Why Does the Stress-Energy Tensor Conservation Lead to a Surface Integral?

[mason]

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 T^{μ[itex]\nu[/itex]}_{,[itex]\nu[/itex]}=0 to prove the following results for a bounded system (ie. a system for which T^{μ[itex]\nu[/itex]}=0 outside of a bounded region
a)
[itex]\frac{\partial}{\partial t}[/itex][itex]\int[/itex]T^{0[itex]\alpha[/itex]}d³x=0

Homework Equations

T is a symmetric tensor so T^{μ[itex]\nu[/itex]}=T^{[itex]\nu μ[/itex]}

The Attempt at a Solution

The Integral is over spatial variables so I brought the integral inside making
[itex]\frac{\partial}{\partial t}[/itex][itex]\int[/itex]T^{0[itex]\alpha[/itex]}d³x
=[itex]\int[/itex][itex]\frac{\partial}{\partial t}[/itex]T^{0[itex]\alpha[/itex]}d³x
=[itex]\int[/itex]T^{0[itex]\alpha[/itex]}_,0d³x
and then I would say I use the identity given to say T^{0[itex]\alpha[/itex]}_,0=0

In the solution manual though, Schutz says the identity gives us that
T^{0[itex]\alpha[/itex]}_,0=-T^j0_,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 T^{0[itex]\alpha[/itex]}_,0=0 from the identity T^{μ[itex]\nu[/itex]}_{,[itex]\nu[/itex]}=0 ?

 

[TSny]

Hello, Mason. Welcome to PF!

Does anybody know why I can't just say that T^{0[itex]\alpha[/itex]}_,0=0 from the identity T^{μ[itex]\nu[/itex]}_{,[itex]\nu[/itex]}=0 ?

Keep in mind the Einstein summation convention: a repeated index denotes summation over that index. So, the left hand side of T^{μ[itex]\nu[/itex]}_{,[itex]\nu[/itex]}=0 is actually the sum of 4 terms.

In the solution manual though, Schutz says the identity gives us that
T^{0[itex]\alpha[/itex]}_,0=-T^j0_,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?