(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the identity [tex]T^{\mu \nu}_{ ,\nu} = 0[/tex] to prove the following results for a bounded system (ie a system for which [tex]T^{\mu \nu} = 0[/tex]

outside a bounded region of space),

[tex]\frac{\partial}{\partial t}\int T^{0\alpha}d^{3}x = 0[/tex]

2. Relevant equations

3. The attempt at a solution

The integral obviously gives 4 equations (one for each [tex]\alpha[/tex]) which must all be = 0.

I tried just working on the first and passing the partial derivative into the integral and writing [tex]\frac{\partial}{\partial t} T^{0 0} = -\left(\frac{\partial}{\partial x} T^{0 1} +\frac{\partial}{\partial y} T^{0 2} + \frac{\partial}{\partial z} T^{0 3} \right)[/tex]

This gives

[tex]- \int T^{0 i}_{,i}d^{3}x[/tex]

which doeesnt really seem to be getting me anywhere - also the same reasoning wouldn't work for the other equations because they must all be partially differentiated wrt t and this is only relevant for [tex]T^{0 0}[/tex] in [tex]T^{\mu \nu}_{ ,\nu} = 0[/tex]

Another direction I thought of was to use Gauss' law but then there is no outward normal one-form and so maybe not....

There are another two parts to this question but I thought that if I had an idea of how to do the first part I could figure the others out by myself.

Thanks for any replies :)

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# Homework Help: Energy-stress tensor integration proof (from schutz ch.4)

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