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Advanced Physics Homework Help
Calculating Energy-Momentum Tensor in GR
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[QUOTE="Markus Kahn, post: 6287814, member: 654025"] [B]Homework Statement:[/B] 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}##. [B]Relevant Equations:[/B] 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. [/QUOTE]
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Calculating Energy-Momentum Tensor in GR
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