I Energy-momentum tensor as energy density

[Ranku]

Can the energy-momentum tensor of matter and energy be cast in terms of energy density of matter and energy, similar to how the energy-momentum tensor of vacuum energy can be cast in terms of the energy density of vacuum energy?

 

[Orodruin]

No. The energy density is just one of the components of the energy-monentum tensor. A general ideal fluid can be described by assigning a rest frame and an energy density and pressure in that frame.

 

[Ibix]

Are you asking if you can write $T_{\mu\nu}=\rho g_{\mu\nu}$ similar to the way the cosmological constant enters the field equations as $\Lambda g_{\mu\nu}$? No, not in general, since you wouldn't be able to describe any mechanical properties of the matter except its energy density.

 

[Ranku]

Are you asking if you can write $T_{\mu\nu}=\rho g_{\mu\nu}$ similar to the way the cosmological constant enters the field equations as $\Lambda g_{\mu\nu}$? No, not in general, since you wouldn't be able to describe any mechanical properties of the matter except its energy density.

Mechanical property of matter as in pressure of matter? If so, then over cosmological scale, where pressure can be ignored, can we have $T_{\mu\nu}$~$\rho g_{\mu\nu}$?

 

[Ibix]

No. The stress-energy tensor of a pressureless fluid has, in coordinates where the fluid is at rest, only one non-zero component. That can't be proportional to the metric.

 

[Orodruin]

More specifically, the stress-energy tensor of an ideal fluid is given by $T^{\mu\nu} = (\rho + p) U^\mu U^\nu - p g^{\mu\nu}$ (in +—- convention), where $\rho$ is the energy density, $p$ the pressure, and $U$ the 4-velocity of the fluid’s rest frame. Setting the pressure to zero would result in $T^{\mu\nu} = \rho U^\mu U^\nu$.

Note that the pressure is generally not negligible in cosmology. This is only the case for a matter gas at low temperature.

 

[kimbyd]

Note that the pressure is generally not negligible in cosmology. This is only the case for a matter gas at low temperature.

And density. The way I like to think of it is that normal matter doesn't contribute to any pressure between galaxies. It did experience pressure in the very early universe, but there hasn't been any pressure from normal matter on cosmological scales for a very long time.