How to Prove the Energy-Momentum Tensor Identity?

[WannabeNewton]

Homework Statement

Show that $$\frac{1}{2}\frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k}dV = \int_{V}T^{jk}dV$$.

Homework Equations

The Attempt at a Solution

$\partial _{t}T^{t\nu } = -\partial _{i}T^{i\nu }$ from conservation of energy - momentum. $\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial_{t}\partial_{t}T^{tt})x^{j}x^{k}$ 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 $\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial _{i}\partial _{m}T^{im})x^{j}x^{k}$ and by using the product rule on $\partial _{i}\partial _{m}(T^{im}x^{j}x^{k})$ to solve for the right hand side of the previous equation I get $$\partial_{t} \partial_{t}(T^{tt}x^{j}x^{k}) = \partial _{i}\partial _{m}(T^{im}x^{j}x^{k}) - 2\partial _{i}(T^{ij}x^{k} + T^{ik}x^{j}) + 2T^{jk}$$ and this is where I am stuck. I don't know if what I am doing after this is exactly correct. For instance, $$-2\int_{V}\partial _{i}(T^{ij}x^{k})dV = -2\int_{\partial V}(T^{ij}x^{k})dS_{i} = 0$$ as per Stoke's Theorem and because $T^{ij}$ 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 $T^{ik}$ also in the parentheses and for the first expression I did $$\int_{V}\partial_{m} \partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{V}\partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{\partial V}(T^{im}x^{j}x^{k})dS_{i} = 0$$ for the same reason as before so that $\int_{V}\partial _{t}\partial _{t}(T^{tt}x^{j}x^{k})dV = \frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k} = 2\int_{V}T^{jk}dV$. 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.

 

[sgd37]

try using the fact that

$$\int d^3 x T^{ij} = \int d^3 x [ \partial_k (T^{ik} x^j ) - ( \partial_k T^{ik} ) x^j ]$$

 

[WannabeNewton]

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.

 

[sgd37]

I have attached the section of my cambridge notes covering this formula

 

[WannabeNewton]

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.