Stress-energy quadrupole identity
= Stress-energy quadrupole identity
{title2=$\ddot I^{ij}=2\int T^{ij}\,d^3x$}
For a symmetric <stress-energy tensor> with <compact support> and $\partial_bT^{ab}=0$ in <Minkowski spacetime>, let $I^{ij}=\int T^{00}x^ix^j\,d^3x$. Two integrations by parts give $\dot I^{ij}=\int(T^{0i}x^j+T^{0j}x^i)\,d^3x$ and the displayed identity. Compact support removes the boundary fluxes. The support should stay bounded over the time interval used when differentiating the integrals.