Zeroth Law and Constant Surface Gravity

The zeroth law of black-hole mechanics says that the surface gravity is constant over each connected component of a stationary black-hole horizon, provided the symmetry and matter hypotheses needed by the chosen proof are satisfied. Constancy is not automatic for an arbitrary null surface, and the numerical value is meaningful only after the horizon generator has been normalized. This is a classical geometric statement; identifying surface gravity with temperature requires separate quantum-field reasoning.

Required background. The Einstein Equation supplies the field equation used in the energy-condition proof.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary checks; Causality and Global Structure distinguishes Killing and event horizons; Flat-Background Hilbert Stress-Energy from Metric Variation supplies the dominant energy condition’s flux interpretation; and Black-Hole Geometry and Mechanics supplies generator normalization and horizon taxonomy.

Surface gravity on a Killing horizon

Let $\mathcal H$ be a smooth Killing horizon generated by a Killing field $\chi^a$. On $\mathcal H$,

$$\nabla_a(\chi^2)=-2\kappa\chi_a, \qquad \chi^b\nabla_b\chi^a=\kappa\chi^a.$$

The first equation defines surface gravity $\kappa$; the second says that the Killing parameter is nonaffine along the null generators. In an asymptotically flat stationary black hole,

$$\chi^a=t^a+\Omega_H\phi^a,$$

where $t^a$ is normalized to unit time translation at infinity and $\phi^a$ has $2\pi$-periodic orbits. Without this normalization, $\chi^a\mapsto c\chi^a$ gives $\kappa\mapsto c\kappa$.

The Killing equation already implies $\mathcal L_\chi\kappa=0$: surface gravity is constant along each generator. The nontrivial part of the zeroth law is equality between different generators on a connected horizon cross-section. Bardeen, Carter, and Hawking state the law and its stationary setting in Bardeen, Carter, and Hawking 1973, Open PDF, pp. 165–167, especially equations (2.11)–(2.13).

Field-equation proof

Assume four-dimensional Einstein gravity, a smooth connected Killing-horizon component, and the dominant energy condition on the horizon. No asymptotic flatness is needed for the local constancy argument, although it is normally used to fix the scale of $\chi^a$.

Because the horizon geometry is invariant under $\chi^a$, the null generators have zero expansion and shear. Raychaudhuri’s equation then gives

$$R_{ab}\chi^a\chi^b=0.$$

Using Einstein’s equation, this becomes $T_{ab}\chi^a\chi^b=0$; the cosmological and trace terms vanish under double contraction with the null vector. The dominant energy condition says that $-T^a{}_b\chi^b$ is future causal. It is also orthogonal to $\chi^a$, so on the horizon it must be proportional to $\chi^a$. Consequently $R_{ab}\chi^b$ is proportional to $\chi_a$.

The Killing identity yields the horizon relation

$$\chi_{[a}\nabla_{b]}\kappa =-\chi_{[a}R_{b]c}\chi^c.$$

The right-hand side now vanishes. Hence the derivative of $\kappa$ has no component tangent to a cross-section. Together with $\mathcal L_\chi\kappa=0$, this proves $\nabla_a\kappa=0$ in all directions tangent to $\mathcal H$. Surface gravity is constant on the connected component.

The argument’s logic matters. Stationarity eliminates expansion and shear; the field equation converts Ricci focusing to matter flux; the dominant energy condition strengthens zero null-null flux to a vector proportionality; and the Killing identity converts that proportionality into transverse constancy. Dropping any layer leaves a gap rather than a weaker version of the same proof. Gourgoulhon and Jaramillo derive the Killing-horizon identities and zeroth-law alternatives in Gourgoulhon and Jaramillo 2006, §§5.3 and 10.2.

Geometric alternatives

There are proofs that trade the field equation and energy condition for stronger geometric structure. For a static Killing horizon, reflection symmetry and hypersurface orthogonality control the transverse derivative of $\kappa$. A stationary axisymmetric proof can instead use circularity and a $t$–$\phi$ reflection symmetry. A connected bifurcate Killing horizon also admits a purely geometric constancy argument: the generator vanishes on the bifurcation surface, and the Killing identity transports its derivative data along both branches.

These are alternative hypothesis packages, not extra assumptions that must all be imposed together. A nondegenerate stationary black hole often admits a bifurcate extension even when the physical exterior under consideration contains only the future branch. An extremal horizon has $\kappa=0$ and no ordinary bifurcation surface, so the bifurcate proof does not cover it; the field-equation or symmetry route must be assessed separately.

For weakly isolated horizons, an analogous zeroth law follows from the boundary condition $\mathcal L_\ell\omega_a=0$ and the allowed constant rescalings of $\ell^a$. That result is quasi-local and does not require a stationary exterior, but it concerns a specified isolated-horizon structure rather than every null boundary.

Limits and checks

Connectedness cannot be omitted. Schwarzschild–de Sitter has distinct black- hole and cosmological Killing horizons whose surface gravities are separately constant but generally unequal. The zeroth law does not assign one value to all components of a multi-horizon spacetime.

The Killing condition cannot be replaced by an arbitrary null generator. A position-dependent rescaling $\ell'^a=f\ell^a$ changes the nonaffinity to

$$\kappa_{(\ell')}=f\kappa_{(\ell)}+\mathcal L_\ell f,$$

which can vary between generators. Such a rescaled field is generally not Killing. This counterexample shows why constancy belongs to a normalized symmetry or isolated-horizon class, not to the underlying null hypersurface alone.

For Schwarzschild, the asymptotically normalized result $\kappa=1/(4GM)$ is constant over the sphere and has dimension inverse length. For Kerr,

$$\kappa=\frac{r_+-r_-}{2(r_+^2+a^2)},$$

which is likewise constant, reduces to the Schwarzschild value as $a\to0$, and vanishes in the extremal limit $r_+\to r_-$. These are checks of the theorem in specific families, not proofs for the general class.

Check your understanding

Two disconnected Killing-horizon components are generated by the same coordinate vector field and have constants $\kappa_1$ and $\kappa_2$ with $\kappa_1\neq\kappa_2$. Does this violate the zeroth law? What happens under a single constant rescaling $\chi^a\mapsto c\chi^a$?

Solution

There is no violation: the theorem asserts constancy on each connected component, not equality among disconnected components. Under one global constant rescaling, both values become $c\kappa_1$ and $c\kappa_2$. Their ratio is unchanged, so normalization cannot generally make unequal component surface gravities equal.

Next, Equilibrium First Law converts neighboring stationary solutions into a variational relation among charges and horizon area.

References

• Bardeen, J. M., B. Carter, and S. W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. DOI.