What is the value of the variation stress energy tensor?

Question

If we are living in a portion of space-time where the metric is very close to flat space and we know that the stress energy tensor is negligible at this portion of space-time is it ok to assume that $frac{delta T_{munu}}{delta g^{munu}}$ is also negligable?

CR Drost · Answer

No, not in general. Things are only negligible for certain purposes. For example, a field that obeys the Klein-Gordon equation has the stress-energy tensor:$$T^{munu} = frac{hbar^2}{m} left (g^{mu alpha} g^{nu beta} + g^{mu beta} g^{nu alpha} - g^{munu} g^{alpha beta} right ) partial_{alpha}barpsi partial_{beta}psi - g^{munu} m c^2 barpsi psi$$
The derivative of this thing contains several important terms like $m c^2 bar psi psi$ which are very important in certain contexts.

What is the value of the variation stress energy tensor?

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