Is the Stress-Energy tensor invariant in all RFs? If not (which is highly probable) how does it change?

EDIT: does the Einstein equation help? Since (without $\Lambda$) $$ R_{\alpha\beta} -\frac{1}{2}Rg_{\alpha\beta}= kT_{\alpha\beta} $$

and since the metric depends on the coordinate system, is this the connection?

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    $\begingroup$ A tensor is by definition covariant. Stress-energy tensor of what? $\endgroup$
    – Qmechanic
    Commented Jul 27, 2019 at 11:46
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    $\begingroup$ A tensor is "invariant" but the components of a particular representation of it, in a particular coordinate system, depend on the coordinate system. Or to put it a different way, The "laws of physics" don't depend on what coordinate system you choose to describe them. $\endgroup$
    – alephzero
    Commented Jul 27, 2019 at 12:03
  • $\begingroup$ @Qmechanic I actually do not know of which object I would calculate the stress energy tensor, but in general a cosmological one, like a star either resting or moving $\endgroup$ Commented Jul 27, 2019 at 12:48
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    $\begingroup$ Are you asking how tensor components transform under coordinate transformations? $\endgroup$
    – Qmechanic
    Commented Jul 27, 2019 at 15:40
  • $\begingroup$ $R_{\alpha\beta}$, $g_{\alpha\beta}$, and $T_{\alpha\beta}$ all transform in exactly the same way, according to the transformation rule for components of a rank-2 covariant tensor. That is basically the point of tensor equations. If all terms transform similarly, the equation remains valid in all frames. But the fact that $T_{\alpha\beta}$ transforms like $g_{\alpha\beta}$ doesn’t mean that one is directly related to the other. $\endgroup$
    – G. Smith
    Commented Jul 27, 2019 at 16:23

1 Answer 1


The components of the stress-energy tensor are different in different reference frames. The components of any rank-2 covariant tensor transform according to the following rule:

$$T_{\alpha’\beta’}=\frac{\partial x^{\alpha}}{\partial x^{\alpha’}}\frac{\partial x^{\beta}}{\partial x^{\beta’}}T_{\alpha\beta}.$$


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