Trace of the stress energy tensor

[dsaun777]

The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?

 

[Nugatory]

The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?

The trace of any tensor is invariant under coordinate transformations.

 

[PeterDonis]

is it true that the trace of all of these matter situations is invariant?

As @Nugatory said, the trace of any tensor is invariant under coordinate transformations. However, the traces of the stress-energy tensors describing different kinds of stress-energy will not all be the same. For example, the trace of the stress-energy tensor describing electromagnetic radiation is zero; but the trace of the stress-energy tensor describing dust is nonzero. And the trace of the stress-energy tensor describing a perfect fluid is also nonzero, but different from that of dust.

 

[Orodruin]

And the trace of the stress-energy tensor describing a perfect fluid is also nonzero, but different from that of dust.

Dust is a particular case of a perfect fluid with $w=0$. A perfect fluid has the trace $\rho(1+3w)$ of the stress-energy tensor, which can a priori take any value unless one starts introducing additional requirements on the equation of state parameter $w$.

 

[dsaun777]

Dust is a particular case of a perfect fluid with $w=0$. A perfect fluid has the trace $\rho(1+3w)$ of the stress-energy tensor, which can a priori take any value unless one starts introducing additional requirements on the equation of state parameter $w$.

And w being pressure/ density?

 

[Orodruin]

And w being pressure/ density?

Yes, in the rest frame of the fluid.

 

[PeterDonis]

Dust is a particular case of a perfect fluid with $w=0$.

Yes, I should have been clearer, by "perfect fluid" I meant "perfect fluid with nonzero pressure", which would correspond to $w \neq 0$.