Surface gravity and stationary horizon laws

Keep signature (−,+,+,+) and G=c=1, so surface gravity has inverse-length units. In an asymptotically flat stationary axisymmetric black hole, normalize the stationary Killing vector t to have norm minus one at infinity and the axial generator phi to have closed orbits of parameter period two pi. The horizon Killing generator is their combination with the horizon angular velocity. It is null on the horizon, even though t alone need not be null there for a rotating hole.

Along the horizon, the Killing generator satisfies a nonaffine geodesic equation whose coefficient is kappa. Multiplying the generator by a positive constant multiplies kappa by the same constant. This changes a time normalization, not the geometry or a physical temperature defined with a fixed asymptotic clock. A dynamical event horizon need not be a Killing horizon, and alternative dynamical surface-gravity definitions require extra choices. The stationary definition must not be treated as a unique number for every evolving horizon.

The zeroth law states that kappa is constant across an appropriate connected stationary Killing horizon. One sufficient derivation uses the Einstein equation with the dominant energy condition; other geometric versions use additional symmetry or bifurcation assumptions. Constancy is not obtained merely from assigning the name horizon to a surface. For the uncharged Kerr family, the outer and inner radii and rotation parameter determine kappa, angular velocity, and area explicitly.

The first law relates infinitesimal changes between nearby stationary solutions, with mass and angular momentum defined using the normalized asymptotic symmetries. Controlled physical-process versions also exist, using perturbative assumptions and suitable initial and final conditions. Neither form is a general nonlinear equation for an arbitrary instantaneous area and mass. A finite change must be integrated with the changing coefficients or computed from the full solutions, not obtained by multiplying one initial kappa by a large area jump.

In the static Schwarzschild exterior, let f equal one minus two M over r. The proper acceleration needed to remain at fixed r is $a_{\mathrm{stat}}=M/(r^2\sqrt f)$. Its redshifted value is M divided by r squared, tending to kappa only in the horizon limit. It is neither constant at all radii nor the acceleration of a freely falling observer. The stationary Killing time and the observer's proper time are different clocks.

The connection to temperature adds quantum assumptions and a state choice. A Schwarzschild equilibrium state has a redshifted thermal interpretation, while a collapse state with outgoing Hawking flux is not the same global equilibrium bath. Extremal limits, where kappa tends to zero while area can remain nonzero, require particular care and are not captured by casually extrapolating every nonextremal derivation. Stationary mechanics supplies exact geometric relations within its domain; it does not settle the microscopic entropy or the endpoint of evaporation.