 Stress-energy quadrupole identity

For a symmetric stress-energy tensor with compact support and ${\partial }_b{T}^{ab}=0$ in Minkowski spacetime, let $I^{ij}=\int T^{00}{x}^i{x}^j{d}^{3}x$. Two integrations by parts give ${\dot{I}}^{ij}=\int (T^{0i}{x}^j+T^{0j}{x}^i)d^{3}x$ and the displayed identity. Compact support removes the boundary fluxes. The support should stay bounded over the time interval used when differentiating the integrals.