It is an interesting fact that the surface gravity plays a similar role in the theory of stationary black holes as the temperature does in ordinary thermodynamics. Since the latter is constant for a body in thermal equilibrium, the result. The zeroth law can be established by different means: Each of the following alternatives is sufficient to prove that is uniform over the Killing horizon generated by .
(i) Einstein’s equations are fulfilled with matter satisfying the dominant energy condition.
(ii) The domain of outer communications is either static or circular.
(iii) is a bifurcate Killing horizon.
(i) The original proof of the zeroth law rests on the first assumption . The reasoning is as follows: First, Einstein’s equations and the fact that vanishes on the horizon (see above), imply that on . Hence, the one-form 35 is perpendicular to and, therefore, space-like or null on . On the other hand, the dominant energy condition requires that is time-like or null. Thus, is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, on . The result that is uniform over the horizon now follows from the general property36
(ii) By virtue of Eq. (6) and the general Killing field identity , the zeroth law follows if one can show that the twist one-form is closed on the horizon :37 which then proves the second version of the first zeroth law.38
(iii) The third version of the zeroth law, due to Kay and Wald , is obtained for bifurcate Killing horizons. Computing the derivative of the surface gravity in a direction tangent to the bifurcation surface shows that cannot vary between the null-generators. (It is clear that is constant along the generators.) The bifurcate horizon version of the zeroth law is actually the most general one: First, it involves no assumptions concerning the matter fields. Second, the work of Rácz and Wald strongly suggests that all physically relevant Killing horizons are either of bifurcate type or degenerate [146, 147].
© Max Planck Society and the author(s)