Redshift z

[quantumtopology]

I don't think angular distance would be proportional to z^2 in hyperbolic space, it wouldn't be a linear relation to begin with.

 

Forget this lapsus, obviously a function with z^2 is not linear

 

A question. You know k=-1 is also a solution of FRW metric in a expanding universe, do you absolutely rule it out?

 

Would you answer this?

 

 

 

quoted from Modest:

"Given a volume of space (a sphere) in a homogeneous universe the escape velocity of a particle at the sphere's edge is related to the density of the sphere by:" end quote

 

I am not talking about a sphere, I'm talking about hyperbolic space.

In infinite hyperbolic static space GR does no longer applies when distance tends to infinite as Einstein himself realized in 1917, his only way out was either the cosmological constant or accept expansion, wich he did in 1931, even then he always favored the k>0 solution.

 

"The problem with infinity"

http://www.spsnational.org/radiations/2008/ecp_bigbang2.pdf

 

 

quoted from Modest

"If the universe is static then V=0" end quote

 

Why? do you mean Volume of a hyperbolic universe is 0? I don't understand.

 

quoted from Modest

"I must have missed where you showed that. Can you point me to it." end quote

 

It's been stated in this thread several times that the Tolman test only applies to Euclidean space.

 

 

from the wikipedia on Tolman test

" Different physicists have claimed that the results support different models.

In a simple (static and flat) universe....."

 

Regards

 

Qtop

[coldcreation]

 

Why would objects move along geodesics in a homogenous, isotropic gravitational field with no gradient, with a metric tensor that remains the same from point to point (i.e., everywhere)?

 

The principle of extremal aging

 

Geodesics as extremal curves.

 

Thanks for the links.

 

First of all, the above links deal with a Lorentzian manifold (and geodesics as extremal curves). This is a special case, subclass, of a pseudo-Riemannian manifold. When considering the more general case of a pseudo-Riemannian manifold the situation is different. The Lorentzian manifold may not hold in the general case of a homogenous and isotropic manifold of the pseudo-Riemannian type. Obviously the question above is related to the most general case; the curvature of the entire spacetime manifold (not just a submanifold of the type surrounding massive bodies), or an inhomogeneity (obstruction) of the more general case.

 

Lorentzian manifolds are important since they offer special-case physical applications to general relativity. A principal assumption of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold, but unlike Riemannian manifolds (with positive-definite metrics), a Lorentzian manifold has the signature (p,1) or (1,q), which allows tangent vectors to be classified into timelike, null or spacelike; meaning that a Lorentzian manifold is time-orientable. Source. This works fine, and is applicable for local systems in the framework of GR, where distances and time between events are small (e.g., the path taken by a falling rock, an orbiting satellite, or planetary orbits), or when dealing with paths objects take when they are not free (when their movement is constrained for different reasons). In other words your link describes geodesic motion of point particles under the influence of gravity.

 

My question was with reference to a global field where the metric tensor has the same value at each point in the field (a globally homogenous and isotropic spacetime). There is no reason why general relativity should be formulated using a special-case Lorentzian manifold, where geodesics describe the motion of point particles under the influence of gravity when considering the large-scale topological features of the universe. On the contrary, general relativity should be formulated using the more general Riemannian and pseudo-Riemannian manifolds. The geometric properties of the spacetime depends on the metric chosen. You get different results with different metrics.

 

 

The peculiarity that arises with a Lorentzian metric when generalized, is that the path of a free-particle (a geodesic) is the longest path between two events. Whereas, in the more general case of a Riemannian manifold geodesics are locally distance-minimizing paths. (). This is what interest me because as we observe the universe, the only way we can, in the look-back time, light travels toward us (or converges on all observers) on the shortest geodesic path between the source and the observer (not the longest), in line with (no pun intended) the more general metric geometry of Riemann (which originated with the discoveries of Gauss: see Gaussian curvature).

 

 

Here are a couple of links that may be of interest:

 

Geodesics in semi–Riemannian Manifolds: Geometric Properties and Variational Tools

 

In this document, not section 4.2. Variational principles for static and stationary space-times: extrinsic and intrinsic approaches. The discussion is quite complex, and may or may not support what I claim (some sections seem to and others not). My guess is that this type of research is a step in the right direction, but that assumptions on the global metric (under an intrinsic hypotheses) more related to the geometry of the manifold are required.

 

Solitons on pseudo-Riemannian Manifold: Stability and Motion

 

This work I have not looked at in detail yet. Neither of these papers are discussed in Explanation 2.

 

CC

[modest]

A question. You know k=-1 is also a solution of FRW metric in a expanding universe, do you absolutely rule it out?

Would you answer this?

 

It is not ruled out. In L-CDM omega-K is 1.005 +/- .006 meaning K could be positive, zero, or negative, but that the radius of curvature is very large relative to the visible universe.

 

quoted from Modest:

"Given a volume of space (a sphere) in a homogeneous universe the escape velocity of a particle at the sphere's edge is related to the density of the sphere by:" end quote

 

I am not talking about a sphere, I'm talking about hyperbolic space.

 

You asked me to show how collapse is possible in infinite space. I provided a method of solution. Did you not follow?

 

In infinite hyperbolic static space GR does no longer applies when distance tends to infinite as Einstein himself realized in 1917

 

In an open universe one must set boundary conditions. A reasonable and appropriate boundary condition is for the metric to degenerate at infinity. Einstein did not like the idea on philosophical grounds.

 

This is wholly, completely, and entirely different from saying collapse is impossible in an open manifold.

 

quoted from Modest

"If the universe is static then V=0" end quote

 

Why? do you mean Volume of a hyperbolic universe is 0?

 

No, I mean this:

the

escape velocity

[V] of a particle at the sphere's edge is related to the density of the sphere by...

 

Why?

 

I derive the equation at the link I gave. It's a safe bet the why's are answered there.

 

quoted from Modest

"I must have missed where you showed that. Can you point me to it." end quote

 

It's been stated in this thread several times that the Tolman test only applies to Euclidean space.

 

Has it has been supported by a link or some force of logic?

 

 

from the wikipedia on Tolman test

" Different physicists have claimed that the results support different models.

In a simple (static and flat) universe....."

 

That wiki gives the surface brightness for flat space does not imply that a surface brightness test in hyperbolic space is impossible.

 

[math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature.

 

~modest

[coldcreation]

The distance between any and all sets of two points will decrease with time.

 

Neither is there a center of gravity for a closed, finite universe.

 

Given a volume of space (a sphere) in a homogeneous universe the escape velocity of a particle at the sphere's edge is related to the density of the sphere...

 

...this relationship is true regardless of the size of the sphere (you can take the limit as volume approaches infinity and the relationship will be the same) because the density of any volume is the same.

 

The idea that the universe is a finite sphere is one big massive assumption. The idea that the distance between any and all sets of two points will decrease with time is based on that same massive assumption.

 

In addition, there is no edge to sphere, if we're talking about the geometry of the universe (so placing an object near one makes no sense). It looks like you're left with a distasteful boundary condition. If your talking about the universe being a real sphere then there would be a center to the universe. So I take it you're referring to non-Euclidean geometry. Nothing is GR or basic non-Euclidean geometry states that two parallel lines will intersect; only that they will converge, and that they could do so indefinitely in an space with no boundary.

 

To claim the universe must be finite excludes the possibility that it might actually be infinite.

 

My point is that the universe can be infinite spatiotemporally yet still display spherical geometry (appearing spherically symmetric). The sphere in geometry simply represents a plane with constant curvature. A plane can be thought of as a sphere with infinite radius. The sphere is the only imbedded surface without boundary or singularities with constant positive mean curvature. (Source)

 

If you remove that dubious assumption then the whole argument about escape velocity vanishes, i.e., it vanishes when when you consider an infinite universe. In other words your assumption that the "relationship is true regardless of the size of the sphere" cannot be considered tenable, except perhaps on philosophical grounds (if you accept the boundary condition that is).

 

 

CC

[quantumtopology]

Has it has been supported by a link or some force of logic?

That wiki gives the surface brightness for flat space does not imply that a surface brightness test in hyperbolic space is impossible.

I gave you the link to the wiki, but if you choose to ignore it thre is nothing else I can do.

 

Richard Tolman devised a test that stated that in an expanding universe with arbitrary geometry, the surface brightness of a set of identical objects will decrease by (1+z)^4.

That was to be compared to a static flat universe that decrease by (1+z), if you are able to show me what would be the power of (1+z) by wich it would decrease in a hyperbolic static universe then we'll have made some progres.

 

 

[math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature.

 

Do you honestly think that is true?

 

From the wiki again:

 

"The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, z, is expressed in terms of the comoving distance, χ as:

 

Da= X/(1+z)

 

Where r(χ) is defined as:

sin X for k>0 , X for k=0 and sinh X for K<0

 

End quote from wiki

 

So even in the assumed expanding cosmology your equation is not true regardless curvature, except for k=0.

 

Regards

 

Qtop

[coldcreation]

 

Richard Tolman devised a test that stated that in an expanding universe with arbitrary geometry, the surface brightness of a set of identical objects will decrease by (1+z)^4.

That was to be compared to a static flat universe that decrease by (1+z),

 

Good point.

 

It's easy to rule out a static model with the Tolman surface brightness test when the assumption is based on a flat Euclidean static model.

 

It's much more difficult (impossible?) to rule out a static model, by the same method, that has a non-Euclidean metric curvature of the manifold, since surface brightness could very well decrease by (1+z)^4 due to curvature.

 

Note also that Olber's paradox is resolved nicely by the same argument. In a static universe with nonzero global curvature the night sky is dark, exactly as in an expanding universe. :hihi: To claim, as in the wiki link, that Olber's paradox is "evidence for a non-static universe such as the current Big Bang model" is laughable. :D

 

 

CC

[quantumtopology]

It's much more difficult (impossible?) to rule out a static model, by the same method, that has a non-Euclidean metric curvature of the manifold, since surface brightness could very well decrease by (1+z)^4) due to curvature.

 

Exactly, and in fact it wouldn't even have to reach the 4th power, due to differences in luminosity, the actual result of the test as given by Sandage who is not precisely a suspicious antimainstream cosmologist [astro-ph/0106566] The Tolman Surface Brightness Test for the Reality of the Expansion. IV. A Measurement of the Tolman Signal and the Luminosity Evolution of Early-Type Galaxies

the observed power in (z+1) by which brightness decreases is around 2.5. Which is perfectly reachable in a hyperbolic universe due to the curvature and that is in perfect agreement with the dimming of SNIa.

The conclusion is that probably The Tolman Test cannot differentiate an accelerating expanding flat universe from a static hyperbolic universe, for the same reasons it is likely that Angular diameter distance-redshift relation cannot do it either. We have to look elsewhere.

 

Note also that Olber's paradox is resolved nicely by the same argument. In a static universe with nonzero global curvature the night sky is dark, exactly as in an expanding universe. :hihi: To claim, as in the wiki link, that Olber's paradox is "evidence for a non-static universe such as the current Big Bang model" is laughable. :D

 

Right, but remember we are dealing now with space curvature, not spacetime.

 

Regards

Qtop

[quantumtopology]

[math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature.

 

Modest, I guess this is the formula that relates Luminosity distance with Angular size distance.

You have to admit that relation is totally based on the assumtion of a LCDM model so it cannot have relevance in this discussion.

Anyway I would suggest we do not use Angular size to tell between models, you know is plagued with all kind of observational problems above certain z , besides you've said on different occasions that Type Ia SNe data is much better as a distance ruler , in fact you said:"Supernova light curves are much, much, much better standard candles than angular size." in post 25 of the de sitter thread.

[modest]

Ok, guys. The number of unsupported strange claims per post is increasing exponentially. We really need to follow,

 

Hypography Site Rules

1. In general, back up your claims by using links or references.
   
2. If you make strange claims, please provide proof or at least backup of some kind.
   

 

 

...Given a volume of space (a sphere) in a homogeneous universe...

 

The idea that the universe is a finite sphere is one big massive assumption....

 

Please recognize the difference between "volume of space in a homogeneous universe" and "the universe is a finite sphere". Also, had you read the derivation that I linked you would have read this:

The sphere does *not* represent the whole universe, but just an arbitrarily designated portion.

 

First of all, the above links deal with a Lorentzian manifold (and geodesics as extremal curves). This is a special case, subclass, of a pseudo-Riemannian manifold. When considering the more general case of a pseudo-Riemannian manifold the situation is different. The Lorentzian manifold may not hold in the general case of a homogenous and isotropic manifold of the pseudo-Riemannian type. Obviously the question above is related to the most general case; the curvature of the entire spacetime manifold (not just a submanifold of the type surrounding massive bodies), or an inhomogeneity (obstruction) of the more general case.

 

The following link gives the metric in the [++++] and [+++-] forms. The physical interpretation is the same either way.

 

http://www.phil-inst.hu/~szekely/PIRT_Budapest/ft/Realdi_ft.pdf

 

[math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature.

Do you honestly think that is true?

 

From the wiki again:

 

"The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, z, is expressed in terms of the comoving distance, χ as:

 

Da= X/(1+z)

 

Where r(χ) is defined as:

sin X for k>0 , X for k=0 and sinh X for K<0

 

End quote from wiki

 

Both me and wiki are correct. [math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature and angular diameter distance to an object at redshift z depends on curvature. To support my claim:

Closely related to the angular-diameter distance is the luminosity distance, which is the distance objects appear to have based on their observed brightness. The two relations are connected by the reciprocity relation, that their ratio is DL/DA=(1+z)

²

where z is the redshift. This relation is independent of spatial curvature, and is a simple consequence of the conservation of photon number.

angular-diameter distance - not, physical, reciprocity relation

If you are having trouble reconciling the above quote with wiki then let me know and I'll explain.

 

So even in the assumed expanding cosmology your equation is not true regardless curvature, except for k=0.

 

This is not true.

 

It's easy to rule out a static model with the Tolman surface brightness test when the assumption is based on a flat Euclidean static model.

 

It's much more difficult (impossible?) to rule out a static model, by the same method, that has a non-Euclidean metric curvature of the manifold, since surface brightness could very well decrease by (1+z)^4 due to curvature.

 

Expansion increases the apparent size and decreases luminosity decreasing surface brightness. Negatively curved space makes objects appear smaller and dimmer—the factors cancel.

 

~modest

[quantumtopology]

Ok, guys. The number of unsupported strange claims per post is increasing exponentially. We really need to follow,

 

I certainly agree with the rules and try to follow them (sometimes I might forget). Point me to any claim I didn't offer some link to back up in my last posts, let me know if links from wikipedia are not considered valid.

Anyway I find a bit suspicious that you bring up this just know, I don't think what is being claimed currently is any stranger than what has been claimed all along the thread. You might try and read post number 1 from 2005.

Modest, I consider myself a quite fair person, when I thougt I had to refute unfounded claims I have, even if they were favoring my own thesis, but somehow I glimpse certain stubbornness on your part to concede a single point, you seem to forget everything that doesn't directly support "the sacred LCDM model" and believe me that I give you a lot of credit for discussing our "strange claims" with us, so far you are the only mainstream cosmology supporter I know that is willing to do it. But I think you do it because you genuinely think science is not religion, is this right?

Sorry for the speech

 

 

Both me and wiki are correct. [math]D_A = D_L/(1+z)^2[/math] is true regardless of curvature and angular diameter distance to an object at redshift z depends on curvature. To support my claim:

angular-diameter distance - not, physical, reciprocity relation

If you are having trouble reconciling the above quote with wiki then let me know and I'll explain.

 

This is not true.

Did you read my last post? In the answer you quote I thought you were using the usual formula(my fault), thus my answer in the last post-theo one inmediately previous to your last.

 

Regards

Qtop

[modest]

I certainly agree with the rules and try to follow them (sometimes I might forget). Point me to any claim I didn't offer some link to back up in my last posts, let me know if links from wikipedia are not considered valid.

Anyway I find a bit suspicious that you bring up this just know, I don't think what is being claimed currently is any stranger than what has been claimed all along the thread. You might try and read post number 1 from 2005.

Modest, I consider myself a quite fair person, when I thougt I had to refute unfounded claims I have, even if they were favoring my own thesis, but somehow I glimpse certain stubbornness on your part to concede a single point, you seem to forget everything that doesn't directly support "the sacred LCDM model" and believe me that I give you a lot of credit for discussing our "strange claims" with us, so far you are the only mainstream cosmology supporter I know that is willing to do it. But I think you do it because you genuinely think science is not religion, is this right?

Sorry for the speech

 

I just know, as I'm sure you do, how easily these discussion turn into "yes it is", "no it isn't", "yes-huh", "nu-uh"... etc. I'm sure I'd be as guilty of perpetuating it as anyone. Being sure we back up our claims is good fire prevention as far as that goes.

 

Did you read my last post? In the answer you quote I thought you were using the usual formula(my fault), thus my answer in the last post-theo one inmediately previous to your last.

 

I didn't see it. I must have been writing when you posted.

 

Modest, I guess this is the formula that relates Luminosity distance with Angular size distance.

You have to admit that relation is totally based on the assumtion of a LCDM model so it cannot have relevance in this discussion.

 

I'd say that it's based on FLRW, not ΛCDM. ΛCDM is one specific example of the FLRW solutions. The relationship would be true of all those solutions—which is to say—any GR universe which is well-approximated with isotropy and homogeneity.

 

Anyway I would suggest we do not use Angular size to tell between models, you know is plagued with all kind of observational problems above certain z , besides you've said on different occasions that Type Ia SNe data is much better as a distance ruler , in fact you said:"Supernova light curves are much, much, much better standard candles than angular size." in post 25 of the de sitter thread.

 

I agree that angular diameter distance is subject to a large error margin. Luminosity distance by way of SN-Ia offer the better constraint. Mostly where D_A is mentioned today is in relation to the CMB.

 

It is, however, I think clear from the data that angular diameter distance decreases above about z=1.5,

 

 

while the data isn't accurate enough to constrain the deceleration parameter, especially when lambda is included, it does at least show the general trend toward increasing angular diameter—something which hyperbolic space alone would not have. It would have rather the opposite.

 

~modest

[quantumtopology]

while the data isn't accurate enough to constrain the deceleration parameter, especially when lambda is included, it does at least show the general trend toward increasing angular diameter something which hyperbolic space alone would not have. It would have rather the opposite.

 

Can you back that claim?

 

Didn't we agree on not using Angular size due to its lack of accuracy at the distances where the trend toward increasing angular diameter begins, and yet you post it again.

 

Qtop

[modest]

Can you back that claim?

Sure.

Spatial curvature may either increase or decrease the angular-diameter distance, with a spherical geometry focusing light rays and making objects appear larger, and a hyperbolic geometry having the opposite effect.

angular-diameter distance

Hyperbolic geometry decreases angular diameter over euclidean.

Didn't we agree on not using Angular size due to its lack of accuracy at the distances where the trend toward increasing angular diameter begins, and yet you post it again.

 

In the other thread I recall saying that angular diameter could not constrain q, but clearly the angular size does increase rather than decrease. As the quote above says, hyperbolic geometry would be expected to have the opposite effect. It would not agree well with observation.

 

~modest

[coldcreation]

Sure.

angular-diameter distance

Hyperbolic geometry decreases angular diameter over euclidean.

 

 

In the other thread I recall saying that angular diameter could not constrain q, but clearly the angular size does increase rather than decrease. As the quote above says, hyperbolic geometry would be expected to have the opposite effect. It would not agree well with observation.

 

~modest

 

Yup, that sure is counter intuitive.

 

The article does say this though right after the sentence you quote: "However there are no objects in the Universe which have a fixed physical size and exist across a wide range of distances, so in practice this counter-intuitive behaviour is not observed."

 

If indeed this effect is not observed it is entirely based on an idea, i.e., it is expected by a theory.

 

Intuitively, then, the next sentence should read: 'Spatial curvature may either increase or decrease the angular-diameter distance, with a spherical geometry focusing light rays and making objects appear smaller, and a hyperbolic geometry having the opposite effect" (making objects appear larger).

 

This is where the problem resides, since focusing light rays in a spherical universe means that geodesics converge.

 

The luminosity distance, on the other hand (the distance objects appear to have based on their observed brightness) is observed empirically. This shows a strong reduction in the surface brightness of distant objects; redshift "reduces the overall luminosity of an object." That would seem to be consistent with hyperbolic geometry, where clocks would appear to tick more slowly with distance, and objects would be further away than expected in a Euclidean universe (i.e., they appear further than their true distance).

 

So, the current application of the angular-diameter distance (which is not observed) to measure cosmic microwave background anisotropies (where it is thought to give the characteristic scale of the peak structure in the angular power spectrum) is really not an empirical measurement of curvature at all. It is only based on an expectation. How then can this provide "the best evidence that the Universe has a flat spatial geometry"?

 

Remember, distant SNe Ia appear to be further than expected, as if spacetime were 'stretched' out further with increasing distance. Distant SNe Ia would appear unexpected dim, giving the impression they are further away than their redshifts indicate. That would appear to be a hyperbolic signature (from the perspective of an observer in a static globally curved spacetime scenario). Wether this is a sign of hyperbolicity or sphericity in an expanding universe depends on how one adjusts the parameters to account for observational data.

 

Time dilation, then, in the look-back time, would be consistent with objects appearing further away than they actually are.

 

 

I am thoroughly confused.:)

 

It looks like my ducks were lined up after all: Perhaps diagram A was the correct answer (meaning modest and quantumtopology would have been incorrect from the get-go, and coldcreation right).

 

 

Modest, what gives?

 

If angular diameters are not observed (as stated in your link) then what does the observational evidence show: would distances appear further or closer than they actually are? I think your answer will be that is depends on the model. But that means that observational bias is unavoidable, does it not? Everything hinges on the meaning of redshift z in a given model. Depending on its meaning (and depending on the sign of curvature) objects can be further or closer than the Euclidean (actual, proper, or real) distance.

 

The question is, if the universe is static and curved (assuming we don't know yet the sign of curvature), where redshift is due to the propagation of EMR along geodesics, then how can we know whether the curvature is hyperbolic of spherical, based on observations (if not by time dilation based on rise times of standard candles)?

 

 

[Edit:] Perhaps the best solution for the time being is to hypothesis curvature (as a cause of redshift z) without specifying the signature (positive or negative) until a consistent model-based interpretation of distances can be corroborated on empirical grounds (as modest has previously claimed); thereby predictions can be tested, not just against observations, but against the standard model (Lambda-CDM).

 

Questions of global stability would still remain a domain of active research for both geometries. The solution for global stability of a spherical Einsteinian model with Gaussian curvature has been attained in Explanation 2 (pending confirmation of validity by those qualified). The case for hyperbolicity still needs to be explored. That is the direction of research I will now embark upon. The results of this investigation will be posted in Explanation 2 (whether a hyperbolically curved universe can remain stable, or not). This should delay the posting of Explanation 2 for a period of 48 to 72 hours, I would suspect. :) [End edit]

 

 

CC

[quantumtopology]

I am thoroughly confused.:)

 

The question is, if the universe is static and curved (assuming we don't know yet the sign of curvature), where redshift is due to the propagation of EMR along geodesics, then how can we know whether the curvature is hyperbolic of spherical, based on observations (if not by time dilation based on rise times of standard candles)?

CC, we need to reach some agreement here if this discussion is to make some sense. When I speak of hyperbolicity I am referring to the 3D space, not to global curvature which is 4D, but you seem to be talking about global 4D espace curvature, now global curvature is something much complex to determine, in the expanding scenario for instance is not determined by the model since it depends on density(that is not yet determined) and besides is dynamical,not constant, due to the scale factor, so it is something not even addressed by the experts, only in speculative form, and is not fixed by GR field equations either. there are too many parameters unknown, GR is "only" clearly determining the local curvature originated by a certain local stress-energy tensor configuration.

 

In the static scenario, global curvature should be at least constant but unless you consider a finite model like Einstein's universe, global curvature is not straighforward at all in an infinite universe, since global density is really hard to determine, if you use GR without the cosmological constant in a static infinite universe(infinite radius) you get Minkowski spacetime that has null curvature as is shown in http://www.spsnational.org/radiation...p_bigbang2.pdf pages 27 and 28 in equations 14 to 18.

 

So in a non-expanding scenario, without cosmological constant, if you faithfully follow GR to its last consequences you get an infinite universe with null curvature.

That is the only mathematically sound solution with the conditions of a non-expanding space and keeping the original Einstein equation without lambda.

So the instability issue only arises when lambda is introduced in order to avoid the problem with infinity.Since in a finite static model any local density or pressure deviation from equilibrium would lead to either contraction or expansion.

 

Do you agree with the abovementioned?

 

Qtop

[coldcreation]

...When I speak of hyperbolicity I am referring to the 3D space, not to global curvature which is 4D, but you seem to be talking about global 4D espace curvature, [...] it is something not even addressed by the experts, only in speculative form, and is not fixed by GR field equations either. there are too many parameters unknown, GR is "only" clearly determining the local curvature originated by a certain local stress-energy tensor configuration.

 

Qtop, when I speak of hyperbolicity I am either referring to a point (Gaussian point), a line (geodesic), a plane (hyperbolic paraboloid), a 3-dimensional space (Minkowski space with time), or a 4-dimensional spacetime.

 

According to Einstein's general relativity the universe is a four-dimensional space-time continuum. To remove a dimension is fine when the goal is to schematize the real world, like figures A, B, C etc.

 

Representations of the physical structure of the universe in less than 4 dimensions, or where space is separated time are simply models or projections or a special case of a more general solution, respectively. Gravity corresponds to the curvature of a spacetime (or space-time) manifold.

 

The ultimate goal, with the help of GR (and its incorporation of non-Euclidean geometry) is to describe the 4-dimensional world in which we live (the physical universe). Anything less than 4-D falls short of that goal.

 

 

In the static scenario, global curvature should be at least constant but unless you consider a finite model like Einstein's universe, global curvature is not straighforward at all in an infinite universe, since global density is really hard to determine, if you use GR without the cosmological constant in a static infinite universe(infinite radius) you get Minkowski spacetime that has null curvature as is shown in http://www.spsnational.org/radiation...p_bigbang2.pdf pages 27 and 28 in equations 14 to 18.

 

There is not just one interpretation to be made from Einstein's universe. There are many.

 

Your link seems to be down at the moment, so I can't comment on it. One thing I can say is that Minkowski spacetime is a convenient mathematical framework that expresses Einstein's special relativity. Minkowski space is used to describe systems over finite distances within the Newtonian limit, without significant gravitational effects. In the case where gravitational affects are significant, where light travel on geodesic paths, where time dilation and gravitational redshifts occur, and when dealing with the large-scale topology of the universe (where distances a vast and gravity is non-negligable), spacetime becomes curved and one must abandon Minkowski spae-time, in favor of general relativity (set in a four-dimensional spacetime continuum).

 

So I don;t see arriving at GR through the back door of a flat Minkowski spacetime (or visa versa) at infinity or anywhere else.

 

 

So in a non-expanding scenario, without cosmological constant, if you faithfully follow GR to its last consequences you get an infinite universe with null curvature.

 

I would love to see a peer reviewed source for that if you have one. (Please double check the link above, or post the correct link if the one above explains that.)

 

Off hand, I can assure you that a universe with null curvature cannot exist.

 

 

That is the only mathematically sound solution with the conditions of a non-expanding space and keeping the original Einstein equation without lambda.

So the instability issue only arises when lambda is introduced in order to avoid the problem with infinity.Since in a finite static model any local density or pressure deviation from equilibrium would lead to either contraction or expansion.

 

Do you agree with the abovementioned?

 

No. I disagree. For example it was found that the original Einstein universe, with or without lambda, was unstable. I also disagree that a finite static model any local density or pressure deviation from equilibrium would lead to either contraction or expansion. Recall that a finite universe has no boundary according to Einstein. So the notion of 'finite' in this sense lacks physical meaning. Of course no boundary does no obligatorily imply infinity. But it sure points in that direction.

 

 

CC

[quantumtopology]

Qtop, when I speak of hyperbolicity I am either referring to a point (Gaussian point), a line (geodesic), a plane (hyperbolic paraboloid), a 3-dimensional space (Minkowski space with time), or a 4-dimensional spacetime.

 

According to Einstein's general relativity the universe is a four-dimensional space-time continuum. To remove a dimension is fine when the goal is to schematize the real world, like figures A, B, C etc.

 

Representations of the physical structure of the universe in less than 4 dimensions, or where space is separated time are simply models or projections or a special case of a more general solution, respectively. Gravity corresponds to the curvature of a spacetime (or space-time) manifold.

 

The ultimate goal, with the help of GR (and its incorporation of non-Euclidean geometry) is to describe the 4-dimensional world in which we live (the physical universe). Anything less than 4-D falls short of that goal.

 

Fine. who said otherwise?

 

 

 

There is not just one interpretation to be made from Einstein's universe. There are many.

Like for example? I can't see your point here.

 

Your link seems to be down at the moment.

Try now, you need to read it to see my point: http://www.spsnational.org/radiations/2008/ecp_bigbang2.pdf

 

 

So I don;t see arriving at GR through the back door of a flat Minkowski spacetime (or visa versa) at infinity or anywhere else.

See link

 

 

Off hand, I can assure you that a universe with null curvature cannot exist.

I didn't imply that it could in my post. I'm telling you the outcome of applying GR equations in a static, infinite set.

 

 

Q

No. I disagree. For example it was found that the original Einstein universe, with or without lambda, was unstable.

 

Can you give me a reference on this, where it states that GR equations without lambda give an unstable universe? see my link

 

I also disagree that a finite static model any local density or pressure deviation from equilibrium would lead to either contraction or expansion. Recall that a finite universe has no boundary according to Einstein. So the notion of 'finite' in this sense lacks physical meaning. Of course no boundary does no obligatorily imply infinity. But it sure points in that direction.

 

On this point I think both Modest and I disagre with you. But seems to be common knowledge, and I haven't been able to find any flaw here.

 

Regards

 

Qtop