Angular Momentum in Collapse

Angular momentum changes collapse by constraining how matter can contract, not by supplying a universally defined local “centrifugal force.” The useful quantities are the angular momentum at infinity, quasi-local angular momentum on a closed surface, and the distribution of specific angular momentum through the matter. Keeping those levels distinct explains both rotational support and the ways transport can defeat it.

Required background. The Einstein equation supplies the constraint and evolution equations whose rotational charges are being compared.

Helpful background. Black-Hole Geometry and Mechanics develops Kerr horizons and spin bounds; Black-Hole Perturbation Theory separates rotational modes from gauge; Formulations of General Relativity explains the initial-value variables; Relativistic Matter and Compact Objects supplies rotating-fluid equilibria; and Computational Relativity develops conserved diagnostics in evolutions.

Angular momentum at infinity and on a surface

Assume an asymptotically flat, four-dimensional spacetime with a spacelike slice $\Sigma$, unit future normal $u^a$, induced metric $h_{ab}$, and extrinsic curvature $K_{ab}=-h_a{}^c h_b{}^d\nabla_cu_d$. With the usual asymptotic falloff and parity conditions that make rotational charges finite, choose an asymptotic rotation $\phi^a$. The ADM angular momentum is the large-sphere limit

$$J_{\mathrm{ADM}}[\phi] =\frac{1}{8\pi G}\lim_{S\to\infty} \oint_S (K_{ab}-Kh_{ab})\phi^a s^b\,dA .$$

On a finite closed two-surface $S\subset\Sigma$, the same integral defines a useful quasi-local $J_S[\phi]$ when $\phi^a$ is a specified tangential axial vector whose flow has closed orbits of parameter period $2\pi$. This fixes the normalization of $\phi^a$; the orbits’ proper lengths need not equal $2\pi$. Without exact axisymmetry, choosing an approximate axial vector is an additional prescription, so different reasonable choices need not agree. The geometric hypotheses and normalization behind axisymmetric angular-momentum inequalities are set out in Dain 2012, §§2–3, pp. 6–20, PDF.

For matter with stress tensor $T_{ab}$, an axial Killing field gives the conserved current $j^a=T^a{}_b\phi^b$. Integrating $\nabla_a j^a=0$ between two slices converts the change of matter angular momentum into flux through their boundary. Gravitational waves can also transport angular momentum, but no generally covariant local gravitational density exists; the associated flux is defined at null infinity, a horizon, or through a chosen perturbative/extraction construction.

The sign can be checked without memorizing a surface formula. Define the Eulerian matter momentum $S_a=-h_a{}^bT_{bc}u^c$. With the stated sign of $K_{ab}$, the momentum constraint is

$$D_b(K^{ab}-Kh^{ab})=8\pi G S^a.$$

If $\phi^a$ is an exact axial Killing field of the spatial metric, integration and the Killing equation give

$$J_{S_{\rm out}}-J_{S_{\rm in}} =\int_{\Sigma_{\rm between}}S_a\phi^a\,dV.$$

For a regular center the inner contribution vanishes. This identifies the positive matter angular momentum with the positive surface charge. In a nonsymmetric slice an additional volume term involving $(K^{ab}-Kh^{ab})D_b\phi_a$ remains; the matter integral alone is then not the total angular momentum. Exactly axisymmetric gravitational waves can carry energy but no net angular momentum about the symmetry axis. Therefore an axial-spin loss in an exactly axisymmetric isolated fluid calculation must be explained by matter flux, horizon transfer, or numerical error, not by an axial gravitational-wave torque.

Support, transport, and dimensionless spin

For a fluid element in an exactly axisymmetric, inviscid, isentropic flow away from shocks, the relativistic specific angular momentum $j=h_{\mathrm{fluid}}u_\phi$ is advected along the worldline. Here $h_{\mathrm{fluid}}=(\rho+p)/\rho_0$ is enthalpy per unit rest mass, $\rho_0$ is rest-mass density, and $u_\phi=(u_{\mathrm{fluid}})_a\phi^a$, with $u_{\mathrm{fluid}}^a$ the fluid four-velocity rather than the slice normal. The isentropic Euler equation reads $u_{\mathrm{fluid}}^a\nabla_a[h_{\mathrm{fluid}}(u_{\mathrm{fluid}})_b]+\nabla_bh_{\mathrm{fluid}}=0$. Contracting with the axial Killing field removes both the azimuthal pressure derivative and the antisymmetric Killing-gradient term, yielding $u_{\mathrm{fluid}}^a\nabla_a j=0$. A parcel with larger $j$ generally encounters rotational support at a larger circumferential radius. Collapse therefore depends on the cumulative distribution

$$M_*(<j)=\int_{j(x)<j}\rho_*\,d^3x,$$

not only on the total $J$. Here $\rho_*\,d^3x=\rho_0(-u_{\rm fluid}\cdot u)\,dV$ is the baryon-mass measure on the slice. Low-$j$ material may form a central horizon while high-$j$ material remains in a torus. Magnetic and viscous stresses or non-axisymmetric pressure forces can redistribute angular momentum. Shocks require flux jump conditions; an axisymmetric ideal-fluid shock does not by itself provide an external axial torque. These distinctions matter when interpreting an apparent change of the numerical $j$ distribution.

The dimensionless ratio $\chi=J/(GM^2)$ is meaningful only after specifying the mass and the region whose angular momentum is used. For a stationary Kerr black hole, $M$ is the horizon/ADM mass and $\lvert\chi\rvert\leq1$. Initial matter data with $J/(GM_{\mathrm{ADM}}^2)>1$ do not violate that bound: the complete configuration is not already a Kerr horizon, and excess angular momentum can remain outside or be radiated. In Shibata’s axisymmetric collapse models, conservation of rest mass, $J$, and the specific-angular-momentum spectrum were numerical diagnostics rather than assumed identities; see Shibata 2000, §2.1, Eqs. (2.15)–(2.18), pp. 9–10 and §4.2, pp. 15–16.

What a spin budget can and cannot predict

A defensible budget tracks at least $M_{\mathrm{ADM}}$, total $J$, baryon mass, $M_*(<j)$, horizon mass and spin after formation, and matter/radiation fluxes across the outer boundary. It checks the spherical limit $J\to0$, reverses the sign of the rotation, and distinguishes coordinate radius from circumferential or areal radius. A balance failure may signal boundary loss, atmosphere torque, imperfect symmetry, or inadequate resolution rather than new physics.

Angular momentum alone does not decide the endpoint. The equation of state controls pressure and shocks; the rotation law controls the low-$j$ core; non-axisymmetric modes generate torques; and the collapse time determines whether those modes have time to grow. Thus “more spin prevents a black hole” is false as a general statement. More accurately, at fixed matter model and initial-data family, increasing a declared spin parameter can delay prompt central collapse, alter the threshold, or redirect mass into an exterior disk.

Check your understanding

An axisymmetric evolution forms a horizon with mass $M_H=0.9M$ and angular momentum $J_H=0.7GM^2$. Compute its dimensionless spin and explain why comparing $J_H$ with the initial total $J$ requires more information.

Solution

The horizon spin is

$$\chi_H=\frac{J_H}{GM_H^2}=\frac{0.7}{0.9^2}\simeq0.864,$$

which is subextremal. A comparison with the initial total requires the angular momentum still carried by exterior matter, matter flux through the outer boundary, and the convention used to define $J_H$. Exact axisymmetry sets the net axial gravitational-wave flux to zero; it does not set the exterior matter’s angular momentum to zero. Thus $J_H=J_{\mathrm{initial}}$ is not implied by axisymmetry.

References

• Dain, Sergio. “Geometric Inequalities for Axially Symmetric Black Holes.” Classical and Quantum Gravity 29 (2012): 073001. doi:10.1088/0264-9381/29/7/073001.
• Shibata, Masaru. “Axisymmetric Simulations of Rotating Stellar Collapse in Full General Relativity: Criteria for Prompt Collapse to Black Holes.” Progress of Theoretical Physics 104 (2000): 325–358. doi:10.1143/PTP.104.325.