Basics of the Stress Energy Tensor

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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[itex]\nu[/itex],[itex]\nu[/itex]=0 to prove the following results for a bounded system (ie. a system for which Tμ[itex]\nu[/itex]=0 outside of a bounded region
a)
[itex]\frac{\partial}{\partial t}[/itex][itex]\int[/itex]T0[itex]\alpha[/itex]d3x=0


Homework Equations


T is a symmetric tensor so Tμ[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]T0[itex]\alpha[/itex]d3x
=[itex]\int[/itex][itex]\frac{\partial}{\partial t}[/itex]T0[itex]\alpha[/itex]d3x
=[itex]\int[/itex]T0[itex]\alpha[/itex],0d3x
and then I would say I use the identity given to say T0[itex]\alpha[/itex],0=0

In the solution manual though, Schutz says the identity gives us that
T0[itex]\alpha[/itex],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[itex]\alpha[/itex],0=0 from the identity Tμ[itex]\nu[/itex],[itex]\nu[/itex]=0 ?
 

Answers and Replies

  • #2
TSny
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Hello, Mason. Welcome to PF!

Does anybody know why I can't just say that T0[itex]\alpha[/itex],0=0 from the identity Tμ[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 Tμ[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
T0[itex]\alpha[/itex],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?
 

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