The Universe filled with a "homogeneous perfect fluid"

[MathematicalPhysicist]

On page 353 of Schutz's textbook he writes the following:

As in earlier chapters, we idealize the universe as filled with a homogeneous perfect fluid.

So it seems that the ether is replaced by a "homogeneous perfect fluid".
It seems the medium which fills the universe is not ether but "homogeneous perfect fluid".
But in that case what characteristics did the proposed ether have that a homogeneous perfect fluid doesn't and vice versa?
So it seems that physics still needs a medium to fill the universe. I thought to myself that there's only empty space, matter and radiation.

 

[Ibix]

A homogeneous perfect fluid just means all the matter/dark matter/radiation/dark energy in the universe, which is assumed to be uniform density and pressure everywhere and leads to the FLRW solution. If you perturb the uniformity a bit the perturbations grow and eventually form stars and galaxies in the overdense regions.

This has nothing to do with ether, which was an attempt to explain electromagnetism. You can have GR without a uniform density fluid, but you don't get the perfect FLRW solution. Our universe is, of course, not a perfect FLRW spacetime, although it's a decent approximation at large scales.

 

[Dale]

On page 353 of Schutz's textbook he writes the following:So it seems that the ether is replaced by a "homogeneous perfect fluid".
It seems the medium which fills the universe is not ether but "homogeneous perfect fluid".
But in that case what characteristics did the proposed ether have that a homogeneous perfect fluid doesn't and vice versa?
So it seems that physics still needs a medium to fill the universe. I thought to myself that there's only empty space, matter and radiation.

The perfect fluid is the empty space, matter, and radiation. It has nothing to do with the ether.

 

[MathematicalPhysicist]

You can have GR without a uniform density fluid, but you don't get the perfect FLRW solution. Our universe is, of course, not a perfect FLRW spacetime, although it's a decent approximation at large scales.

What do you get instead?

 

[Dale]

What do you get instead?

Minkowski spacetime.

 

[MathematicalPhysicist]

BTW I have another question which is related to relativistic radiation.
Why is the equation of state for highly relativistic particles is of the form $p=\frac{1}{3}\rho$?
Is there some sort of derivation of this form from something else?

 

[MathematicalPhysicist]

Minkowski spacetime.

This is the sapcetime of SR, surely not of a gravitational theory which globally doesn't look like the Minkowski spacetime.

 

[Dale]

This is the sapcetime of SR, surely not of a gravitational theory which globally doesn't look like the Minkowski spacetime.

It is also a vacuum solution to the Einstein Field Equations. It is a legitimate (though boring) spacetime for GR.

 

[martinbn]

What do you get instead?

You can get a nonisotropic universe for example.

 

[Ibix]

What do you get instead?

Minkowski, Schwarzschild, Kerr, Godel, Oppenheimer-Snyder... Or a less symmetric and more realistic spacetime. It depends what stress-energy distribution you decide to put in.

 

[MathematicalPhysicist]

You can get a nonisotropic universe for example.

Isn't it called anisotropic?

 

[vanhees71]

If you start from the cosmological principle, i.e., a Friedmann-Lemaitre-Robertson-Walker spacetime, from the Einstein equations it is clear that the energy-momentum tensor is that of an ideal fluid. This is due to the maximal symmetry of this spacetime model. You are however still free to choose the equation of state, and indeed according to our current "Cosmological Standard Model", based on observations like the fluctuations of the cosmic micro-wave background radiation and redshift-distance relations for type-1 supernovae, we have a mixture of different ideal fluids in the universe: matter (the known "baryonic" matter and dark matter), radiation/photons, and dark energy, each with a specific equation of state.

Of course, that's only a much coarse-grained overall picture. The coarse graining needed to get a homogeneous and isotropic universe is over scales of around 250 Mio light years according to

https://en.wikipedia.org/wiki/Cosmological_principle

Below that scale you have "structure formation", i.e., you must take into account the "local deviations" from homogeneity and isotropy as observed in the various structures on different scales like galaxy groups and (super)clusters, galaxies, etc.

https://en.wikipedia.org/wiki/Observable_universe#Large-scale_structure

and you have to model the "cosmic fluid" with more refined methods too like kinetic theory.

 

[Orodruin]

This is the sapcetime of SR, surely not of a gravitational theory which globally doesn't look like the Minkowski spacetime.

SR is a special case of GR, so yes, GR includes Minkowski space as a possible vacuum solution.

 

[MathematicalPhysicist]

SR is a special case of GR, so yes, GR includes Minkowski space as a possible vacuum solution.

How would we know which spacetime is ours?

 

[Orodruin]

How would we know which spacetime is ours?

Same way as we learn anything in physics. Experimentation and observation. It is easy to lose track of the fact that physics is an empirical science in the midst of beautiful theories ...

 

[MathematicalPhysicist]

Same way as we learn anything in physics. Experimentation and observation. It is easy to lose track of the fact that physics is an empirical science in the midst of beautiful theories ...

So what did we find?
I heard once that our universe is flat, i.e. $k=0$ but that's in Robertson-Walker metric; I assume there are other metrics where our universe can still be flat, don't we have such metrics?

 

[Orodruin]

So what did we find?
I heard once that our universe is flat, i.e. $k=0$ but that's in Robertson-Walker metric; I assume there are other metrics where our universe can still be flat, don't we have such metrics?

Not that satisfy the assumptions imposed by the cosmological principle.

 

[Ibix]

The cosmological principle gets you the Friedmann equations and that gives you an FLRW universe. It matches observation (cosmological redshift, CMB) remarkably well on the large scale. It's catastrophically wrong on the small scale, (the universe is not homogeneous, as a quick glance around you will confirm) so we add perturbations to the uniformity so that over dense regions collapse and form structures.

You then go out and measure things like redshift versus distance, CMB fluctuations, galaxy distributions and other things. These give you values for the variables (like $k$) in the models.

 

[MathematicalPhysicist]

BTW I have another question which is related to relativistic radiation.
Why is the equation of state for highly relativistic particles is of the form $p=\frac{1}{3}\rho$?
Is there some sort of derivation of this form from something else?

Did everyone skipped my question in post number 6 in the quote above?

 

[Orodruin]

Did everyone skipped my question in post number 6 in the quote above?

These are my lecture notes on special relativity. Your question is treated in section 12.2.1.

 

[Dale]

So what did we find?
I heard once that our universe is flat, i.e. $k=0$ but that's in Robertson-Walker metric; I assume there are other metrics where our universe can still be flat, don't we have such metrics?

Note that flat here is spatially flat, but the universe is still curved in spacetime.

 

[MathematicalPhysicist]

Note that flat here is spatially flat, but the universe is still curved in spacetime.

So it's flat in the coordinates $(x,y,z)$ but curved in $(t,x,y,z)$, am I correct?
I wonder, can the universe change to different spacetimes over the course of time?
In which case it can be at some times Minkowski at others Godel, etc.
I mean the universe might not have a constant metric which it acts all the time.

 

[MathematicalPhysicist]

These are my lecture notes on special relativity. Your question is treated in section 12.2.1.

Thanks I'll give it a careful look.

 

[ergospherical]

I think what Dale is referring to is that the the hypersurfaces orthogonal to $\partial/\partial t$ have spatial (intrinsic) curvature proportional to $k/a^2$, but their extrinsic curvature $K_{ij} = \dfrac{1}{2N}(\dot{h}_{ij} - D_i N_j - D_j N_i$) instead goes as $H^2$.

 

[Dale]

So it's flat in the coordinates (x,y,z) but curved in (t,x,y,z), am I correct?

Yes, that is correct.

I wonder, can the universe change to different spacetimes over the course of time?

No, spacetime includes all of time, so it cannot change to different spacetimes over time. However, you could have the situation where the shape of spacetime has certain easily identified features that are different along the time direction. Think for example of a solid that is a hemisphere attached to a cube.

https://i.stack.imgur.com/CwPD6.jpg

Where we are considering “time” to be vertical and “space” to be horizontal slices. Then your “space” changes from 2D squares to 2D circles over 1D “time”. But the “spacetime” shape is the whole hemisphere+cube 3D shape.

 

[vanhees71]

Did everyone skipped my question in post number 6 in the quote above?

For (interaction free) massless particles there is no scale in the description. Correspondingly the Lagrangian of the corresponding field theory is invariant under scaling transformations, and this implies through Emmy Noether's theory a conservation law, and the conservation law boils down to the fact that the covariant trace of the energy-momentum tensor of the field is 0, but this trace is $T^{\mu}_{\nu}=\epsilon-3P=0$.

 

[martinbn]

I wonder, can the universe change to different spacetimes over the course of time?

Dale answered this, but i think that you may mean something else. You can have a spacetime, say with coordinates $(t,x,y,z)$ such that the portion with $t<0$ is Minkowski, an the rest is not. Or any number of changes like that. For isntance take the line element to be $ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)$, where the function $a(t)$ is dentically 1 for negative $t$ and something complicateted for later time. Of course sich examples will violate the energy conditions and will be unrelistic, but there are such spacetimes.

 

[mitochan]

So it seems that physics still needs a medium to fill the universe. I thought to myself that there's only empty space, matter and radiation.

You may be able to regard matter and radiation fill the universe though they are not homogeneous.
For an example the space is filled with cosmic background photons. quantum field of particles extends to all the universe.

 

[horacio torres]

Same way as we learn anything in physics. Experimentation and observation. It is easy to lose track of the fact that physics is an empirical science in the midst of beautiful theories ...

physics is an empirical science in the midst of beautiful theories ...

Beautiful line

Horacio

 

[Fractal matter]

On page 353 of Schutz's textbook he writes the following:So it seems that the ether is replaced by a "homogeneous perfect fluid".
It seems the medium which fills the universe is not ether but "homogeneous perfect fluid".
But in that case what characteristics did the proposed ether have that a homogeneous perfect fluid doesn't and vice versa?
So it seems that physics still needs a medium to fill the universe. I thought to myself that there's only empty space, matter and radiation.

Yes, it's a hypothesis. It appears it has many qualities of superfluid He3.