How Are Pressure and Density Defined in the Comoving Coordinate System?

[bloby]

To use Einstein fields equations you have to choose a coordonate systeme(c.s.). The FRW metric is the metric in a c.s. in which the matter appears isotropic and homogeneouse. This c.s. is the comoving c.s. in which galaxies keep their own time independent coordinate. The R-H-S of Einstein equations must be expressed in the same c.s. Now the energy-momentum tensor takes the form of a perfect fluid at rest. How are defined pressure and density? In a "comoving volume" the quantity of matter will remains the same...

 

[Chalnoth]

To use Einstein fields equations you have to choose a coordonate systeme(c.s.). The FRW metric is the metric in a c.s. in which the matter appears isotropic and homogeneouse. This c.s. is the comoving c.s. in which galaxies keep their own time independent coordinate. The R-H-S of Einstein equations must be expressed in the same c.s. Now the energy-momentum tensor takes the form of a perfect fluid at rest. How are defined pressure and density? In a "comoving volume" the quantity of matter will remains the same...

Well, as you said, the pressure and density depend upon the coordinate system. Just as a particularly dumb example, I can express water as being 1g/cm^3, or 1000kg/m^3. In the end, though, the relationship between pressure and energy density is determined by the type of matter you have. For example, normal matter and dark matter effectively experience zero pressure on cosmological scales (e.g. there is no pressure between galaxies).

By contrast, light has a pressure equal to one third its energy density.

The relationship between pressure and energy density is actually what determines how the matter evolves within a comoving volume. For instance, with normal matter, which has no pressure on cosmological scales, an expanding volume always has the same average amount of matter, and so the total amount doesn't change, and the density just decreases along with the increase in volume.

Photons, however, are different. Because they experience pressure, expansion actually removes energy from the system, so while an expanding gas of photons may keep the same number of photons per unit volume, the individual photons themselves reduce in energy. This is the cosmological redshift.

Now, as you correctly point out, this does depend upon the coordinate system you use. Everything I have said in this post assumes FRW coordinates, the ones you talked about. I could, in principle, use any other coordinate system I chose, and things would appear somewhat different in those coordinate systems, but none of the actual physics would change. One can thus think of this description of one possible way of talking about it, but not the only correct way: just as before, I can correctly talk about water having 1g/cm^3 density, or 1000kg/m^3 density. Neither is more valid than the other, though one may be more useful in certain contexts than the other.

 

[bloby]

For instance, with normal matter, which has no pressure on cosmological scales, an expanding volume always has the same average amount of matter, and so the total amount doesn't change, and the density just decreases along with the increase in volume.

Thanks for your answer.

I don't understand why the volume is expanding. In the comoving coordinates it remains the same doesn't it? The expansion of the volume comes frome the scale factor, it is the volume mesured with the metric. Why is the density defined by this volume and not the "comoving" volume?

 

[Chalnoth]

Thanks for your answer.

I don't understand why the volume is expanding. In the comoving coordinates it remains the same doesn't it? The expansion of the volume comes frome the scale factor, it is the volume mesured with the metric. Why is the density defined by this volume and not the "comoving" volume?

Well, as I mentioned, there are multiple ways of looking at it. Yes, you could consider the expansion from the perspective of coordinates that don't change with the expansion, but in that case what you see is that atoms and things made of atoms get smaller with time. But it is much more intuitive for us to think of atoms as being stable in size, so instead we see the coordinate grid itself as expanding, which is what I mean by the volume expanding.

 

[bapowell]

bloby,
There might be some confusion regarding terminology here. When Chalnoth says 'comoving volume', he means some region of space that is expanding along with the universe -- he's not necessarily implying that one must be using comoving coordinates. We can still measure a comoving volume in static coordinates, so that each dimension is getting larger in proportion to the scale factor, a(t). Or we could measure the comoving volume in comoving coordinates, in which case each dimension is constant in time.

 

[bloby]

I think my problem is more basic. I just read in Weinberg that the density and pressure in the energy-momentum tensor are scalars mesured in locally inertial frames. I have to work and think about it...

 

[marcos54]

What is an "expanding gas of photons"? I'm a mere attorney/dilettante, but that would seem to be some fast gas. Is that a hypothetical term of art?

 

[Chalnoth]

What is an "expanding gas of photons"? I'm a mere attorney/dilettante, but that would seem to be some fast gas. Is that a hypothetical term of art?

If you have a hollow box at a certain temperature, then the interior of the box gets filled with a gas of photons. If this box is allowed to get larger with time, then you'd have an expanding gas of photons inside.

Don't be fooled by the speeds of the individual components of the gas: the individual molecules in air, for instance, move at around 1.5 times the speed of sound in air.

 

[bloby]

If someone does the same mistake: the density in the energy-momentum tensor is the density measured in locally inertial frames because the Einstein equations must reduce to the Newton equation in a locally inertial frame for weak fields.