Redshift z

[modest]

What I meant to say was a Lorentzian manifold is a special case of a pseudo-Riemannian manifold (not of a Riemannian manifold). I neglected to add "pseudo."

 

And a pseudo-Riemannian manifold is a generalization of a Riemannian manifold.

 

That, I'm pretty sure, was the only error.

 

It is a core principle of GR that spacetime can be modeled with a Lorentzian manifold. Riemannian manifolds are locally Euclidean. There are three dimensions of space and one of time in our universe and they carry opposite signatures. Clearly a Lorentzian manifold, and only a Lorentzian manifold, is appropriate. The paragraph about Lorentz frames makes no sense.

 

~modest

[modest]

Modest, I think you are completely right about the nomenclature here and it's good of you to correct it, but I also think that we know what CC meant, you said it yourself, when talking about Lorentzian manifolds he meant Minkowski spacetime

 

Also, my example was of a Lorentzian manifold with constant positive curvature. If he were objecting to Minkowski spacetime then he would be objecting to the wrong thing.

 

~modest

[coldcreation]

It is a core principle of GR that spacetime can be modeled with a Lorentzian manifold. Riemannian manifolds are locally Euclidean. There are three dimensions of space and one of time in our universe and they carry opposite signatures. Clearly a Lorentzian manifold, and only a Lorentzian manifold, is appropriate. The paragraph about Lorentz frames makes no sense.

 

I understand your concerns. General relativity can indeed be (and is) modeled with a Lorentzian manifold. In many circumstances Lorentzian manifolds are appropriate (perhaps for all submanifolds). Those circumstances revolve around local events. There is little doubt that the universality of local Lorentz covariance, together with the equivalence principle, describes local gravitational phenomena quite well.

 

There is nothing wrong with a Lorentz manifold being a special case of a more general pseudo-Riemannian manifold.

 

If it can be established that the mass-energy density of the universe affects the topology of a global spacetime manifold, it will follow that the global manifold must have nonzero metrical curvature. Of course, the existence of nonzero metrical curvature at local points of the manifold does not imply nonzero global spacetime curvature, nor does imply a global topology. Lorentzian manifolds, in fact, do not give you a global topology.

 

The other thing I wanted to say is that global topology of a homogeneous field of constant curvature does not tell you anything about the local spacetime curvature induced by local inhomogeneities (stars, galaxies, etc.). It can only tell you about its own intrinsic properties locally. For example. Let's say the global topology was spherical (K = 1) like the earth. That topology tells you nothing about how many mountains and valleys there are locally, or anything about the elevations of the mountains. All the topology tells you is that spacetime tends to flatness locally. I noticed that when I went to the beach a couple days ago and looked at the horizon of the Meditteranean. Likewise, judging from a local region (the observers frame of reference), the mountain and valleys tell you nothing about the global topology.

 

So it is very possible, that we have a Lorentzian manifold everywhere-locally (that tells you how objects move in spacetime and how spacetime is curved) and a pseudo-Riemannian topology globally (that describes a Gaussian manifold of constant curvature). This reasoning would apply to either positively or negatively curved topology, but not to a flat pseudo-Riemannian manifold simply because there would be no cosmological redshift is a flat universe, and so would not be consistent with observations.

 

Oh, and because a Lorentzian manifold tells you nothing about the global topology, it can't determine whether or not spacetime is flat or not on large-scales. That is one reason why a Lorentzian manifold permits a flat Minkowskian universe (amongst other shapes). And indeed, it is customary to treat the general relativistic manifold as an ordinary topological space with the same topology as a 4-dimensional Euclidean spacetime.

 

And that is why a Lorentzian manifold must be considered a special case of a pseudo-Riemannian manifold. A metric with a Lorentzian signature gives only local attributes of the manifold. It does not tell us the overall global topology, as would a pseudo-Riemannian metric. The two together are not incompatible. They can cohabitate.

 

A global pseudo-Riemannian manifold of constant Gaussian curvature does not tell objects how to move locally (since it is locally similar to a Euclidean space, no motion is induced locally due the Gaussian curvature. :)). For that, a Lorentzian manifold is required locally. That is one reason why it is so difficult to extend general relativity to cosmology. We should not be misled by believing that a local physical manifold corresponds to the global topology.

 

It is entirely possible (or inevitable) that the local manifold has a different topology than the global physical manifold. With this in mind, it is worthwhile to consider very carefully whether a physically meaningful local spacetime topology is necessarily the same as the topology of the global 4-dimensional systems of coordinates. Note, a submanifold of a global pseudo-Riemannian manifold is not obligatorily a pseudo-Riemannian manifold with the same metric (nor does it even need to be a pseudo-Riemannian manifold at all). The submanifold(s) may very well posses a Lorentzian signature yet be 'contained' inside a more general global manifold (just as the mountains and valleys on earth are 'contained' on a spherical globe).

 

There certainly are no a priori requirements that a particular global structure can be uniquely determined by a given set of local experiences. If we restrict ourselves to a class of naively realistic local models consistent with the observable predictions of general relativity, there remains an ambiguity in the conceptual framework with regards to the global topology. The situation is complex due to the fact that the field equations of general relativity permit a wide range of global solutions. Some of these solution are unphysical (depending on initial condition, boundary condition, etc.). And so restrictions need to be imposed. The field equations, in this sense, do not represent a complete theory, since these restrictions cannot be inferred from the field equations. Incompleteness is a feature of all physical laws expressed as sets of differential equations, since a wide range of possible formal solutions can generally be extrapolated from such equations. This, by no means is detrimental to relativity. It just requires that at least one external principle (or constraint) be added to yield definitive results. (See for example, this on the topic).

 

At present, general relativity does not yield unique predictions about the topological shape of the global manifold. Rather, (once the unphysical solutions are weeded out) it imposes particular conditions on the allowable shapes. The simplest ('well-behaved') global solutions consistent with both general relativity and empirical evidence appear to be that of a pseudo-Riemannian manifold of constant positive or negative Gaussian curvature (yes, with Lorentzian submanifolds). Admittedly, I leave open the possibility that the sign of curvature K can be either or (1 or -1) so as to avoid committing to specific distant correlations, pending a complete model, and empirical verification. But only one of these two possibilities should eventually emerge as a viable topology consistent with physical laws. Obviously, the interpretation of a field theory such as general relativity with a globally flat background spacetime manifold would no longer hold.

 

 

For the above reasons, any thought experiment (in my opinion) that deals only with local phenomenon (with some unphysical extrapolations to the global) is virtually irrelevant for the topic at hand. My contention with this remark is that the global topology, only, sheds light on a possible mechanism for the stability of the cosmos, and for the cause of redshift z. Again, I do not exclude the possibility that the universe is expanding according to Lambda-CDM. I simply point out that there is a viable alternative, totally in line with GR, that needs to be explored further.

 

PS. The paragraph you refer to (which "makes no sense"), albeit perhaps poorly written, simply means that a Lorentzian manifold can be interpreted as representing a locally curved spacetime. It was not a critique, if that's what you were looking for (maybe that's why it made no sense). I was actually agreeing with you there. :)

 

Regards,

 

CC

[quantumtopology]

I insist on not getting caught in terminology nitpicking :)

[quantumtopology]

Hi, guys

I have a doubt about the cosmological principle, he thing is that as it is applied to standard cosmology it only affects the spatial dimensions, not the time one. The principle that includes the 4 dimensions being the "perfect" cosmological principle.

The fact that the standard principle only affects the spatial part has always seemed a little artificial to me, righ after Minkowski declared spacetime a fused entity.

 

So here is my doubt: from the standard principle we derive spatial homogeneity, and therefore cosmologists expect to find more homogeneity of the universe the larger the scale. But the larger the distance we observe, the further back in time we are looking, so it would seem that observed spatial homogeneity implies the same amount of temporal homogeneity in a relation that must be kept the larger the scale we look at, it seems a clear consequence of the finiteness of the speed of light, that the larger the spatial map we can watch the larger the temporal map we see too.IOW, our telescopes allow us to see spacetime rather than space.

I'd really like to know what the trouble is with this reasoning. Anybody?

 

Regards

QTop

[coldcreation]

Gentlemen,

 

Returning to the topic at hand, I've illustrated the situation expressed above, whereby the global topology is pseudo-Riemannian. The local spacetime curvature represents Lorentzian submanifolds.

 

See Figure PRM-LSM

 

Figure PRM-LSM is a schematic diagram representing the global topology, and the local geometric structure of a general relativistic spacetime continuum (a cross-section equatorial slice through the visible universe). This topology is a global four-dimensional maximally symmetric simply connected non-reductive homogeneous and isotropic pseudo-Riemannian manifold of constant negative Gaussian curvature, with everywhere-local Lorentzian submanifolds. 

 

 

The explanation follows...

 

 

CC

[modest]

Oh, and because a Lorentzian manifold tells you nothing about the global topology, it can't determine whether or not spacetime is flat or not on large-scales.

 

Believe me, you are misunderstanding entirely, and I believe the quote above might be the source of that misunderstanding. Do a google search for 'topology of spacetime' or 'spacetime topology' and notice the first paragraph of the first link:

 

Spacetime topology, the topological structure of spacetime, is a subject studied primarily in general relativity. This physical theory models gravitation as a Lorentzian manifold (a spacetime) and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.

 

To say that a Lorentzian manifold can tell you nothing of topology or global curvature is contrary to the fundamentals of spacetime in cosmology.

 

The examples I've given,

• of a curved surface... a sphere
  
• of flat spacetime... a hollow shell
  
• of global curvature... a Lorentzian manifold of constant positive curvature
  

have each been objected to for reasons that seem very obfuscatory, and your defense of those objections have made no sense to me.

 

~modest

[coldcreation]

[snip]

The examples I've given,

• of a curved surface... a sphere
  
• of flat spacetime... a hollow shell
  
• of global curvature... a Lorentzian manifold of constant positive curvature
  

have each been objected to for reasons that seem very obfuscatory, and your defense of those objections have made no sense to me.

 

:)

 

First, I recapitulate: you argue that the staticity claim is invalid on the grounds that general relativity permits only one global manifold, that of Lorentzian signature. Your claim is based on the rather local idea that "spacetime tells matter how to move—matter tells spacetime how to curve." Your conclusion is that material objects cannot remain stable in a curved spacetime regime; that curved spacetime in GR implies instability. So if the global topology is curved, all the matter in the universe (and indeed the universe itself) must either expand or collapse. That is, "things get closer together (or further apart) in space over time."

 

I have given arguments entirely consistent with general relativity that contradict your claim, some in this thread and others here.

 

I argue that the staticity claim is valid on the grounds that general relativity permits a globally homogeneous pseudo-Riemannian manifold of constant intrinsic Gaussian curvature, that does not impel material objects to be displaced, i.e, the universe remains free of global instability. Objects such as galaxies move relative to local spacetime curvature (gravitational interactions), in accord with GR. What we have is a globally static (stationary) homogeneous pseudo-Riemannian manifold of the most general type, with locally inhomogeneous Lorentzian submanifolds.

 

 

Second, I don't even need to object. The problem, for me, is not the extendibility or non-extendibility of a Lorentzian spacetime to the global topology of an expanding universe. My concern is not that of an expanding Lorentzian manifold. I have no need to defend or refute the claim.

 

The real problem for me was to find a global solution that would be consistent with a non-expanding universe. As it turned out, the most general pseudo-Riemannian manifold is both consistent with general relativity, and consistent with a static manifold that is nevertheless curved globally.

 

 

As for your examples above, I've explicitly stated that the spherical model is still on the table. The flat spacetime is not, since there would be no redshift z in a non-expanding flat universe. As for your third, this is simply a special case of a pseudo-Riemannian manifold, so I really have no problem with it, per say. Just as I have no problem with special relativity, under its domain of purview. Of course, your presentation of a Lorentzian manifold of constant positive curvature, if I recall, was in the context of an anti-de Sitter spacetime, which is of little interest here, since it's an expanding manifold dominated by dark energy.

 

I had given you a link from some of the remarks above were referenced. Here it is again if you missed it the first time around: Source.

 

This time I quote (rather than pilfer :)):

 

We only expect Lorentz frames to be local, but we do need them to be big enough to cover at least some amount of spacetime. [...]

 

[...] we don't expect to be able to define anything like a global Lorentz frame for the entire universe, so there is no such natural expectation of being able to define a global principle of conservation of momentum. This is an example of a general fact about relativity, which is that conservation laws are difficult or impossible to formulate globally.

 

[...] Given an event P, we can now classify all the causal relationships in which P can participate. In Newtonian physics, these relationships fell into two classes: P could potentially cause any event that lay in its future, and could have been caused by any event in its past. In a Lorentz spacetime, we have a trichotomy rather than a dichotomy. There is a third class of events that are too far away from P in space, and too close in time, to allow any cause and effect relationship, since causality's maximum velocity is c. (1998-2009 Benjamin Crowell)

 

See too this link which attempts to resolve these problems, and others: Topology Change in Classical General Relativity

 

 

I'm not saying the case is closed. It is still open. To date, the global topology of the universe has remained elusive, despite the extension (or sewing together) of local Lorentzian manifolds to form a kind of global topology. All now seems to hinge not on gravity, or the metric structure of the gravitational field, but on dark energy, since it's thought to be the dominant component driving an accelerated expansion. Radial velocity (or motion) essentially determines the topology (i.e, the metric with the scale factor are describing the topology).

 

I'm simply trying to find a solution to the problem of global topology in the case where expansion is non-operational (but still consistent with observations). It would have presented a serious problem had no known manifold (consistent with GR) been compatible with the model.

 

It is interesting to point out, again, that the global topology defined as a homogeneous pseudo-Riemannian manifold of constant curvature solves, rather elegantly, both the problem of stability and redshift z.

 

 

You see? :unsure:

 

 

CC

[modest]

First, I recapitulate: you argue that the staticity claim is invalid on the grounds that general relativity permits only one manifold, that of Lorentzian signature.

 

I have never made that argument.

 

Your claim is based on the rather local idea that "spacetime tells matter how to move—matter tells spacetime how to curve."

 

That is not a "local" idea. That is an idea of general relativity—both local and global. I should explain the basics of what is happening here.

 

General relativity is a theory that matches spacetime with the matter / energy content of the universe. One side of the Einstein equation describes the curvature of spacetime and the other side describes the matter/energy/momentum content. So, to put it schematically:

 

spacetime --> Ricci Curvature --> Einstein Equation <-- stress energy tensor <-- matter/energy

 

GR matches the left hand side which is a Lorentz manifold with the right hand side which is the matter / energy content of the universe. This is the same whether it is local or global—it is the same theory either way. On the spacetime metric are what is called geodesics. They are the shortest path between two points in spacetime. Matter and energy which are under the influence of gravity alone follow geodesics.

 

Matter follows time-like geodesics and light (or other radiation) follows null geodesics. Thus: spacetime tells matter and energy how to move. This is universally true, whether globally or locally or any other -ly. It *is* the theory. If you want to say that light follows null geodesics but matter does not follow time-like geodesics then you are in disagreement with general relativity and you will need a new theory of gravity. That's fine, but you must explicitly say that is what you are doing.

 

What you are describing at the moment is *entirely* inconsistent with GR. In globally curved spacetime time-like geodesics converge or diverge.

 

I argue that the staticity claim is valid on the grounds that general relativity permits a globally homogeneous pseudo-Riemannian manifold of constant intrinsic Gaussian curvature, that does not impel material objects to be displaced, i.e, the universe remains free of global instability.

 

A Lorentzian manifold has the signature -,+,+,+. One of time and three of space. Specifically what pseudo-Riemannian manifold are you considering?

 

Objects such as galaxies move relative to local spacetime curvature (gravitational interactions), in accord with GR. What we have is a globally static (stationary) homogeneous pseudo-Riemannian manifold of the most general type, with locally inhomogeneous Lorentzian submanifolds.

 

What does "of the most general type" mean? Euclidean and Minkowskian space are the most general types of pseudo-Riemannian manifolds.

 

The problem, for me, is not the extendibility or non-extendibility of a Lorentzian spacetime to the global topology of an expanding universe.

 

You could do a google search for "Lorentz inextensibility".

 

The real problem for me was to find a global solution that would be consistent with a non-expanding universe. As it turned out, the most general pseudo-Riemannian manifold is both consistent with general relativity, and consistent with a static manifold that is nevertheless curved globally.

 

See condition 5 in: What Is a Reasonable Spacetime -- Andrzej Krolak -- 1985

 

Could you say, mathematically, what spacetime are you considering? Or, just, what are the number of dimensions and what are their sign?

 

Of course, your presentation of a Lorentzian manifold of constant positive curvature, if I recall, was in the context of an anti-de Sitter spacetime, which is of little interest here, since it's and expanding manifold dominated by dark energy.

 

Anti-de sitter space is negatively curved.

 

I had given you a link from some of the remarks above were referenced. Here it is again if you missed it the first time around: Source.

 

This time I quote (rather than pilfer :)):

 

That's talking about Lorentz frames—as in: special relativity and flat spacetime. I don't know how or why it would be relevant.

 

~modest

[coldcreation]

GR matches the left hand side which is a Lorentz manifold with the right hand side which is the matter / energy content of the universe. This is the same whether it is local or global—it is the same theory either way. On the spacetime metric are what is called geodesics. They are the shortest path between two points in spacetime. Matter and energy which are under the influence of gravity alone follow geodesics.

 

While what your write is correct, the correctness does not imply that that solution is representative of the physical world on cosmological scales. Nor does it imply this is the only solution consistent with the general theory of relativity. I will elaborate on this below.

 

 

Matter follows time-like geodesics and light (or other radiation) follows null geodesics. Thus: spacetime tells matter and energy how to move. This is universally true, whether globally or locally or any other -ly. It *is* the theory. If you want to say that light follows null geodesics but matter does not follow time-like geodesics then you are in disagreement with general relativity and you will need a new theory of gravity. [...]

 

Saying that something is "universally true" is a perilous assumption.

 

Do we need a new theory of gravity: Probably not. Changing a term (or reinterpreting a term) in an equation such as the Einstein field equation, does not obligatorily mean that the entire edifice of general relativity needs to be replaced by a new theory of gravity. Nor, even, does it imply that general relativity needs to be modified fundamentally.

 

Much hinges on the time coordinate in the context of the field equation. Now this term is interpreted as a cosmic time (a kind of absolute time). This is the standard time coordinate for specifying the FLRW solution of Einstein's equations. But time t needs not be interpreted in such a fashion. Indubitably, this cosmic time interpretation has led to the concept of a scale factor that changes with time. When the concept of time is treated as a dimension (attached to the 3 space dimensions), the increments of which vary with increasing distance from an observer (i.e., when the manifold has the properties of a four-dimensional spacetime continuum in the pure general relativistic sense), the stability of the universe becomes much less problematic. But that doesn't resolve the entire issue at hand.

 

Much hinges, too, on the physical properties of a globally homogeneous field of constant curvature: a pseudo-Riemannian manifold, or Riemannian manifold of constant sectional Gaussian curvature. These types of field differ, for obvious reasons, from the inhomogeneous gravitational fields in the vicinity of gravitating bodies or systems. The former is the same at all points in spacetime. The latter changes with distance (they are not homogeneous or constant, points in the field differ). A universe described by Riemannian or pseudo-Riemannian manifold of constant curvature can encompass these local inhomgeneities in the form of submanifolds. The same theory of gravity (GR) is all that is required, nothing new is needed. (See the last link in this post for an analytic study on this topic).

 

 

What you are describing at the moment is *entirely* inconsistent with GR. In globally curved spacetime time-like geodesics converge or diverge.

 

What I am describing is entirely based on general relativity. Again, the problem can be resolved. Objects are not obligatorily accelerated in any particular direction (geodesically) in a global manifold of constant curvature, since acceleration is in all directions (all points, and all directions, are equivalent). Locally, of course, the situation is different (and well tested).

 

 

A Lorentzian manifold has the signature -,+,+,+. One of time and three of space. Specifically what pseudo-Riemannian manifold are you considering? Could you say, mathematically, what spacetime are you considering? Or, just, what are the number of dimensions and what are their sign?

 

I'm considering a four-dimensional non-reductive homogeneous pseudo-Riemanninan (and Riemannian) manifolds of constant positive Gaussian curvature, and of constant negative Gaussian curvature. Whether the pseudo-Riemannian signature is positive, negative, (2,2) or with Lorentzian signature of (n -1,1), or other, I do not know yet (the Ricci-flat case, however, is excluded). I've been studying the question intensively lately, but the theme is quite complex. Perhaps not insurmountable though. But much remains to be known about these class of objects.

 

For anyone interested, I would look at this work for example: Reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

 

See too: Homogeneous pseudo-Riemannian structures of linear type

 

 

And: An Integral Equation for Spacetime Curvature in General Relativity

 

 

The following two links discuss some of the problems between local and global spacetime in GR:

 

Local and Global Properties of the World

 

Cosmic Topology

 

 

 

Perhaps, most importantly, (Qtop, you'll be interested in what follows) I would take a very close look at this: Homogeneous cosmological solutions of the Einstein equation (Ernst Fischer, 2009). Here is the FULL TEXT in PDF format.

 

Abstract: Homogeneous solutions in the framework of general relativity form the basis to understand the properties of gravitation on global scale. Presently favoured models describe the evolution of the universe by an expansion of space, governed by a scale function, which depends on a global time parameter. Dropping the restriction that a global time parameter exists, and instead assuming that the time scale depends on spatial distance, leads to static solutions, which exhibit no singularities, need no unobserved dark energy and which can explain the cosmological red shift without expansion. In contrast to the expanding world model energy is globally conserved. Observations of high energy emission and absorption from the intergalactic medium, which can scarcely be understood in the ‘concordance model’, find a natural explanation.

 

This is what I would like to discuss in the upcoming posts...

 

 

 

CC

[quantumtopology]

CC, that paper from Fischer indeed interests me, even if I don't completely understand it. It has similarities both with the Einstein model with lambda integrated in the energy tensor as negative pressure, and with de Sitter's with the redshift due to a factor multiplying the time displacement part of the metric. What would you say about it? It would be negatively curved, right?

[modest]

GR matches the left hand side which is a Lorentz manifold with the right hand side which is the matter / energy content of the universe. This is the same whether it is local or global—it is the same theory either way. On the spacetime metric are what is called geodesics. They are the shortest path between two points in spacetime. Matter and energy which are under the influence of gravity alone follow geodesics.

 

While what your write is correct, the correctness does not imply that that solution is representative of the physical world on cosmological scales. Nor does it imply this is the only solution consistent with the general theory of relativity. I will elaborate on this below.

 

I’m not sure what you mean by "solution". What I was saying is true of the theory of general relativity itself and not an attribute of specific solutions of GR.

 

Saying that something is "universally true" is a perilous assumption.

 

Right, but I’m saying that it is universally true according to general relativity. It is deductively true that all matter and radiation under the influence of gravity alone follow geodesics according to the general theory of relativity. It’s just how the theory works. It’s certainly admissible to put forward an hypothesis which is contrary to that principle, but it would no longer be consistent with general relativity.

 

Do we need a new theory of gravity: Probably not. Changing a term (or reinterpreting a term) in an equation such as the Einstein field equation, does not obligatorily mean that the entire edifice of general relativity needs to be replaced by a new theory of gravity. Nor, even, does it imply that general relativity needs to be modified fundamentally.

 

You may as well say "changing a term in Newton's law of gravity makes the universe static" or "reinterpreting a term in quantum mechanics makes it consistent with general relativity". We would need a specific solution or example demonstrating the result otherwise it is just assumptions.

 

Much hinges on the time coordinate in the context of the field equation. Now this term is interpreted as a cosmic time (a kind of absolute time). This is the standard time coordinate for specifying the FLRW solution of Einstein's equations. But time t needs not be interpreted in such a fashion. Indubitably, this cosmic time interpretation has led to the concept of a scale factor that changes with time.

 

Time is that which a clock measures. In the general theory of relativity, time is that which a clock measures. In the FLRW metric, time is that which a clock measures. “Cosmic time” is nothing more or less than that which is measured by a clock positioned isotropically and moving inertially. In describing spacetime Einstein says,

 

Just as in Euclidean geometry the space-concept refers to the position-possibilities of rigid bodies, so in the general theory of relativity the space-time-concept refers to the behavior of rigid bodies and clocks.

 

"Time" is always that thing which gets measured by a clock. It is a matter of definition, not of interpretation.

 

What I am describing is entirely based on general relativity. Again, the problem can be resolved. Objects are not obligatorily accelerated in any particular direction (geodesically) in a global manifold of constant curvature, since acceleration is in all directions (all points, and all directions, are equivalent). Locally, of course, the situation is different (and well tested).

 

Right, I see your train of thought. The premise is that all points are equal, or "equivalent". They all describe spacetime equally from their point of view—they all see it isotropically. But, there is no basis to move from that premise to the conclusion that objects have no relative acceleration.

 

You could use the balloon analogy. Cover the balloon in dots and fill it with air. Each dot sees, from its perspective, that all other dots are moving away from it at a speed proportional to distance. Each dot considers itself at rest. Just because each dot sees the same thing—isotropic expansion—and each dot thinks it is, itself, at rest doesn't mean the distance between dots remains constant.

 

To make the analogy a little more real, consider a homogeneous nebula that is dense enough to collapse gravitationally. The nebula can be very large—even infinitely sized—it doesn't matter. The point is that during the collapse each particle in the nebula will see things isotropically. Each particle will see the other particles moving towards it with a speed proportional to distance while each particle considers itself at rest. This will be true for any particle which in no way negates the actual physical decrease in distance between the particles.

 

In terms of spacetime the same conclusion holds:

 

-source

 

From A's perspective (the top spacetime diagram), A is at rest while B has a radial velocity. From B's perspective (the bottom spacetime diagram), B is at rest while A has a radial velocity. There is an actual relative velocity between galaxies A and B. As you say, all points and all directions are equivalent, but it is a mistake to extrapolate from that situation that the velocity is somehow non-real.

 

I'm considering a four-dimensional non-reductive homogeneous pseudo-Riemanninan (and Riemannian) manifolds of constant positive Gaussian curvature, and of constant negative Gaussian curvature. Whether the pseudo-Riemannian signature is positive, negative, (2,2) or with Lorentzian signature of (n -1,1), or other, I do not know yet (the Ricci-flat case, however, is excluded).

 

To represent the universe with a Riemannian manifold of positive definite curvature is to say that the tangent space is Euclidean. This would be a violation of GR's equivalence principle which requires that an inertial observer must result in a co-moving local Minkowski space. This is not only demanded physically (because we know SR works rather than Newtonian mechanics), but it is built into the very foundation of general relativity.

 

Look at Einstein's derivation of General Relativity in 1916 section 4 pages 154 and 155 "The Relation of the Four Co-ordinates to Measurement in Space and Time".

 

For infinitely small four-dimensional regions the theory of relativity in the restricted sense is appropriate...

 

...If a rigid rod is imagined to be given as the unit measure, the co-ordinates, with a given orientation of the system of co-ordinates, have a direct physical meaning in the sense of the special theory of relativity. By the special theory of relativity the expression

 

[math]ds^2 = - dX_1^2 + dX_2^2 + dX_3^2 + dX_4^2[/math]

 

then has value which is independent of the orientation of the local system of co-ordinates, and is ascertainable by measurements of space and time.

 

The tangent space is Minkowskian of signature -,+,+,+. Only in a Lorentz metric is this true. This is why it is a fundamental principle of General Relativity that spacetime can be represented with a Lorentzian manifold. Look at Wiki's article on pseudo-riemannian manifolds,

 

A principal assumption of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3,1) (or equivalently (1,3)).

 

What they are saying is true. Look at condition 5 of the link I gave in my last post concerning "what is a reasonable spacetime",

 

(5) [math]\overline{g}[/math] is a [math]C^{\infty}[/math] Lorentz metric i.e. a smooth semi-Riemannian metric on M of signature (+,-,-,-)

 

I think it would be good to recognize that this is true, or (at least) not to assume it is false in the face of so much evidence. I don't mean for this to be a big issue, it's just that I wrote a rather-long post talking about the relative nature of gravitational potential and the response was to reject the example I used because it was a Lorentzian manifold. "Lorentzian" is pretty much just saying that the spacetime has a space part and a time part, so the objection seemed really rather odd to me.

 

:shrug:

 

~modest

[quantumtopology]

“Cosmic time” is nothing more or less than that which is measured by a clock positioned isotropically and moving inertially.

What do you mean by positioned isotropically. Isotropy in our universe is considered pretty much empirically confirmed and I thought according to the copernican principle and cosmological principle (no special place principle) that means isotropy does not depend on position. IOW, if we had isotropy in one location and not in others there would be no homogeneity and the place with isotropy should be at the center, in a special or privileged observing point.

[modest]

Absolutely

 

I think I was actually, and quite confusingly, thinking of how it is positioned in spacetime to be isotropic. For example, placing a clock vertically on a conformal spacetime diagram would assure that it sees the universe isotropically. Positioning it at an angle would assure that it doesn't see the universe isotropically.

 

In terms of three dimensional space, I absolutely agree—it doesn't matter where it's placed so long as it is comoving with the Hubble flow

 

~modest

[coldcreation]

I’m not sure what you mean by "solution". What I was saying is true of the theory of general relativity itself and not an attribute of specific solutions of GR.

 

I was referring to solutions of the Einstein field equations.

 

You wrote: "GR matches the left hand side which is a Lorentz manifold with the right hand side which is the matter / energy content of the universe. This is the same whether it is local or global."

 

Whether the underlying physics that describes spacetime curvature locally can be extrapolated to the global is an assumption that depends on which type of global manifold is considered. But that's not all. The assumption also depends on the choice of the coordinate system (e.g., comoving or not).

 

In the case of the FLRW solutions the cosmic time coordinate is dubious. Rather than providing a justification for the interpretation that a homogeneous spacetime continuum is continuously curved globally (where time variations are a function of spatial distance dependent on the Gaussian curvature of a dimension four manifold), the cosmic time component, on all distance scales, simply provides a justification for the interpretation that the scale factor is different at all epochs. The "size" of the universe cannot stay the same. The universe is thus unstable.

 

That's fine, but it must be recognized that this hypothetical model is an interpretation derived from a conjured extrapolation based on an assumption (actually several assumptions, if we include initial conditions at cosmic time t = 0).

 

The root of the problem lies in the derivation of FLRW equation(s) that utilizes comoving coordinates and synchronous universal cosmic time t.

 

 

Right, but I’m saying that it is universally true according to general relativity. [...]

 

It cannot be known whether something (or anything) is "universally true" in any physical theory. It's certainly ambitious to postulate such.

 

Suppose we set ourselves a more modest goal (:)).

 

Let's just say it's an assumption that matter and radiation under the influence of gravity alone follow geodesics according to the general theory of relativity globally, exactly as locally. And let's just say that it is a gross assumption that the dynamics of the universe itself (or in its entirety, whatever that means) understands and obeys the same principle.

 

 

Right, I see your train of thought. The premise is that all points are equal, or "equivalent". They all describe spacetime equally from their point of view—they all see it isotropically. But, there is no basis to move from that premise to the conclusion that objects have no relative acceleration.

 

Yes there is!

 

The simple reason is that, if as you imply, there is a slope to the globally homogeneous manifold that cause objects to converge or diverge geodesically towards or away (radially) from one another, then the universe cannot be homogeneous. That follows for the assumption mentioned above (that t is cosmic time). Certain observers will be higher or lower in the field at any given cosmic time. If you remove that notion, it follows from general relativity—and the principles of differential geometry inherent in the theory—that a Riemannian (or pseudo-Riemannian) manifold can be curved negatively or positively (due to the mass-energy content) yet all points are equivalent. That is because the curvature is constant. There are no coordinate systems, in this respect, that can be regarded as privileged.

 

All observers are at the same potential (or height). There is no directional inward or outward bound slope that impels objects to gravitate towards or away from one another, since there is an equal force (curvature is the same) in all directions. The "inward" (attractive) tendency or force is compensated (exactly canceled) by an "outward" (attractive) tendency or force. In another way, all objects in the manifold experience an acceleration that comes from every direction due to the constant curvature.

 

 

Thought experiment question: If you sprinkle the surface of a large sphere of constant radius (and constant curvature) with gravitating objects (say galaxies) initially at rest, what happens? Do all galaxies eventually merge into one big mass at some place on the surface?

 

I would hardly think so!

 

The effects of acceleration that would cause objects to merge, or form clusters, would indeed be due to intrinsic properties of local gravitational fields in the vicinity of the objects. It would not be due to the global curvature of the manifold. Proof of that assertion would be that, although objects are confined to the sphere, they do not move along true geodesics (great arcs, or sections of a great circle, in this case).

 

Non-intuitively enough, the same would hold true for objects sprinkled onto a hyperbolic surface of constant curvature (all points are saddle-points). And, most importantly, the same would hold true, too, for objects sprinkled into a homogeneous four-dimensional manifold of continuous curvature. (Side note: the exact same would hold true for the case of a flat Minkowski space, or even a Newtonian/Euclidean space).

 

 

Your assumption...

In curved spacetime the potential is not constant along space which is why there is a gravitational force. There is a slope to the potential.

...is based entirely on local dynamics. But extrapolating this concept of local dynamics to global dynamics may not be justified.

 

 

You could use the balloon analogy. Cover the balloon in dots and fill it with air. Each dot sees, from its perspective, that all other dots are moving away from it at a speed proportional to distance. Each dot considers itself at rest. Just because each dot sees the same thing—isotropic expansion—and each dot thinks it is, itself, at rest doesn't mean the distance between dots remains constant.

 

Similarly, in a homogeneous gravitational field scenario; just because each observer sees redshift z and time dilation that increases with distance (due to spacetime curvature) it doesn't mean that the global field is sloped downwards (or that the magnitude of curvature increases or decreases) in any particular direction proportionally with distance. Each observer can consider herself, and the objects in the manifold, as at rest relative to the background gravitational field.

 

The observer needs not conclude that all objects are moving away from her at a speed proportional to distance.

 

 

...consider a homogeneous nebula that is dense enough to collapse gravitationally. The nebula can be very large—even infinitely sized—it doesn't matter. [...] Each particle will see the other particles moving towards it with a speed proportional to distance while each particle considers itself at rest. This will be true for any particle which in no way negates the actual physical decrease in distance between the particles.

 

Likewise, in the curved spacetime scenario (without assuming all objects would gravitationally collapse into a point somewhere, or everywhere), each observer will see very distant (high-z) objects as if they were immersed deeply inside a gravitational well. Yet each observer considers herself at a higher elevation in the field (or at a saddle point). This by no means negates the physical possiblility that the homogeneous gravitational field within which they reside is intrinsically curved.

 

 

[...] As you say, all points and all directions are equivalent, but it is a mistake to extrapolate from that situation that the velocity is somehow non-real.

 

Likewise, in the homogeneously curved spacetime (all points and all directions are equivalent), it would be a mistake to assume that curvature is not real (or intrinsic to the entire field).

 

Too, it would be a mistake to extrapolate that the intrinsic Gaussian curvature impels all objects in a homogeneous manifold to coalesce.

 

 

[...] "Lorentzian" is pretty much just saying that the spacetime has a space part and a time part, so the objection seemed really rather odd to me.

 

Pseudo-Riemannian manifolds are a basic ingredient of general relativity, within which the presence of a gravitational field is attributed to a nonzero curvature of the underlying pseudo-Riemannian metric structure. The metric structure is in turn subject to the Einstein field equations. The problem is that, depending on the interpretation of specific terms in the equation, the physical outcome can vary considerably.

 

The current interpretation of these terms (e.g., the cosmic time t and lambda) has created the difficulty of finding a homogeneous gravitational field in general relativity. The usual tendency is to treat spacetime in terms of the coordinates of an underlying Minkowski space-time. As it stands now, general relativity does not admit any spacetime with all the global properties that would expected of a uniform gravitational field. Said differently, there is no global solution to the Einstein field equations that uniquely and satisfactorily embodies Newtonian ideas about a uniform field, i.e, the desired properties for a uniform gravitational field of constant curvature in GR cannot all be satisfied at once (without a metric that becomes degenerate).

 

My next post (unless I get sidetracked) will be an attempt to further explain how this problem can be resolved in the context of general relativity, and how such can be physically interpreted within the context of cosmology.

 

 

CC

[quantumtopology]

Absolutely

 

I think I was actually, and quite confusingly, thinking of how it is positioned in spacetime to be isotropic. For example, placing a clock vertically on a conformal spacetime diagram would assure that it sees the universe isotropically. Positioning it at an angle would assure that it doesn't see the universe isotropically.

 

In terms of three dimensional space, I absolutely agree—it doesn't matter where it's placed so long as it is comoving with the Hubble flow

 

~modest

 

This is tricky, if we strictly follow relativity and consider that Lorentz invariance is valid in a curved spacetime, then there is no acceptable concept of "cosmic time" or any global time scale, as time changes with distance. That seems reasonable if we take note of the fact that when we look at spatial distances we are observing also the past or look-back time.

This admittedly is not the case in standard cosmology where a certain coordinate choice is privileged (the one that originates the FRW metric) and assumed to posses special features such as spatial homogeneity in time spacelike hypersurface slices.And therefore a preferred "cosmic time". According to GR, there is nothing wrong with choosing a coordinates for practical and convenience reasons, as all coordinate choices should give us the same results in physical experiments. But then we must be coherent and we shouldn't decide that the purely spatial isotropy and homogeneity we seem to get with a coordinate choice is physically relevant if it doesnot appear with other coordinate choices as GR general covariance demands.

 

Besides, there is reasons to suspect that the apparently "only spatial" isotropy and homogeneity of the universe with the FRW metric is a coordinate artifact, apart from the fact that it doesn't show up with other coordinates, it demands a certain uniform velocity to be comoving with a supposed flow, and in doing this it is giving us a way to make "closed experiments" of spatial isotropy (see: http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/experiments.html#Tests_of_isotropy_of_space ) that allows us to distinguish different inertial frames, this is forbidden by the SR principle.

 

Regards

QTop

[quantumtopology]

By the way Modest, what do you think of the homogenous solution of the link CC gave? http://www.springerlink.com/content/k550472m97528233/fulltext.pdf

Actually, I think it is not a "homogenous solution" in the sense we normally refer to: as "spatially only" homogenous solution. However is not inhomogenous as its equations give a contant density not dependent on radius. I'm trying to find the flaw, do you see some obvios flaw?

 

Regards

QTop