I've been reading about scalar fields in the context of general relativity, and I found this page: https://en.wikipedia.org/wiki/Stress-energy_tensor#Scalar_field. It says that the stress-energy tensor for a scalar field is

$$ T^{ \mu \nu} = \frac {\hbar^2}{m} (g^{ \mu \alpha}g^{ \nu \beta} + g^{ \mu \beta}g^{ \nu \alpha} - g^{ \mu \nu}g^{ \alpha \beta}) \partial_\alpha\overline{\phi} \partial_\beta\phi - g^{ \mu \nu}mc^2\overline\phi\phi$$

Now, I don't know how accurate this is, because I haven't been able to find decent references or another source. However, assuming that this equation is true, is there an analogous equation for the stress-energy tensor of an arbitrary vector field?


I forgot to mention that I know there is a stress-energy tensor for the electromagnetic field; I wondered if this held for any other fields.

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    $\begingroup$ The expression quoted is not about an arbitrary scalar field, it is about the particular Klein-Gordon scalar field (mass, no interaction). For the arbitrary field, you should take its Lagrangian and take variation with respect to metric. No one can give you universal formula since Lagrangians can be different (for examples, see Maxwell, Proca, gauge bosons). $\endgroup$ – firtree Aug 24 '14 at 19:35
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    $\begingroup$ @firtree minor detail: the expression quoted is for a massive and complex free scalar field, hence the $m$ and the $\bar{\phi}$ quantities. But that aside, you are correct. $\endgroup$ – Alex Nelson Aug 26 '14 at 21:50
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    $\begingroup$ @HDE226868 The very next section of the Wikipedia article, that is, Hilbert stress–energy tensor, gives a recipe to calculate the SET for any field with given field Lagrangian. The Hilbert SET is the one used in GR usually. If you wonder whether some SET exists at all for an arbitrary field, yes it does, see Noether's theorem. $\endgroup$ – firtree Aug 27 '14 at 0:31

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