Quantum Gravity

BLACK-HOLE THERMODYNAMICS AND HAWKING RADIATION 207local behaviour of correlation functions by inspecting their time developmentfrom the past into the future (Fredenhagen and Haag 1990).As an intermediate step towards full quantum gravity, one might consider theheuristic ‘semiclassical’ Einstein equations discussed in Section 1.2; see (1.35).This enables one to take into account back-reaction effects on the semiclassicallevel. The evaluation of 〈T ab 〉—which requires regularization and renormalization—isa difficult subject on its own (Frolov and Novikov 1998). The renormalizedvalue for 〈T ab 〉 is essentially unique (its ambiguities can be absorbed incoupling constants) if certain sensible requirements are imposed; cf. Section 2.2.4.Evaluating the components of the renormalized 〈T ab 〉 near the horizon, one findsthat there is a flux of negative energy into the hole. This follows from the Unruhvacuum described above. Clearly this leads to a decrease of the mass. Thesenegative energies represent a typical quantum effect and are well known fromthe—accurately measured—Casimir effect. This occurrence of negative energiesis also responsible for the breakdown of the classical area law in quantum theory.The negative flux near the horizon lies also at the heart of the ‘pictorial’representation of Hawking radiation that is often used; see, for example, Parikhand Wilczek (2000). In vacuum, virtual pairs of ‘particles’ are created and destroyed.However, close to the horizon one partner of this virtual pair might fallinto the black hole, thereby liberating the other partner to become a real particleand escaping to infinity as Hawking radiation. The global quantum field exhibitsquantum entanglement between the inside and outside of the black hole, similarto the case of the accelerated observer discussed above.In the case of an eternal Schwarzschild black hole, where both past and futurehorizons exist, there exists a distinguished quantum state which describes theequilibrium of the black hole with thermal radiation at Hawking temperature.This state is also called the ‘Hartle–Hawking vacuum’. It is directly analogousto the Minkowski vacuum in (7.8).One can also give explicit expressions for the Hawking temperature (7.11) inthe case of rotating and charged black holes. For the Kerr solution, one hask B T BH = κ () −12π =2 GM1+ √G2 M 2 − a 2 8πGM < 8πGM . (7.15)Rotation thus reduces the Hawking temperature. For the Reissner–Nordströmsolution (describing a charged spherically symmetric black hole), one hask B T BH =8πGM(1 − G2 q 4 )r 4 +