Angular momentum at the black hole threshold

Near the black hole threshold in phase space, the black hole mass as a function of the initial data shows the “critical scaling” $M\simeq C(p-p_{*}{)}^{\gamma }$, where $p$ labels a family of initial data, $p_{*}$ is the value of $p$ at the threshold, and the critical exponent $\gamma$ is universal for a given matter model. The black hole charge $Q$ obeys a similar law. To complete the picture, we include the angular momentum as a perturbation. For the black hole angular momentum $\overrightarrow{L},$ we find the oscillating behavior $\overrightarrow{L}\simeq \mathrm{Re}\left[\right(\overrightarrow{A}+i\overrightarrow{B}\left)\right(p-p_{*}{)}^{\mu +i\omega }]$. The assumptions of the calculation hold for $p=\rho /3\mathrm{}$ perfect fluid matter, and we calculate $\mu \simeq 0.799$ and $\omega \simeq 0.231$.