Suppose you are given the Lagrangian of a scalar field [tex]\Phi(t)[/tex](adsbygoogle = window.adsbygoogle || []).push({});

[tex] \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ).[/tex]

By introducing covariant notation with [tex]\eta_{\mu \nu} = (1,-1,-1,-1)[/tex] this reads as

[tex]\mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi- V(\Phi ).[/tex]

Let us now switch to GR, i.e. $\eta^{\mu \nu} \rightarrow [tex]g^{\mu \nu}[/tex]. The Lagrangian remains the same since covariant derivative of a scalar field is the same as normal derivative, i.e. [tex]\nabla_{\mu} \Phi = \partial_{\mu} \Phi[/tex]. I derive the energy-momentum-Tensor [tex]T^{\mu \nu}[/tex] by varying [tex]g_{\mu\nu}[/tex] in the action

[tex]S = \int \mathcal{L} \sqrt{-g}\; dx^4,[/tex]

i.e.

[tex]\delta S = \frac{1}{2}\int T^{\mu \nu} \delta g_{\mu\nu} \sqrt{-g}\; dx^4.[/tex]

So we obtain with the variation [tex]\delta g_{\mu\nu}[/tex]

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Energy momentum tensor of a scalar field by varying the metric

Loading...

Similar Threads - Energy momentum tensor | Date |
---|---|

A Dark Matter and the Energy-Momentum Tensor | Mar 12, 2018 |

I How do we get invariant curvature from momentum and energy? | Dec 9, 2017 |

I Energy-momentum tensor | Apr 23, 2017 |

I Energy-Momentum Tensor from the Action Principle | Feb 17, 2017 |

I Stress as momentum flux | Feb 14, 2017 |

**Physics Forums - The Fusion of Science and Community**