Why is the stress-energy tensor defined with those components?

[Proof.Beh]

Hi,

Why we define the matter tensor to this form '

' on general relativity?
Is it a physical supposal only? Or we can explain a mathematical proof for it?

Thanks.
-----------
Mr Beh

 

[pervect]

I think this expression ignores the pressure terms, so it's not quite right as written.

The Lagrangian density representing matter, though, is just a scalar field, $-\rho_0$ according to my recent readings. As to why a scalar field is used for matter. I think that's because that's a convenient approximation. We model matter by some distribution of energy - when you stretch matter, doing work, this energy is stored somewhere. Rather than obsess about exactly where and in what form this energy is stored, the usual approximation is to say that this energy is uniformly distributed - this gives a scalar field approximation.

GR also allows one to use stress-energy tensors for other sorts of fields (Dirac fields, and spin-1 fields for example). The ultimate point is that any sort of Lagrangian field is going to have a stress-energy tensor, so any sort of Lagrangian field theory is going to be representable in Einstein's equation by it's associated stress-energy tensor.

 

[Proof.Beh]

Thanks for your beautiful answer,

But note that these approximations (according to you) that we see at relativity and others, such as field theory of Nordstrom, for probability of QM are valid and we must try to solve the their problemes instead of extension QM (& string theory).

All the best,
Mr Beh