# Energy-Momentum Tensor Question

Diracobama2181
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 $$g_{\mu \rho}g_{v p}T^{\rho p}=g_{\mu \rho}g_{v p}(\Pi^{\rho}\partial^p-g^{\rho p}\mathcal{L})$$?

Gold Member
Due to my poor study symmetry of the first term, e.g.
$$\Pi^\mu \partial^\nu = \Pi^\nu \partial^\mu$$
was not clear to me. What are ##\Pi,\partial## ?

Diracobama2181
Due to my poor study symmetry of the first term, e.g.
$$\Pi^\mu \partial^\nu = \Pi^\nu \partial^\mu$$
was not clear to me. What are ##\Pi,\partial## ?
Apologies, that should be $$\Pi^{\mu}\partial^{v}\phi$$ where $$\phi$$ is a field and $$\Pi^{\mu}=\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)}$$. Originally had this posted to the High Energy subforum since this was originally a quantum field theory question, but someone moved it here.

anuttarasammyak