Redshift z

[modest]

Modest your graph from post 482 of look-back time versus angular diameter distance is from a expanding universe but in your opinion, does it agree with diagram C?

 

Yes, it absolutely does.

 

But, and this is what I think is confusing you, light travel time distance is not the usual measure of distance that determines the curvature of space. Light travel time is a mixture of space and time which, if anything, would tell a person about the curvature of spacetime.

 

To say:

1. look-back time (ie light travel time) ends up looking like diagram C
   
2. proper distance (ie comoving distance) ends up looking like diagram B
   

Has two very different meanings. When cosmologists talk about the curvature of *space* they are referring to #2. CC has been diagramming #1.

 

There is a very easy way to demonstrate this if someone knows how spacetime diagrams work. CC has been diagramming the green dots and the curvature of space is usually understood as the blue dots:

 

 

This is a spatially hyperbolic universe, so the blue dots would end up looking like diagram A. In the concordance model the horizontal slices of constant time would be flat. Space would be flat. The blue dots would then be like diagram B in the concordance model. The green dots would still be like diagram C.

 

In any expanding model the green dots will plot like diagram C because distances (of any sort) used to be smaller when the universe was younger (because things were closer together). This does not, however, show the curvature of space.

 

So, I think the confusion is that cosmologists are not using look-back time when talking about the curvature of space, and also, from CC's posts, I don't think he is talking about the curvature of space but rather the curvature of spacetime... which is something really rather different. The spacetime diagram of a Schwarzschild mass in my last post shows the curvature of spacetime near a massive body. The geodesics converge near the body and diverge further from it. It is negative curvature of spacetime.

 

Negative spacetime curvature corresponds to an attractive gravitational force and positive spacetime curvature corresponds to a repulsive gravitational force.

 

As far as I can tell, CC is now saying that spacetime (*not* space) could be globally positively curved. With this new idea his interpretations about time dilation and redshift will find a lot of agreement from me. Redshift and time dilation should act pretty much like he says in his second to last post in such a universe. Redshift and time dilation would indeed approach infinity at the horizon in such a universe. I would agree that such a universe would have a de Sitter effect.

 

The real problem, that I see, is that spherical spacetime manifolds would have a repulsive force. The things inhabiting such a universe would not be static. Two perfect examples would be de Sitter space and inflation. Both have positively curved spacetime that translates directly to a repulsive, scattering effect.

 

~modest

[quantumtopology]

More basic questions:

in the redshift versus angular size graph from the http://arxiv.org/PS_cache/astro-ph/p.../9812018v1.pdf link, with the diagonal straight line that says "Euclidean" , does this graph refer to spacetime or space? If the former I assume you take the superior right triangle (where the observations lie) to be positive curvature and the inferior left triangle negative curvature, is that right? If referes to to space(3D)... well I guess that can't be since then the observations should fall in the "Euclidean" line if you think space is flat.

 

Regards

[modest]

More basic questions:

in the redshift versus angular size graph from the http://arxiv.org/PS_cache/astro-ph/p.../9812018v1.pdf link, with the diagonal straight line that says "Euclidean" , does this graph refer to spacetime or space?

The Euclidean line is meant to be a static and spatially euclidean universe. It represents the expected angular size / redshift relationship assuming that redshift is proportional to distance and angular size is inversely proportional to distance (both reasonable assumptions in a static, euclidean universe).

 

I may have been mistaken in my assertion that the data would fall below that line in a spatially hyperbolic universe. I was thinking that the x axis was distance, but it isn't. It's redshift and in a hyperbolic universe there would be no reason to assume redshift is proportional to distance. So, I guess I can't categorically say that without expansion a spatially positively curved universe would have data above the line and a spatially negatively curved universe would be below the line.

 

That would have been true if the x axis were a measure of proper distance, but it's not, so I would tend not to think of that plot as a measure of spatial curvature (or that of spacetime).

 

If the former I assume you take the superior right triangle (where the observations lie) to be positive curvature and the inferior left triangle negative curvature, is that right? If referes to to space(3D)... well I guess that can't be since then the observations should fall in the "Euclidean" line if you think space is flat.

 

Regards

 

Not sure I follow. In a static universe the angle that objects of a fixed size take up in the sky is inversely proportional to the distance of the object if space is Euclidean, less than that if space is has negative curvature and greater than that if it has positive curvature.

 

~modest

[quantumtopology]

The question of my last post gets answered with Modest's last post.

 

 

When cosmologists talk about the curvature of *space* they are referring to #2. CC has been diagramming #1.

 

Right. Now I understand.

 

Negative spacetime curvature corresponds to an attractive gravitational force and positive spacetime curvature corresponds to a repulsive gravitational force.

 

I'm not sure about this. If in our universe gravitational force is attractive ,does it mean it is a spacetime with negative curvature? I believe not. Please clear this up.

 

As far as I can tell, CC is now saying that spacetime (*not* space) could be globally positively curved. With this new idea his interpretations about time dilation and redshift will find a lot of agreement from me. Redshift and time dilation should act pretty much like he says in his second to last post in such a universe. Redshift and time dilation would indeed approach infinity at the horizon in such a universe. I would agree that such a universe would have a de Sitter effect.

 

Right, that's because it would be very similar to a de Sitter universe, wouldn't it? In the wikipedia is defined as a lorentzian manifold with constant positive curvature described by the hyperboloid of one sheet.

A lot of confusion can arise from considering hyperbolic space as a space of negative curvature (certainly this confused me) when in fact what defines hyperbolicity is not curvature but the parallel postulate that in hyperboic space doesn't hold.

 

 

The real problem, that I see, is that spherical spacetime manifolds would have a repulsive force. The things inhabiting such a universe would not be static. Two perfect examples would be de Sitter space and inflation. Both have positively curved spacetime that translates directly to a repulsive, scattering effect.

 

Exactly what happens in the de Sitter universe due to the positive cosmological constant. As we have discussed in the last post of the sitter effect thread.

 

 

So, we can conclude diagram C seems like a spacetime with positive curvature, that could be spatially(3D) hyperbolic but not necesarily, and that would fit any expanding universe, and a de Sitter universe (wheter or not we consider it static or expanding,physically wouldn't matter as Modest showed in the other thread). Correct me if you don't agree with this conclusion or I'm forgetting something ok?

 

Regards

[modest]

I'm not sure about this. If in our universe gravitational force is attractive ,does it mean it is a spacetime with negative curvature? I believe not. Please clear this up.

 

Actually, up until about 6 billion years ago the overall (global and average) force in our universe was attractive after that point it became repulsive. That is to say, gravity was trying to pull distant galaxies together after which point gravity started trying to push them apart. The universe before ~ 6 Gys was decelerating and a after ~6 G yrs started accelerating. After the onset of acceleration the overall gravitational force became repulsive.

 

If you think about this for a bit you'll be left with the question: what provided the initial velocity of expansion at T=0 if gravity was, in fact, fighting that expansion for the first few billion years? I'm not sure anyone knows the answer to that.

 

But, to directly answer your question, yes. A constant Riemann curvature tensor < 0 corresponds to an attractive force [this would be a universe with no cosmological constant and matter]. R > 0 is a repulsive force [a universe dominated by the cosmological constant (ie de Sitter space)] and R=0 is an empty universe devoid of both the cosmological constant and matter. The last case is actually another way of saying that gravity has no influence in a universe without mass or lambda. It would be a special relativity universe.

 

Teh concept of curvature of spacetime in GR is extremely complicated and I am hardly qualified, so I'm sure I'm grossly simplifying things.

 

Right, that's because it would be very similar to a de Sitter universe, wouldn't it?

 

:lol: If there were no matter then it would be exactly a de Sitter universe. Our universe is turning into a de Sitter universe as the density of matter gets smaller.

 

In the wikipedia is defined as a lorentzian manifold with constant positive curvature described by the hyperboloid of one sheet.

 

Yup. Also:

 

The de Sitter Spacetime

 

A lot of confusion can arise from considering hyperbolic space as a space of negative curvature (certainly this confused me) when in fact what defines hyperbolicity is not curvature but the parallel postulate that in hyperboic space doesn't hold.

 

Your first impression was correct. Hyperbolic space has negative curvature. Spherical space has positive curvature. The parallel postulate holds in neither. In hyperbolic space parallel lines diverge and in spherical space they converge.

 

Exactly what happens in the de Sitter universe due to the positive cosmological constant. As we have discussed in the last post of the sitter effect thread.

 

In a de Sitter universe things scatter. It has accelerated expansion (which is to say that the speed between two things accelerates with time). The cosmological constant provides a repulsive force to gravity.

 

So, we can conclude diagram C seems like a spacetime with positive curvature, that could be spatially(3D) hyperbolic but not necesarily

 

I don't think look-back time can be directly translated to curvature. We can at least say that a de Sitter universe would diagram like C in look-back time. But, like I've said, most any expanding model would. But, a diagram of look-back time is certainly not a direct measure of curvature.

 

~modest

[quantumtopology]

I found a clarifying paper that solves the aparent paradoxes of Doppler and Cosmological interpretation of redshift z that makes clear neither one is more correct than the other, is just a matter of reference frame choice (this is regarding post 474).

[0803.2701] Interpretation of the Cosmological Metric

 

 

About CC's comment in post 470:

"in the same way you don't need static solutions to the Einstein field equations for local dynamics, you don't need static solution of the field equations for global dynamics."

 

It`s not as simple as it may seem, we should agree at least that in the local scenario where GR has been proved to be true to this day (mercury precession, light bending around the sun, gravitational shift of spectrum, etc...) , where expansion doesn't have measurable effect so we can say this local scale is static, there is no need of corrections or "static solutions" for GR equations, this indeed leads us to question whether in the case of a static universe one should need "any static solution" at all or even to wonder if that kind of solutions makes sense.

Something that should make us think is the fact that in GR energy is only locally conserved.

 

Ultimately GR does not serve to specify a cosmological, global dynamics system as it is shown from 1917 first try by Einstein on , since there are many possible solutions expanding and non-expanding that fit GR as shown by Friedmann. Although needless to say every solution universe we want to postulate should respect GR.

 

Actually the FRW metric is independent in this sense from GR, is a metric constructed for any isotropic and homogenous universe that can be applied for instance to a Special relativity universe like Milne's model.

 

So in the end any cosmological hypothesis is determined by observation (and the interpretation given to that observation) thus we base our cosmological model in an observation: redshift z (wich leads to other observations like CMBR but needs the former).

And the interpretation we make of this redshift. Here is where all comes down to.

Sadly there is no easy way for us to discern if the redshift we measure from objects outside our Local group of galaxies is due to velocity or to hyperbolicity of space(3D) or a changing frame effect that stretches the wavelenght, or else.

[coldcreation]

About CC's comment in post 470: "in the same way you don't need static solutions to the Einstein field equations for local dynamics, you don't need static solution of the field equations for global dynamics."

 

It`s not as simple as it may seem, ...

 

Yes, you are correct. It is not that simple. I've actually been working out some of the details of the staticity argument. It's taken several days so far. That's why I haven't posted recently. It might take a little longer. But I pretty much have the first draft ready, and I will post it shortly.

 

There is a difference between what I wrote above; that local and global stability are founded on the same dynamics. That sentence referred more to dynamics of galaxy cluster being similar to dynamics of the solar system, or superclusters being dynamically similar to the Local Group. Meaning local physics is global physics. But that did not address directly the question of the stability of the entire universe (however big it is).

 

The main difference, to give you a hint of what will come, is that for massive bodies we are dealing with locally curved spacetime where the fields of objects interact dynamically. Globally what we have is an isotropic gravitational field.

 

Certainly there is a difference and it is the distinction between the two that brings to light stability, not solely similarity as mentioned above. Both types of field are purely general relativistic in nature. And the beauty is that the cosmological constant is not required. I will attempt to show that the introduction of lambda into the field equations (by Einstein) to render stable a spherical universe was not just ad hoc, but unwarranted.

 

I hope to make that clearer in my next post; and it's going to be a lengthy one. I have to do some touching up and a little more research on some tricky aspects of the problem (particularly those to do with nonnegative isotropic spherically symmetric curved spacetimes).

 

 

CC

[coldcreation]

...

 

 

 

Globally Homogenous and Isotropic Curved Spacetime

and the Mechanism for Stability

 

 

 

...positive curvature provides a repulsive force. What's to keep these galaxies from flying apart? GR says they would want to.

 

Positive curvature would be the opposite of a Schwarzschild solution. It'd be the complete opposite[...]

 

The problem with your interpretation, as I see it, would be the repulsive 'force'. Any object with some distance from an observer would want to follow an inertial path away from that observer.

 

And:

 

Negative spacetime curvature corresponds to an attractive gravitational force and positive spacetime curvature corresponds to a repulsive gravitational force.

 

As far as I can tell, CC is now saying that spacetime (*not* space) could be globally positively curved. [...]

 

The real problem, that I see, is that spherical spacetime manifolds would have a repulsive force. The things inhabiting such a universe would not be static. Two perfect examples would be de Sitter space and inflation. Both have positively curved spacetime that translates directly to a repulsive, scattering effect.

 

 

This is becoming very interesting and complex. I think we are getting closer to understanding the full implications of a general relativistic cosmology (but we're not quite there yet). Let's sum up and simplify the problem:

 

A few points, and a few questions, here, remain outstanding. I will attempt to answer these questions below, but not without difficulty. I will generally tend to avoid should, would, and could, for simplification. So excuse the affirmative mannerisms. Also, I have not provided the mathematical proofs below. Some of these are well known, particularly regarding locally symmetric manifolds with nonnegative isotropic curvature. I have provided several links that may serves the purpose. But there are others too (that I have yet to decipher).

 

The important point to stress is that much is known about isotropic curved spacetime manifolds, yet within the framework of general relativity these concepts have not been applied to the full extent necessary to solve the issues of staticity, i.e., interpretations of GR have been lacking. It will be shown why (conceptually and qualitatively) below. The attempt will be to show where and why these misinterpretations have occurred and to demonstrate how these have lead to the instability associated with the current solutions to the Einstein field equations which in turn lead to a cosmology that is in noncompliance with both the general principles of Einstein's theory of gravity and that which is actually transpiring in the physical universe (or potentially transpiring).

 

 

What follows is a very rough draft:

 

 

• 
  
  First, in brief, and in chronological order (historically) extending back to Newton, it must be established if and how a universe with a nonzero value of gravity (subsequently interpreted as curvature by Einstein) can remain stable against collapse or dispersion (i.e., free of gravitational perturbations that would cause wholesale expansion or collapse of the entire universe). I will argue that stability is the only viable option consistent with GR.
  
  
   
  
  
   
  
  
• Second, and related to the first, is the question as to whether stability can be achieved without a cosmological constant-like repulsive force (or vacuum pressure) to counter gravity. The answer will be yes, it is possible for stability to be maintained globally without vacuum pressure to counter gravity.
  
  
   
  
  
   
  
  
• The next important question would be whether electromagnetic radiation (EMR) would appear redshifted with the associated time dilation factor as it propagates through a
  homogenous
  and
  isotropic
  gravitational field when viewed from the stationary inertial rest-frame of an observer. The latter refers to how we (or any observer) would 'see' the EMR spectrum of luminous objects through a telescope, or
  Holmdel Horn Antenna
  :eek2:). I shall claim that EMR would indeed be redshifted.
  
  
   
  
  
   
  
  
• Again, related to the above, and in answer to the questions of
  modest
  , is the (tricky) question as to whether such a curvature (assuming that to be the cause of redshift z) would cause material particles to actually
  scatter
  radially (as assumed would occur in a de Sitter universe) or whether the effect is only an apparent scattering. The answer can only be that the "scattering" is spurious. It is not a real motion. It looks like a radial motion only because of the misinterpretation of redshift as a Doppler (or Doppler-like effect) and the misunderstanding of the differences between local gravity field and globally isotropic fields (discussed below).

  
  

 

 

 

As stated in the OP at the outset of this thread:

 

There are two possible interpretations for cosmological redshift z that show wavelength independence over 19 octaves of the spectrum:

 

(a) a change in the scale factor to the metric; implying the expansion of quasi-Euclidean space and the relative recession of objects in it, i.e., the radius of the universe changes with time t. This is an unstable manifold.

 

(:artgallery: the general relativistic curvature of spacetime; implying a non-expanding, stationary non-Euclidean manifold through which EMR propagates along geodesic paths.

 

 

 

Of course, the situation would be more complicated if both were operational: a possibility which cannot yet be excluded. In other words, redshift z could be due to both spacetime curvature and expansion, where the degree of curvature and the radial motion would combine to give us the observed redshift. The universe would be both non-Euclidean and expanding (albeit at a slower rate). I have not yet excluded this possibility. Though, I have yet to envisage how the two effects could be disentangled to give us an idea of the actual geometry and velocity separately. Let's hope this is not the case. If it is, then one could still conclude that the universe is much older than now assumed, and that running the clock backwards would not automatically lead to a big bang.

 

 

The question is not whether spacetime is curved positively or negatively, spherically or hyperbolically. Certainly observations should determine that. The key point is that a Euclidean manifold cannot exist in a universe where general relativity plays the leading role. Global curvature is a departure from linearity, the sign of which (or the intensity of which) depends entirely on measurements made by any observer, and according to the rest-frame of all observers. If one observer familiar with GR finds that spatial increments appear smaller and clocks appear to slow down with distance (or visa versa), she will be entitled to assume the universe is curved. All observers should find the same.

 

 

The illustration below, Figure 2D, represents a non-expanding Einstein-de Sitter-like universe. This is, as modest mentioned above, a schematic of an inverse Schwarzschild solution. (It is not a white hole, which would be the actual the interpretation an inverted Schwarzschild black hole). This is just another option like A or C (we'll call it D). I will refer to this diagram to explain the above problems, below:

 

 

 

Figure 2D

A Gaussian Spacetime Manifold with an attitude. This manifold represents the negative square root solution to the Schwarzchild-like metric, where the horizon is located at the outside circle, and the stationary observer at O. The manifold is homogenous, isotropic and static.

 

 

Note: this manifold is similar to Figure 1C reproduced below for convenience. The difference is 2D is not a planar cross section as 1C. The projection of 2D over to a 1C type cross section would simply be that the spherical shells would appear, from O, slightly more compact toward the horizon and less compact towards O. In both cases photons converge to all points, and from all directions. So it really makes no difference whether the diagram shows a downward curve of an upward curve. The result is the same (with perhaps minor differences in distance) when represented on a cross section polar grid as figure 1C below.

 

This manifold too is non-expanding. Redshift z would appear to be caused by a de Sitter-like effect as EMR 'climbs' up the slope to the observer located at a Lagrange-like point of the field: a kind of gravitational redshift with the associated time dilation factor.

 

But interpretation of spectral shifts as a gravitational redshift would be invalid. The difference is that this is not a local gravity field. There is no a slope or well to climb in or out (inverted or not). This is an isotropic field where every point is equal (on average), the potential is the same everywhere (excluding local inhomogeneity induced deviations and evolution in the look-back time). This means that (contrarily in the case of a locally curved manifold) material particles will not move according the gradient or potential, towards or away from an observer at O. This is one reason why reduced dimension diagrams are deceptive. It is a gravitational redshift-like effect that occurs over large distance as EMR traverses a homogenous and isotropic field.

 

In a homogenous and isotropic gravitational field of the global kind we are discussing, the angular isotropic coordinates do not faithfully represent distances within the manifold, nor do the radial coordinates faithfully represent radial distances. Spatiotemporal increments change with distance from the observer in the look-back time. On the other hand, distances in the constant time (or cosmic time) hyperslices would be represented trivially without distortion. The latter is of no real interest here (since there is nothing to measure nothing to observe in cosmic time). So, the look-back time is the ONLY way to measure curvature (by combining astronomical measurements: angular diameter distance versus redshift, luminosity distance versus redshift, distance modulus versus redshift, lookback time versus redshift, etc.).

 

 

 

 

Figure 1C

Figure 1C represents a cross-section of a non-expanding globally homogenous and isotropic four-dimensional spherically symmetric geometrically curved Riemannian (or pseudo-Riemannian) general relativistic spatiotemporal manifold (i.e., a cross section of the visible universe in reduced dimension).

 

 

 

A static spherically symmetric spacetime based on a metric theory of gravitation: general relativity

 

 

The Einstein static manifold with non-negative isotropic curvature (Figure 1C) is both locally and globally symmetric (unlike local gravity fields surrounding massive objects). In another way, an Einstein manifold with positive isotropic curvature obligatorily has constant sectional curvature. However, note that Figure 2D appears to have an additional curvature (or a relaxation in the curvature) towards the horizon. This implies that rather than being finite spatiotemporally, this manifold is infinite (though, too, Einstein's model can also be interpreted as infinite). The sphere will never 'close' i.e., there is no boundary, as in classical positive sectionally curved manifolds. This is permissible intuitively, though I have yet to understand why such a geometry would manifest itself (if indeed it does), except by some symmetry inherent within the physical laws. Or, perhaps this curvature appears as part of a maximally extended solution to the Einstein field equations: a version of the Schwarzschild metric refering to the idea that the spacetime would not have any 'edges. In other words, for any possible trajectory of a photon following a geodesic in the spacetime manifold, it should be possible to continue this path arbitrarily far into the particle's future or past (or until it hits the mirror of a telescope).

 

See too for example: S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151, 1–21 (2010).

 

 

Figure 2D is not the inverse of black hole (i.e., it is not a white hole), nor would this universe obligatorily be expanding. A de Sitter effect would be operational in this manifold, but z cannot be considered a gravitational redshift (any more than expanding space can be considered a Doppler effect). Material particles or galaxies would not actually scatter, they would just appear that way because the spectral features of light from the source would be redshifted.

 

Why would EMR be redshifted in a static manifold of the type 2D? Global isotropic curvature hinges on light travel time. Recall, the observer 'sees' only the look-back time. If she could 'see' the universe 'now' (with cosmic time) she would not be able to measure curvature. She would be embedded in a nonzero gravitational field for sure; a homogenous and isotropic field. But spacetime would appear to her (in cosmic time) as if it were Euclidean. She would see no cosmological redshift (i.e., there would be no loss of energy or time dilation associated with the the photon), and the CMB would be everywhere the same temperature (if she could even detect it).

 

 

The reason there would be no observable curvature (in a homogenous and isotropic gravity field at any given cosmic time) is because the concept of global curvature depends on the velocity of light relative the rest-frame of an observer as she witnesses events.

 

In the real world EMR would follow a geodesic path, even in an isotropic field as if circling a sphere. Material particles would not follow a geodesic path since the value of curvature is everywhere the same, on average, or the gravitational potential is everywhere the same, i.e., without 'slope' (hence they would not scatter).

 

This is why I mentioned above that the global curvature is only apparent to an observer (when the velocity of light is c) just like the scattering of material particles is only apparent (i.e., it is spurious). Quantumtopology caught that. Yes, it is a spurious curvature in a sense, but it is real to photon that loses energy during travel time and it's real to the observer who knows GR and interprets it literally.

 

The original Einstein model (without a boundary) and the original de sitter universe are stable even without lambda, for the very reasons described above. There was never any reason to introduce a dubious pressure term in the first place, at least not for a universe bathed in an isotropic and homogenous gravitational field, such as present in the universe we inhabit. The instability was a leftover from Newtonian gravitation, but with onset of geometry as the key mechanism behind the gravitational interaction the notion of instability should have vanished, and should have done so once it was realized that a globally symmetric isotropic gravity field exerts no net force on a material particle. Whereas there will be a change in the spectrum of a photon will change in a static gravitational field, as measured by local observers in their proper Euclidean rest-frame. Source. That is because the isotropic field curvature is direction independent, unlike fields of massive objects (i.e., isotropic field curvature 'direction' depends only on the observer's local frame).

 

Again, locally, gravitational fields surrounding massive objects are different. They are not isotropic. They are not uniform in all directions. Local fields have properties that vary systematically depending on direction and the location and motion of an observer.

 

 

The distinction between local and global curvature

 

 

The distinction between local and global curvature (between intrinsic curvature and extrinsic curvature, respectively) is key to the issue of stability, since the former causes objects to accelerate (or free-fall) and the latter does not (i.e., though the observer could choose, erroneously, to interpret observations is such a manner, motion would just be apparent, not real). Intrinsic curvature is detectable to the 'inhabitants' of a surface (and not just outside observers), e.g., by measuring the sum of the interior angles of a triangle to determine if it was greater than or less than 180 degrees.

 

An extrinsic curvature, cannot be detected by an observer unless she can study the three-dimensional space surrounding the surface on which she resides. And by measuring distances in four-dimensional spacetime she can determine the magnitude of 'global' curvature, or extrinsic curvature (i.e., the departure from linearity with distance respective to her Euclidean rest-frame). This according to GR is pseudo-Riemannian manifold. All points in all directions are indistinguishable since they will all have the same magnitude of curvature when measured from their local rest-frame.

 

Again by contrast, intrinsic curvature is defined at each point in manifold (each point correspond to a unique magnitude dependent on the gravitational potential at those points), where the magnitude of curvature depends on mass. It is the local curvature of a submanifold away from a general or global manifold.

 

In contrast to local curvature, the global extrinsically curved spacetime manifold is detectable not by the way in which objects physically move, but by the way objects 'appear' to move. Extrinsic curvature is detectable not by the way photons actually move, but by the way photons 'appear' to move on a geodesic as they converge towards the observer (spectral shifts are displaced toward the less refrangible (red) end of the spectrum). Measurements will show an apparent (spurious) radial departure from linearity that increases with distance.

 

Modern cosmology has complicated the issues by means of presuming that space itself is actually expanding while maintaining a quasi-Euclidean geometric structure. In other words, the apparent radial motion is not a true motion (locally) though space, but with space. This is certainly not Newtonian motion but nor is it a motion implicit in general relativity (GR says nothing about expanding space). It is an extrapolation based on redshift z which, when interpreted as a Doppler shift, was in noncompliance with Einstein's relativity, since it implied a center (or privileged coordinate system: a point) away from which everything would radially be displaced. And, as galaxies continued on their expansion through space a velocity would be attained equal to (and greater than) the speed of light c. In order to avoid these absurd logical contradictions, problems or paradoxes, amongst others, a nonconventional precept of space had to be invented: the metric expansion of physical space itself. (GR says or implies nothing of the kind).

 

Here, rather than the metric expansion of space, we have the metric curvature of spacetime (the topic of this thread). The beauty of this concept is that when applied to cosmology there arises no contradictions, problems or paradoxes, since metric curvature of spacetime follows directly and literally from Einstein's general relativity: there is no separation of space and time, no center of the universe or privileged reference frame, gravitational time dilation, gravitational redshift, and the gravitational time delay are operational locally and extended globally, but based on the observers rest-frame in the look-back time, rather than as dependent on altitude in a gravitational well. The similarity between these effects (locally and globally) becomes significant when dealing with the speed of light and large distances. At the core of this presentation are Einstein's equations, which describe the relation between the geometry of a four-dimensional, semi-Riemannian spacetime manifold, and the energy-momentum present within spacetime.

 

 

The stability or staticity of the universe follows directly from geometric arguments:

 

 

(1) Locally, phenomena that in Newtonian mechanics are ascribed to the action of the force of gravity (such as free-fall objects, orbital motion, and trajectories), correspond to inertial motion within a curved spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Rather, gravity corresponds to changes in the properties of spacetime, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy-momentum of matter. And the resulting local dynamics are virtually (but not entirely: thanks to Herr Einstein) indistinguishable form classical mechanics. In sum, gravitational perturbations arise in both interpretations; of gravity as an attractive force and as a geometric property of spacetime, and continues to do so in the latter since gravitational acceleration of masses are induced by other masses: the local combined gravity fields of n-body systems are directionally dependent, i.e., they posses anisotropy, inhomogeneities, non-isotropic features. The local velocity fields of objects can be explained by their Newtonian motions. So locally both density inhomogeneity and field anisotropy play a key role in the dynamic instability. As it so happens, these same features also play a key role in the observed stability and longevity of gravitating systems (along with mean motion resonance, and so on). But let's look beyond that for the moment.

 

(2) Globally, both density homogeneity and gravitational field isotropy play a key role in the dynamic stability of the universe.

 

The laws formulated within the general relativistic framework take on the same form in all coordinate systems—they exhibit general covariance. Though invariants are less powerful for distinguishing locally non-isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds.

 

 

Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

 

 

Globally, the distinction had to be made between phenomena that in Newtonian mechanics were ascribed to the action of the force of gravity. Certainly, this force corresponded to inertial motion within a curved spacetime (in general relativity) locally. But globally the manifold is isotropic, and the universe thought to be homogenous. There is no preferred direction. All points and all directions are indistinguishable. Isotropically curved spacetime has the same intensity or magnitude regardless of the location, and exerts the same action regardless of how objects move locally. In a homogenous and isotropic gravitational field there is no 'force' or 'slope' that causes the deflection or motion of objects.

 

 

______________

 

 

Note that in Figure 2D the shape of the curvature appears to change form out towards the horizon. This is still a spherically symmetric isotropic curved manifold consistent with Einstein's GR, where redshift z is a curved spacetime phenomenon, and material particle-scatter-free.

 

The problem is analogous to the doppler redshift vs the expansion of space redshift, they are not the same. One is a due to a real motion through space, and the other a relative effect cause by the expansion of space (based on, in my opinion, an invalid interpretation of GR). In our case, curvature is not a local curvature (as spacetime would be curved locally, surrounding massive objects), it is not a curvature in that sense. This is a global effect from the point of view of an observer, form her inertial coordinate system as she observes the universe in the look-back time.

 

So the interpretation of radial velocity (expansion) is spurious as Hubble suspected.

 

 

Wouldn't an isotropic gravitational field have the same intensity or degree of curvature equal to zero regardless of the direction of measurement?

 

No. Measurements would indicate nonzero curvature. Because, even though EMR propagates at the speed of light c through an isotropic gravity field, and the field exerts the same action regardless of how the test particle is oriented, from the inertial perspective of an observer EMR necessarily loses energy and is time dilated increasingly so with distance.

 

In a particular sense, the global gravitational field loses the purely geometrical interpretation that is characteristic of GR locally. But it is still pure general relativity. It become a truly relative experience, for all observers and from all locations, whether in motion or at 'rest.'

 

The assumption that a manifold with constant isotropic curvature has an intrinsic attractive or repulsive tendency cannot be justified on the basis of GR without introducing an ad hoc term into the equations. There is no need for the term (lambda), (i.e., not only is it not required but it would be a blunder to include it), there is no need to assume that a static Einsteinian manifold of constant isotropic curvature is unstable against expansion of collapse, since (1) there is no gravitational well, (2) no center of gravity, (3) no center of the universe, (4) no boundary, (5) no change in gravitational potential, or 'slope', (6) no cosmic time, (7) no absolute space, (8) no great attractor, (9) no privileged direction, (10) no freely falling reference frame (or hey are all freely falling), (11) no accelerated reference frame, and (12) no privileged reference frame in an isotropically curved general relativistic spacetime continuum.

 

 

When gravity is treated as an attractive force the Newtonian problem of stability is inevitable. However, when considering a globally isotropic gravity field as a geometric phenomenon in accord with GR the problem vanishes. Whereas locally Newtonian gravitation resembles GR to a very close approximation, on the large scale (as in the strong gravity field case, and close to speed of light propagation) the divergence between attraction concept and geometry interpretation of gravity becomes large (since a globally attractive force would otherwise imply that spacetime is not homogenous and isotropic).

 

 

Static Solutions of the Field Equations

 

 

It's interesting to note, on the side, that according to: Isotropic Static Solutions of the Field Equations in Einstein's Theory of Gravitation

 

In the classic Schwarzschild's solution of the gravitational field of a mass particle it is well known that a system of isotropic coordinates exist in which the velocity of light is the same in every direction at any point exterior to the particle. [...]

 

From the field equations within matter in general we can also prove incidentally that the Einstein static universe is the only solution for a closed static space filled with matter which is kept at constant pressure everywhere without assuming the spherical symmetry property of the universe to start with.

 

the Schwarzschild solution is the most general spherically symmetric, vacuum solution of the Einstein field equations.

 

 

Isotropic coordinates:

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often useful. The defining characteristic of an isotropic chart is that its radial coordinate (which is different from the radial coordinate of a Schwarschild chart) is defined so that light cones appear round. This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances. On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart.

 

Isotropic charts are most often applied to static spherically symmetric spacetimes in metric theories of gravitation such as general relativity [...] For isolated spherically symmetric solutions of the Einstein field equation, at large distances, the isotropic and Schwarzschild charts become increasingly similar to the usual polar spherical chart on Minkowski spacetime.

 

 

Another analogy: Cosmic time is the time coordinate commonly used in the big bang theory. It is defined for a homogeneous, expanding universe. With this time it can be deduced where the comoving galaxies are located at any given cosmic time. NOW, for example, galaxies at the horizon are thought to be 47 billion light years away (not 14 Gly). The analogy with the topic here is that similarly to the idea that NOW is where galaxies 'really' are, we say, NOW spacetime globally is homogenous and isotropic, but what we observe is a spacetime that is not homogenous and isotropic, since we find ourselves centered (just as any observer would) smack dab at the center of the celestial sphere, and at the top of a Lagrange-like point (as would appear form O in Figure 2D). Of course we are not at a Lagrange-like point. Or if we are, then all observers are too (which makes not sense). The only possible interpretations are that either the universe is expanding according to LCDM, or the universe is non-expanding (as non-contracting) and the observed redshift is a geometric phenomenon based on a very literal interpretation of GR, when viewed from the observer's privileged rest-frame. All rest-frames are equivalent, i.e., all observers will find themselves centered in a curved spacetime (at the apparent minima of the globally isotropic gravitational field).

 

 

Conclusion

 

The main problem, historically, has been perhaps one of intuition, or nonintuitivity regarding curved spacetimes. The problem today is not so much one of physics or mathematics involving complex geometries, since there is no new physics or new geometry in the outline above. It seems simply a problem of adapting the full breadth of general relativity and its geometric implications to empirical observations (and visa versa).

 

 

One of the potentially erroneous interpretations that lead to the big bang theory was that a global manifold must have intrinsic curvature which, like local fields, would impel objects (e.g., nebulae) to move along a geodesic path leading inexorably to blanket catastrophic collapse: thus the introduction of lambda (a 'force' that would counter gravity). It seemed convenient that or gross nondiscriminatory expansion would alleviate the problem, and it did for a few decades. But the point is that Einstein needed not include the infamous term in his equations. Expansion was convenient because it essentially did away with the fudge factor which had removed the beauty of Einstein's original equations, just as dark energy removes the beauty of contemporary cosmology. But as it turns out, Einstein had discarded lambda for the wrong reason. It wasn't due to instability (and so unnecessary) that it had to be removed, it was because it was never required from the start. The beauty had always been there, and still is...

 

 

In retrospect it's easy to see how Einstein could have misunderstood the cosmological (i.e., geometrical) implications of his own equations. After all, little was known of the properties of isotropic gravitational fields in a homogenous universe, and how such a gravitational field would act on matter within it. What complicated matters further was Einstein's postulation of a Riemannian manifold that was spherically closed and spatiotemporally finite (but that's only the short story). These properties seemed to imply a center of gravity, or center of the universe, to which all geodesic motion would converge. So the problem, though different than Newton's, ended with the same conclusion: something was missing that would render stable the universe against gravitational collapse. Yet nothing was missing at all.

 

Then again, I could very well be mistaken. This mechanism for stability might not be operational in the real world. Of course that would not by inference exclude other hypotheses. In order to disprove this assertions it would simply have to be demonstrated that (a) the global field is not curved, i.e., the universe is Euclidean or quasi-Euclidean, (:smart: the manifold is curved but objects would follow a geodesic of the field curvature (as locally) resulting in instability, © that light throughout the entire EMR spectrum would not be affected as it propagates through a global curved spacetime continuum (and so on).

 

 

 

Here is an interesting read. More on this later:

 

On Isotropic Coordinates and Einstein’s Gravitational Field

 

 

To be continued...

 

 

Coldcreation, Barcelona, June 9-11, 2010

[quantumtopology]

That sure was a lengthy post! :hihi:

Some quick points as I don't have much time now:

 

The global curvature, It's not totally clear if you consider it apparent (not physically real)

or not. From the title of the post I'd say you doubt if it's positive (thus nonnegative) but you won't allow it to be flat either since that woud make it Special Relativity.

An isotropic gravity is difficult to make compatible with curvature. The key probably lies in this you quote from the wikipedia :

"For isolated spherically symmetric solutions of the Einstein field equation, at large distances, the isotropic and Schwarzschild charts become increasingly similar to the usual polar spherical chart on Minkowski spacetime"

 

Regards

[quantumtopology]

The main problem with a static positive curvature universe is that the ony possible solutions seem to be Einstein's and de Sitter's.

 

You are leaning now towards an Einstein-like universe without cosmological constant and infinite, now I don't know if that is even mathematically feasible.

 

And I'm stil puzzled by the apparent versus real nature of the global curvature, if it is apparent, how can it affect the photon?

 

More things, an extrinsic curvature would only be apparent from an ambient with one more dimension than the spacetime manifold, that is from 5 a dimensional view, not from our point of view, and any way you would have to justify that extra dimension that seems superfluous.

 

Regards

[modest]

CC, your latest post was indeed very long and I might have time tonight to reply, but if not then maybe after the weekend. Sorry for the brevity an all that.

 

I found a clarifying paper that solves the aparent paradoxes of Doppler and Cosmological interpretation of redshift z that makes clear neither one is more correct than the other, is just a matter of reference frame choice (this is regarding post 474).

[0803.2701] Interpretation of the Cosmological Metric

 

I read the abstract, but not the paper. The premise is well-known. Of course an empty FLRW niverse can be transformed into the coordinates of SR and of course the two coordinate choices will have different coordinate-dependent consequences. The physical (or, observable) consequences are, however, the same either way.

 

For example, the whole idea of super-luminary velocity only applies if you define distance and time in a certain way.

 

There's a very qualified physicist on these boards that often quotes a quote of a rather fitting line on the issue:

 

Steven Weinberg heads the chapter on the principle of general covariance with a quote of Lewis Carrol, from Alice's Adventures in Wonderland:

"Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her and to wonder what was going to happen next."

 

and a post I wrote on the topic: Re: "the whole mess will start over again ..."

[quantumtopology]

Modest, the essence of the paper I linked could be summarized with that quote from Alice so I take it that you agree with it.

:hihi:

[quantumtopology]

CC, Perhaps a major problem with what you suggest is that GR doesn't seem to have solutions at infinity for static universes that are not empty and that is why Einstein was so desperate to find this kind of solution with his 1917 model, he was not able, that is why he had to add the cosmological constant, not only looking for a static solution (which seemed obvius at the time just from watching nearby objects velocities) but also, and I think this haven't been stressed enough to avoid infinity, so his model is spatially bound whith a finite radius.

 

So if what you propose is an infinite universe, without lambda but with positive mass density, you have to sort out this tough problem with infinity or consider GR to have only local scope.

 

Regards

[modest]

Modest, the essence of the paper I linked could be summarized with that quote from Alice so I take it that you agree with it.

:)

 

I should think so. Without reading the paper, I can only assume it makes the correct conclusions from the premise, and the premise I certainly do agree with. You can glean my thoughts on the issue from: Re: "the whole mess will start over again ..."

 

And, as it relates to redshift, I can't imagine a better, or more concise, correct, or easy-to-follow link than:

 

What Causes the Hubble Redshift? Are the light waves "stretched" as the universe expands, or is the light doppler-shifted because distant galaxies are moving away from us?

 

EDIT---> I should add, I certainly agree with your statement "neither one is more correct than the other, is just a matter of reference frame choice" 100% :hihi:

<---- EDIT

 

 

ColdCreation, I just sat down to read your latest post...

 

~modest

[coldcreation]

The main problem with a static positive curvature universe is that the ony possible solutions seem to be Einstein's and de Sitter's.

What's wrong with that? What about variations of the two?

 

You are leaning now towards an Einstein-like universe without cosmological constant and infinite, now I don't know if that is even mathematically feasible.

Whenever mathematics deals with infinity problems arise.

 

 

More things, an extrinsic curvature would only be apparent from an ambient with one more dimension than the spacetime manifold, that is from 5 a dimensional view, not from our point of view, and any way you would have to justify that extra dimension that seems superfluous.

We live in a submanifold of a manifold.

 

The global curvature, It's not totally clear if you consider it apparent (not physically real).

 

And I'm still puzzled by the apparent versus real nature of the global curvature, if it is apparent, how can it affect the photon?

 

It is because it affects the photon that it is apparent. Said differently, it is an apparent affect on the photon from the observers frame. It is thus an apparent and physically real affect.

 

Let's say we measure something, e.g., the distance to a galaxy far removed, and we deduce curvature. We can conclude something happens, either at the source, during travel time, or here in the vicinity of our planet that distorts the image of the galaxy. In a sense this is an optical effect, like looking through a magnifying glass. The image is real. The effect is real, but the curvature is only apparent: the galaxy is not physically distorted in it's own frame of reference. We can figure that out by correcting for the distortion.

 

In the case of global curvature the effect is apparent. A distortion appears to be taking place between the observer and the source. The tricky part is to determine whether the effect is spurious or not. Sometimes (almost always in cosmology) there can be differing interpretations for the same observed phenomena. For example, the apparent superluminal velocities of objects at the horizon of an expanding universe can be interpreted as an effect generated by the curvature of spacetime as the photons propagate towards the observer. The latter does not imply that spacetime at those distance objects in infinitely curved, or that they are traveling faster than light in an expanding neo-Newtonian or pseudo-relativistic space.

 

Affects are real and they are apparent to the observer, relative to her rest-frame.

 

The observer located at one of those distant galaxies at the superfluous edge of the universe, typing a few key enthralled words on her computer sitting atop a her Lagrangian-like point peering out into the peaceful heavens on her pixelated screen would see the Milky Way as a tiny spec of real estate breaking the speed of light too. Either that or she could conclude that the Milky Way is immersed in a gravitational potential well so deep that the photons emanating from the luminous objects that make up the Galaxy barely escape in time before the gates of hell close for good. :phones:

 

Both scenarios are apparent (we see the light) but we know that these are only relative affects that have all to do with the propagation of light (EMR) and how its spectrum if affected as the source objects are radially moving as viewed from an inertial frame of reference, or how the spectrum is affected from the time the photon embarks on its long journey through the homogenous and isotropic reaches of intergalactic and intercluster spacetime continuum to the time it registers as data on our computer screens.

 

These affects are apparent and they are real. But these observed affects needs to be interpreted properly within the framework of a world-model, of physical laws, of Einstein's general principle of relativity.

 

 

CC

[modest]

If one observer familiar with GR finds that spatial increments appear smaller and clocks appear to slow down with distance (or visa versa), she will be entitled to assume the universe is curved. All observers should find the same.

 

I don’t believe this is necessarily the case. I would say that curvature is a sufficient cause for redshift and time dilation, but not a necessary cause. The simplest counterexample would be two particles moving away from one another in Minkowski spacetime. They would each see the other as time dilated and redshifted, but the metric is by definition flat.

 

But interpretation of spectral shifts as a gravitational redshift would be invalid. The difference is that this is not a local gravity field. There is no a slope or well to climb in or out (inverted or not). This is an isotropic field where every point is equal (on average), the potential is the same everywhere (excluding local inhomogeneity induced deviations and evolution in the look-back time). This means that (contrarily in the case of a locally curved manifold) material particles will not move according the gradient or potential, towards or away from an observer at O.

 

Your conclusion doesn’t follow from the premise. I will grant you that a homogeneous and isotropic metric should treat every observer the same, but that would in no way imply that the physical distance between them cannot be increasing.

 

Consider the Earth. If I am standing on the equator then there is about 100 kilometers actual distance between two lines of longitude. Standing where I am, I will notice that another observer closer to one of the poles will only have a few kilometers distance between those same lines of longitude. My conclusion will either be that real distances are getting larger near the poles or that the lines of longitude are converging near the poles.

 

The other observer will, for this analogy, consider themselves on an equator just like I considered myself on an equator. The lines of longitude and latitude get redrawn so that each observer considers themselves at an equator. When they look at me they likewise see lines of longitude converging as if I am closer to a pole. Real distances are growing relative to longitude and latitude distances from both observer’s perspective. The situation is reciprocal.

 

When Einstein first encountered de sitter’s universe his interpretation was that the observer saw things as if she were surrounded by a distant infinitely-dense ring of matter. In other words, it is as if the observer is sitting atop an isotropic Lagrange point. In this way, it is very much the opposite of a Schwarzschild solution. The mass is not at the center, but at the edges. Each observer sees the universe this way. From Sally’s perspective, Tom is near her horizon getting accelerated toward it. From Tom’s perspective, Sally is near his horizon and getting accelerated toward it.

 

Why would this imply that the physical distance is not increasing? In both frames of reference, both observers agree that the distance is increasing as the other observer is accelerated away.

 

So, the look-back time is the ONLY way to measure curvature (by combining astronomical measurements: angular diameter distance versus redshift, luminosity distance versus redshift, distance modulus versus redshift, lookback time versus redshift, etc.).

 

In a de Sitter universe we would expect the angular diameter vs. redshift to appear as the black line. The concordance model which fits observation nicely is the blue line:

 

 

This is a problem.

 

Figure 2D is not the inverse of black hole (i.e., it is not a white hole),

 

A white hole is the time-reversal of a black hole. In both cases the mass is at the center. A white hole is not what I was meaning when saying the opposite of a Schwarzschild solution.

 

nor would this universe obligatorily be expanding. A de Sitter effect would be operational in this manifold, but z cannot be considered a gravitational redshift (any more than expanding space can be considered a Doppler effect). Material particles or galaxies would not actually scatter, they would just appear that way because the spectral features of light from the source would be redshifted.

 

But, then you would have light following geodesics, but not matter. Matter needs to follow inertial geodesics. In a globally curved manifold where R>0 this means things 'fall' away from one another just as much as in a globally curved manifold where R<0 things would fall toward one another. It isn't just light that gets affected by gravity.

 

Why would EMR be redshifted in a static manifold of the type 2D?

 

I think you might be using "static" in a way that it is not usually used. The Schwarzschild metric is static, but this does not mean clouds of isotropic gas cannot collapse to form an isotropic star as described in the Schwarzschild metric. A 'static' metric does not imply that distances between things will not increase or decrease with time.

 

Likewise, de Sitter's universe can have a static metric (de Sitter space), but it is know from solving General Relativity's equations of motion that a test particle in that universe will be accelerated away from the observer with a force proportional to distance.

 

 

I'm sorry... I'm getting called away from the computer. I'll read and reply to the rest hopefully shortly.

 

I do think we are getting closer to a mutual understanding :phones:

 

~modest

[modest]

Ok, I read the rest of the post. The theme is clear.

 

Being as unassuming as possible, I will just say that it cannot be consistent with general relativity (or the idea of curved spacetime in general). You are essentially saying that matter is not affected by gravity in a isotropic and homogeneous universe yet light is affected.

 

If the universe has constant global curvature then it will behave as if it has constant global curvature. Curved spacetime means that geodesics (the shortest distance between spacetime points) deviate from linearity. Precisely how much they deviate by constant global curvature is described exactly and completely by "On the Deviation of Geodesics and Null-Geodesics, Particularly in Relation to the Properties of Spaces of Constant Curvature and Indefinite Line Element" 1933 J. L. Synge.

 

From the idea of geodesic deviation the normal behavior of FLRW universes and a de Sitter universe in particular are derived and affirmed: [gr-qc/9709060] Deviation of geodesics in FLRW spacetime geometries. Sections 3.3 and 4.21 in particular and quoting part of the conclusion:

 

Conclusion:

 

One way of solving the EFE [Einstein field equations] is to treat them as algebraic equations relating Rabcd to Rab and Cabcd, then solving the GDE [geodesic deviation equation] (which characterizes relative acceleration due to spacetime curvature) to determine both the spacetime geometry and its properties. In the case of a FLRW model, this can be carried out explicitly, as shown above: integrating the GDE ( cf. Eqs. (53), (86) and (95) ) allows complete characterization of all interesting geometrical features of the exact FLRW geometry in an elegant manner | determining the timelike evolution, spacelike geometry, and null ray properties, which in turn determine the basic observational properties. The Newtonian analogue of some of this has been given by Tipler 27, 28].

 

Your idea breaks the fundamental postulate of GR—the equivalence principle. The nature of curved spacetime is the equivalence between accelerated inertial reference frames and gravity. To say that spacetime is globally curved yet there is no accelerated inertial reference frames just doesn't make sense. By the very definition, global curvature would have accelerated inertial reference frames, and again by definition matter would want to follow those inertial reference frames.

 

The other direction you take breaks the fundamentals of spherical geometry. In such a geometry both Tom and Sally see parallel geodesics converge with distance. Just because the curvature is global doesn't mean the action can't be reciprocal. Essentially: both Tom and Sally can consider themselves on the top of the world while the other is downhill. This is not a problem.

 

~modest