Energy-momentum tensor radiation-dominated universe.

[binbagsss]

I'm looking at 'Lecture Notes on General Relativity, Sean M. Carroll, 1997'

Link here:http://arxiv.org/pdf/gr-qc/9712019.pdf

Page 221 (on the actual lecture notes not the pdf), where it generalizes that the energy-momentum tensor for radiation - massive particles with velocities tending to the speed of light and EM radiation- can be expressed in terms of the field strength.

So it says that at such speeds , the particles become indistinguishable from the speed of light as far as the equation of state is concerned.

My Question:

How are the energy-momentum tensor and equation of state related? How does it follow from this fact that the energy-momentum tensor of the particles takes the same form as photons do?

Thanks in advance.

 

[ChrisVer]

Hmm... The energy momentum tensor gives you the continuity equation by taking the conservation of energy equation (8.20) in the page you referred to.
On the other hand the equation of state relates the radiation density to the pressure eq (8.21).
And that's how you can solve the continuity equation.

Then because the equation of state does not distinguish between highly-relativistic particles and massless particles (by the choice of [itex]w[/itex]), the continuity equation solution won't change.

 

[PeterDonis]

How are the energy-momentum tensor and equation of state related?

The equation of state is a relationship between different components of the stress-energy tensor; it allows you to simplify the equations by not having to separately solve for every component.

What Carroll is saying is that the energy-momentum tensor for highly relativistic particles has (at least to a good enough approximation) the same relationship between components as the energy-momentum tensor of radiation. So you can use the same solutions of the relevant equations for both.

 

[binbagsss]

The equation of state is a relationship between different components of the stress-energy tensor;

In what way exactly?

 

[PeterDonis]

In what way exactly?

The stress-energy tensor of a perfect fluid is ##T_{ab} = \left( \rho + p \right) u_a u_b + p g_{ab}##, where ##u## is the 4-velocity of the fluid and ##g## is the metric. Basically this says that the SET is a 4 x 4 matrix which, in the rest frame of the fluid, is diagonal, with elements ##\left( \rho, p, p, p \right)##. Here ##\rho## is the energy density and ##p## is the pressure (as measured in the rest frame of the fluid).

The equation of state is a relationship between ##\rho## and ##p##, so it reduces the number of free parameters in the above from two to one; with it, you can express everything in terms of one variable.