Gravitational collapse

[arfa brane]

It doesn't collapse because Omega is 1. The expansion of space, not spacetime curvature.

But Einstein's equation predicts the expansion of space, the same equation predicts gravitational collapse.

It might be convenient to classify these two things as separate, but they are both described by the same equation. The equation describes how spacetime is curved by matter.
In that case, what does this equation say about curvature?

$$
\frac {\ddot V} { V} {\mid _{t = 0}} = -{1\over 2} (\rho + P_x + P_y + P_z) $$

?

 

[Markus Hanke]

But Einstein's equation predicts the expansion of space, the same equation predicts gravitational collapse.

Are you talking about the Einstein Field Equations themselves here, or about one of its solutions ?

 

[hansda]

But you're not talking about cosmological expansion.

I am comparing cosmological expansion(CE) with GC(gravitational collapse).

You do know that expansion and contraction, which is what you brought up, are not the same things. Right?

Expansion and contraction are obviously not the same thing but they can be opposite.

 

[hansda]

It doesn't collapse because Omega is 1. The expansion of space, not spacetime curvature.

Is it that in cosmological expansion(CE), space and spacetime are different?

Is it that in CE, only space is stretched while spacetime remains unaffected?

In that case there must some creation of space in CE.

 

[hansda]

Cosmological expansion involves the stretching of space, gravitational collapse is matter moving through space.

Isnt it a fiction that in Cosmological Expansion, space/spacetime is stretched without any mass; whereas in Gravitational Collapse mass moves through space without any stretching of space/spacetime?

 

[Tach]

Isnt it a fiction that in Cosmological Expansion, space/spacetime is stretched without any mass;

What gives you this fringe idea? Cosmological Expansion isn't "fiction", it is observable , as explained to you several times.

 

[arfa brane]

Are you talking about the Einstein Field Equations themselves here, or about one of its solutions ?

I'm talking about both the field equations and any solution or 'consequence'.

I'm trying to understand Baez's approach, and the equation I posted is what he uses. I understand it's very simplified (but that's why it's interesting).

 

[brucep]

But Einstein's equation predicts the expansion of space, the same equation predicts gravitational collapse.

It might be convenient to classify these two things as separate, but they are both described by the same equation. The equation describes how spacetime is curved by matter.
In that case, what does this equation say about curvature?


$$
\frac {\ddot V} { V} {\mid _{t = 0}} = -{1\over 2} (\rho + P_x + P_y + P_z) $$

?

The Cosmological expansion is a discussion about the expansion of space and the evolution of the universe. The Cosmological metric is the meter stick GR provides for evaluating specific elements of the expansion. The WMAP experiment results predict our universe is spatially flat. OMEGA = 1 to a very small error bar. Guth's Eternal Inflation predicts OMEGA = 1. Regardless the starting point OMEGA must go very quickly to 1. The expanding cosmological space is Euclidean. Spacetime curvature is local phenomena not cosmological phenomena.

 

[arfa brane]

Spacetime curvature is local phenomena not cosmological phenomena.

If the universe is spatially flat then it has no curvature. So curvature is a global phenomenon.

This makes sense because expansion (zero curvature) and collapse (local nonzero curvature) are described by the same equation.

 

[brucep]

If the universe is spatially flat then it has no curvature. So curvature is a global phenomenon.

This makes sense because expansion (zero curvature) and collapse (local nonzero curvature) are described by the same equation.

Spacetime curvature is local 4 dimensional geometric phenomena described by GR. The cosmological metric of GR describes a 3 dimensional expanding Euclidean space. All the metric solutions to Einstein's Field are wonderful for describing the physics that falls in the solutions domain of applicability. The Cosmological metric is a tool for evaluating the evolution of our universe. The following link is a bit about the metric and how it's used.
http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_04.pdf

 

[arfa brane]

Spacetime curvature is local 4 dimensional geometric phenomena described by GR.

Are you saying that curvature is not something that applies to the universe in general? You can't use EFE to determine the overall curvature of the universe?

 

[Markus Hanke]

Are you saying that curvature is not something that applies to the universe in general? You can't use EFE to determine the overall curvature of the universe?

No, that is not what he is saying.
I suppose this discussion boils down to a very simple fact - so far as the EFEs are concerned, you get out what you put in. If you start with a vanishing SEM tensor, you get a ( local ) vacuum solution, like e.g. the Schwarzschild metric. If you start with a local SEM tensor ( e.g. a homogeneous fluid in a sphere ), you get a local interior metric. If you start with a global SEM tensor ( e.g. Goedel dust ) you get a cosmological solution.

What I am trying to point out is simply that there are different classes of solutions to the EFEs; gravitational collapse would be described by the interior Schwarzschild metric, which is a local solution, whereas the universe as a whole is described by the FLRW metric, which is a cosmological solution. These are not the same, even though there are certain similarities.

Does this help ?

 

[arfa brane]

plus ça change: Einstein's general theory applies to non-inertial (accelerating) frames, the special theory applies only to inertial frames. Gravitational collapse requires a general theory because test particles in free-fall are accelerating.

Hence, it might be true that the speed of light (relative to accelerating frames) isn't really a restriction on the motion of infalling bodies.

pendant que:

What I am trying to point out is simply that there are different classes of solutions to the EFEs; gravitational collapse would be described by the interior Schwarzschild metric, which is a local solution, whereas the universe as a whole is described by the FLRW metric, which is a cosmological solution. These are not the same, even though there are certain similarities.

Does this help ?

Yes, I know about different metrics. Nonetheless I think Baez is trying to show (at the links I provided earlier) that there is a way to consider EFE which covers all the differences. This is just to formulate a solution that relates changes in volume to changes in radius (admittedly of a spherically symmetric object). That is, even at first order and with homogenous, pressureless matter densities, most of the consequences of GR are still apparent.

 

[Markus Hanke]

Nonetheless I think Baez is trying to show (at the links I provided earlier) that there is a way to consider EFE which covers all the differences.

Well, both of the metrics are of course solutions to the same set of equations. The EFE itself is the physical law that underlies the dynamics which space-time is constrained to, so it does cover both cases. It is really just a question of initial conditions leading to different metrics in different cases.

 

[brucep]

Are you saying that curvature is not something that applies to the universe in general? You can't use EFE to determine the overall curvature of the universe?

You can apply anything you want but that doesn't mean it will render useful physics. The universe is flat. If it was curved 'in the spatial dimensions' then our universe would be either open or closed. It's not according to the WMAP results. Spacetime geometry is local phenomena. One of the strengths of GR is it tells us that tangent to the gravitational manifold we can attach a flat spacetime manifold where most all the local physics can be accomplished using the mathematics of SR. This would suggest that the average spacetime curvature over the full extent of our universe would be very close to nil. Really not the way we need to use the physics of GR to evaluate the cosmological evolution of our universe. Hence the metric solution for cosmological expansion, or contraction, doesn't include the metric component of spacetime curvature. In the metric this is the component for spacetime curvature 2M/r. All the metrics we use to evaluate spacetime events from invariant local proper coordinates or frame dependent remote coordinates include the spacetime curvature component. The evolution of the universe is evaluated over distances that couldn't be considered local by any stretch of the imagination. Hence the cosmological metric solution that is put in the link. GR works really good.

 

[arfa brane]

Spacetime geometry is local phenomena.

So the Hubble volume isn't local?

Hence the metric solution for cosmological expansion, or contraction, doesn't include the metric component of spacetime curvature.

What is the metric component of spacetime curvature?

The evolution of the universe is evaluated over distances that couldn't be considered local by any stretch of the imagination. Hence the cosmological metric solution that is put in the link. GR works really good.

Baez says that Einstein's equation (sic) boils down to pressure in four dimensions, at http://math.ucr.edu/home/baez/einstein/node8.html, he discusses how adding the vacuum SET to the equation(s) (that's the local inertial SET) speaks to cosmological evolution.

He also says: "For the vacuum to not pick out a preferred notion of `rest', its stress-energy tensor must be proportional to the metric. "

What metric is he referring to?

 

[hansda]

Spacetime curvature is local 4 dimensional geometric phenomena described by GR. The cosmological metric of GR describes a 3 dimensional expanding Euclidean space. All the metric solutions to Einstein's Field are wonderful for describing the physics that falls in the solutions domain of applicability. The Cosmological metric is a tool for evaluating the evolution of our universe. The following link is a bit about the metric and how it's used.
http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_04.pdf

It seems GR is local whereas cosmology is Newtonian.

 

[Tach]

It seems GR is local whereas cosmology is Newtonian.

this is just as false as the other fringe ideas you've been pushing.

 

[hansda]

So the Hubble volume isn't local?

I think Hubble Volume is local.

What is the metric component of spacetime curvature?

I think it is described here.

 

[hansda]

this is just as false as the other fringe ideas you've been pushing.

Have you read the link i provided? This is as per the link and not my conclusion.