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JD_PM

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- Homework Statement
- Prove that the stress-energy tensor is given by the functional derivative of the action with respect to ##\delta g^{\mu \nu}##

$$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$

- Relevant Equations
- $$ \mathcal{L} = (\partial_{\mu} \alpha) h^{\mu} (\phi) $$

$$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$

Prove that the stress-energy tensor is given by the functional derivative of the action with respect to ##\delta g^{\mu \nu}##

$$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$

Tong suggests (around 25:30) we could get the desired tensor by performing a transformation ##\delta \phi = \alpha \phi##, where ##\alpha## is not a constant; ##\alpha := \alpha(x)##.

Thus the action is

$$ \mathcal{L} = (\partial_{\mu} \alpha) h^{\mu} (\phi) $$

Then the change of the action is

$$\delta S = \int d^4 x \delta L = -\int d^4 x \alpha (x) \partial_{\mu} h^{\mu}$$

But I do not really know how to proceed from here. Could you please give me a hint?Any help is appreciated.

Thank you.

$$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$

Tong suggests (around 25:30) we could get the desired tensor by performing a transformation ##\delta \phi = \alpha \phi##, where ##\alpha## is not a constant; ##\alpha := \alpha(x)##.

Thus the action is

**not**invariant and we get$$ \mathcal{L} = (\partial_{\mu} \alpha) h^{\mu} (\phi) $$

Then the change of the action is

$$\delta S = \int d^4 x \delta L = -\int d^4 x \alpha (x) \partial_{\mu} h^{\mu}$$

But I do not really know how to proceed from here. Could you please give me a hint?Any help is appreciated.

Thank you.