The ADM-source mass and angular-momentum parameters

In the preceding end expansion, $M_0>0$ is the coefficient called ADM mass in the Radu source, $S=(S^1,S^2,S^3)$ is its ADM angular momentum vector, and $J=\mathrm{\mid }S\mathrm{\mid }=\sqrt{{\sum }_i(S^i{)}^2}$ is the nonnegative magnitude, not a signed component and not the Kerr parameter $a$. These are the source's fixed charge and orientation conventions, as specified by its asymptotic chart and the leading $g_{0i}$ coefficient. No equality with a newly introduced alternative surface-integral convention is asserted or needed here.

The subextremal parameter condition for the quantitative theorem is $J<M_0^2$. In the mass-one normalization used by that theorem, the normalized mass is exactly $1$ and its angular magnitude is the dimensionless $J\mathrm{/}{M}_0^2$, thereafter denoted $J$ by the source. The quantitative theorem must be applied to the rescaled mass-one data and their actual geometric constant; numerical smallness of an unnormalized dimensionful angular momentum is not its condition.

The finite vector $S$ determines its Euclidean norm uniquely, and positivity of $M_0$ makes $J\mathrm{/}{M}_0^2$ defined. Dimensional rescaling of the stationary end uses $t^{\mathrm{\prime }}=t\mathrm{/}{M}_0$, $x^{\mathrm{\prime }}=x\mathrm{/}{M}_0$ and $g^{\mathrm{\prime }}=M_0^{-2}g$ expressed in these coordinates, with normalized stationary field $T^{\mathrm{\prime }}=M_{0}T={\mathrm{\partial }}_{t^{\mathrm{\prime }}}$. The leading temporal coefficient becomes $-1+2\mathrm{/}{r}^{\mathrm{\prime }}$ and the leading mixed coefficient has vector $S\mathrm{/}{M}_0^2$, by substituting $x=M_0{x}^{\mathrm{\prime }}$. Thus the stated mass-one labels match the source normalization. This checks the parameter bookkeeping only; bounds for a geometric constant must still be checked on those rescaled data, not transported without calculation.