Virial identity for the four-dimensional gravitational Hartree equation ($I^{\prime \prime }(t)=16E(u)$)

For a finite-variance solution in four dimensions,satisfies $I^{\prime \prime }(t)=16E(u)$. The identity uses the degree $-2$ homogeneity of the Newtonian kernel. Negative-energy finite-variance data consequently blow up in finite time by the convexity argument.