I know that the components of stress energy tensor are: energy density, energy flux, momentum density and momentum flux.

But can I explicitly calculate the form of stress energy tensor in any complicated situation?

Or explicit form is known only for few systems? How do I do it?


First, I suppose here that we are not considering here gravitational fields. (the explanation will be given below). So, we are considering matter fields, radiation fields, etc..

You have first to write a Lagrangian density $\mathcal L_{NG}$ for your problem (here NG stands for non-gravitational). For instance, the electromagnetic field has the Lagrangian density $\mathcal L_{NG} = -\frac{1}{4}g^{\mu\rho} g^{\nu\lambda} F_{\mu\nu} F_{\rho\lambda}$

Then, you may obtain the Hilbert stress-energy tensor :

$$(T_{\mu\nu})_{NG} = \dfrac{-2}{\sqrt{-g}} \dfrac{\partial (\sqrt{-g}\mathcal L_{NG}) }{\partial g^{\mu\nu}} \tag{1}$$

The formula $(1)$ may be justifyed by considering, that, ultimately, the stress-energy tensor for non-gravitational fields, $(T_{\mu\nu})_{NG}$, is the source of gravitation ($g_{\mu\nu}$), so, even, if, in a problem, you are not considering gravitational couplings, you may write a Lagrangian density with an explicit metrics $g^{\mu\nu}$ , and then calculate the Hilbert stress energy-tensor. One avantage is that this stress-energy tensor is automatically symmetric.

Note that, then, the formula $(1)$ does not appy to the lagrangian density of the gravitational fields itself


Because the Hamiltonian – the total energy – determines the evolution of all dynamical degrees of freedom, via the Hamilton equations, and because all the other components of the stress-energy tensor are related to the energy density by Lorentz transformations, it follows that knowing the precise formula for the stress-energy tensor is equivalent to knowing everything about the laws of physics that govern the evolution of the given physical system in time.

If you know what the basic degrees of freedom (time-dependent variables) are and how they evolve, you know how to write the stress-energy tensor. If you don't know one, you don't know the other.

Physics studies numerous theories and they're sometimes rather complex – and each of them has a stress-energy tensor. Some of these theories are "fundamental", some of them are just "approximate" or "effective", and so on. Covering all of them and the form of their stress-energy tensors would be pretty much equivalent to explaining all of physics.


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