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Energy - Momentum tensor identity

  1. Aug 10, 2011 #1

    WannabeNewton

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    1. The problem statement, all variables and given/known data
    Show that [tex]\frac{1}{2}\frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k}dV = \int_{V}T^{jk}dV [/tex].


    2. Relevant equations
    3. The attempt at a solution
    [itex]\partial _{t}T^{t\nu } = -\partial _{i}T^{i\nu }[/itex] from conservation of energy - momentum. [itex]\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial_{t}\partial_{t}T^{tt})x^{j}x^{k} [/itex] since the x's are fixed coordinates of their respective volume element inside the source. So using the equality from conservation of energy - momentum I get [itex]\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial _{i}\partial _{m}T^{im})x^{j}x^{k}[/itex] and by using the product rule on [itex]\partial _{i}\partial _{m}(T^{im}x^{j}x^{k})[/itex] to solve for the right hand side of the previous equation I get [tex]\partial_{t} \partial_{t}(T^{tt}x^{j}x^{k}) = \partial _{i}\partial _{m}(T^{im}x^{j}x^{k}) - 2\partial _{i}(T^{ij}x^{k} + T^{ik}x^{j}) + 2T^{jk}[/tex] and this is where I am stuck. I don't know if what I am doing after this is exactly correct. For instance, [tex]-2\int_{V}\partial _{i}(T^{ij}x^{k})dV = -2\int_{\partial V}(T^{ij}x^{k})dS_{i} = 0[/tex] as per Stoke's Theorem and because [itex]T^{ij}[/itex] has to vanish at the boundary of the source so that the pressure differs smoothly from the source to the outside but I don't think I applied Stoke's Theorem correctly here. I did the same with the [itex]T^{ik}[/itex] also in the parentheses and for the first expression I did [tex]\int_{V}\partial_{m} \partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{V}\partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{\partial V}(T^{im}x^{j}x^{k})dS_{i} = 0[/tex] for the same reason as before so that [itex]\int_{V}\partial _{t}\partial _{t}(T^{tt}x^{j}x^{k})dV = \frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k} = 2\int_{V}T^{jk}dV[/itex]. Could anyone tell me where and how I used Stoke's Theorem wrongly here and how I am supposed to correctly use it in the context of this problem? Thanks in advance.
     
    Last edited: Aug 10, 2011
  2. jcsd
  3. Aug 10, 2011 #2
    try using the fact that

    [tex] \int d^3 x T^{ij} = \int d^3 x [ \partial_k (T^{ik} x^j ) - ( \partial_k T^{ik} ) x^j ] [/tex]
     
  4. Aug 10, 2011 #3

    WannabeNewton

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    I tried that but I couldn't find any to use it to get to the answer? Could you show me how you would use it for one of the expressions or something along the lines? Thanks for the reply.
     
  5. Aug 10, 2011 #4
    I have attached the section of my cambridge notes covering this formula
     

    Attached Files:

  6. Aug 10, 2011 #5

    WannabeNewton

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    Oh thank you very much. I didn't even think of integration by parts lol. Talk about a much simpler way of getting rid of surface terms.
     
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