- #1

Diracobama2181

- 74

- 2

- TL;DR Summary
- What would be the energy-momentum tensor for lowered indices?

I know the tensor can be written as $$T^{\mu v}=\Pi^{\mu}\partial^v-g^{\mu v}\mathcal{L}$$ where $$g^{\mu v}$$ is the metric and $$\mathcal{L}$$ is the Lagrangian density, but how would I write $$T_{\mu v}$$? Would it simply be $$T_{\mu v}=g_{\mu \rho}g_{v p}T^{\rho p}$$? And if so, is there a way to simplify $$g_{\mu \rho}g_{v p}T^{\rho p}=g_{\mu \rho}g_{v p}(\Pi^{\rho}\partial^p-g^{\rho p}\mathcal{L})$$?