Quantum Gravity

CANONICAL QUANTIZATION OF THE SCHWARZSCHILD BLACK HOLE 211Variation of the action with respect to φ yields as usual the Gauss constraintG = P ′ Γ ≈ 0 . (7.25)The constraint (7.24) generates radial diffeomorphisms for the fields R, L, andtheir canonical momenta. It does not generate diffeomorphisms for the electromagneticvariables. This can be taken into account if one uses the multiplier˜φ = φ − N r Γ instead of φ and varies with respect to ˜φ (Louko and Winters-Hilt 1996), but for the present purpose it is sufficient to stick to the above form(7.24).The model of spherical symmetric gravity can be embedded into a whole classof models usually referred to as ‘two-dimensional dilaton gravity theories’. Thisterminology comes from effective two-dimensional theories (usually motivatedby string theory), which contain in the gravitational sector a scalar field (the‘dilaton’) in addition to the two-dimensional metric (of which only the conformalfactor is relevant). An example is the ‘CGHS model’ defined in (5.60) withinwhich one can address the issues of Hawking radiation and back reaction. Thismodel is classically soluble even if another, conformally coupled, scalar field isincluded. The canonical formulation of this model can be found, for example,in Louis-Martinez et al. (1994) and Demers and Kiefer (1996). The dilaton fieldis analogous to the field R from above, while the conformal factor of the twodimensionalmetric is analogous to L.Consider now the boundary conditions for r →∞. One has in particularL(r, t) → 1+ GM(t) ,R(r, t) → r, N → N(t) , (7.26)ras well asP Γ (r, t) → q(t), φ(r, t) → φ(t) . (7.27)From the variation with respect to L, one then finds the boundary term ∫ dtNδM. In order to avoid the unwanted conclusion N = 0 (no evolution at infinity),one has to compensate this term in advance by adding the boundary term∫− dt NMto the classical action. Note that M is just the ADM mass. The need to includesuch a boundary term was recognized by Regge and Teitelboim (1974); cf.Section 4.2.4. Similarly, one has to add for charged black holes the term∫− dt φqto compensate for ∫ dtφδq, which arises from varying P Γ . If one wished instead toconsider q as a given, external parameter, this boundary term would be obsolete.