# Energy - Momentum tensor identity

## Homework Statement

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

## 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.

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try using the fact that

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