Explain Dark Matter: Verlinde's Theory Tested

[resurgance2001]

Ok - I have qualified this question with the word stupid so please go gently with me. Imagine you are talking to a rather stupid child when answering please.

My question is about dark matter. Every time I do a search for "proof of dark matter", I get the same answer, which is that using the known laws of gravity the rotation of the galaxies can't be explained unless we 'invent' the concept of dark matter. So why is it automatically assumed that the existing laws of gravity are correct? I heard that Verlinde's theory of gravity just passed its first test. I can't remember exactly what had been tested, but it sounded a bit like a Schwarzschild kind of situation, with a non rotating mass. The point is that Verlinde's theory is able to acurately (so I've been told) the behaviour of the matter without resorting to dark matter.

 

[Orodruin]

So why is it automatically assumed that the existing laws of gravity are correct?

It is not. Theories of modified gravity have been around for as long as dark matter.

Verlinde's theory of gravity

From what I have understood, Verlinde's theory reproduces the Einstein field equations. This makes it equivalent to GR for the purposes of dark matter.

I can't remember exactly what had been tested, but it sounded a bit like a Schwarzschild kind of situation, with a non rotating mass. The point is that Verlinde's theory is able to acurately (so I've been told) the behaviour of the matter without resorting to dark matter.

You will need to provide references for those statements. "I heard that" is not a very good reference.

 

[resurgance2001]

It is not. Theories of modified gravity have been around for as long as dark matter.From what I have understood, Verlinde's theory reproduces the Einstein field equations. This makes it equivalent to GR for the purposes of dark matter.
You will need to provide references for those statements. "I heard that" is not a very good reference.

Thanks I will go back and find where I was reading all this. It was the bit about Verlinde's theory doing away with dark matter that caught my attention. I will go back and do some more research. Thanks again and Happy New Year!

 

[Chronos]

Erik Verlinde has earned more than a piddling of controversy for his gravity ideas. Lubos Motl's comments might be of interest: http://motls.blogspot.com/2016/11/verlindes-de-sitter-mond-is-highly.html

 

[resurgance2001]

Ok - I have found a reference: https://www.google.co.uk/amp/phys.org/news/2016-12-verlinde-theory-gravity.amp?client=safari

The test was in the context of gravitational lensing and yes he claims that this works without the need for dark matter.

 

[resurgance2001]

Erik Verlinde has earned more than a piddling of controversy for his gravity ideas. Lubos Motl's comments might be of interest: http://motls.blogspot.com/2016/11/verlindes-de-sitter-mond-is-highly.html

Ok - I have glanced through the article, but the author clearly has an axe to grind. The only reason I paid attention to the story about Verlinde's theory was that it has just passed an actual experimental test. I don't know, but as soon as I saw something doing away with dark matter it just seemed appealing. My original question was whether there is any independent physical evidence for the existence of dark matter other than it being the only way to explain the movement of galaxies etc based on existing known laws of gravity. I guess the answer is there is not as of yet any actual evidence for it because it is dark.

 

[Chronos]

Indirect evidence is the best we have managed to date, given experimental results tend to affirm the idea DM does not interact with ordinary matter to a measurable extent by means other than gravity. Motl is not the only critic - e.g., https://arxiv.org/abs/1009.5414, Gravity is not an entropic force and here; https://arxiv.org/abs/1108.5240, Conservative entropic forces. Sabine Hossenfelder also comments here http://backreaction.blogspot.com/2010/03/gravity-is-entropy-is-gravity-is.html

 

[Bandersnatch]

Every time I do a search for "proof of dark matter", I get the same answer, which is that using the known laws of gravity the rotation of the galaxies can't be explained unless we 'invent' the concept of dark matter

There are additional two, independent reasons to suspect dark matter - the bullet cluster and the power spectrum of BAOs. Taken together, you've got three different sets of data that can be explained by one idea. That's a pretty convincing case for dark matter, IMO.

 

[resurgance2001]

Thanks Bandersnatch. I had not heard of the bullet cluster or the power spectrum of BAO's before. I will do some research about these.

A part of my question comes from the way that I am currently being taught cosmology where the existence of dark matter seems to be taken for granted and that it is a dogma someone should not even dare to question. I have seen references in recent papers to dark energy and then even phantom energy. At this point it almost seems like having introduced one ad hoc correction to a theory which unfortunately by its very definition is undetectable, we are now free to start inventing the existence of any number of invisible undetectable entities that will make the equations work. The very name 'phantom' sets off alarm bells ringing in my head. Is this really science. Is there an experiment that can be done to test the existence of dark energy or phantom energy?

More and more as I have been recently studying GR there 'seem' to be the inclusion of ad hoc additions to explain away 'apparent' inconsistencies. For example, the recession of galaxies at speeds greater than the speed of light.

The dogma there is that it is the space itself expanding. I wanted to get into that discussion on the forum but the significant discussion has already been closed. That make it hard for someone who is trying to learn this stuff. It's almost like saying GR is sacrosanct, accept the dogma.

Notwithstanding, as I said above I will read up on the two things you have mentioned right now.

 

[resurgance2001]

Indirect evidence is the best we have managed to date, given experimental results tend to affirm the idea DM does not interact with ordinary matter to a measurable extent by means other than gravity. Motl is not the only critic - e.g., https://arxiv.org/abs/1009.5414, Gravity is not an entropic force and here; https://arxiv.org/abs/1108.5240, Conservative entropic forces. Sabine Hossenfelder also comments here http://backreaction.blogspot.com/2010/03/gravity-is-entropy-is-gravity-is.html

Thanks Chronos for sharing these links. My current level of maths and quantum mechanics is not sufficient yet for me to follow all the arguments. However the abstract for the first paper serves as a useful starting point for me to learn more. Cheers

 

[Bandersnatch]

Consider including these links in your reading:

http://background.uchicago.edu/~whu/metaanim.html - visualises the meaning of the power spectrum graph
http://space.mit.edu/home/tegmark/movies.html - shows how certain parameters of the theory affect the predicted shape of the graph (including dark matter)

The following is a complete summary of the bullet cluster discovery, with links to papers, press releases, animations, etc.:
http://chandra.harvard.edu/photo/2006/1e0657/media/

Sean Carroll often writes in his blog (http://www.preposterousuniverse.com/blog/) about dark matter, the evidence for it, and how it fares against ideas like modified gravity. His writing provides a good insight into the scientific process.

 

[resurgance2001]

Thanks Bander - I will look at those also. I get really confused about power spectrums also. I don't know why. It's probably just that the term is new to me.

 

[Drakkith]

A part of my question comes from the way that I am currently being taught cosmology where the existence of dark matter seems to be taken for granted and that it is a dogma someone should not even dare to question. I have seen references in recent papers to dark energy and then even phantom energy. At this point it almost seems like having introduced one ad hoc correction to a theory which unfortunately by its very definition is undetectable, we are now free to start inventing the existence of any number of invisible undetectable entities that will make the equations work. The very name 'phantom' sets off alarm bells ringing in my head. Is this really science. Is there an experiment that can be done to test the existence of dark energy or phantom energy?

More and more as I have been recently studying GR there 'seem' to be the inclusion of ad hoc additions to explain away 'apparent' inconsistencies. For example, the recession of galaxies at speeds greater than the speed of light.

The dogma there is that it is the space itself expanding. I wanted to get into that discussion on the forum but the significant discussion has already been closed. That make it hard for someone who is trying to learn this stuff. It's almost like saying GR is sacrosanct, accept the dogma.

I'm on a tablet and not my normal computer, so I can't elaborate like I want to. I'd just like to say that though current theories on dark matter may seem "ad hoc", they are the best we have currently. That's why you find dark matter everywhere and not MOND or another modified theory of gravity. Also remember that less than a century ago we were proposing brand new particles like the proton, which almost certainly rubbed some folks the wrong way.

In fact, we already know of particles that interact so weakly with normal matter that a million or a billion or something stream through you every single second and you don't even know it! They are called neutrinos! They only interact via gravity and the weak force, not electromagnetism like protons, electrons, and neutrons do.

So really, a brand new type of matter, one that interacts only through gravity, isn't that far fetched. It would mean that matter (dark and normal) is composed of various particles that can interact with each other using anywhere between 1 to 4 of the fundamental forces of nature. We already have 2 to 4 covered, what's so difficult to accept about a particle only using 1?

 

[resurgance2001]

I'm on a tablet and not my normal computer, so I can't elaborate like I want to. I'd just like to say that though current theories on dark matter may seem "ad hoc", they are the best we have currently. That's why you find dark matter everywhere and not MOND or another modified theory of gravity. Also remember that less than a century ago we were proposing brand new particles like the proton, which almost certainly rubbed some folks the wrong way.

In fact, we already know of particles that interact so weakly with normal matter that a million or a billion or something stream through you every single second and you don't even know it! They are called neutrinos! They only interact via gravity and the weak force, not electromagnetism like protons, electrons, and neutrons do.

So really, a brand new type of matter, one that interacts only through gravity, isn't that far fetched. It would mean that matter (dark and normal) is composed of various particles that can interact with each other using anywhere between 1 to 4 of the fundamental forces of nature. We already have 2 to 4 covered, what's so difficult to accept about a particle only using 1?

Thanks - I've got over my suspicions for the moment. What I am trying to get my head around now is acoustic peaks in the CMP. I keep seeing all these blasted 'power spectra' and I am trying to see which features are supposedly caused by the acoustic peaks, and which are due to quantum fluctuations during inflation. Or do they all contribute to each other - as in do the quantum fluctuations during inflation lead to over and under densities which in turn get amplified by the acoustic oscillations this leading to the overall 'bumpy' or 'squigly' shape of the CMB power spectrum? Happy New Year!

 

[Drakkith]

Thanks - I've got over my suspicions for the moment. What I am trying to get my head around now is acoustic peaks in the CMP. I keep seeing all these blasted 'power spectra' and I am trying to see which features are supposedly caused by the acoustic peaks, and which are due to quantum fluctuations during inflation. Or do they all contribute to each other - as in do the quantum fluctuations during inflation lead to over and under densities which in turn get amplified by the acoustic oscillations this leading to the overall 'bumpy' or 'squigly' shape of the CMB power spectrum? Happy New Year!

Hmm. I'm not sure. See if this article helps: https://www.cfa.harvard.edu/~deisenst/acousticpeak/acoustic_physics.html

It doesn't talk about inflation, but it does talk about where the peaks come from and what they lead to.

 

[resurgance2001]

Hmm. I'm not sure. See if this article helps: https://www.cfa.harvard.edu/~deisenst/acousticpeak/acoustic_physics.html

It doesn't talk about inflation, but it does talk about where the peaks come from and what they lead to.

Thanks again.

 

[RUTA]

I'm on a tablet and not my normal computer, so I can't elaborate like I want to. I'd just like to say that though current theories on dark matter may seem "ad hoc", they are the best we have currently. That's why you find dark matter everywhere and not MOND or another modified theory of gravity. Also remember that less than a century ago we were proposing brand new particles like the proton, which almost certainly rubbed some folks the wrong way.

In fact, we already know of particles that interact so weakly with normal matter that a million or a billion or something stream through you every single second and you don't even know it! They are called neutrinos! They only interact via gravity and the weak force, not electromagnetism like protons, electrons, and neutrons do.

So really, a brand new type of matter, one that interacts only through gravity, isn't that far fetched. It would mean that matter (dark and normal) is composed of various particles that can interact with each other using anywhere between 1 to 4 of the fundamental forces of nature. We already have 2 to 4 covered, what's so difficult to accept about a particle only using 1?

Here are some quotes from Carroll at http://www.preposterousuniverse.com/blog/2015/07/07/why-is-there-dark-matter/

None of these properties is, by itself, very hard to satisfy if we’re just inventing new particles. But if we try to be honest — asking “What would expect to see, if we didn’t know what things actually looked like?” — there is a certain amount of tension involved in satisfying them all at once. Let’s take them in turn.

So should we be surprised that we live in a universe full of dark matter? I’m going to say: yes. The existence of dark matter itself isn’t surprising, but it seems easier to imagine that it would have been hot rather than cold, or dissipative rather than dissipationless. I wouldn’t count this as one of the biggest surprises the universe has given us, since there are so many ways to evade these back-of-the-envelope considerations. But it’s something to think about.

 

[resurgance2001]

That's interesting Ruta. Why is it surprising that it is cold rather than hot? I have to confess I don't know what dissipative means in this context. I will have to look it up.

Another question: is dark energy thought to be evenly distributed about the universe. Would there be any way of knowing. I am working on a question for an assignment at the moment that asks how Ho would be affected if we were living in a super cluster void. I am saying that using the first of the Friedmann equations I would expect it to be lower because the matter density would be much lower. But in the concordance model Ohema Lamda is about 0.7 which nearly four times Ohmega m (matter including baryon and dark) . If the dark energy is evenly distributed then as I said above I would expect Ho to be smaller, but if we don't know how dark energy is distributed then this might have a cancelling effect.

 

[RUTA]

That's interesting Ruta. Why is it surprising that it is cold rather than hot? I have to confess I don't know what dissipative means in this context. I will have to look it up.

Another question: is dark energy thought to be evenly distributed about the universe. Would there be any way of knowing. I am working on a question for an assignment at the moment that asks how Ho would be affected if we were living in a super cluster void. I am saying that using the first of the Friedmann equations I would expect it to be lower because the matter density would be much lower. But in the concordance model Ohema Lamda is about 0.7 which nearly four times Ohmega m (matter including baryon and dark) . If the dark energy is evenly distributed then as I said above I would expect Ho to be smaller, but if we don't know how dark energy is distributed then this might have a cancelling effect.

The reason it's surprising that DM is cold is that ordinary matter was very hot at the big bang and has cooled via radiation (reheating when it collapses into stars). If DM had been created as hot as ordinary matter at the big bang, then it should still be hot because it doesn't radiate. And, it can't transfer its heat energy via collisions with colder ordinary matter because it doesn't interact via electromagnetism (it will just pass right through ordinary matter). So it's very difficult for DM to dissipate its heat energy, yet it's cold.

Your second question is nontrivial b/c the LCDM model is highly idealized -- even matter is uniformly distributed in that model. There are no realistic solutions to GR for galaxies, clusters and voids to my knowledge. And if you use perturbation theory, you can miss important nonlinear contributions. Here, for example, is an argument from Donald G. Saari (distinguished professor and director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine) that his discrete N-body simulation of stars in a galaxy "differs significantly from that of its parts or [continuum] approximations. This casts doubt about a standard argument claiming massive amounts of dark matter." https://sinews.siam.org/Details-Page/dynamics-and-the-dark-matter-mystery (see references therein).

 

[resurgance2001]

Yes - of course that makes sense. Very interesting. But then since the dark matter can't actually be directly detected why do they go for a cold dark matter model and not a hot one?

 

[PAllen]

Thanks Bandersnatch. I had not heard of the bullet cluster or the power spectrum of BAO's before. I will do some research about these.

A part of my question comes from the way that I am currently being taught cosmology where the existence of dark matter seems to be taken for granted and that it is a dogma someone should not even dare to question. I have seen references in recent papers to dark energy and then even phantom energy. At this point it almost seems like having introduced one ad hoc correction to a theory which unfortunately by its very definition is undetectable, we are now free to start inventing the existence of any number of invisible undetectable entities that will make the equations work. The very name 'phantom' sets off alarm bells ringing in my head. Is this really science. Is there an experiment that can be done to test the existence of dark energy or phantom energy?

More and more as I have been recently studying GR there 'seem' to be the inclusion of ad hoc additions to explain away 'apparent' inconsistencies. For example, the recession of galaxies at speeds greater than the speed of light.

The dogma there is that it is the space itself expanding. I wanted to get into that discussion on the forum but the significant discussion has already been closed. That make it hard for someone who is trying to learn this stuff. It's almost like saying GR is sacrosanct, accept the dogma.

Notwithstanding, as I said above I will read up on the two things you have mentioned right now.

The questions you raise about GR can readily be discussed here, if you go about it right - and they have been discussed many times. In terms of the range of opinions acceptable here, I hold and defend the view that expansion of space in cosmology is an inessential interpretation added to the math of GR that is unnecessary in the same sense as a aether theory is inessential to SR; like a aether theory, its proponents find it useful and it can never be proved wrong.

In particular, superluminal recession rates are readily demonstrated in SR where there is no curvature and the metric can trivially be made time independent. This should really be discussed in different thread; I just raise it to point out that there is no problem raising such questions here in an appropriate way.

On the other hand, the wrong way to go about such things is accusing numerous people of blindly following dogma.

 

[RUTA]

Yes - of course that makes sense. Very interesting. But then since the dark matter can't actually be directly detected why do they go for a cold dark matter model and not a hot one?

It has to stay bound to galaxies and clusters. If it's hot, it will not stay gravitationally bound.

 

[phinds]

Yes - of course that makes sense. Very interesting. But then since the dark matter can't actually be directly detected why do they go for a cold dark matter model and not a hot one?

Because if it were hot we COULD detect it through its radiation, but there is none so it has to be cold.

 

[phinds]

It has to stay bound to galaxies and clusters. If it's hot, it will not stay gravitationally bound.

Huh? I don't follow that at all. Why would being hot prevent it from staying gravitationally bound?

 

[Vanadium 50]

Because if it were hot we COULD detect it through its radiation

Not necessarily. The prototypical HDM is neutrinos. Neutrinos don't radiate.

Hot means "relativistic". If they move that fast, they exceed escape velocity for galaxies and clusters.

 

[phinds]

Not necessarily. The prototypical HDM is neutrinos. Neutrinos don't radiate.

Good point

Hot means "relativistic". If they move that fast, they exceed escape velocity for galaxies and clusters.

Aha. Well that explains that. Thanks.

 

[resurgance2001]

The questions you raise about GR can readily be discussed here, if you go about it right - and they have been discussed many times. In terms of the range of opinions acceptable here, I hold and defend the view that expansion of space in cosmology is an inessential interpretation added to the math of GR that is unnecessary in the same sense as a aether theory is inessential to SR; like a aether theory, its proponents find it useful and it can never be proved wrong.

In particular, superluminal recession rates are readily demonstrated in SR where there is no curvature and the metric can trivially be made time independent. This should really be discussed in different thread; I just raise it to point out that there is no problem raising such questions here in an appropriate way.

On the other hand, the wrong way to go about such things is accusing numerous people of blindly following dogma.

Hi Phinds - thanks for your reply. When you say super luminal recession rates are readily demonstrated in SR can you explain this more? I thought that was the whole point, that in SR it is not possible for anything to go faster than the speed of light, no?

 

[phinds]

Hi Phinds - thanks for your reply. When you say super luminal recession rates are readily demonstrated in SR can you explain this more? I thought that was the whole point, that in SR it is not possible for anything to go faster than the speed of light, no?

calling @PAllen [not phinds since the post being responded to is his not mine]

@resurgance2001, recession is not proper motion. Proper motion cannot exceed c, recession can be any value at all and objects at the outer area of our Observable Universe are currently receding from us at about 3c.

 

[PAllen]

Hi Phinds - thanks for your reply. When you say super luminal recession rates are readily demonstrated in SR can you explain this more? I thought that was the whole point, that in SR it is not possible for anything to go faster than the speed of light, no?

This is a major weakness in how cosmology is presented in a number of textbooks. To clarify, a few definitions are in order:

1) Relative velocity: The tangent vectors of two world lines are compared. This is an unambiguous operation in SR, no matter how far apart in space and time two vectors are, because parallel transport is path independent. This corresponds to the coordinate velocity of one in the instantaneous rest frame of the other. In GR, this operation is only unambiguous if the two world line tangents are coincident, or at least sufficiently close for spacetime to be treated as flat. Otherwise, relative velocity is simply undefinable or at least, inherently ambiguous - vary your parallel transport path and you get wildly different relative velocities. Thus, velocity comparison over large distances is not a possible operation at all in GR. However, two additional observations are worth making:

a) The result of comparison between a lightlike vector and a timelike vector is path independent and the result is always c, with total generality in GR. Thus, the relative speed of a light and any material body is identically c, with no exceptions in GR.

b) Even though relative velocity of distant world lines is fundamentally ambiguous in GR, this ambiguity does not allow superluminal relative velocities because comparison of any timelike vectors over any possible parallel transport path is always less than c. Thus, a precise statement is that the precise relative speed is ambiguous, but still always less than c. Different comparison paths produce different values, but they are all less than c.

2) Celerity: given some foliation of spacetime, the rate of change in proper distance measured on this foliation between one world line and another by the proper time time on one of them. This is slighly more general that the most common definition of celerity in SR, allowing general use in GR. Cosmological recession rates are a particular instance of celerity: the standard cosmological foliation is used, and the world lines are the comoving ones. In SR, there is no upper bound to the value of a celerity, and the same is true in GR. A trivial example in SR is to use the foliation of a solar inertial frame, use the sun's world line as a reference world line, and the world line of a rocket traveling near c relative to the sun as the one whose celerity we measure. Then as the rocket travels one light year in the chosen foliation, its corresponding proper time can be made arbitrarily small, with corresponding arbitrarily large celerity. Of more relevance to cosmology, you can introduce cosmological style coordinates in SR (Milne coordinates) that have many features of FLRW solutions; in fact, this coordinate system is simply the result of the zero density limit of FLRW solutions. The resulting spacetime is simply flat Minkowski space, foliated by hyperbolic slices. Recession rate computed in these coordinates in flat spacetime (pure SR) grows with distance for 'comoving' world lines, with no upper bound, just as in realistic FLRW solutions.

There are many differences between such a fake flat spacetime cosmology and a realistic one, but superluminal recession rate is not one of these differences. The unfortunately common claim that this is a distinction is a fundamental category error: comparing relative velocities in SR with recession rates in an FLRW solution.

An important difference, for example, is that given a congruence of world lines with isotropy and homogeneity and positive expansion scalar in flat spacetime, the foliation of common proper time from initial coordinate singularity (in flat spacetime, it is only coordinate singularity not a true singularity) must have a unique hyperbolic geometry. In curved spacetime, any constant curvature geometry is possible for different mass densities, including flat spatial slices such as our universe appears very close to.

 

[resurgance2001]

This is a major weakness in how cosmology is presented in a number of textbooks. To clarify, a few definitions are in order:

1) Relative velocity: The tangent vectors of two world lines are compared. This is an unambiguous operation in SR, no matter how far apart in space and time two vectors are, because parallel transport is path independent. This corresponds to the coordinate velocity of one in the instantaneous rest frame of the other. In GR, this operation is only unambiguous if the two world line tangents are coincident, or at least sufficiently close for spacetime to be treated as flat. Otherwise, relative velocity is simply undefinable or at least, inherently ambiguous - vary your parallel transport path and you get wildly different relative velocities. Thus, velocity comparison over large distances is not a possible operation at all in GR. However, two additional observations are worth making:

a) The result of comparison between a lightlike vector and a timelike vector is path independent and the result is always c, with total generality in GR. Thus, the relative speed of a light and any material body is identically c, with no exceptions in GR.

b) Even though relative velocity of distant world lines is fundamentally ambiguous in GR, this ambiguity does not allow superluminal relative velocities because comparison of any timelike vectors over any possible parallel transport path is always less than c. Thus, a precise statement is that the precise relative speed is ambiguous, but still always less than c. Different comparison paths produce different values, but they are all less than c.

2) Celerity: given some foliation of spacetime, the rate of change in proper distance measured on this foliation between one world line and another by the proper time time on one of them. This is slighly more general that the most common definition of celerity in SR, allowing general use in GR. Cosmological recession rates are a particular instance of celerity: the standard cosmological foliation is used, and the world lines are the comoving ones. In SR, there is no upper bound to the value of a celerity, and the same is true in GR. A trivial example in SR is to use the foliation of a solar inertial frame, use the sun's world line as a reference world line, and the world line of a rocket traveling near c relative to the sun as the one whose celerity we measure. Then as the rocket travels one light year in the chosen foliation, its corresponding proper time can be made arbitrarily small, with corresponding arbitrarily large celerity. Of more relevance to cosmology, you can introduce cosmological style coordinates in SR (Milne coordinates) that have many features of FLRW solutions; in fact, this coordinate system is simply the result of the zero density limit of FLRW solutions. The resulting spacetime is simply flat Minkowski space, foliated by hyperbolic slices. Recession rate computed in these coordinates in flat spacetime (pure SR) grows with distance for 'comoving' world lines, with no upper bound, just as in realistic FLRW solutions.

There are many differences between such a fake flat spacetime cosmology and a realistic one, but superluminal recession rate is not one of these differences. The unfortunately common claim that this is a distinction is a fundamental category error: comparing relative velocities in SR with recession rates in an FLRW solution.

An important difference, for example, is that given a congruence of world lines with isotropy and homogeneity and positive expansion scalar in flat spacetime, the foliation of common proper time from initial coordinate singularity (in flat spacetime, it is only coordinate singularity not a true singularity) must have a unique hyperbolic geometry. In curved spacetime, any constant curvature geometry is possible for different mass densities, including flat spatial slices such as our universe appears very close to.

I woul really need to see this in another source, preferable textbook before I can fully follow the argument. I haven't come across the word foliation before, or celerity. I looked both up but I am a slow learner and I usually need to read three or four different accounts of the same thing before I get it. I will look in Schultz and see if I can find it there. But if you can direct me to any other sources that would be much appreciated. Thanks