Kerr geometric parameters and SI unit restoration

In the SI coordinate convention $x^0=c{t}_{\mathrm{S}\mathrm{I}}$, the Kerr metric's coordinate $t$ is the length coordinate $x^0$ (not the time in seconds), and $r$ is a length; $\theta ,\varphi$ are dimensionless. The metric parameters $M$ and $a$ in the Kerr exterior definition are geometric lengths: the associated physical mass is $m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=c^{2}M\mathrm{/}{G}_N$, and the physical angular momentum is $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=c^{3}Ma\mathrm{/}{G}_N$. Thus $a=J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}c)$ is the specific angular-momentum parameter. The horizon parameter ${\mathrm{\Omega }}_H=a\mathrm{/}(r_{+}^2+a^2)$ has units of inverse length; the physical angular frequency is $c{\mathrm{\Omega }}_H$ in inverse seconds.

For the local Hamilton-Jacobi solutions on an axis-avoiding coordinate rectangle, adopt the following action normalization. Let $S_{\mathrm{S}\mathrm{I}}:U\to \mathbb{R}$ be a $C^1$ reduced action with units $\mathrm{J}\mathrm{s}=\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, and define $S:=(G_N\mathrm{/}{c}^3)S_{\mathrm{S}\mathrm{I}}$. Thus $[S]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$. The momentum covector is $p=dS$; on the separated branches $S=-Et+L\varphi +S_r(r)+S_{\theta }(\theta )$, define $E:=-p_t$ and $L:=p_{\varphi }$. The real constant $\mu$ is the normalization parameter in $g^{ab}{p}_a{p}_b=-{\mu }^2$, and the angular separation constant is $Q:=p_{\theta }^2+{\cos }^2\theta [a^2({\mu }^2-E^2)+\frac{L^2}{{\sin }^2\theta }]\mathrm{.}$ These are the constants of the local separation theorem, with its domain and radicand restrictions. Since $t,r$ have length units and the angles are dimensionless, $[p_t]=[p_r]=\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$ and $[p_{\theta }]=[p_{\varphi }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$. The inverse-metric contraction has units ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$, so this convention assigns $[\mu ]=[E]=\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$, $[L]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$, and $[Q]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^4$. These assignments apply also when a constant vanishes. They fix an action normalization; they are not dimensions forced by the unnormalized Hamilton-Jacobi equation alone. If a positive rest mass $m_0$ is assigned in a massive-particle interpretation, its corresponding positive parameter is $\mu =G_N{m}_0\mathrm{/}{c}^2$; that interpretation is additional model data, not an existence or motion claim here.

These conversions restore units in the stated framework. They do not assert numerical parameter values, a particular astrophysical source, or an observation.

SI unit and coordinate conventions of the classical gravitational framework

Subextremal Kerr exterior in length coordinates

Kerr horizon angular velocity and the ergoregion

Local Kerr Hamilton-Jacobi separation