- #1

LCSphysicist

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- Homework Statement
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- Relevant Equations
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I want to get the stress energy tensor of a scalar field using the Hilbert method (namely, ##T^{\mu v} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu v}}##)

$$S = \int \frac{1}{2}(\partial_\mu \phi \partial^{\mu} \phi - m^2 \phi ^2)\sqrt{-g}d^4x$$

$$= \int \frac{1}{2}(\partial^{v} \phi \partial^{\mu} \phi g_{v \mu} - m^2 \phi ^2)\sqrt{-g}d^4x$$

$$\delta S / \delta g_{a b} =$$

$$ \int \frac{1}{2}(\partial^{a} \phi \partial^{b} \phi )\sqrt{-g} d^4x + \int \frac{1}{2}(\partial^{v} \phi \partial^{\mu} \phi g_{v \mu} - m^2 \phi ^2)\frac{\sqrt{-g} g^{a b}}{2}d^4x$$

Where i have used ##\delta \sqrt{-g} = \sqrt{-g} g^{x y} \delta g_{x y} / 2##

$$T^{a b} = \frac{1}{2}(\partial^{a} \phi \partial^{b} \phi) + \frac{1}{2}(\partial^{v} \phi g_{v \mu} \partial^{\mu} \phi- m^2 \phi ^2) g^{a b}$$

This is not what i was expecting...

$$S = \int \frac{1}{2}(\partial_\mu \phi \partial^{\mu} \phi - m^2 \phi ^2)\sqrt{-g}d^4x$$

$$= \int \frac{1}{2}(\partial^{v} \phi \partial^{\mu} \phi g_{v \mu} - m^2 \phi ^2)\sqrt{-g}d^4x$$

$$\delta S / \delta g_{a b} =$$

$$ \int \frac{1}{2}(\partial^{a} \phi \partial^{b} \phi )\sqrt{-g} d^4x + \int \frac{1}{2}(\partial^{v} \phi \partial^{\mu} \phi g_{v \mu} - m^2 \phi ^2)\frac{\sqrt{-g} g^{a b}}{2}d^4x$$

Where i have used ##\delta \sqrt{-g} = \sqrt{-g} g^{x y} \delta g_{x y} / 2##

$$T^{a b} = \frac{1}{2}(\partial^{a} \phi \partial^{b} \phi) + \frac{1}{2}(\partial^{v} \phi g_{v \mu} \partial^{\mu} \phi- m^2 \phi ^2) g^{a b}$$

This is not what i was expecting...