Quantum field theory with and without conical singularities: black holes with a cosmological constant and the multi-horizon scenario

Abstract

Boundary conditions and the corresponding states of a quantum field theory depend on how the horizons are taken into account. There is ambiguity as to which method is appropriate because different ways of incorporating the horizons lead to different results. We propose that a natural way of including the horizons is to first consider the Kruskal extension and then define the quantum field theory on the Euclidean section. Boundary conditions emerge naturally as consistency conditions of the Kruskal extension. We carry out the proposal for the explicit case of the Schwarzschild-de Sitter manifold with two horizons. The required period β is the interesting condition that it is the lowest common multiple of 2π divided by the surface gravity of both horizons. Restricting the ratio of the surface gravity of the horizons to rational numbers yields finite β. The example also highlights some of the difficulties of the off-shell approach with conical singularities in the multihorizon scenario; and serves to illustrate the much richer interplay that can occur among horizons, quantum field theory and topology when the cosmological constant is not neglected in black hole processes.

Concept of temperature in multi-horizon spacetimes: analysis of Schwarzschild–De Sitter metric

“…The most direct manner in which this issue can be settled is by introducing a global coordinate patch which can cover the whole manifold, though it is also possible to reach the same conclusions by considering the single-valuedness for the two Kruskal extensions defined in Eqs. (17) and (19) [46]. In the case of single horizon, we have seen that the Kruskal coordinates are adequate for this purpose.…”

Black hole emission of vector particles in (1+1) dimensions

“…It should be noted that simple expression (21) obtained is different from one given in [27], [28]. Also the temperature (21) corresponding to two horizons is not the sum of two temperatures associated with event and cosmological horizons.…”

Some general properties of the renormalized stress-energy tensor for static quantum states on (n + 1)-dimensional spherically symmetric black holes

“…All other components of the RSET are automatically regular across the cosmological horizon. Using (35,37), we can build up a table of conditions similar to Table 1, but involvingZ rather than Z . Combining all the possible behaviours at the event and cosmological horizons, one could build up a long list of different combinations, although most of these will not be physically relevant.…”

“…The reason for this is that the state which represents the emission of Hawking radiation by a black hole formed by gravitational collapse in de Sitter space is not static [34][35][36][37][38][39][40][41].…”

“…For static, spherically symmetric, asymptotically flat black holes with a non-extremal event horizon, it is likely that the Unruh vacuum will be a static state and therefore covered by our approach. However, it is known that for black holes with extremal horizons [26][27][28][29][30][31][32][33], or with both event and cosmological horizons [34][35][36][37][38][39][40][41], the equivalent of the Unruh vacuum is not a static state. For non-static states, the conservation equations are considerably more complex and correspondingly less information is accessible from their solution, apart from in two space-time dimensions (a similar situation arises if one studies the solutions of the conservation equations on a Kerr black hole [42]).…”