# Basics of the Stress Energy Tensor

## Homework Statement

Hello I'm trying to self study A First Course in General Relativity (2E) by Schutz and I've come across a problem that I need some advice on.

Here it is:
Use the identity Tμ$\nu$,$\nu$=0 to prove the following results for a bounded system (ie. a system for which Tμ$\nu$=0 outside of a bounded region
a)
$\frac{\partial}{\partial t}$$\int$T0$\alpha$d3x=0

## Homework Equations

T is a symmetric tensor so Tμ$\nu$=T$\nu μ$

## The Attempt at a Solution

The Integral is over spatial variables so I brought the integral inside making
$\frac{\partial}{\partial t}$$\int$T0$\alpha$d3x
=$\int$$\frac{\partial}{\partial t}$T0$\alpha$d3x
=$\int$T0$\alpha$,0d3x
and then I would say I use the identity given to say T0$\alpha$,0=0

In the solution manual though, Schutz says the identity gives us that
T0$\alpha$,0=-Tj0,j for a reason that completely eludes me and then used gauss' law to convert it to a surface integral, then said that since the region of integration is unbounded the integral can be taken anywhere (ie outside of the bounded region where T=0).

Does anybody know why I can't just say that T0$\alpha$,0=0 from the identity Tμ$\nu$,$\nu$=0 ?

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TSny
Homework Helper
Gold Member
Hello, Mason. Welcome to PF!

Does anybody know why I can't just say that T0$\alpha$,0=0 from the identity Tμ$\nu$,$\nu$=0 ?
Keep in mind the Einstein summation convention: a repeated index denotes summation over that index. So, the left hand side of Tμ$\nu$,$\nu$=0 is actually the sum of 4 terms.

In the solution manual though, Schutz says the identity gives us that
T0$\alpha$,0=-Tj0,j
Note that this equation can't be correct. There is a lone ##\alpha## index on the left, but no ##\alpha## index on the right. Did you copy this equation correctly?