Exploring the Role of Perfect Fluid Solutions in General Relativity

[TrickyDicky]

Fluid solutions of the EFE are very useful in GR and pervade all of the theory's applications be it cosmological (FRW dusts for matter-dominated models, radiation fluids for radiation-dominated ones...) or astrophysical (stars interior models, etc).

Now, I know there is a component of idealization in all these models, but it must have some base on the physical universe we observe. I mean our universe either behaves globally as a perfect fluid or it doesn't, right?
So for GR fluid solutions, to what extent is being a perfect fluid a physical feature of our universe as opposed to an unphysical one, i.e. coordinate-dependent?

 

[Mentz114]

Nearly all the known perfect fluid solutions of the EFE are unphysical in some way - but they are all coordinate independent. Aren't all solutions coordinate independent ?

I don't know if I understand your question.

 

[pervect]

Lets say you want to find the gravitational field of a cube of matter. You can do so, but it won't be a fluid solution - the stress energy tensor to support this shape will have pressures that are not isotropic.

However, the forces that allow matter to hold its shape are pretty weak, and as you scale up the cube, eventually you'll find that you can't hold the shape anymore with known materials, the material is too weak. I'm not sure exactly where this happens, but I doubt you can build a cubical planet, much less a cubical star out of known materials.

So I'd say that what insures the goodness of the fluid approximation is that eventually the forces that allow non-fluid matter to hold its shape are just too weak to be important on a large scale.

And large scales are just where you need GR, rather than Newtonian gravity.

 

[TrickyDicky]

Nearly all the known perfect fluid solutions of the EFE are unphysical in some way - but they are all coordinate independent. Aren't all solutions coordinate independent ?

I don't know if I understand your question.

Sure, solutions of tensorial equations ought to be coordinate-independent.
What i meant was that the perfect fluid behaviour of these solutions seems to depend(if we trust general covariance of GR which we should) on how the spacelike hypersurfaces are sliced up wrt the timelike worldlines. Only when the slicing results in orthogonality bewtween the spatial hypersurfaces and the timelike geodesics we have vanishing vorticity of the congruence of fundamental observers and the dust of galaxies in cosmology or the pertinent paricles for other fluid solutions. Any other slicings will lose that property of perfect fluids and also will get you different degrees of inhomogeneity and expansion(contraction). I know all these features are considered to belong to the boundary conditions rather than to the physical consequences of GR theory proper, and therefore many claim those properties are outside the scope of what is affected by general covariance but still I found difficult to understand how something that seems so "physical" as behaving as a perfect fluid or not seems to depend on how we setup our coordinates on spacetime. It seems to go against the general principle that coordinate-dependent features are not physical.

 

[PeterDonis]

What i meant was that the perfect fluid behaviour of these solutions seems to depend(if we trust general covariance of GR which we should) on how the spacelike hypersurfaces are sliced up wrt the timelike worldlines.

What do you mean by "perfect fluid behavior"? The fact that, in a particular frame (the "rest frame" of the fluid), the SET of a perfect fluid takes a particularly simple form is not a "behavior"; it's just a mathematical fact that helps you to make calculations. An observer who is not at rest in the fluid's rest frame will see "behavior" that is different from that seen by an observer who is at rest in the fluid's rest frame; but that's because the two observers are in different states of motion relative to the fluid, which is a coordinate-independent statement.

 

[TrickyDicky]

What do you mean by "perfect fluid behavior"? The fact that, in a particular frame (the "rest frame" of the fluid), the SET of a perfect fluid takes a particularly simple form is not a "behavior"; it's just a mathematical fact that helps you to make calculations. An observer who is not at rest in the fluid's rest frame will see "behavior" that is different from that seen by an observer who is at rest in the fluid's rest frame; but that's because the two observers are in different states of motion relative to the fluid, which is a coordinate-independent statement.

I'm not talking about different observers with different states of motion, those particular frames are still using as reference the fundamental "rest frame" of a particular solution that allows that slicing. I'm talking about different slicings of the spacetime manifold in which there is not any possible "rest frame" of the fluid because there is no observers that have timelike worldlines orthogonal to the spatial hypersurfaces. Of course those would be different solutions of the EFE. But what I'm finding odd is that the same theory gives rise to such physically different solutions.
I know general covariance is generally taken to be only an "up to diffeomorphism" principle, but still heuristically I'm troubled by that, thus my reflection. I realize that it is not an strictly coordinate-dependent issue as I was saying, but rather a solution-dependent thing.

 

[PeterDonis]

I'm talking about different slicings of the spacetime manifold in which there is not any possible "rest frame" of the fluid because there is no observers that have timelike worldlines orthogonal to the spatial hypersurfaces.

In other words, you're not talking about a perfect fluid. You're talking about a different solution, as you note next.

But what I'm finding odd is that the same theory gives rise to such physically different solutions.

Why does this seem odd? It seems perfectly natural to me. You start with different assumptions and get different solutions. What's odd about that?

 

[TrickyDicky]

Why does this seem odd? It seems perfectly natural to me. You start with different assumptions and get different solutions. What's odd about that?

Yeah, this is highly subjective, we have to work with what we have, I was thinking out loud and meant odd comparing it with an idealized GR more like what Einstein had in mind before he came up with the EFE, something like a theory with a unique solution that made superfluous additional different geometric constraints and boundary conditions. But I guess reality is more complex than that.

 

[PAllen]

Yeah, this is highly subjective, we have to work with what we have, I was thinking out loud and meant odd comparing it with an idealized GR more like what Einstein had in mind before he came up with the EFE, something like a theory with a unique solution that made superfluous additional different geometric constraints and boundary conditions. But I guess reality is more complex than that.

It is certainly true that Einstein expressed disappointment that boundary conditions were needed to get solutions matching our universe.

 

[ikjyotsingh]

A perfect fluid solution is a very good idealization

With respect to the original question: "So for GR fluid solutions, to what extent is being a perfect fluid a physical feature of our universe as opposed to an unphysical one, i.e. coordinate-dependent?"

There are two answers to this, which may echo what has already been said on this issue in these forums.

1. A fluid solution assumes that on large scales the universe behaves like a continuum of particles. The alternate is to consider kinetic theory, but, there is a limiting region extending beyond quantum fluctuations, where this continuum approximation is quite valid. In the early universe, where conditions are largely unstable due to high temperatures, one must consider viscous effects in the energy-momentum tensor as well.

2. A perfect fluid distribution is a NECESSARY requirement of the standard model of cosmology which assumes that our universe today is modeled by a six-dimensional isometry group, that is invariant with respect to spatial rotations and translations, and is hence represented by the Friedmann-LeMaitre-Robertson-Walker metrics. The perfect fluid distribution is necessary because of the spatial rotational invariance, the isotropy condition. This is not just a mathematical assumption, the universe is indeed observed to be 99.999% isotropic based on observations of the CBR.

Hope that helps.

Ikjyot Singh Kohli