How is the stress energy tensor obtained? In most textbooks, it's simply stated as

$$T^\mu{}_\nu=(\rho+P)U^\mu U_\nu-P\delta^\mu{}_\nu$$

I can see why this makes sense for a comoving observer at rest wrt. the perfect fluid. But I don't understand how the general case is arrived at. Would someone kindly explain?


The answer to this depends on what you're starting from. If you know the Einstein tensor, then you can find the stress-energy tensor from the Einstein field equations. If you know the Lagrangian density, then you can find the stress-energy tensor by variation with respect to the metric. If you know the rate at which energy-momentum is being transported along four orthogonal axes, then that corresponds to the stress-energy tensor.

  • $\begingroup$ Thank you, Ben, would you mind telling me what the Lagrangian density is? And/or notes on how I might do the variation wrt the metric? Also, are there any links about the transport along 4 orthogonal axes -- I am not at all familiar with this approach? Thanks again! $\endgroup$ – Alex Oct 20 '13 at 10:02
  • $\begingroup$ @Alex: For transport of momentum, see the WP article, en.wikipedia.org/wiki/Stress-energy_tensor . I don't have a good reference handy for the variation with respect to the metric. $\endgroup$ – user4552 Oct 20 '13 at 14:29

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