- #1

- 112

- 14

- Homework Statement
- During the derivation of the Einstein field equations via the variation principle we defined the energy momentum tensor to be

$$T_{\mu\nu} \equiv -2\frac{1}{\sqrt{-g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.$$

Assume now that

$$S_M = \int d^4x \sqrt{-g}(\nabla_\mu\phi\nabla^\mu\phi-\frac{1}{2}m^2\phi^2),$$

where ##\phi## is the scalar field. $Calculate ##T_{\mu\nu}##.

- Relevant Equations
- All given above

My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears

$$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$

Now we can apply the variation:

$$\begin{align*}

\delta S_M

&= \int d^4x (\delta\sqrt{-g}) (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2) + \int d^4x \sqrt{-g} \delta g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi\\

&= \int d^4x (-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu} )(g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2) + \int d^4x \sqrt{-g} \delta g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi\\

&= \int d^4x \sqrt{-g}\left(- \frac{1}{2}g_{\mu\nu} [g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2] + \nabla_\mu\phi\nabla_\nu\phi \right)\delta g^{\mu\nu}\\

&= \int d^4x \frac{\sqrt{-g}}{2}\left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right)\delta g^{\mu\nu}.

\end{align*}$$

We then have

$$\frac{\delta S_M}{\delta g^{\mu\nu}} = \frac{\sqrt{-g}}{2}\left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right) \quad \Rightarrow \quad T_{\mu\nu} = - \left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right)$$

Is this sensible? I'm not really sure if I performed the variation correctly, so I would appreciate if someone could give it a look.

$$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$

Now we can apply the variation:

$$\begin{align*}

\delta S_M

&= \int d^4x (\delta\sqrt{-g}) (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2) + \int d^4x \sqrt{-g} \delta g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi\\

&= \int d^4x (-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu} )(g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2) + \int d^4x \sqrt{-g} \delta g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi\\

&= \int d^4x \sqrt{-g}\left(- \frac{1}{2}g_{\mu\nu} [g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2] + \nabla_\mu\phi\nabla_\nu\phi \right)\delta g^{\mu\nu}\\

&= \int d^4x \frac{\sqrt{-g}}{2}\left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right)\delta g^{\mu\nu}.

\end{align*}$$

We then have

$$\frac{\delta S_M}{\delta g^{\mu\nu}} = \frac{\sqrt{-g}}{2}\left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right) \quad \Rightarrow \quad T_{\mu\nu} = - \left(\nabla_\mu\phi\nabla_\nu\phi+\frac{1}{2}g_{\mu\nu}m^2\phi^2 \right)$$

Is this sensible? I'm not really sure if I performed the variation correctly, so I would appreciate if someone could give it a look.