Extremality and Final-State Bounds

A candidate remnant must satisfy more than charge conservation. Horizon geometry, energy conditions, and the requirement that a regular black-hole exterior exist impose inequalities among area, angular momentum, charge, and mass. These bounds are strongest when their hypotheses are kept visible: a Kerr–Newman formula for a stationary electrovacuum remnant, a quasi-local inequality for a stable axisymmetric marginal surface, and a perturbative gedanken experiment are related but not interchangeable theorems.

Helpful background. The area law and stationary solutions are developed in Black-Hole Geometry and Mechanics; near-extremal response uses Black-Hole Perturbation Theory; constraint charges build on Formulations of General Relativity; numerical inequalities are tested in Computational Relativity; and loss fluxes follow Gravitational Radiation and Waveforms.

Stationary and quasi-local bounds

In this section, the Kerr–Newman formulae use geometrized Einstein–Maxwell units $G=c=4\pi\epsilon_0=1$: $M$ and $Q$ have dimensions of length, while $J$ has dimensions of length squared. For a stationary Kerr–Newman black hole with ADM mass $M$, angular momentum $J$, and electric charge $Q$, regularity of the horizon requires

$$M^2\ge Q^2+\frac{J^2}{M^2}.$$

Equality is extremality. A genuinely dimensionless normalized gap is

$$\delta_{\rm ext}=1-\frac{Q^2}{M^2}-\frac{J^2}{M^4}\ge 0.$$

With representative SI quantities, define $\chi=cJ_{\rm SI}/(GM_{\rm SI}^2)$ and $\zeta=Q_{\rm SI}/(\sqrt{4\pi\epsilon_0G}\,M_{\rm SI})$; then $\delta_{\rm ext}=1-\chi^2-\zeta^2$. Inserting dynamical quasi-local quantities into this stationary-family expression does not automatically produce a theorem.

A genuinely quasi-local statement is available under different hypotheses. Let $S$ be a smooth, closed, orientable, axisymmetric, stably outermost marginally outer trapped surface in four-dimensional Einstein–Maxwell theory, with the axisymmetric stability and electromagnetic data specified in the theorem. Assume the non-electromagnetic matter satisfies the dominant energy condition and take nonnegative cosmological constant. For vanishing magnetic charge, its area, total quasi-local angular momentum $J$, and electric charge $Q$ obey

$$A^2\ge 16\pi^2\left(4J^2+Q^4\right).$$

Here $J$ includes the electromagnetic surface contribution as well as the gravitational Komar term; inserting the gravitational term alone is not the theorem’s prescription. With both electric and magnetic charges, replace $Q^4$ by $(Q_E^2+Q_M^2)^2$. The exact definitions and stability hypotheses appear in Gabach Clément, Jaramillo, and Reiris 2013, Theorem 1.1, § 2.1, Eqs. (7)–(9), and §§ 3–4. The result applies to the selected horizon cross-section; it is not a global uniqueness theorem and does not determine its mass.

Proof architecture

The quasi-local argument begins with the MOTS stability inequality. One selects an axisymmetric test function adapted to the norm of the axial Killing field. Einstein’s equation and the dominant energy condition turn the matter terms into nonnegative contributions. Maxwell constraints express the electric and magnetic fluxes through $S$ as charges, while the normal connection encodes angular momentum. After reducing the geometric functional to the orbit space, a variational inequality shows that its minimum at fixed $(J,Q)$ is attained by the extreme Kerr–Newman throat data. Rearrangement yields the area bound.

Every step carries a boundary of validity. Without axisymmetry the required quasi-local $J$ is not canonically supplied by the theorem. If the surface fails the theorem’s stable-outermost condition, the proof lacks the required sign in its stability inequality. This is a condition on an outward deformation, not a requirement that the surface be the globally outermost MOTS on an entire slice. Negative cosmological constant and non-Maxwell gauge fields change the functional. A toroidal inner MOTS therefore does not contradict the stated result; it may fail stability, topology, or matter hypotheses.

For absorption by a nearly extremal stationary hole, a complementary proof uses first- and second-order variational identities. First-order energy conditions alone leave an apparent window for overcharging or overspinning. The canonical-energy term at second order accounts for self-force and finite-size backreaction and closes that window for perturbations satisfying the assumptions. Sorce and Wald derive the controlling inequalities and their near-extremal expansion Sorce and Wald 2017, § IV, Eqs. (109)–(122), pp. 29–32, PDF. The dynamical censorship interpretation and its capture inequalities are developed in Overspinning and Overcharging Thought Experiments; here the same result is used as a remnant-consistency bound.

Merger budgets and implications

In a merger, use charge conservation $Q_f=Q_1+Q_2-Q_{\rm escaped}$, radiative energy and angular-momentum balance, the final horizon area, and a declared quasi-local spin. The escaped charge is carried by charged matter; electromagnetic waves themselves carry energy and angular momentum but no net electric charge. Any surviving charged exterior material must also be accounted for. Under the classical null-energy, predictability, and regularity assumptions, the area theorem compares suitable event-horizon cuts: if an early cut intersects the two disjoint event-horizon components with areas $A_1$ and $A_2$, a sufficiently late cut of their joined event horizon has $A_f\ge A_1+A_2$. This is not an additive theorem for arbitrary numerical apparent horizons or MOTSs on unrelated slices. Combining independently measured quantities yields several residuals; no single residual should be forced to vanish by defining one output from the others.

These inequalities exclude an inconsistent measured remnant, but they do not prove that arbitrary initial data settle to Kerr–Newman. They also do not rescue a simulation whose errors are larger than its distance from extremality. Near equality, convergence of $A$, $J$, and $Q$ and preservation of constraints must be demonstrated separately.

Transfer exercise

An axisymmetric stable electrovacuum MOTS has $J=2$ and $Q=1$ in geometric units. Find its minimum allowed area. Would observing $A=10\pi$ satisfy the theorem?

Solution

The bound gives $A\ge4\pi\sqrt{4J^2+Q^4}=4\pi\sqrt{17}$. Since $4\sqrt{17}\pi\approx16.49\pi$, the value $10\pi$ violates the inequality. One should first test the numerical measurement and then check the theorem’s hypotheses; a failure of axisymmetry or stability means the theorem cannot be applied, not that a regular extremal bound has been defeated.

References

• Gabach Clément, María E., José Luis Jaramillo, and Martín Reiris. “Proof of the Area-Angular Momentum-Charge Inequality for Axisymmetric Black Holes.” Classical and Quantum Gravity 30 (2013): 065017. doi:10.1088/0264-9381/30/6/065017.
• Sorce, Jonathan, and Robert M. Wald. “Gedanken Experiments to Destroy a Black Hole II: Kerr–Newman Black Holes Cannot Be Overcharged or Overspun.” Physical Review D 96 (2017): 104014. doi:10.1103/PhysRevD.96.104014.