Conservation of Energy-Momentum Tensor

[ELESSAR TELKONT]

Homework Statement

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

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

Homework Equations

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

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?

 

[Jhero]

Response from Scientist:

Your attempt at the solution is almost correct. However, there are a few errors in your calculations.

First, in the step where you have written \partial_{0}\partial_{n}T^{0n} as \partial_{0}\partial_{n}T^{0n}+\partial_{0}\partial_{n}T^{0n}, this is incorrect. The correct expression is \partial_{0}\partial_{n}T^{0n}+\partial_{n}\partial_{0}T^{n0}.

Second, in the step where you have written \partial_{1}\partial_{0}T^{10} as \partial_{1}\partial_{0}T^{01}, this is also incorrect. The correct expression is \partial_{1}\partial_{0}T^{10}+\partial_{0}\partial_{1}T^{01}.

Finally, in the step where you have written \partial_{0}\partial_{n}T^{0n}=-\partial_{0}\partial_{n}T^{0n}, this is incorrect. The correct expression is \partial_{0}\partial_{n}T^{0n}+\partial_{n}\partial_{0}T^{n0}.

By correcting these errors, you should be able to arrive at the desired result of \partial_{0}^{2}T^{00}=\partial_{m}\partial_{n}T^{mn}. Keep in mind that when dealing with partial derivatives, the order in which they are taken matters. Make sure to double check your calculations and keep track of the indices carefully. Hope this helps!