Generalized Painlevé-Gullstrand descriptions of Kerr-Newman black holes

Generalized Painlevé-Gullstrand coordinates for stationary axisymmetric spacetimes are constructed explicitly; and the results are applied to the Kerr-Newman family of rotating black hole solutions with, in general, non-vanishing cosmological constant. Our generalization is also free of coordinate singularities at the horizon(s); but unlike the Doran metric it contains one extra function which is needed to ensure all variables of the metric remain real for all values of the mass, charge, angular momentum and cosmological constant. PACS numbers: 04.70.Bw, 04.20.Jb 1. Overview The Doran metric[1] is another coordinatization of the Kerr-Newman solution[2, 3] which describes a charged rotating black hole. The Doran metric can be considered to be the extension of the Painlevé-Gullstrand(PG)[4, 5] description of a black hole from spherically symmetric to axisymmetric spacetime. These descriptions have the advantage of being free from coordinate singularities at the horizon(s). The Kerr solution, which was discovered much later, is not a straightforward generalization of the Schwarzschild solution. Similarly the explicit Doran metric is a comparatively recent description, but it has found its way into several investigations of black hole physics. Constant-time Doran slicings of the ergosurface in non-extremal black holes have been demonstrated to be free of conical singularities at the poles[6]. Calculations of Hawking radiation[7], and also neutrino asymmetry due to the interaction of fermions and rotating black holes[8] have also made explicit use of the Doran metric. Other authors have proposed to utilize the Doran metric to extend or generalize spherically symmetric results to the context of rotating black holes [9, 10, 11]. Recently it has been demonstrated[12] that there is an obstruction to the implementation of flat PG slicings for spherically symmetric spacetimes; and the insistence on spatial flatness can lead to PG metrics which involve complex metric variables. The corresponding vierbein fields are then not related to those of the standard spherically symmetric metric by physical Lorentz boosts. Since the Doran metric contains spherically symmetric PG solution as a special case, it will be afflicted with the same problems (this will be explicitly illustrated later on). In discussions of black hole evaporation using the Parikh-Wilczek method[13], the insistence on spatially flat PG coordinates can give rise to spurious contributions which are ambiguous and problematic both to the computation of the tunneling rate and to the universality of the results. In a more general context, the appearance of complex quantities in the Doran metric causes unnecessary complications, and gives rise to difficulties and ambiguities in the physical interpretations. These problems can be avoided altogether by using a less restrictive form of constant-time slicing which generalizes the Doran metric. In this brief note we demonstrate how this goal can be realized explicitly. Although it is possible to introduce many parameters through the freedom of local Lorentz transformations which relate vierbein one-forms with the same metric, our generalized PG description is “optimal” in that only one additional function, A(r, θ), is needed, and introduced, to avoid all the troubles. Throughout this short note geometric units G = c = 1 and the (− + ++) convention for the spacetime signature are adopted. 2. Generalized Painleve-Gullstrand metrics for stationary axisymmetric spacetimes We seek a generalization of the Doran metric by first assuming that the vierbein one-forms, eGD (A = 0, 1, 2, 3), can be expressed as eAGD = {A (dtP − δtdθ) , Bdr + C (dtP − δtdθ) +D (dφP − δφdθ) , ρdθ, E (dφP − δφdθ)} , (1) Generalized Painlevé-Gullstrand descriptions of Kerr-Newman black holes 2 with the PG coordinates defined as dtP = dt+ Jdr + δtdθ dφP = dφ+Kdr + δφdθ. dtP and dφP being exact differentials require δt ≡ ∂J ∂θ dr and δφ ≡ ∂K ∂θ dr. The most general stationary axisymmetric metric depends on five arbitrary functions of r and θ, and it has been discussed by Chandrasekhar [14, 15]. The metric is compatible with the vierbein eAax = { Asdt, B −1 s dr, ρdθ, Cs(dφ − Ωdt) } . (2) To produce the same metric, eGD and e A ax must be related by a Lorentz transformation. This can be realized by requiring ΛB = e A GD · [ eax ]−1 to satisfy Λ ηΛ = η. This procedure fixes the functions B,C,D,E, J,K in terms of the variables in eax as J = ± √ A2−A2 s AAsBs , B = C JAB s , D = − 2 s Ω C , K = Ω A s B s J , C = √ A2 −As + C2 sΩ, E = AsCs ABBs . (3) However the function A(r, θ) is so far still unrestricted and completely free. In fact it expresses the freedom in the choice of the local Lorentz frame, as can be seen from the explicit relation