Redshift z

[quantumtopology]

I

I think this is where the misunderstanding originates: In the standard expanding model the curvature parameter takes on values +1, 0, or –1 for positive curvature, Euclidean or flatness, and negative curvature, and is usually related to the scale factor or size of the universe and redshift z. It could be argued that this an archaic way of looking at curvature, and more generally, of looking at the universe.

 

That's what I wrote in my last post, perhaps you missed it. :phones:

[modest]

Modest, there is still a misunderstanding.

 

According to GR, mass-energy density increases curvature of spacetime in a 'negative' direction (hyperbolically); not a 'positive' direction as you write.

 

I'm sorry, I don't have time to respond to everything.

 

The curvature that I have been speaking to and that cosmologists speak to with omega-K is a curvature of space, not spacetime. The addition of energy density increases the curvature of space in the positive direction.

 

The curvature of spacetime becomes far more complicated and doesn't translate exactly to what we're talking about (eg adding the angles of triangles).

 

I think this is where the misunderstanding originates: In the standard expanding model the curvature parameter takes on values +1, 0, or –1 for positive curvature, Euclidean or flatness, and negative curvature, and is usually related to the scale factor or size of the universe and redshift z. It could be argued that this an archaic way of looking at curvature, and more generally, of looking at the universe.

 

There's no need for obfuscation. We can describe exactly the physical things we are talking about. If you'd like to re-brand the metric expansion of space as the negative curvature of time it makes no difference. We're talking about things physically moving apart. Likewise with the curvature of space, we're talking about cosmic triangles that are, in our universe, flat. The angles add to 180.

 

~modest

[quantumtopology]

To go further on the criticism to the standard interpretation of CMB radiation, besides what CC posted about fluctuations, about 2 years ago somethin called "dark flow" (as if we didn't have enogh with dark energy and dark matter) was discovered by Kashlisnky's team, and even though at first it was criticized due to some statistical issues, just a couple of months ago they have published a paper confirming it in "The astrophysical journal letters".

The meaning of this flow is being interpreted in the most phantastic ways :"tug of other universe", "influences prior to inflation", etc. Anything before reconsidering the origin of CMB, not one voice pointing to more logical views,like: "perhaps the CMB is local and we are not watching another universe". Maybe people including scientists are more inclined to science-fiction in cosmology issues.

 

Mysterious 'Dark Flow' May Be Tug of Other Universe : Discovery News

Dark flow - Wikipedia, the free encyclopedia

A New Measurement of the Bulk Flow of X-Ray Luminous Clusters of Galaxies

[coldcreation]

The curvature that I have been speaking to and that cosmologists speak to with omega-K is a curvature of space, not spacetime. The addition of energy density increases the curvature of space in the positive direction.

 

The curvature of spacetime becomes far more complicated and doesn't translate exactly to what we're talking about (eg adding the angles of triangles).

 

There's no need for obfuscation. We can describe exactly the physical things we are talking about. If you'd like to re-brand the metric expansion of space as the negative curvature of time it makes no difference. We're talking about things physically moving apart. Likewise with the curvature of space, we're talking about cosmic triangles that are, in our universe, flat. The angles add to 180.

 

~modest

 

This thread is about two differing interpretations for the same observations. Both of these interpretations, to some extent, are based on Einstein's general relativity, albeit, one more so than the other.

 

The big bang theory is a simple cosmological model, derived from the basic framework of Newton’s theory of gravity. It is a quasi-Euclidean, neo-Newtonian, pseudo-special relativistic system. The dual nature of the expanding model remains unsatisfactory—one the one hand it considers gravitational phenomena as a local deviation of the spacetime metric, whereas, on the largest scales the associated gravitational influences are reduced to trivial insignificance.

 

The big bang theory considers space and time separately in determining the space-time coordinates in other coordinate systems, e.g., those of other galaxies. But general relativity states that the world of events forms a four-dimensional continuum. Modern cosmology describes a Newtonian world of events as a dynamic inertial picture changing in time and hurled onto the background of three-dimensional space, rather than as a stationary view on the background of a four-dimensional spacetime continuum where Einstein’s gravitation plays the key role.

 

Certainly, the so-called expansion is considered a relative expansion, or relative motion. Nevertheless, this potentially spurious apparent motion is only of special relativistic character. The general relativistic nature of the expanding model remains highly ambiguous in that the spacetime structure of the universe remains unknown, and curvature of space depends on a ‘deceleration parameter’ or the velocity of recession.

 

 

The question this thread poses is straightforward: How, do we make the distinction between an expanding Newtonian inertial system and a stationary general relativistic universe, when a beam of light is affected (curved) in a gravitational field exactly as if the source of a beam were traveling (away from us) at great velocity? Are we dealing with an inertial problem, or a gravitational problem? Clearly, the solution must come from general relativity and must differ drastically from the Newtonian solution when dealing with large distances or strong gravitational fields. The laws of gravitation, just as the laws of nature, ought to be devised for all possible coordinate systems, while the laws of classical Newtonian mechanics (and big bang cosmology) are valid only in inertial systems (see Einstein, Infeld, 1938, 1961, pp. 207-222)

 

 

We should try to find a better representation of bodies behaving in a way expected by Euclidean geometry. If, however, we should not succeed in combining Euclidean geometry and physics into a simple and consistent picture, we should have to give up the idea of our space being Euclidean and seek a more convincing picture of reality under more general assumptions about the geometrical character of our space

(Einstein, Infeld, 1938, 1961, p. 225-226)

 

 

 

______________________

 

 

 

Placing the standard model aside for a moment, let's take a look at the diagram below (a variation of manifold A above) which represents a stationary general relativistic universe in a simple and consistent manner that discards the notions of Newtonian inertial systems and Euclidean geometry (manifold B above). The assumption about the geometrical character of spacetime curvature is in direct accord with the postulates of general relativity in that the total mass-energy density of the universe induces a nonzero value for curvature: Just as mass-energy induces a hyperbolically curved spacetime locally, mass-energy induces a hyperbolically curved spacetime globally, from the rest-frame of any observer. The field is homogenous and isotropic. There is a loss of energy and time dilation associated with photons as they propagate across the the homogenous gravitational field that manifests itself as redshift z.

 

 

 

 

Figure A:

Hyperbolic non-expanding spacetime manifold in reduced dimension (oblique angle).

 

 

The observer is centrally located (at the origin), and is surrounded by successive spherical shells which represent distances of one billion light years. The deviation from linearity increases with distance from the origin and reaches its maxima at the horizon (represented by the outer edge of the manifold).

 

 

Observations:

 

 

1. In this type of spacetime manifold clocks located at distant supernovae and their host galaxies would appear (in the look-back time) to be running slower than local clocks.
    
    
   
2. The large shells of radiation and material emitted by distant SNe Ia would appear to have a greater area than they would in a geometrically flat spacetime, making the source look very faint.
    
    
   
3. The visible universe would appear larger and deeper, than would be the case in a Euclidean or geometrically spherical universe (the circumference of the visible universe would appear larger than 2πr).
    
    
   
4. Distant SNe Ia would appear unexpected dim, giving the impression they are further away than their redshifts indicate (cosmological triangulation measurements would reveal less than 180 degrees).
    
    
   
5. 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 and increasingly ‘stretched’ with increasing distance.
    
    
   
6. Spacetime would appear to be hyperbolically curved, in accord with Einstein's general postulate of relativity. This is would be the type of manifold described by Lobachevsky (1823): a hyperbolic space that would find itself embedded in the Riemann space of constant ‘negative’ curvature (1854-1866).
   

 

 

Edit: This type of spacetime could easily be mistaken for a pseudo-Newtonian space that is expanding inertially (accelerating even), just as an expanding space could easily be mistaken for a stationary hyperbolic spacetime continuum governed by general relativity. :phones:

 

 

CC

[CraigD]

Until someone can explain how the Universe ( in whatever space time form) got here, how the electron and proton generate a field with the single strongest force in the Universe without losing any energy, we're pissing in the wind.

Though explaining how the universe got here is difficult and controversial, thousands of decent mathematical physics students have know for several centuries how to explain how any massive body – be it made of atoms or something more exotic, or of unexplained composition – exerts gravitational force on other bodies without doing any work, and thus neither loosing or gaining energy, simply via the classical mechanical definitions of force, work, and energy.

 

The formula for work is [math]W = F \cdot \Delta D[/math], where [imath]F[/imath] is force and [imath]\Delta D[/imath] is change in position. Note that, regardless of the magnitude of the force, if the change in position coinciding with the direction of the force is zero, work is also zero. If work is non-zero, the energy loss or gain is accounted for by a change in the kinetic energy of the gravitationally interacting bodies. This bookkeeping is usually discussed in the context of the concept of gravitational potential energy.

 

As CC asks

What have your comments to do with redshift z?

I also don’t see much connection between your post and this thread, LB. If you’ve no objection, I’ll move our posts to a new physics thread.

[modest]

To go further on the criticism to the standard interpretation of CMB radiation, besides what CC posted about fluctuations, about 2 years ago somethin called "dark flow" (as if we didn't have enogh with dark energy and dark matter) was discovered by Kashlisnky's team, and even though at first it was criticized due to some statistical issues, just a couple of months ago they have published a paper confirming it in "The astrophysical journal letters".

The meaning of this flow is being interpreted in the most phantastic ways :"tug of other universe", "influences prior to inflation", etc. Anything before reconsidering the origin of CMB, not one voice pointing to more logical views,like: "perhaps the CMB is local and we are not watching another universe". Maybe people including scientists are more inclined to science-fiction in cosmology issues.

 

Mysterious 'Dark Flow' May Be Tug of Other Universe : Discovery News

Dark flow - Wikipedia, the free encyclopedia

A New Measurement of the Bulk Flow of X-Ray Luminous Clusters of Galaxies

 

There's a thread about that hiding around here somewhere. I'll look for it when I have time.

 

Placing the standard model aside for a moment, let's take a look at the diagram below (a variation of manifold A above) which represents a stationary general relativistic universe in a simple and consistent manner that discards the notions of Newtonian inertial systems and Euclidean geometry (manifold B above)....

 

Figure A:

Hyperbolic non-expanding spacetime manifold in reduced dimension (oblique angle).[/center]

 

Your posts seem full of misinformation. That diagram is not a spacetime diagram. There is no time axis. It is a diagram of space (2 dimensions of space). As a GR universe it would be called a slice of constant cosmic time. Because it is hyperbolic and isotropic we know its energy density is less than the critical density. It is not a substitute for cosmic redshift.

 

~modest

[coldcreation]

Your posts seem full of misinformation. That diagram is not a spacetime diagram. There is no time axis. It is a diagram of space (2 dimensions of space).

 

That illustration is a representation in reduced dimension of a spacetime manifold. The present (now) is located at the origin (at the observer), and the past, is everywhere else. The outer edge of the illustration represents the furthest distance and the furthest point in the look-back time, a relative location and relative time (compared to what would be expected in a Euclidean spacetime). So the time axis could be considered any line that tends to and intersects at the origin. It is a spacetime map.

 

I will attempt to draw one of these diagrams with an impression of volume in order to represent a third spatial dimension. But that might take a while. Your time axis, unfortunately will not look much different than in the current Fig. A above. It will be a line (or lines) extending form the outer edge to the center, which represent the look-back time from O. Notice that there is a time dilation factor which would be directly proportional to the spatial curvature, as viewed from O. Notice too that redshift would increase nonlinearly with distance as viewed from O.

 

Your comment of inconsistency would have been correct had everywhere on the manifold been the same universal time or cosmic time. That is not the case above though (this is no god's eye view of the universe "now"). That is because of the limit of the velocity of light c.

 

Figure A is meant to show what an observer 'sees' from any relative rest-frame O, which could be anywhere in the universe, i.e., all observer would see the universe as if they were embedded at the 'center' of a hyperbolic spacetime manifold. Note; this model would be hyperbolically spherically symmetric.

 

 

 

As a GR universe it would be called a slice of constant cosmic time. Because it is hyperbolic and isotropic we know its energy density is less than the critical density. It is not a substitute for cosmic redshift.

 

Even if your criticism were justified it would still be besides the point (obfuscation-like :eek2:), since you know the intention of the diagram A is to portray in reduced dimension a model of spacetime, one which is non-expanding, or one that is expanding, as both models would be virtually indistinguishable, form the view-point of any observer at the present time (always located at the origin). That is the point of this entire thread.

 

As a GR universe Fig. A would be called, not a slice of constant cosmic time, but a slice of a 4-dimensional universe (3 space, 1 time) which conforms to certain principles (e.g., the cosmological principle, in that it would be homogenous and isotropic yet evolving in the look-back time as seen from O).

 

The physics in the two universes under study in this thread (one expanding and one not) are virtually indistinguishable. A characteristic feature of Figure A is that every part of it should contain enough information to reconstruct the entire image when projected into four dimensions (though I've likely exaggerated or underestimated the hyperbolicity of the manifold. In that sense it is not a mathematically rigorous model). But the light-line are represented (light time for unit distance can be (and should be) interpreted form the diagram. Each successive 'spherical' shell (which would be represented concentric circle in a Euclidean universe) here represents the time it would take light to travel in one year, from the reference-frame O. Clearly these are shown in Fig. A to dilate increasingly in the look-back time (as a function of distance).

 

The light lines are represented in Fig. A by one-dimensional lines (again intersecting at O). It is evident that the area surrounding these lines appears larger with distance, thus the incoming photons as measured from O travel a geodesic (each time line is thus a like a meter stick, the increments would increase non-linearly with distance). This is the look-back-time visible to all observers via their light lines. This illustration thus corresponds to the domain of spacetime (not just space).

 

Of course, in the real world, measurements depend upon physical assumptions, i.e., the assertion that one recognizes the object in question, and that the class of objects is homogeneous enough that its members can be used for meaningful spatial distance and temporal estimates. That is why SNe Ia and the results of the data attained by this class of objects (e.g., redshift and light-curves) is fundamental to our understanding of the spacetime manifold.

 

Fig. A represents a hyperbolic and isotropic spacetime. In the case where the model hypothesizes expansion (which is not the intention of Fig, A, but it could be interpreted that way), its energy density would be, as you write, less than the critical density. Hyperbolicity would not be a substitute for cosmic redshift. However, in the non-expanding case, hyperbolicity would be a substitute, or the raison d'être, of redshift z, and there would be no need for a critical density, at least not in the sense of the standard model.

 

If you see any other apparent inconsistencies or misinformation I will gladly attempt to clarify to the best of my abilities. :phones:

 

 

 

CC

[quantumtopology]

There's a thread about that hiding around here somewhere. I'll look for it when I have time.

 

You have a prodigious memory, I did a quick search and indeed there is a thread :

"Distance to the Furthest Visible Objects" from 2008 in which the first press release about "dark flow" is mentioned by a poster called Essay, but his mention went unnoticed and nobody discussed it further.

[modest]

That illustration is a representation in reduced dimension of a spacetime manifold. The present (now) is located at the origin (at the observer), and the past, is everywhere else. The outer edge of the illustration represents the furthest distance and the furthest point in the look-back time, a relative location and relative time (compared to what would be expected in a Euclidean spacetime). So the time axis could be considered any line that tends to and intersects at the origin. It is a spacetime map.

 

Ok, that's good. You can remove one of the spatial dimensions and add time. You end up with a normal spacetime diagram like I posted before:

 

(this is an empty, hyperbolic universe)

 

What you are talking about is the surface of the past lightcone—the green dots. The problem is that such a representation is not hyperbolic. It translates to C on your diagram:

 

 

You have a prodigious memory, I did a quick search and indeed there is a thread :

"Distance to the Furthest Visible Objects" from 2008 in which the first press release about "dark flow" is mentioned by a poster called Essay, but his mention went unnoticed and nobody discussed it further.

 

Yup, I see Lemit did reference the paper there, but I was actually thinking of this thread:16525.

 

~modest

[coldcreation]

Ok, that's good. You can remove one of the spatial dimensions and add time. You end up with a normal spacetime diagram like I posted before:

 

What you are talking about is the surface of the past lightcone—the green dots. The problem is that such a representation is not hyperbolic. It translates to C on your diagram:

 

The diagram you posted is not a good example of what is happening in the real world. The idea here is not to make a projection onto a flat surface, which would end up looking like an illustration by M. C. Escher of a hyperbolic sphere, where distances appear to become smaller with increasing distance from the origin (at the center). While those types of projections are an accurate description of hyperbolic geometry they are misleading in that they end up looking spherical (positively curved). That is the problem with projections. Fig. A is not a projection of that type.

 

Clearly, if you look at a hyperbolic surface a saddle shape, or a Pringles potato chip shape and flatten it out, then look at it from above, the edges (if they don't break :phones:) are stretched outwards, along with the grid pattern. That is why C is misleading, and A is the more accurate representation.

 

The other problem with the diagram you posted are that the is a set of line (the 'horizontal' ones that remain 'parallel' or equally spaces one from the other). That would not be the case when considering a spacetime manifold of the type required to agree with observations, of the Figure A type above.

 

I'm working on a new illustration with an extra dimension, that will show successive spherical shells, in volume. The central sphere is the smallest (quasi-Euclidean section). The spheres become larger with greater relative distance (just as in Fig. A above, the space between each circle increases nonlinearly).

 

Hopefully with that diagram you should be able to grasp the situation; one that corresponds to that which is observed in the cosmos.

 

 

CC

[quantumtopology]

The idea here is not to make a projection onto a flat surface, which would end up looking like an illustration by M. C. Escher of a hyperbolic sphere, where distances appear to become smaller with increasing distance from the origin (at the center). While those types of projections are an accurate description of hyperbolic geometry they are misleading in that they end up looking spherical (positively curved). That is the problem with projections. Fig. A is not a projection of that type.

That is exactly what happened to me when I first chose diagram C, for some reason I was thinking about a projection on a Euclidean space -probably I've seen too many Escher pictures-:phones:

Modest , I've seen the thread you link; interesting , if a little confusing for me ,though.

The way I see it the Kashlinsky work is a new problem for L-CDM model, and as that is being fought by many cosmologists like ned wright and others that try to dismiss it quickly.

[coldcreation]

There are several reasons why there is still confusion about curvature being negative or positive in the illustration A and C above, besides the fact that 'direction' of 'curvature' is not always intuitive: The illustrations A, B and C represent Polar coordinate systems, or polar-like constructions: coordinate systems in reduced dimension where each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

 

Yet, since distance, in diagrams A, B and C is determined by the photon travel time (look-back time) the time element is inextricably linked with the spatial coordinates (space and time are inseparable). In another way, A, B and C represent three different models of a spacetime continuum and how the manifold would be seen by an observer at her arbitrary reference-frame. That is why distances between each 'spherical' shell (or circle) are expressed in light years (the unit of length). Figure A above could just as well have been plotted using the gigaparsec as the unit of length. The difference would have been fewer 'spherical' shells. For example I could have drawn 4 shells which would give us a distance to the horizon of about 13 billion light years (about four gigaparsecs, since each circle would represent about 3.26 billion light years).

 

So what we have in Figure A is a reduced dimension polar coordinate systems (that represents 4-dimensional manifold), where the azimuth of a point is the angle between the positive x-axis (unlabeled in the illustration) and the projection of the vector onto the xy-plane (the component of the vector in the xy-plane). That is because the circles represent spherical shells, where an angular measurement in the spherically symmetric coordinate system conforms to a vector from the observer's view point (the origin) projected perpendicularly from a 3-dimesnional spherical shell onto a reference plane, where time is a function of distance.

 

 

 

Diagrams A, B and C should not be considered (though they may look like) or confused with classical stereographic projections, which are conformal and perspective-like, but do not have equal area, or are not equidistant, such as the classic Escher hyperbolic spheres.

 

 

Figure A above is not this type of projection. That is why the problem inferred to by modest is unjustified, yet the error is comprehensible. Perhaps I could have been clearer in the initial explanation, so as not to be confused with the type of illustration below:

 

 

The projection principle for the Gnomonic, Stereographic and Orthographic projection. Source

 

 

Stereographic projections are conformal and perspective-like but are not of equal area, or equidistant.

 

Again, diagrams A, B and C, or Figure A above, are not stereographic projections.

 

In geometry, the stereographic projection is a particular mapping (function) that projects, e.,g., a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

 

The stereographic projection is a manner of picturing the sphere as the plane, but not without some inevitable compromises.

 

Though stereographic projections are conformal azimuthal projections (all meridians and parallels are shown as circular arcs or straight lines) parallels and meridians intersect at right angles. In the polar aspect the meridians are equally spaced straight lines, the parallels are unequally spaced circles centered at the pole. Spacing gradually increases away from the pole.

 

The Gnomonic projection, a type of stereographic projection, displays all great circles as straight lines. So the shortest route between two locations in reality corresponds to that on the manifold. This is achieved by projecting with respect to the center of a sphere (perpendicular to the surface), onto a tangent plane. Though the least distortion occurs at the tangent point, only half of the sphere can be projected onto a finite map.

 

The gnomonic projection is neither conformal nor equal-area. The scale increases rapidly with the distance from the center. Area, shape, distance and direction distortions are extreme, but all circles (the shortest routes between points on a sphere) are shown as straight lines. For the Gnomonic projection, the perspective point (like a source of a light beam), is the centre of the sphere. For the stereographic this point is the opposite pole to the point of tangency, and for the orthographic the perspective point is an infinite point in space on the opposite side of the sphere. (Same source as above.)

 

 

So clearly, the differences between projections of the gnomonic type and the type described by Figure A are exposed and clarified. Hopefully that will alleviate the misunderstandings. Somehow though, I doubt it will. This problem has not subsided, despite the elaborate explanations by those directly involved from the inception of non-Euclidean geometry itself, onward, and GR has not seemingly resolved the issues. Quite the contrary. The phenomenon of curved spacetime remains for the most part, and not just for the laymen, a difficult thing to imagine, particularly in four dimensions. That is partly because no reduced dimension illustration is completely non-problematic.

 

The key point to remember, as far as diagram A is concerned, is that it represents that which is observed in the past light-cone of any observer (in the form of a non-Euclidean spacetime continuum). It is a deduction base on a particular interpretation of the empirical evidence, nothing else. It is a conclusion based on the idea that if the universe is non-expanding, this is what must be happening.

 

And as a final point, I'd like to emphasizes once more; the universe is not really topologically or geometrically hyperbolic. It only would appear that way, and would so to all observers, regardless of their location. That would be so because the universe, in actuality, would be homogenous and isotropic gravitationally, at any given time t.

 

So the apparent (or relative) curvature would be due to the limit of the velocity of light c, the loss of energy of each photon by a factor of (1 + z), and the time dilation factor (the apparent increase with distance in the observed timescales of an object relative to the rest-frame of that object) of cosmological redshift z by the factor (1 + z) that determines the metric curvature (and visa versa). Thus, the observer should 'see' the universe from his/her location as if embedded centrally on the surface (in reduced dimension) of a hyperbolic paraboloid, where both spatial and temporal increments appear to increase non-linearly with increasing distance in the look-back time.

 

 

CC

[quantumtopology]

And as a final point, I'd like to emphasizes once more; the universe is not really topologically or geometrically hyperbolic

 

Hmmm, I missed this, in what posts did you emphasized this point?

 

It only would appear that way, and would so to all observers, regardless of their location. That would be so because the universe, in actuality, would be homogenous and isotropic gravitationally, at any given time t.

 

So, what geometry would the universe have in your opinion?

[coldcreation]

 

And as a final point, I'd like to emphasizes once more; the universe is not really topologically or geometrically hyperbolic. It only would appear that way, and would so to all observers, regardless of their location. That would be so because the universe, in actuality, would be homogenous and isotropic gravitationally, at any given time t.

 

Hmmm, I missed this, in what posts did you emphasized this point?

 

So, what geometry would the universe have in your opinion?

 

Ok, I took my finger off my queen, but I'd like to take that move back, before I lose it. :phones:

 

So I take back what was written above and restate: The global field would be hyperbolically curved, since gravity is everywhere present, and at all times t.

 

I've been saying all along that the observed curvature (if indeed redshift z is evidence of curvature) is a relative effect, an apparent one, that does not depend on the location of the observer at any given time t. That means there is no center of the universe, in conformity with the cosmological principle. Though every observer is entitled to consider herself centered on the manifold, in reality there would be no center, no maximum or minimum of the field (except those associated with infinity and those of each observational rest-frame). It's a relative effect, because all observers are entitled to call their frame of reference unique, locally Euclidean, a minima of the global field potential, etc.

 

Take Figure A2 below, for example. The principle is the same as figure A above: The observer is located at the center of spherical shells (each is 0.5 gigaparsec unit distance) that extend to the horizon. There is also a superimposed hyperbolic paraboloid (yes, for a little exaggerated artistic license) also centered on the observer at the origin.

 

 

 

 

Figure 2A

 

 

 

Figure A2 is a representation of a non-Euclidean spacetime manifold. The observer (judging from redshift z, SNe Ia light curves, by comparing apparent and absolute magnitudes of SNe Ia, and so on), concludes that the universe is hyperbolically curved or is expanding, or is both non-Euclidean and expanding. Note the time dilation with increasing distance from the observer rest-frame.

 

The point is that this is a view from the perspective of any observer, wherever she is located in the universe at this time. That means that an observer located, say, at a distance of 2 gigaparsecs will see the universe exactly as if she were located at the origin (the center of Figure A2). Likewise, as observer situated on the outer extremity of the hyperbolic paraboloid, as viewed from the origin, will see the universe as if she was (not at the extremity of this manifold) at the origin, again, centrally located. So the curvature is a relative effect. It is a continuous, homogenous, isotropic curvature, with no center, maxima or minima, and it would extent to infinity. One might easily conclude, in the case where it is believed the universe to be non-expanding, that the universe is infinite in all directions, and infinite spatiotemporally both in the past and in the future, i.e., there would be no beginning and no end.

 

So being homogenous and isotropic gravitationally, the cosmos has no edge, boundary or center of gravity, and no unique or privileged rest-frame. It would be infinite without bounds, a truly remarkable place, to say the least. :)

 

 

 

CC

[coldcreation]

Continued from above...

 

 

The situation described here, as to whether the observations could be construed as an real intrinsic curvature of spacetime, or simply a relative effect depending on the rest-frame of an observer, is almost analogous to the concept of Doppler shifts vs. expanding space.

 

Recall that redshift was originally interpreted (by Hubble, Humason and Slipher) as due to the Doppler effect. Hubble later discovered a correlation between the increasing redshifts and the increasing distance of nebulae. It was quickly realized that redshift z could be explained by a mechanism other than both the de Sitter effect or Doppler shifts. There was a correlation between redshift and distance (the Hubble's law) that seemed required by cosmological models on the basis of general relativity, consistent with a metric expansion of space (Eddington, A., The Expanding Universe: Astronomy's 'Great Debate', 1900–1931, 1933). The idea was that photons propagating through the expanding space are stretched, resulting in cosmological redshift (as opposed to a Doppler effect): where the velocity boost (i.e. the Lorentz transformation) between the emitting source and the observer is not due to a classical momentum and energy transfer, but alternatively, the photons increase in wavelength (redshift) as the space through which they propagate expands. The observational consequences of this type of redshift can be derived using the equations from general relativity that describe a homogeneous and isotropic universe.

 

 

There is a correlation if one were to consider a non-expanding manifold, where redshift is due to the curvature of spacetime. Whether the observations could be construed as a real intrinsic curvature of spacetime, or simply a relative effect depending on the rest-frame of an observer (i.e., a mechanism that describes the correlation between increasing redshifts with increasing distance), resulting from photons traveling through a curved spacetime continuum, as viewed by any observer's rest-frame. It seems too, the requirement that this cosmological model must be derived from general relativity, consistent with a metric curvature of space.

 

The idea is that photons propagating through spacetime lose energy (as if stretched) due to the non-linear regime through which they propagate, resulting in cosmological redshift z (as opposed to a Doppler effect or the expansion of space): where the photons increase in wavelength (redshift) as the space through which they propagate curves hyperbolically. The question is whether that effect would be due to an intrinsic property of the curvature of the spacetime continuum (the general curvature of spacetime), or whether the effect would be solely a relative one based on the location of the observer (a spurious local effect). The answer is not at all obvious, though, the observational consequences (in either cases) are real, and can be derived using the equations from general relativity that describe a homogeneous and isotropic universe, in accord with the cosmological principle.

 

Einstein's equations describe the relation between the geometry of a 4-dimensional manifold representing spacetime (correspond to a curved geometry) and the energy-momentum contained in that spacetime (a Lorentzian manifold). Gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths (the geodesic) that 'objects' (e.g., photons) will naturally follow. The curvature is caused by the energy-momentum of matter, or the total mass-energy density of the universe, in the case under study here. It follows that the physical consequences, both locally and globally, are far from spurious.

 

Indeed, to discard the concept of curved spacetime (and that, whether geometrically hyperbolic or spherical; though a spherical universe seems ruled out by observation) would be to revert back to a Newtonian world at worst, or a special relativistic world at best. Neither of these can be considered a viable option.

 

 

Here below is another way (albeit non-problematic-free) to visualize what could be transpiring in a non-expanding (or even expanding) universe, as viewed by any observer from the center of the image, looking outwards through the depths of the cosmos.

 

 

 

 

Figure 3A: a general relativistic spacetime continuum

 

 

 

Figure 3A represents a cross section in reduced dimension of a hyperbolic spacetime manifold with the same characteristic features and Figure 2A and A above. There is a superimposition of an oblique polar grid where the observer is at the center (the origin O) that cuts though consecutive spherical shells surrounding the origin. The 'concentric' circles in the polar grid represent about one billion light years each that extend to the horizon. The spherical shells represent distances of about 0.5 gigaparsecs each, extending to the visual horizon. The observer is looking at her past light-cone, so this is an image of the look-back time, from O.

 

Notice the apparent distances appear to increase with distance from the observer.

 

The goal here is to portray a 4-dimensional globally symmetric general relativistic spacetime of a stationary universe with a hyperbolically curved continuum. But one could easily interpret the same observed features as a non-static, expanding quasi-Euclidean, pseudo-special relativistic neo-Newtonian world of events as a dynamic inertial picture changing in time and hurled onto the background of three-dimensional space.

 

 

 

CC

[quantumtopology]

But one could easily interpret the same observed features as a non-static, expanding quasi-Euclidean, pseudo-special relativistic neo-Newtonian world of events as a dynamic inertial picture changing in time and hurled onto the background of three-dimensional space.

 

So easily that actually this is the way 99% of people interptret it. But we like it the hard way, don't we? :phones:

So why would we? That's what this thread is all about.

Maybe it's time to move on a step further and talk about what is behind such a spacetime.

If we agree that spacetime is another name for mas-energy and that Einstein field equations equate the geometry of spacetime and the attributes of matter and energy. then some change must be introduced in the stress-energy tensor in order to allow for our interpretation of spacetime curvature and redshift z.

It was Einstein himself who admitted that the right side of his field equation was the weakest with its mix of constants and concepts. Specially we are in lack of a vacuum stress-energy tensor, that is, a better description of the vacuum more in consonance with what is known in quantum mechanics.

It is my belief that any interpretation of the nature of spacetime and redshift z must be based in a better understanding of matter and radiation.

[coldcreation]

So easily that actually this is the way 99% of people interptret it. But we like it the hard way, don't we?

 

Yea, it's fun to speculate, to challenge ideas, to maintain a skeptical approach in the face of disputable or ambiguous evidence. The latter is what Hubble did well into the 1950's. Evidence that Hubble was still questioning whether the expansion was real, or not, can be seen in one of his 1953 diagrams (Darwin Lecture) where he noted in uppercase, “NO RECESSION FACTOR.” His use of the Doppler formula for calculating “apparent velocities” was strictly for “convenience” and simplicity. He very shrewdly left open the possibility that at some time in the future, when observational data would yield less “dubious” results, that an “ultimate interpretation” (as yet “unknown”) might be equally compelling, if not more so. That time may be now, with the advent of SNe Ia data. :Crunk:

 

 

If we agree that spacetime is another name for mas-energy and that Einstein field equations equate the geometry of spacetime and the attributes of matter and energy. then some change must be introduced in the stress-energy tensor in order to allow for our interpretation of spacetime curvature and redshift z.

 

I don't think we need to go that far. :rolleyes:

 

 

CC