# Conservation of Energy-Momentum Tensor

1. Apr 13, 2009

### ELESSAR TELKONT

1. The problem statement, all variables and given/known data

1) Use conservation of Energy-Momentum Tensor to show that

$$\partial_{0}^{2}T^{00}=\partial_{m}\partial_{n}T^{mn}$$

2. Relevant equations

$$\partial_{\nu}T^{\mu\nu}=0$$

3. The attempt at a solution

$$\partial_{\nu}T^{\mu\nu}=0$$

$$\partial_{\mu}\partial_{\nu}T^{\mu\nu}=\partial_{\mu}0=0$$

$$\partial_{0}\partial_{\nu}T^{0\nu}+\partial_{1}\partial_{\nu}T^{1\nu}+\partial_{2}\partial_{\nu}T^{2\nu}+\partial_{3}\partial_{\nu}T^{3\nu}=0$$

$$\partial_{0}\partial_{0}T^{00}+\partial_{0}\partial_{n}T^{0n}+\partial_{1}\partial_{\nu}T^{1\nu}+\partial_{2}\partial_{\nu}T^{2\nu}+\partial_{3}\partial_{\nu}T^{3\nu}=0$$

$$\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{\nu}T^{1\nu}-\partial_{2}\partial_{\nu}T^{2\nu}-\partial_{3}\partial_{\nu}T^{3\nu}$$

$$\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{0}T^{10}-\partial_{2}\partial_{0}T^{20}-\partial_{3}\partial_{0}T^{30}-\partial_{1}\partial_{n}T^{1n}-\partial_{2}\partial_{n}T^{2n}-\partial_{3}\partial_{n}T^{3n}$$

$$\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{1}\partial_{0}T^{01}-\partial_{2}\partial_{0}T^{02}-\partial_{3}\partial_{0}T^{03}-\partial_{m}\partial_{n}T^{mn}$$

$$\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{0}\partial_{1}T^{01}-\partial_{0}\partial_{2}T^{02}-\partial_{0}\partial_{3}T^{03}-\partial_{m}\partial_{n}T^{mn}$$

$$\partial_{0}^{2}T^{00}=-\partial_{0}\partial_{n}T^{0n}-\partial_{0}\partial_{n}T^{0n}-\partial_{m}\partial_{n}T^{mn}$$

$$\partial_{0}^{2}T^{00}=-2\partial_{0}\partial_{n}T^{0n}-\partial_{m}\partial_{n}T^{mn}$$

This result have an extra term and a negative sign respect the disired result. What am I doing wrong?