^3}} = \frac{{dh}}{{d\sigma}}[/imath] , which is equivalent to [imath]\frac{dh}{d\sigma}\frac{\partial \sigma}{\partial x^1} = \frac{dh}{d\sigma}\frac{\partial \sigma}{\partial x^2} = \frac{dh}{d\sigma}\frac{\partial \sigma}{\partial x^3} = \frac{dh}{d\sigma}[/imath] , which is equivalent to [imath]\frac{{\partial \sigma}}{{\partial x^1}} = \frac{{\partial \sigma}}{{\partial x^2}} = \frac{{\partial \sigma}}{{\partial x^3}} = 1[/imath] , which is equivalent to [imath]\sigma = x^1 + x^2 + x^3[/imath].

So his metric has spatial components [imath]g_{ii}= \frac{a^2}{a^2-\left({x^1+x^2+x^3}\right)^2} [/imath] , which are

not

the spatial components of a metric for a space of constant positive curvature.

That argument is from differential calculus, one of the prerequisites for understanding differential geometry. Any claim that it is "extending the local description in the form of differential properties to an extended volume" is invalid and merely demonstrates that the claimant is clutching at straws in trying to defend the indefensible.

[2] I did not derive those equations. I took them from Fischer's paper and augmented them by adding the character-string "(0)" where appropriate. I did that in order to reveal what his expressions actually meant. He seems to understand that they are only values-at-a-point, but by attributing them to me he appears to be denying responsibility for their nature of not being differential equations but being only values-at-a-point and hence not being solvable.

[3] The invalidity of this statement is easily established:

Given [imath]h\left({\sigma}\right)=\frac{a^2}{a^2 - \sigma ^2}[/imath] so that [imath]h\left(0 \right)=1[/imath] and [imath]h'\left(0 \right)=0[/imath] then as the origin is moved along a finite line segment the 'comoving' values remain at [imath]h=1[/imath] and [imath]h'=0[/imath]. Therefore, along a finite straight-line segment of length [imath]L[/imath] from the original point, integration of [imath]h'=0[/imath], with the initial condition [imath]h\left(0\right)=1[/imath] yields [imath]h\left(\sigma\right)=1[/imath] for [imath]0\le \sigma \le L[/imath], which contradicts the original definition for [imath]h[/imath].

Therefore it is not true that "relations over finite distances can be obtained only by integration along some path, regarding the values of the functions f and h and their derivatives as co-moving information". Obtaining such relations requires an initial choice of coordinate system, and an adherence to that coordinate system as the integration is performed.

Here Cruel2Bkind tries to extend the solution at s=0 to extended spatial sections in contradiction to the rules of differential geometry.[1] Integration of the field equations in curved space to obtain geodesic lines requires a continuous shift of the coordinate system, so that the direction of change remains tangential to the actual direction of motion.[2]

[1] Such a response to being informed of an error in one's calculations is not only inappropriate but also invalid. Nothing in what I wrote has anything to do with an attempt "to extend the solution at s=0 to extended spatial sections".

Besides which there is no such thing as a solution to a differential equation that only applies at one value of the independent variable. There are no such rules in differential geometry because there are no such non-extendable solutions of differential equations applicable to only single points.

[2] Integration of the field equations delivers the components of the metric tensor, it does not yield the geodesic lines. When the metric components are found, geodesics can be obtained without any such "continuous shift of the coordinate system".

The expressions for the Ricci components

[imath]R_{00}=\frac{{3{f'}^2 }}{{4fh}}-\frac{{3f''}}{{2h}}-\frac{{3f'h'}}{{4h^2 }}[/imath]

[imath]R_{ij}=-\frac{h''}{2h}+\frac{{3{h'}^2}}{{4h^2}}-\frac{f''}{2f}+\frac{{f'}^2}{{4f^2}}+\frac{f'h'}{2fh} \; \; \; (i,j=1,2,3)[/imath] & [imath](i \ne j)[/imath]

[imath]R_{ii}=-\frac{2h''}{h}+\frac{{3{h'}^2}}{{2h^2}}-\frac{f''}{2f}+\frac{{f'}^2}{{4f^2}}-\frac{f'h'}{4fh} \; \; \; (i=1,2,3)[/imath]

were all obtained by standard algebraic manipulation techniques of expanding Einstein summation, differentiating quotients, applying the distributive law to products involving bracketed summations and simplifying by adding/subtracting like terms. Obviously being only manipulation methods employed to simplify complex expressions, like those used by Fischer himself to get his equation (15) which is the same as the equation for [imath]R_{ii}[/imath] above, criticisms such as "extend(ing) the solution at s=0 to extended spatial sections" are inappropriate, invalid, and inept.

The equations for [imath]R_{00}[/imath] and [imath]R_{ij}[/imath] are no more than what Fischer should have found had he been diligent.

Again here the application to extended space-time of the local definition of curvature with its differential connections between neighboring points leads to nonsensical results like the existence of horizons and singularities. The consequences, which Cruel2Bkind draws from his wrong interpretation of the field equations, are not valid.[1] The solution is homogeneous with equal properties at every point of the manifold.[2]

The assumption that the energy density must be negative is wrong.[3] It may be caused by the sign error in my eq.(19).[4] But apart from the fact that the time scale changes with distance, the solution of the field equations discussed in my paper is identical with Einstein’s static universe, where the coefficient ? is chosen just in the way to obtain the correct positive energy density in the Newtonian limit.[5] It is a valid solution, which is in agreement with the observed properties of space-time, not only with respect to red shift, but which also is stable without requiring something like dark energy.

[1] The inept misrepresentation that there is an "application to extended space-time of the local definition of curvature with its differential connections between neighboring points" seems to be nothing more than an attempt to divert attention from gross errors in conceptualization and calculation. Moreover in developing his own solution for [imath]f[/imath] Fischer should criticise himself for the same reasons if he has any claim to consistency. But the fact is that there such a criticism is quite inappropriate. He could have leveled a valid criticism as explained below, but he has missed his chance of doing that. Merely claiming that I have a "wrong interpretation of the field equations", without providing detailed analysis of every mathematical equation I have given, does not suffice to warrant the claim of invalidity. However it is quite apparent that Fischer has a rather confused interpretation of the field equations. The stress-energy tensor is purely determined by matter/radiation. All of its components must have realistic values; if it transpires that the field equations have solutions only if one or more of those components have values that could not occur in the real universe, then the metric that gave rise to those field equations has no application to the real universe.

[2] I have already proven that the energy density is not homogeneous. Mere contradiction does not suffice as disproof.

[3] There was no "assumption" that the energy density must be negative. That result was derived from his connection coefficients by legitimate mathematical procedures. It requires a legitimate response.

[4] It was not caused by any sign error in his paper. In referring to choosing the sign of [imath]\kappa[/imath] so as to "obtain the correct positive energy density in the Newtonian limit" is he revealing that he was guilty of assuming what he was trying to prove.

However I must admit to making an error in my previous post on 21 April 2011. The solution for [imath]f[/imath] given therein does not satisfy the field equation [imath]R_{ij}=0[/imath] (for [imath]i \ne j[/imath] and [imath]i,j \ne 0[/imath]). My mistake was in assuming that the pressure could be homogeneously zero and that the equation involving pressure and [imath]f[/imath] could be solved for [imath]f[/imath]. However the fact that the proposed solution for [imath]f[/imath] does not satisfy [imath]R_{ij}=0[/imath] proves that even though the pressure must be zero at the origin (which is an unrealistic value as the pressure of the ubiquitous cosmic microwave background radiation although tiny is not zero) the pressure elsewhere is a function of the distance from the origin; it is not homogeneous.

There are three field equations in three unknowns [imath]f , \varepsilon , p[/imath]

[imath]G_{00}=\kappa T_{00}[/imath] [imath]\equiv[/imath] [imath]\frac{3fh''}{h^2}-\frac{9f{h'}^2}{4h^3} = -\kappa \varepsilon f[/imath]

[imath]G_{ii}=\kappa T_{ii}[/imath] [imath]\equiv[/imath] [imath]\frac{f''}{f}-\frac{{f'}^2}{2f^2}+\frac{h''}{h}+\frac{f'h'}{2fh}-\frac{3{h'}^2}{4h^2}=\kappa ph[/imath]

[imath]G_{ij}=\kappa T_{ij}[/imath] [imath]\equiv[/imath] [imath]\frac{3{h'}^2}{4h^2}-\frac{h''}{2h}+\frac{{f'}^2}{4f^2}-\frac{f''}{2f}+\frac{f'h'}{2fh}=0[/imath]

Of the three equations, it is the last that involves only the unknown [imath]f[/imath] and the known [imath]h[/imath] that must be solved for [imath]f[/imath]. The first two equations give the other two unknowns viz. the pressure and the energy density, neither of which can be homogeneous.

[5] The 'solution', apart from the distance dependency of the time-scale, in his paper may be identical with Einstein’s static universe, but it is not a solution of the corrected field equations developed from his connection coefficients. He has compounded errors to get from his erroneous connection coefficients to expressions for the field equations that resemble those for Einstein's static universe and then claims that because his 'solution', excluding the time-scale, is identical to that of Einstein's static universe, his procedure must be correct. Such efforts leave much to be desired.

One more observation of Fischer's 'work' needs voicing. Consider his expression for the curvature scalar. At the origin where [imath]\sigma =0[/imath] the value for the three-dimensional curvature scalar derived from his connection coefficients is negative. The three-dimensional curvature scalar is found from the expression [imath]3R_{ii}g^{ii} \; , \; \left({i \ne 0}\right)[/imath]. At the otigin where [imath]h'\left(0 \right)=0[/imath] the value is [imath]-\frac{6}{a^2}[/imath], as is clearly seen in Fischer's equation (18). The value for the 3d scalar, using the corrected value for [imath]R_{00}[/imath] and then replacing [imath]h[/imath] by [imath]\frac{a^2}{a^2-{\sigma}^2}[/imath]is:

[imath]{}^{3d}R = \frac{9{h'}^2}{2h^3}-\frac{6h''}{h^2}=- \frac{6\left[{{2a}^2 + 3\sigma ^2}\right]}{a^2 \left[{a^2 - \sigma ^2}\right]}[/imath]

This reveals that the three dimensional curvature scalar is not even constant. In anticipation of the criticism that in claiming it is not constant I am applying the local definition of curvature to extended space-time, let me point out that Fischer's assumption, that because the curvature has a certain value at the origin then it must have that value at every other point, is actually an invalid application of a local value of curvature to extended space-time.

(It should also be noted that Fischer sees no contradiction in extending the value of a differential equation, valid at only one point, to extended space and then extending his 'solution' for [imath]f[/imath], only valid at [imath]\sigma =0[/imath], to extended space.)

So Fischer's space is inhomogeneous in both pressure and energy density, and it has a non-constant negative curvature. However the three dimensional curvature scalar derived from the metrics given near the start of this post is indeed positive and constant.

In an earlier post I proved that if both of a pair of observers see light from each other as red-shifted then they cannot be mutually stationary. The result also follows that if one of a pair of mutally stationary observers sees light from the other as red-shifted then the other must see light from the first as blue-shifted. If a metric gives rise to field equations that result in a situation that mutually stationary observers can see the light from each other as red-shifted then either the metric is invalid, or a mistake has been made in developing the field equations, or a mistake has been made in solving them, or a mistake has been made in solving the null-geodesic equation. I await either a rigorous disproof of these assertions or a valid counter-example.

cruel2Bkind · May 13, 2011

Certainly, in a flat Newtonian, Euclidean or Minkowski universe cosmological redshift and time dilation do not occur (unless the universe is expanding for whatever reason).

Most certainly too, the geometry of curved spacetime (such as that of an Einstein universe) affects the electromagnetic spectrum.[1] It would be ludicrous to imagine a general relativistic universe (of constant positive or negative Gaussian curvature) where the propagation of light remains unaffected from the point of view of an observer as she peers into the deep cosmos... [2]

Einstein's general postulate of relativity (GR) did away with flat spacetime, and in doing so (simultaneously) introduced the notion (dare I say empirical fact) that light is affected by the geometry of space, i.e., spacetime curvature alters spectral lines (leading to redshift and time dilation). [3]

[1] Obviously by such a comment you are confused about Einstein's original model.

[2] Apparently you are not familiar with the mathematics needed to determine whether the frequency of EMR has changed from its emission to its reception.

[3] Find out why Einstein rejected his original static model and regarded his invocation of the cosmological constant as his greatest mistake.

[1] GR makes the general claim. I merely support it.[1]

[2] Sufficient observation time needed to distinguish between the two effects (e.g., to actually 'see' galaxies move radially via an increase in redshift) would be on the order of millions of years. So yes, it is in principle possible, but in all practical purposes impossible.[2]

[1] GR makes no such claim! Check your facts!

[2] Theoretical validity is not contingent on human observation time. You claim to be a science writer but show that you realy do not understand science.

Too many if's. [1]

Everyone agrees that it should, could or would be possible in principle to differentiate between a Doppler effect and a gravitational effect, but I don't think anyone wants to wait millions of years for the answer. :)[2]

EDIT> Judging from redshift alone there is no way to differentiate between cosmological expansion (Doppler effect) and cosmological gravitational redshift (due to the curvature of space). There are however ways to make the correct interpretation, without having to wait millions of years to see the redshift of any given galaxy increase over time. As mentioned above, the Hubble's program (1926-1934) set out to find space curvature from the galaxy count: the Gauss protocol (Hubble, Tolman, 1935, ApJ 82, 302). There are other ways as well, related to angular diameter distance, luminosity distance, etc.

[1] A rather trite comment inclining one to suspect that the commentator has some difficulty in comprehending the argument.

[2] Theoretical validity is not contingent on human observation time.

Your arguments were based on the erroneous proposition k>1. It was erroneous because the situation was not symmetric. In other words, you began you series of calculations with a false premise, an incorrect proposition.

Since your premise (proposition, or assumption) was not correct, the conclusion drawn was in error. Though your argument may have been logically valid, your conclusion was demonstrably wrong, because its first premise was false. A logical analysis might not reveal the error in your argument if that analysis accepted as true the argument's premise.

The truth of your premises must be first be established otherwise what follows can be interpreted as pure nonsense.

... The idea that cosmic redshift has a gravitational cause, and in an homogeneous universe can be expressed as a function of distance alone, ... assuming such a cause of redshift ...
...
So let observer A send a signal to B and wait a time a₁ before receiving the return signal from B.
A immediately resends to B.
After having received the first signal B immediately sends a signal to A and waits for a time b₁ to receive the second signal from A.
Since B sees A as red-shifted then

b₁/a₁ > 1 .

Let k = b₁/a₁. So each will see the other's time as dilated by a factor of k.
...

Obviously the only assumptions I made were that cosmic redshift was caused by spatial curvature and that a pair of far distant mutually stationary observers in a curved space would observe each other as being red-shifted. I then constructed an argument showing that those assumptions lead to a contradiction. If a set of assumptions leads to a contradiction then at least one of the assumptions must be wrong.

Notice that by definition of k, if k=1 then the times each observer has to wait before receiving a return signal from the other observer would have to be equal and in that case B would not see A as red-shifted.

A sends two signals with a time interval of a₁ (measured in A's time frame) between them.

B receives two signals from A with a time interval of b₁ (measured in B's time frame) between them.

By definition of k as the ratio of b₁:a₁, if k=1 then B's elapsed time equals A's elapsed time. In which case B cannot see A as time dilated, and therefore B cannot see A as red-shifted.

"k=1" means there is no observed red-shift. "k>1" means there is an observed red-shift. Since there is the assumption that there is an observed redshift then one must first consider that k>1.

If you still have difficulty in appreciating these initial points in the setting-up of my argument then you may benefit from undertaking a refresher course in junior high-school mathematics.

[1] I don't believe that "cosmic redshift has a curved space cause" any more than I believe that cosmic redshift has a Doppler cause. [1]

It has been merely pointed out that, empirically, the two interpretations are viable to certain extents (see the OP).

[2] Just as it is possible to extrapolate the Doppler effect as the cause of cosmological redshift, the gravitational redshift can be extrapolated as a cause of cosmological redshift. [2] The existence of such a frequency shift has been established and is embodied within Einstein's general theory of relativity.

[1] Quotes from http://coldcreation.blogspot.com/2010/09/redshift-z.html: A General Relativistic Stationary Universe:

There are two possible interpretations for cosmological redshift z : (1) A change in the scale factor to the metric... (2) The general relativistic curved spacetime interpretation (implying a static metric in a stationary universe). In addition to illuminating how redshift z is caused in a globally curved four-dimensional spacetime manifold ...It is emphasized that global curvature plays an essential role in cosmology and provides a natural explanation for various empirical observations. Too, it is exemplified this point of view by considering a novel version of Einstein's 1916-1917 world-model, where cosmological redshift z is directly related to the large-scale structure of the universe.

...

redshift z observed in the spectra of astronomical objects may be due to a curved spacetime phenomenon..

...

The conclusion is that redshift z is caused as electromagnetic radiation propagates along geodesic paths...

This is an attempt to make the geometric attributes of the spacetime manifold itself (the metric) account for the observed redshift z.

...

when gravity is considered a 4-dimensional geometric phenomenon, we assume (for now) that global curvature would not induce the displacement of massive bodies towards one another, but that electromagnetic radiation (EMR) would be affected by such a curvature (as seen by redshift z and time dilation)

...

... This gravitational redshift-like effect (or cosmological redshift z) is produced as light propagates through the global Gaussian curvature of the field, not as light escapes the local field of a massive object.

What concerns us now is the general relativistic notion of cosmological redshift z and time dilation with the look-back time in a static homogeneous and isotropic manifold associated with a pseudo-Riemannian curvature of spacetime. In this case, both redshift z and time dilation are observable effects relative to the rest-frame of an observer as she looks out into a universe that exhibits nonzero Gaussian curvature.

...

... electromagnetic radiation, propagating through a four-dimensional (pseudo-)Riemannian spacetime continuum of constant (positive or negative) Gaussian curvature, will travel a geodesic, while massive bodies will not be confined to propagate along the same geodesic. Objects do not have to follow the shortest path between two points of a great circle (for example). And due to the distortion along the geodesic path of the photon, spectral lines will be redshifted when viewed from the rest-frame of an observer (the greater the distance, the higher the redshift).

...

Conclusion: Redshift z occurs in a non-expanding universe, and increases with distance from an observer. Global stability is maintained against gravitational collapse. The universe does not expand or collapse.

...

The distortion in the path is the cause of cosmological redshift z in a static Einstein universe. There is a loss of energy associated with increasing distance of propagation from the observer in the non-Euclidean manifold: the result is redshift z and time dilation. This is exactly what would be observed from the rest-frame of any observer located anywhere in the Einstein static four-dimensional manifold of constant positive intrinsic Gaussian curvature.

...

Sorry my mistake; because you were advocating a curved-space cause for cosmic red-shift, I thought you advocated a curved-space cause for cosmic redshift.

[2] So you continue to claim, but without any supportive mathematical proof.

The only static model where light is not increasingly redshifted with distance is the flat, Euclidean/Newtonian/Minkowski model.

Thus your thus is invalid is invalid. :)

Obviously you are unaware of the actual structure of Einstein's static model. In that model, light is not increasingly redshifted with distance. Why else would Einstein have abandoned it after hearing of Hubble's observations? Besides which, check out the metric for yourself and have a look at the null-geodesic equation. If you have any real understanding of GR you can then see that there is no "redshift-with-distance" in the ESM.

In Einstein's static model, for a signature of -+++, the metric tensor is diagonal, [imath]g_{00}=-1[/imath] , and no component of the metric tensor is dependent on the time coordinate. Therefore the [imath]\Gamma_{\mu \nu}^0 [/imath] are all zero for all [imath]\mu , \nu[/imath]. Consequently the time component of the null-geodesic equation for the wave-vector [imath]\frac{dk^0}{d\lambda}+\Gamma_{\mu \nu}^0 k^\mu k^\nu=0[/imath] simplifies to [imath]\frac{dk^0}{d\lambda}=0[/imath] ; the frequency does not change; there is no cosmological red-shift in the ESM.

coldcreation · May 15, 2011

[1] Obviously by such a comment you are confused about Einstein's original model.
[2] Apparently you are not familiar with the mathematics needed to determine whether the frequency of EMR has changed from its emission to its reception.
[3] Find out why Einstein rejected his original static model and regarded his invocation of the cosmological constant as his greatest mistake.

Einstein himself was confused about his original static model. Or more precisely, there was a difficulty (that began as early as 1916) in constructing a simplified model of the real universe based on the general principle of relativity. He obviously was not the only one to struggle with the relation between gravitational theory and cosmology. And that confusion persists to some extent still today.

The current understanding of general relativity in the domain of strong gravitational fields, small scales and highly dynamical configurations are not the only areas that remain limited. Perhaps most importantly, there remains a misunderstanding of general relativity in domains where large distances and cosmological time-scales are relevant. New methods for numerical calculations and analysis of the solutions to the Einstein field equations under general assumptions and on large scales will be required for further theoretical progress.

In another way, despite a few rich specific and general results collected over the past 96 years there remains a large and potentially most important part of general relativity (on mathematical-geometrical, theoretical and observational fronts) we have limited or no access to. (See for example Is general relativity essentially understood? Friedrich, H., 2005) (See too Ellis G.F.R., 1998)

Evidently, further studies need to be carried out on all the above fronts to determine whether the frequency of EMR changes from its emission to its reception in an Einstein universe of constant positive Gaussian/Riemannian curvature. Ernst Fischer has shown quantitatively, and Coldcreation qualitatively, how such a process might work. Obviously more research across many fronts to confirm or refute such predictions will be required.

It is very important to review critically the potential successes and failures of cosmological models, despite the high degree of sophistication that some of these currently enjoy (e.g., ΛCDM). The degree of realism of current models may be significantly over-idealized.

Key features in this developing synthesis are the effects of spacetime curvature on matter and radiation; with Einstein's general relativity the principle element in linking together the relation between astronomical observations and cosmological theory.

It is evident too, in light of the problems dealing with analysis of the Einstein field equations and their solutions, that Einstein's rejection of the static model (with or without lambda) may have been premature. That is not to say that the original Einstein model should remain intact without modification, but that further studies need to be implemented to gain a complete (if at all possible) understanding of the essence of the physical universe, its global geometrical structure and its evolution in time.

...according to general relativity, there is no unique method by which vectors at points separated by great distances can be compared in a curved spacetime, i.e., a definition of curved spacetime is the inability to compare vectors at different points. Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift. The interpretation of cosmological redshift z remains open.

Generally, in a curved spacetime manifold, the observed shift in frequency of a photon can be interpreted as a kinematic effect or a gravitational frequency shift (or even both together, superimposed), depending on the choice of coordinates.

[1] GR makes no such claim!

[2] Check your facts!

[3] Theoretical validity is not contingent on human observation time.

[4] You claim to be a science writer but show that you realy do not understand science.

[1] GR can be interpreted in a variety of ways.

[2] In cosmology there are very few facts (if any) to deal with.

[3] Theoretical validity of the type you mentioned above ["If a series of signals could be sent to a galaxy, and be immediately returned on reception by observers there, then the galaxy's relative radial velocity could be determined"] is contingent on human observation time.

[4] Belittling your opponent in order to invalidate an argument is logically fallacious because insults the opponent's personal character and has nothing to do with the logical merits of the arguments or assertions here under review.

PS. I'll be back for the rest of your post.

CC

cruel2Bkind · May 16, 2011

Evidently, further studies need to be carried out on all the above fronts to determine whether the frequency of EMR changes from its emission to its reception in an Einstein universe of constant positive Gaussian/Riemannian curvature. [1] Ernst Fischer has shown quantitatively, and Coldcreation qualitatively, how such a process might work. [2] Obviously more research across many fronts to confirm or refute such predictions will be required.
...
... It is evident too, in light of the problems dealing with analysis of the Einstein field equations and their solutions, that Einstein's rejection of the static model (with or without lambda) may have been premature. [3]...

[1] Einstein constructed his static model in such a way as to have no cosmological redshift. The [imath]g_{00}=-1[/imath] for a metric signature of ' [imath]-+++[/imath] ' ensures that there is no redshift.

[2] As I have been proving in my posts above, Ernst Fischer has shown no such thing. He has not presented a case for a universe of constant positive Gaussian/Riemannian curvature. But he at least recognizes that the Einstein static model does not involve a cosmological redshift.

[3] Since Einstein's model involved a "cosmological fluid" that was homogeneous, and a space that was isotropic at every point, and no redshift, then, given Hubble's results, his model was obviously wrong. So his rejection of it can hardly be called premature.

[1] GR can be interpreted in a variety of ways. [1]

[2] In cosmology there are very few facts (if any) to deal with.

[3] Theoretical validity of the type you mentioned above ["If a series of signals could be sent to a galaxy, and be immediately returned on reception by observers there, then the galaxy's relative radial velocity could be determined"] is contingent on human observation time.[2]
[4] Belittling your opponent in order to invalidate an argument is logically fallacious because insults the opponent's personal character and has nothing to do with the logical merits of the arguments or assertions here under review. [3]

[1] GR makes no claim that there is an inherent inability to distinguish gravitational redshift from doppler redshift. It cannot be interpreted as making such a claim.

[2] Previously:

in post #748 on 29 March 2011 - 02:23 AM, you said:
"Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift. "
in post #751 on 30 March 2011 - 04:02 AM, you said:
"Galaxies are not seen to move. The interpretation of radial motion comes from redshift, which can be interpreted as either a Doppler effect, expansion or stretching of space, or a curved spacetime effect."
in post #757 on 06 May 2011 - 08:15 PM, you said:
"Sufficient observation time needed to distinguish between the two effects (e.g., to actually 'see' galaxies move radially via an increase in redshift) would be on the order of millions of years. So yes, it is in principle possible, but in all practical purposes impossible."

Theoretical physics is concerned with what is in principle possible, it is not constrained by the dictates of practical puposes. If something of relevance to a theory is in principle possible then the formulator of the theory would be quite remiss in neglecting to include considerations of that something-of-relevance.

Such a topic as GR cannot be approached from a perspective that limits itself to what is humanly possible at this time. Although the existence of observational procedures of several millions of years duration, or observers with lifespans of millions of years, may seem very unlikely, nevertheless the possiblity of such observation times must be entertained in order to determine the validity of some hypotheses.

If cosmological redshift was only a Doppler effect or was due to a cosmological "time-dilation with distance" then it would not be expected to increase with time, but if it was due to expansion of space in such a way as to contradict Hogg and Bunn then it would increase over time. However If the cosmological redshift was due to a cosmological time-dilation then the angular width, as seen from earth, of far distant galaxies would not be expected to decrease, but if the redshift was either Doppler or space-stretch then the angular width would be expected to decrease.

So the hypothesis, that "in cosmology there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift", is false.

You have conceded such changes are in principle detectable but dismiss them on the grounds of impracticability. Dismissing the need to consider such in-principle-detectable changes merely because humans cannot detect them, is displaying a fundamental lack of understanding of the nature of theoretical physics.

[3] I did not "belittle" you "in order to invalidate" your "argument". I invalidated your statement, and then made a value judgment of your comprehension of science. In your previous posts you had already indulged in derision of my comprehension of and capabilities in mathematics, logic and GR, in what appeared to be attempts to invalidate my arguments. And you showed no reticence in posting Ernst Fischer's belittling comments. If you don't like it then don't deal in it.

If you agree to be more careful with your language in future then I will have no difficulty in matching your politeness.

cruel2Bkind · May 17, 2011

ERRATA

One more observation of Fischer's 'work' needs voicing. Consider his expression for the curvature scalar. At the origin where [imath]\sigma =0[/imath] the value for the three-dimensional curvature scalar derived from his connection coefficients is negative. The three-dimensional curvature scalar is found from the expression [imath]3R_{ii}g^{ii} \; , \; \left({i \ne 0}\right)[/imath]. At the otigin where [imath]h'\left(0 \right)=0[/imath] the value is [imath]-\frac{6}{a^2}[/imath], as is clearly seen in Fischer's equation (18). The value for the 3d scalar, using the corrected value for [imath]R_{00}[/imath] and then replacing [imath]h[/imath] by [imath]\frac{a^2}{a^2-{\sigma}^2}[/imath]is:

[imath]{}^{3d}R = \frac{9{h'}^2}{2h^3}-\frac{6h''}{h^2}=- \frac{6\left[{{2a}^2 + 3\sigma ^2}\right]}{a^2 \left[{a^2 - \sigma ^2}\right]}[/imath]

The three-dimensional Ricci tensor should have been specified and reference to [imath]R_{00}[/imath] should not have been made as it is not part of the 3d Ricci tensor. The passage should have been:

One more observation of Fischer's 'work' needs voicing. Consider his expression for the curvature scalar. At the origin where [imath]\sigma =0[/imath] the value for the three-dimensional curvature scalar derived from his connection coefficients is negative. The three-dimensional curvature scalar is found from the expression [imath]{}^{3d}R_{\mu \nu}g^{\mu \nu} \; , \; \left({\mu , \nu \ne 0}\right)[/imath] (Einstein summation for [imath]\mu ,\nu[/imath]), where
[math]{}^{3d}R_{ab}=\Gamma_{ab,m}^m - \Gamma_{am,b}^m + \Gamma_{mn}^n \Gamma_{ab}^m - \Gamma_{ma}^n \Gamma_{nb}^m \; , \; \left({a,b,m,n \ne 0}\right)[/math]
is the three-dimensional Ricci tensor. At the origin where [imath]h\left(0 \right)=1 \; ,\; h'\left(0 \right)=0 \; ,\; h''\left(0 \right)=\frac{2}{a^2}[/imath] the value is [imath]-\frac{6h''}{h^2} = -\frac{12}{a^2}[/imath] , as is clearly seen from Fischer's equation (18) and the conditions in the sentence that follows it. The value for the 3d scalar at other points, after replacing [imath]h[/imath] by [imath]\frac{a^2}{a^2-{\sigma}^2}[/imath] , is:

[imath]{}^{3d}R = \frac{9{h'}^2}{2h^3}-\frac{6h''}{h^2}=- \frac{6\left[{{2a}^2 + 3\sigma ^2}\right]}{a^2 \left[{a^2 - \sigma ^2}\right]}[/imath]

coldcreation · May 23, 2011

Obviously the only assumptions I made were that cosmic redshift was caused by spatial curvature and that a pair of far distant mutually stationary observers in a curved space would observe each other as being red-shifted. I then constructed an argument showing that those assumptions lead to a contradiction. If a set of assumptions leads to a contradiction then at least one of the assumptions must be wrong.

Notice that by definition of k, if k=1 then the times each observer has to wait before receiving a return signal from the other observer would have to be equal and in that case B would not see A as red-shifted.
A sends two signals with a time interval of a₁ (measured in A's time frame) between them.
B receives two signals from A with a time interval of b₁ (measured in B's time frame) between them.
By definition of k as the ratio of b₁:a₁, if k=1 then B's elapsed time equals A's elapsed time. In which case B cannot see A as time dilated, and therefore B cannot see A as red-shifted.
"k=1" means there is no observed red-shift. "k>1" means there is an observed red-shift. Since there is the assumption that there is an observed redshift then one must first consider that k>1.

You seem to be confusing the concept of local gravitational redshift and time dilation (of the type that occurs in an inhomogeneous gravitational field of a massive body, where observers are located at different altitudes) with the concept of a global redshift z and time dilation observed in a homogeneous and isotropic curved spacetime manifold. The latter situation is entirely symmetric; meaning that both observers far-removed from one another will observe shifts in spectral lines towards the less refrangible end of the spectrum (redshift z) and the associated time dilation factor. Each observer will see the other (far-removed) as if she was immersed at a lower altitude in a gravitational field.

In other words, cosmological redshift and time dilation occur even though the times each observer has to wait before receiving a return signal from the other observer is equal (just as in a homogeneous expanding universe). This is entirely a relativistic effect, attributed to the propagation of electromagnetic radiation through curved spacetime.

Your claim that no redshift or time dilation would occur is identical to claiming that curved spacetime has no affect whatsoever on the propagation of EMR.

This general relativistic concept can best be appreciated by analyzing the following schematic diagram:

Figure 1Cb

General Relativistic Spacetime Manifold of constant positive Gaussian curvature. This is a cross section of the visible universe, with the observer located at the origin (center). This is a homogeneous and isotropic, stationary, static Einstein universe, where redshift z is caused by a globally curved spacetime. The spherical shells (here seen as concentric circles viewed on an oblique angle centered on the observer at the origin) are each spaced approximately 1.5 Gly (gigalight-years) apart from one another. For the observer at the origin the local region appears Minkowskian (this would translate to an area roughly the size of the Local Group relative to our rest-frame).

This spacetime manifold of constant positive curvature differs from it's Euclidean cousin, in that distances measured from the origin (here is spherical shells) appear to become smaller the further removed. The result is that objects are not at their 'Euclidean distance.' Objects are further than would be indicated judging by redshift alone. The further the object, the more this difference manifests itself. For example, distant SNe Ia would appear unexpectedly dim, giving the impression they are further away than their redshifts indicate.

Light from very remote objects takes longer to reach Earth than would be expected in a flat spacetime—as if time and space (and the light propagating through it) were continually ‘stretched’ with increasing distances. This deviation or departure from linearity has been interpreted as an accelerated expansion, but in the context here, would be interpreted as evidence of spacetime curvature. Locally, spacetime appears flat. The larger the area (or distance) considered, the greater the departure from linearity. Likewise, the further the emitting source, the greater the redshift and time dilation.

At first glance, it would seem that an observer situated at the origin (observer A) occupies a privileged location, but this is not the case. Since the universe in this simplified model of constant curvature is homogeneous and isotropic, all observers will view the universe as if centered at the origin. All events removed from the local vicinity (of any observer, e.g., observer B in your infamous thought experiment) transpire in the look-back time (just as in an expanding universe).

In other words, the situation for both observer A and observer B is symmetric, and yet both will observe spectral lines emitted from the other as shifted towards the red end of the spectrum, with an associated time dilation factor. The further the distance from the observer, the slower clocks will appear to run. There is a difference in the passage of proper time at different locations relative to the photon arrival as described by a metric tensor of spacetime, relative to the past light-cone of each individual observer.

Observer A, located at the origin of her coordinate system (at the center of Figure 1Cb), sees B as redshifted and time dilated. Observer B, located at the origin of her coordinate system (at the center of a schematic diagram similar to Figure 1Cb), sees B as redshifted and time dilated.

This, in effect, translates into an apparent difficulty in determining whether photons are redshifted as a result of expansion (radial motion away from the observer), of redshifted as a results of a curved spacetime phenomenon, since observationally there are equivalent, judging from redshift alone. The universe appears either static and curved, or expanding and flat (or very nearly flat).

It is interesting to notice that in Figure 1Cb, if the redshift is interpreted as a relativistic Doppler effect, the comoving distances marked by spherical shells can be seen as due to expansion, where the scale factor is accelerating near the origin, and expanding more slowly with increasing look-back time. Differentiating between opposing interpretations based on redshift alone is impossible.

In sum, cosmological redshift z observed in a static universe is likely caused by the geodesic trajectory of photons in a curved spacetime. Generally speaking, the large-scale geometric structure of the spacetime continuum distorts the wavelength of every photon in direct proportion to the curvature of space in the elapsed time interval. Redshift is interpreted as a function of both distance and time. According to any observer located at the origin (the rest-frame of an observer) in a static curved spacetime continuum, the distance that separates the source and the observer will not equal the proper distance (or actual distance) between the emitting source and the observer. The wavelength of each photon is lengthened (as if 'stretched') between emission and reception. In another way, the energy of every photon is reduced, degraded, in direct proportion with the spatial distance traveled and the amount of time it takes for the photon to arrive at the observer.

In this static model, redshift occurs not because galaxies are moving away from the observer, nor because of the expansion of space itself, but because of the curvature of spacetime in accord with Einstein's geometric interpretation of gravitation. The curvature of this Einstein manifold is a direct result of the gravitating mass-energy density contained in the universe.

If you still have difficulty in appreciating these initial points in the setting-up of my argument then you may benefit from undertaking a refresher course in junior high-school mathematics.

I have no difficulty in appreciating the initial points in the setting-up of your argument. So your refresher course won't be necessary.

The error in thinking (in your thought experiment) revolves around the time element relative to the observers on the manifold of constant Gaussian curvature. Certainly, all points on the manifold are equivalent (i.e., the situation is symmetric). But the time element is not the same for observers sufficiently separated, as Ernst Fischer has rightly pointed out on many occasions. Every observer has the right to consider herself located at the origin of the coordinate system, and in fact has no choice observationally. Clocks appear to run slower in the vicinity of distant objects (in the look-back time), when compared to local clocks in the rest frame of the observer, by virtue of the intrinsic spacetime curvature of the manifold in which our observers are located. The photon path is a geodesic, i.e., there is a departure from linearity in the photon path. There is a distortion in the space through which the photon propagates that affects the spectral lines, shifting them towards the red end of the spectrum. The popular concept of 'cosmic time' is meaningless in such a universe.

This redshift is not something that would occur in a flat Euclidean, Newtonian or Minkowski universe! In such a universe, the photon wavelength suffers no distortion during travel-time. Redshift does not occur, and time dilation is nonexistent. Intensity, as expected, is diminished inversely and proportionally to the square of the distance.

When the observer immersed in a (pseudo-) Riemannian manifold of constant positive curvature measures distances in the look-back time—based on redshift and supplemented when possible with angular diameter distance, luminosity distance, comoving distance, cosmological proper distance—objects will notappear at the distances expected in a flat Minkowskian universe. (Of course there is no fool proof method for determining distance).

In addition to the increasingly non-Euclidean character of the spatial distribution of objects with distance, there is an associated distortion in the light travel time (temporal intervals) with increasing distance. This is simply the phenomenon of time dilation. Temporal changes in the spectra of distant objects are slower than those of nearby objects., i.e., their spectra exhibit slower temporal evolution (by a factor of (1 + z) than those of nearby objects. Taken at face value, the presence of the time dilation factor of (1 + z) in the SNe Ia data is simply a reduction in the flux density by more than the inverse square law. So the effective distance, from the view-point of any observer, does not behave like a Euclidean distance with increasing redshift.

There is a direct empirical correlation between redshift z as interpreted as a relativistic Doppler effect and the interpretation of redshift z as a curved spacetime phenomenon (with both minor and major differences in physical outcome).

In both the relativistic Doppler interpretation and the spacetime curvature interpretation of redshit z there is an associated time dilation factor, and both have nothing to do with velocity at all. All redshift z gives us⎯when interpreted as a general relativistic phenomenon⎯is a clue as to the relative spatial and temporal separation between us (from the rest-frame of the observer as she peers into the look-back time) and distant objects, when and where the electromagnetic radiation was emitted compared to with the actual distance and time now, i.e., redshift is a measure of curvature, since spatial increments and temporal intervals deviate from linearity with distance in the look-back time.

Note, as explained previously in this thread, it has often been concluded that the Tolman surface brightness test (a test said to distinguish between expanding and static models) is "consistent with the reality of the expansion" (for a flat geometry and uniform expansion over the range of redshifts observed).

This test has generally pitted expanding models against simple static models with a flat geometry. The models proposed here by Ernst Fischer and Coldcreation transpire in a curved spacetime manifold. In qualitative terms, the rate at which photons are received is reduced as each photon travels a geodesic path through curved spacetime. There is both a time dilation factor (1 + z) and an energy loss factor (1 + z) associated with the travel time and distance between the source and the observer. The surface brightness of a standard candle would be dependent of the distance as a function of redshift z, not inversely with the square of its distance (just as in an expanding universe, but without the associated angular-diameter distance requirement for comoving objects in an expanding inertial frame).

Re- the topic of Tolman's surface brightness test (considered by Sandage to be "by far the most powerful as a test between the expanding and static models"): According to the Robertson equation (1938), a non-expanding universe would have only one factor of (1 + z), from the "energy" effect. So the surface brightness of a standard candle would then decrease only as (1 + z) with distance.

Of course, this assumption is erroneous since it assumes a flat manifold. The problem was simply hidden, not dealt with. Had the test been applied to a static manifold of constant positive Gaussian curvature (of the type postulated by Fischer and Coldcreation) the result would likely be consistent with observations, since at least two factors of (1 + z) are required by the curved spacetime interpretation: one factor for the "energy" effect and the other factor for time dilation.

[1] Quotes from http://coldcreation.blogspot.com/2010/09/redshift-z.html: A General Relativistic Stationary Universe:
There are two possible interpretations for cosmological redshift z : (1) A change in the scale factor to the metric... (2) The general relativistic curved spacetime interpretation (implying a static metric in a stationary universe). In addition to illuminating how redshift z is caused in a globally curved four-dimensional spacetime manifold ...It is emphasized that global curvature plays an essential role in cosmology and provides a natural explanation for various empirical observations. Too, it is exemplified this point of view by considering a novel version of Einstein's 1916-1917 world-model, where cosmological redshift z is directly related to the large-scale structure of the universe.
[...]
Sorry my mistake; because you were advocating a curved-space cause for cosmic red-shift, I thought you advocated a curved-space cause for cosmic redshift.
[2] So you continue to claim, but without any supportive mathematical proof.

I simply advocate the possibility of a curved-space cause for cosmological redshift.

The idea is indeed intriguing. It has in fact been developed to some extent over the past 80 or 90 years (De Sitter 1917, Weyl 1921, Hubble 1926, Ellis, G.F.R., 1977 and others), and has not yet been refuted empirically.

Once again, George F. R. Ellis in Is the Universe Expanding? (1977) showed that “spherically symmetric static general relativistic cosmological space-times can reproduce the same cosmological observations as the currently favored Friedmann-Robertson-Walker universes.” In this case the systematic redshifts are interpreted as “cosmological gravitational red shifts.”

One of the beauties, of course, with such an interpretation of cosmological redshift z, is that it avoids an initial singularity (or creation event) where the laws of physics break down near t = 0. As a side effect, it also avoids the need to consider 96% of the universe in the form of dark energy and cold nonbaryonic matter.

So the stationary general relativistic model appears to be the more natural solution to cosmological considerations from the outset.

Recall Hubble's 1926 paper, Extragalactic Nebulae (ApJ 64 321) where it is derived the radius of curvature of an Einstein static model based on the mass density of nebulae. Hubble's uses the theoretical treatment of Haas (Haas, A. 1924, Introduction to Theoretical Physics, London, Constable & Co.). Even though this displacement toward longer wave-lengths is technically not the same as a de Sitter effect, it is still grounded on a non-expanding world-model (i.e., a static universe, in accord with Einstein's 1917 model). There would be a linear relation with distance over small distances (near the observer) with increasing divergence for larger distances.

"The mean density of space can be used to determine the dimensions of the finite but boundless universe of general relativity. De Sitter made the calculations some years ago, but used values for the density, 10^-26 and greater, which are of an entirely different order from that indicated by the present investigations. As a consequence, the various dimensions, both for spherical and for elliptical space, were small as compared with the range of existing instruments.

For the present purpose, the simplified equations which Einstein has derived for a spherical curved space can be used. When

R, V, M

and

ρ

represent the radius of curvature, volume, mass and density, and

k

and

c

and the gravitational constant and the velocity of light[...]

...with reasonable increases in the speed of plates and size of telescopes it may become possible to observe an appreciable fraction of the Einstein universe." (Hubble, Mount Wislon Observatory, September, 1926)

In Einstein’s words, six years before Hubble's seminal paper: “By reason of the relativistic equations of gravitation…there must be a departure from Euclidean relations, with spaces of cosmic order of magnitude, if there exists a positive mean density, no matter how small, of the matter in the universe.” The smallest possible density of matter produces constant positive curvature of space. (1920, see Kerszberg, P. 1989, The Invented Universe, The Einstein-De Sitter Controversy (1916-17) and the Rise of Relativistic Cosmology, p.214).

And it is precisely this departure from flatness (constant positive curvature of space) that affects the entire electromagnetic spectrum (across the full 19 octaves), shifting spectral line toward the red.

How then, based on redshift alone, do we make the distinction between an expanding Newtonian inertial system and a static general relativistic universe when a beam of light is affected in a gravitational field exactly as if the source of a beam were traveling (away from us) at great velocity?

With great difficulty!

Obviously you are unaware of the actual structure of Einstein's static model. In that model, light is not increasingly redshifted with distance. Why else would Einstein have abandoned it after hearing of Hubble's observations? Besides which, check out the metric for yourself and have a look at the null-geodesic equation. If you have any real understanding of GR you can then see that there is no "redshift-with-distance" in the ESM.

In Einstein's static model, for a signature of -+++, the metric tensor is diagonal, [imath]g_{00}=-1[/imath] , and no component of the metric tensor is dependent on the time coordinate. Therefore the [imath]\Gamma_{\mu \nu}^0 [/imath] are all zero for all [imath]\mu , \nu[/imath]. Consequently the time component of the null-geodesic equation for the wave-vector [imath]\frac{dk^0}{d\lambda}+\Gamma_{\mu \nu}^0 k^\mu k^\nu=0[/imath] simplifies to [imath]\frac{dk^0}{d\lambda}=0[/imath] ; the frequency does not change; there is no cosmological red-shift in the ESM.

That is likely why Ernst Fischer has modified the time component (which now changes with distance). The metric tensor needs to be dependent on the time coordinate (which should vary depending on distance). When this dependence is incorporated into the structure of the field equations the outcome can only be that spectral lines are shifted toward the red with increasing distance (along with time dilation, by definition).

CC

coldcreation · May 24, 2011

Ernst Fischer's response to cruel2Bkind (via email exchange with Coldcreation, dated 24 May, 2011)

I will be reposting in a few minutes, in order to fix a few symbols.

CC

coldcreation · May 24, 2011

Ernst Fischer's response to cruel2Bkind (via email exchange with Coldcreation, dated 24 May, 2011)

[Edit: I've tried to fix some of the notations, but they may not all be in the form sent to me by email: sorry if that's the case: CC]

I regard E. Fischer as not having the requisite expertise to make valid assessments of anyone else's competency in, or comprehension of, differential geometry. Fischer would do well to concern himself with his own level of comprehension of differential geometry, and basic differential calculus, before he passes value judgments on the abilities of others.

Although the spatial parts of the formulae used in his paper may be similar to those that appear in text books on differential geometry they are fundamentally different to the spatial line-element for a space of constant positive curvature, in ways in which he is apparently unable to appreciate.

The spatial part of the metric for a hypersherical space-time of radius [imath]a[/imath] in a system of spherical coordinates, where [imath]0 \le x^1 \le a[/imath] is a radial coordinate (but not a true radial length, as it is defined by dividing the circumference of a circle centered on the origin by [...]

This is the same expression as in Fischer's equation (12), only the symbols used to represent variables are different. (As explained in a previous post and repeated below, in Fischer's equations the [imath]\sigma[/imath] in [imath]d\sigma[/imath] has a different meaning to the [imath]\sigma[/imath] in the term [imath]\frac{{a^2}}{{a^2 - \sigma^2}}[/imath] ). However, by the very nature of the restrictions that enabled this simplification, it cannot be used as an expression for the general line-element of the hypersurface of the hypersphere because it excludes transverse-to-radial coordinate infinitesimals and thereby omits the transverse-to-radial contributions to the line-element. Not only does such a restricted formulation present the problem of lack of invariance of the form of the "restricted metric" under transformations to coordinate systems whose origins lie on neighbouring points, but it also presents problems for the definition of the Riemann tensor whether that be given by way of transport around a closed contour or by way of consideration of geodesic deviation. Differential geometry is formulated in terms of neighbourhoods of points and not in terms of points in isolation from their surroundings. Consideration of neighbourhoods is not the same as "extension to extended space". Appreciation of the difference is necessary for a proper comprehension of differential geometry.

The simplified expression cannot define a general line-element in a space of constant positive curvature, and neither can Fischer's equation (12). The status of proper definer of the spatial part of the line-element in a space of constant positive curvature belongs to the three expressions (1), (2) and (3) for the line-element given above.

Obviously cruel2Bkind still struggles with the definition of [imath]\sigma[/imath] in my paper. It denotes the distance between two arbitrary points of a manifold. In Euclidean geometry it is the length of a vector connecting these points. In curved space it is the length of the shortest line connecting the points. In this case there exists no unique vector connecting the points, but only a sequence of infinitesimal vectors of length dσ. But indeed there is an inconsistency in my paper, which may have caused the problem. In eq.(12) I should have used a different symbol in the definition of the spatial line element. The spatial part of the line element is defined by dss²=h(σ)dσ². To calculate the function h(σ) the curvature of circle of radius a has been determined and its value is used in the definition of the line element. This value is independent of the coordinate system, which is used in the derivation of the field equations. The only thing that matters is the relation between the radius of curvature and the density of the matter field. Using a wrong symbol may have caused confusion, but it does in no way change the results of the calculations.

All the considerations to express the line element in different coordinate systems are not relevant. To me the best choice was local rectilinear coordinates, as in this case the numerical effort is lowest due to symmetry. The numerical expressions of the connection coefficients and their derivatives are of equal form for each of the three coordinate directions. The “transverse-to-radial” contributions are of course considered also in this coordinate system, but the mixed derivatives ∂²/∂xi∂xj vanish at σ=0 and thus can be neglected in an infinitesimal neighborhood.

A space is homogeneous if a translation has no effect on equations of motion. A distribution of mass/energy is homogeneous if it has the same value at every point.Within either use of the word 'isotropy' is not a necessary part of 'homogeneity'. A space can be homogeneous and non-isotropic. Homogeneity is not "radial symmetry" at every point. A space may have the property that at every point there is a preferred direction. If that direction is the same for every point in the space then the space may be homogeneous, but it does not have the property of being spherically symmetric about every point. Correct use of the language is essential to a proper understanding of general relativity.

Of course one may define homogeneity as invariance only with respect to some class of transformations, but here I feel this remark as a deliberate word-splitting without any relevance to the actual topic. What is generally discussed as homogeneous solutions of the Einstein field equations, are solutions that have equal properties as viewed from every point of the manifold.

There is nothing inherently "problematic in the definition of a distance between points or in the definition of an infinitesimal distance element in curved space" when that distance is well-defined. However problems do arise when that distance is not well-defined. The concept of something needing to be 'well-defined' in order to be meaningful is a basic concept in mathematics.
I note with interest that Fischer fails to respond to my reasons for regarding [imath]\small{d\sigma^{2}=\frac{a^{2}}{a^{2}-\sigma^{2}}d\sigma^{2}}[/imath] as ridiculous.

The following argument comes from simple algebra, mastery of which is a prerequisite for understanding differential calculus. Comprehension of differential calculus is also a prerequisite for understanding differential geometry:

[...] means something completely different to the [imath]\sigma[/imath] in Fischer's equation (12).
In the first two cases, [imath]\sigma[/imath] necessarily equals zero which means that the space must have only one point and the concepts of connection coefficients and curvature tensors are inapplicable. The last two cases provide evidence for a lack of comprehension of the need for symbols to be unambiguous[/indent]

Failure to appreciate such an argument can be regarded as indicative of a lack of comprehension of the required mathematics.

I regret that by the ambiguous use of the symbol dσ I have led Cruel2Bkind to a completely wrong trace. But as far as only the coefficients of the metric and their derivatives are concerned, the result is not affected by the use of the symbol dσ² instead of dss² in eq.(12).

[1] This is a grossly inappropriate response. What I wrote has absolutely nothing to do with "extending the local description in the form of differential properties to an extended volume". Such an interpretation of what was written raises serious questions about the level of comprehension of the mathematics that must be mastered in order to understand differential geometry. Belittling a critic's comprehension of a topic does not constitute a valid response to a criticism that reveals a flaw in a mathematical presentation.
The flaw is made obvious by the following argument. It requires an appropriate response.

[...]
From Fischer's equations for the Ricci components it must be the case that [...]

So his metric has spatial components [imath]g_{ii}= \frac{a^2}{a^2-\left({x^1+x^2+x^3}\right)^2} [/imath] , which are not the spatial components of a metric for a space of constant positive curvature.[/indent]
That argument is from differential calculus, one of the prerequisites for understanding differential geometry. Any claim that it is "extending the local description in the form of differential properties to an extended volume" is invalid and merely demonstrates that the claimant is clutching at straws in trying to defend the indefensible .

The equations given above are a good example of the inadequate use of equations, which relate differential properties of a manifold. From isotropy of space we have the condition that dh/dσ=0 at σ=0. Thus the only result that can be drawn from the equations is 0=0. Any extensions to finite values of σ give wrong and irrelevant results, as Cruel2Bkind has nicely demonstrated. Again it should be stressed that the formalism of differential geometry delivers only local connections between the geometrical quantities and their derivatives.

In a Euclidean geometry you can describe a circle as a functional relation between x- and y-values. In differential geometry it is described as a line of constant curvature.

[2] I did not derive those equations. I took them from Fischer's paper and augmented them by adding the character-string "(0)" where appropriate. I did that in order to reveal what his expressions actually meant. He seems to understand that they are only values-at-a-point, but by attributing them to me he appears to be denying responsibility for their nature of not being differential equations but being only values-at-a-point and hence not being solvable.

[3] The invalidity of this statement is easily established:

[...]
Therefore it is not true that "relations over finite distances can be obtained only by integration along some path, regarding the values of the functions f and h and their derivatives as co-moving information". Obtaining such relations requires an initial choice of coordinate system, and an adherence to that coordinate system as the integration is performed.

This is again a wrong interpretation. How can one shift the origin of a line element by a finite length L along a straight line in a curved space, which contains no straight lines at all? Shifting in tangential direction in a system of constant curvature would keep h’’ constant.

[1] Such a response to being informed of an error in one's calculations is not only inappropriate but also invalid. Nothing in what I wrote has anything to do with an attempt "to extend the solution at s=0 to extended spatial sections".
Besides which there is no such thing as a solution to a differential equation that only applies at one value of the independent variable. There are no such rules in differential geometry because there are no such non-extendable solutions of differential equations applicable to only single points.
[2] Integration of the field equations delivers the components of the metric tensor, it does not yield the geodesic lines. When the metric components are found, geodesics can be obtained without any such "continuous shift of the coordinate system".

Integrating the field equations delivers the metric tensor. But whatever you chose as a local coordinate system, an isotropic solution must always fulfill the condition that the first derivatives of the tensor components vanish at the origin.

The expressions for the Ricci components

[...]
were all obtained by standard algebraic manipulation techniques of expanding Einstein summation, differentiating quotients, applying the distributive law to products involving bracketed summations and simplifying by adding/subtracting like terms. Obviously being only manipulation methods employed to simplify complex expressions, like those used by Fischer himself to get his equation (15) which is the same as the equation for [imath]R_{ii}[/imath] above, criticisms such as "extend(ing) the solution at s=0 to extended spatial sections" are inappropriate, invalid, and inept.
The equations for [imath]R_{00}[/imath] and [imath]R_{ij}[/imath] are no more than what Fischer should have found had he been diligent.

Solutions of these equations with the condition h’(0)=0 are exactly what I have used to describe a homogeneous solution. But again it must be stressed that applying these conditions to a locally fixed coordinate system, valid solutions are restricted to an infinitesimal environment of the local origin.

[1] The inept misrepresentation that there is an "application to extended space-time of the local definition of curvature with its differential connections between neighboring points" seems to be nothing more than an attempt to divert attention from gross errors in conceptualization and calculation. Moreover in developing his own solution for [imath]f[/imath] Fischer should criticise himself for the same reasons if he has any claim to consistency. But the fact is that there such a criticism is quite inappropriate. He could have leveled a valid criticism as explained below, but he has missed his chance of doing that. Merely claiming that I have a "wrong interpretation of the field equations", without providing detailed analysis of every mathematical equation I have given, does not suffice to warrant the claim of invalidity. However it is quite apparent that Fischer has a rather confused interpretation of the field equations. The stress-energy tensor is purely determined by matter/radiation. All of its components must have realistic values; if it transpires that the field equations have solutions only if one or more of those components have values that could not occur in the real universe, then the metric that gave rise to those field equations has no application to the real universe.

[2] I have already proven that the energy density is not homogeneous. Mere contradiction does not suffice as disproof.

There is no proof at all that energy density is not homogeneous. On the contrary, spatial homogeneity of the energy distribution is the basic proposition to find solutions of the field equations, where universal values of the curvature parameters can be determined and applied to the environment of any arbitrary point. Gross errors in conceptualization may as well be assigned to Cruel2Bkind. It is scarcely understandable, why he so vigorously fights against the well established description of the spatial sector of the geometry in a homogeneous curved space. A quite similar approach was already given by Einstein (see ref.[4] of my paper.)

[3] There was no "assumption" that the energy density must be negative. That result was derived from his connection coefficients by legitimate mathematical procedures. It requires a legitimate response.

[4] It was not caused by any sign error in his paper. In referring to choosing the sign of [imath]\kappa[/imath] so as to "obtain the correct positive energy density in the Newtonian limit" is he revealing that he was guilty of assuming what he was trying to prove.

There is nothing to prove here. The value of [imath]\kappa[/imath] is fixed by the condition that the correct relation between energy density and gravitational attraction results in the Newtonian limit, as it has been used already by Einstein.

However I must admit to making an error in my previous post on 21 April 2011. The solution for [imath]f[/imath] given therein does not satisfy the field equation [imath]R_{ij}=0[/imath] (for [imath]i \ne j[/imath] and [imath]i,j \ne 0[/imath]). My mistake was in assuming that the pressure could be homogeneously zero and that the equation involving pressure and [imath]f[/imath] could be solved for [imath]f[/imath]. However the fact that the proposed solution for [imath]f[/imath] does not satisfy [imath]R_{ij}=0[/imath] proves that even though the pressure must be zero at the origin (which is an unrealistic value as the pressure of the ubiquitous cosmic microwave background radiation although tiny is not zero) the pressure elsewhere is a function of the distance from the origin; it is not homogeneous.
There are three field equations in three unknowns [...]

Of the three equations, it is the last that involves only the unknown [imath]f[/imath] and the known [imath]h[/imath] that must be solved for [imath]f[/imath]. The first two equations give the other two unknowns viz. the pressure and the energy density, neither of which can be homogeneous.

Here again the strange misconception of Cruel2Bkind shows up. The homogeneity of the energy and pressure fields is a proposition. p and ε are given quantities and the field equations relate the curvature and time scale parameters f and h and their local derivatives to these quantities. Integration of the field equations in some local coordinate system gives the elements of the metric tensor, which then can be used to describe changes in the system, be it geodesic motions of particles or photons under these geometrical conditions in a static system or, if one believes in the Big Bang model, to global changes of p and ε, caused by the gravitational interaction of these quantities.

[5] The 'solution', apart from the distance dependency of the time-scale, in his paper may be identical with Einsteinï¿½s static universe, but it is not a solution of the corrected field equations developed from his connection coefficients. He has compounded errors to get from his erroneous connection coefficients to expressions for the field equations that resemble those for Einstein's static universe and then claims that because his 'solution', excluding the time-scale, is identical to that of Einstein's static universe, his procedure must be correct. Such efforts leave much to be desired.

The field equations derived in my paper are a correct solution developed from the connection coefficients, considering homogeneity and isotropy of the matter fields, which require h’=0. That they, apart from the question of time-scale, are identical with Einstein’s static solution, is not proof that they are correct, but should induce Cruel2Bkind to think about the question, if not he is the one that should correct his view of the problem.

One more observation of Fischer's 'work' needs voicing. Consider his expression for the curvature scalar. At the origin where [imath]\sigma =0[/imath] the value for the three-dimensional curvature scalar derived from his connection coefficients is negative. The three-dimensional curvature scalar is found from the expression [imath]3R_{ii}g^{ii} \; , \; \left({i \ne 0}\right)[/imath]. At the otigin where [imath]h'\left(0 \right)=0[/imath] the value is [imath]-\frac{6}{a^2}[/imath], as is clearly seen in Fischer's equation (18). The value for the 3d scalar, using the corrected value for [imath]R_{00}[/imath] and then replacing [imath]h[/imath] by [imath]\frac{a^2}{a^2-{\sigma}^2}[/imath]is:
[imath]{}^{3d}R = \frac{9{h'}^2}{2h^3}-\frac{6h''}{h^2}=- \frac{6\left[{{2a}^2 + 3\sigma ^2}\right]}{a^2 \left[{a^2 - \sigma ^2}\right]}[/imath]
This reveals that the three dimensional curvature scalar is not even constant. In anticipation of the criticism that in claiming it is not constant I am applying the local definition of curvature to extended space-time, let me point out that Fischer's assumption, that because the curvature has a certain value at the origin then it must have that value at every other point, is actually an invalid application of a local value of curvature to extended space-time.
(It should also be noted that Fischer sees no contradiction in extending the value of a differential equation, valid at only one point, to extended space and then extending his 'solution' for [imath]f[/imath], only valid at [imath]\sigma =0[/imath], to extended space.)

So Fischer's space is inhomogeneous in both pressure and energy density, and it has a non-constant negative curvature. However the three dimensional curvature scalar derived from the metrics given near the start of this post is indeed positive and constant.

Here again Cruel2Bkind exchanges the propositions and the consequences. That pressure and energy density are homogeneous is the proposition. The consequence is that the field equations deliver the same value of the local curvature scalar at every point. Again it should be stressed that the formula used to find the relation between the local curvature parameter h’’ and the radius of curvature may not be used to extend the integration of the field equations to finite distances. This would contradict the basis of differential geometry to describe the properties of space only by locally defined parameters.

Ernst Fischer [via email exchange with Coldcreation, dated 24 May, 2011]

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