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Quasilocal energy for rotating charged black hole solutions in general relativity and string theory

We explore the (non)-universality of Martinez's conjecture, originally proposed for Kerr black holes, within and beyond general relativity. The conjecture states that the Brown-York quasilocal energy at the outer horizon of such a black hole reduces to twice its irreducible mass, or equivalently, to \sqrt{A} /(2\sqrt{pi}), where `A' is its area. We first consider the charged Kerr black hole. For such a spacetime, we calculate the quasilocal energy within a two-surface of constant Boyer-Lindquist radius embedded in a constant stationary-time slice. Keeping with Martinez's conjecture, at the outer horizon this energy equals the irreducible mass. The energy is positive and monotonically decreases to the ADM mass as the boundary-surface radius diverges. Next we perform an analogous calculation for the quasilocal energy for the Kerr-Sen spacetime, which corresponds to four-dimensional rotating charged black hole solutions in heterotic string theory. The behavior of this energy as a function of the boundary-surface radius is similar to the charged Kerr case. However, we show that in this case it does not approach the expression conjectured by Martinez at the horizon.


II. BROWN-YORK QUASILOCAL ENERGY
The BY derivation of the quasilocal energy, as applied to a four-dimensional (4D) spacetime solution of Einstein gravity can be summarized as follows. The system one considers is a 3D spatial hypersurface Σ bounded by a 2D spatial surface B in a spacetime region that can be decomposed as a product of a 3D hypersurface and a real line-interval representing time. The time-evolution of the boundary B is the surface 3 B. One can then obtain a surface stress-tensor on 3 B by taking the functional derivative of the action with respect to the 3D metric on 3 B. The energy surface density is the projection of the surface stress-tensor normal to a family of spacelike surfaces such as B that foliate 3 B. The integral of the energy surface density over such a boundary B is the quasilocal energy associated with a spacelike hypersurface Σ whose orthogonal intersection with 3 B is the boundary B. Here we assume that there are no inner boundaries, such that the spatial hypersurfaces Σ are complete. In the case where horizons form, one simply evolves the spacetime inside as well as outside the horizon. Under these conditions, the QLE is defined as: where σ is the determinant of the 2-metric on B, k is the trace of the extrinsic curvature of B, and k 0 is a reference term that is used to normalize the energy with respect to a reference spacetime, not necessarily flat. To compute the QLE for asymptotically flat solutions, we will choose the reference spacetime to be flat as well. In that case, k 0 is the trace of the extrinsic curvature of a two-dimensional surface embedded in flat spacetime, such that it is isometric to B. Interestingly, the foregoing analysis can be applied in a straightforward way to compute QLE of spatial sections of solutions of scalar-tensor theories as well. This is because, for spacetime dimensions higher than two, a scalar-tensor theory can be cast in the "Einstein-Hilbert" form by performing a conformal transformation. The solutions of this conformally related action are given by the Einstein-frame metrics, which themselves are related to solutions of the scalar-tensor theory by a conformal transformation. It was shown in Ref. [4] that the quasilocal mass (QLM), which is a construct related to QLE, is invariant under such a transformation. When there is a timelike Killing vector field ξ µ on the boundary 3 B, such that it is also hypersurface forming, one can define an associated conserved QLM for the bounded system [6,3]: where N is the lapse function related to ξ µ by ξ µ = Nu µ . Further, if ξ · u = −1, then N = 1 and consequently the QLM is the same as the QLE. Unlike QLE, the quasilocal mass is independent of any foliation of the bounded system. Moreover, owing to its conformal invariance, a frame of convenience can be chosen for the computation of QLM without affecting its value. The QLE, in general, is not a conformal invariant. Thus, the frame in which it is evaluated needs to be clearly specified. In this paper, all computations of the QLE will be carried out in the Einstein frame. This is the frame in which Sen analyzed the properties of his rotating charged black hole solution.

III. QLE FOR CHARGED KERR BLACK HOLES
The charged Kerr solution in general relativity represents the spacetime of a rotating, charged black hole. Its spacetime metric and electromagnetic vector potential are given by [9] where Σ = r 2 + a 2 cos 2 θ , The above fields define a black hole solution with mass M, charge Q, and specific angular momentum a.

A. Unreferenced QLE for charged Kerr black holes
Consider a constant stationary-time hypersurface Σ embedded in the charged Kerr spacetime with the metric (3.1). Define the two-surface B to be a surface with some constant value for the Boyer-Lindquist radial coordinate, r = r 0 , embedded in Σ. The line-element on this surface in the slow-rotation approximation, i.e., a ≪ r 0 , is: Terms of order O(a 4 /r 4 0 ) and higher have been neglected. We now calculate the unreferenced QLE at the two-surface B, defined by r = r 0 . We assume that r 0 ≥ r + , where r + represents the outer horizon of the charged Kerr black hole. Using Eq. (A6) of the appendix, the unreferenced QLE ε for the surface r = r 0 can be written explicitly as where we have retained only terms of leading order in the parameter a/r 0 . The above integration can be performed in a straightforward way to yield: Note that in the limit Q → 0, the above expression simplifies to which is the unreferenced QLE of the neutral Kerr black hole [10]. It also has the expected limit when a → 0, in which case Eq. (3.7) gives which is the unreferenced QLE of the Reissner-Nordstrom black holes [3]. Note that no approximations have been made inside the square-root appearing in ε. As r 0 → ∞, we have ε → M − r 0 , which is divergent. This prompts the need for a subtraction term to renormalize the unreferenced QLE. Below, we compute such a reference term.

B. Reference term
To obtain the reference term in the QLE expression, Eq. (2.1), we first find a 2D surface isometric to (3.5), which is embeddable in a flat 3D slice with the line element where R, Θ, and Φ are the spherical polar coordinates. Let the desired 2D surface in the flat slice be characterized by R = f (Θ), where f is a function of the azimuthal angle Θ and the parameters (M, a, r 0 ). Its intrinsic metric is obtained from (3.10) as follows. We assume that Θ = Θ(θ) and Φ = φ. Then, on the two-surface B, we find that R is a function of θ, i.e., R = R(θ), say. Hence, the line-element on B is where an overdot denotes derivative with respect to θ.
Requiring the above line-element to be isometric to (3.5) implies that the following couple of equations be obeyed:Ṙ and Assuming thatṘ 2 ≃ O(a 4 /r 4 0 ) (this condition will be justified a posteriori), the above equations can be combined to yield the following first-order ordinary differential equation: which is easily solved to give: Putting this back in Eq. (3.13), we find R(θ). In the resulting expression, using Eq. (3.15) to substitute for θ in terms of Θ, we finally get The two-surface R = f (Θ) then describes an oblate spheroid, which bulges out near the equator (i.e., near Θ = π/2). Note that the above equations can be used to prove that indeedṘ 2 ≃ O(a 4 /r 4 0 ).

C. Referenced QLE for charged Kerr black hole
In the slow-rotation approximation the intrinsic metric on B, as embedded in flat space, is The extrinsic curvature k 0 of this surface embedded in flat space can be evaluated using the method detailed in the appendix. This in turn can be used to compute the renormalization integral in the QLE (2.1): The referenced QLE is, therefore, obtained to be: As r 0 → ∞, we have which is indeed the Arnowitt-Deser-Misner (ADM) mass [11] of the charged Kerr spacetime. Near the outer horizon r 0 = r + , the energy is (3.21) In this limit, to leading order in a/r + . Above, we have defined for the charged Kerr black hole, which is the generalization of the irreducible mass of a neutral Kerr black hole [12].

IV. THE KERR-SEN SOLUTION
Consider the following string-inspired low-energy effective action in four dimensions: Here G µν is the metric, R is the scalar curvature, F µν = ∂ µ A ν − ∂ ν A µ is the field strength corresponding to the Maxwell field A µ , Φ is the dilaton field, and where B µν is the antisymmetric tensor gauge field. Above we have defined which is the gauge Chern-Simons term. It must be noted that the above theory is one where 6 of the 10 dimensions have been compactified to a six-torus. The massless fields arising from compactification have not been included in the effective action. Only a U(1) component of the full set of non-abelian gauge fields present in the theory has been included above. Consequently, the corresponding solutions carry a U(1) charge only. The metric G µν used here is the metric that arises naturally in the σ-model, and is related to the Einstein metric g µν through the relation: Finally, the action was truncated to contain only those terms that contain two or a lesser number of derivatives. Sen showed that the above theory has rotating charged black hole solutions given by the following field configuration [5]: (r 2 + a 2 cos 2 θ − 2mr)(r 2 + a 2 cos 2 θ) (r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 ) 2 dt 2 − 4mra cosh 2 α 2 (r 2 + a 2 cos 2 θ) sin 2 θ (r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 ) 2 dtdφ + r 2 + a 2 cos 2 θ r 2 + a 2 − 2mr dr 2 + (r 2 + a 2 cos 2 θ)dθ 2 +{(r 2 + a 2 )(r 2 + a 2 cos 2 θ) + 2mra 2 sin 2 θ + 4mr(r 2 + a 2 ) sinh 2 α 2 + 4m 2 r 2 sinh 4 α 2 } × (r 2 + a 2 cos 2 θ) sin 2 θ (r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 ) 2 dφ 2 , (4.5a) Φ = − ln r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 r 2 + a 2 cos 2 θ , (4.5b) (4.5c) (4.5d) B tφ = 2mra sinh 2 α 2 sin 2 θ r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 . (4.5e) The other components of A µ and B µν vanish. The Einstein metric ds 2 E ≡ e −Φ ds 2 is given by where Υ = r 2 + a 2 cos 2 θ + 2mr sinh 2 α 2 , This metric describes a black hole solution with mass M, charge Q, angular momentum J, and magnetic dipole moment µ given by, This is often called the Kerr-Sen black hole solution. The location of the horizon is given by the coordinate singularities, which occur on the surfaces This has the following two roots: which correspond to the outer and inner horizons, respectively. The area of the outer event horizon with the metric given in Eq. (4.6) is found to be Equation (4.9) shows that the horizon disappears unless (4.11) Thus the extremal limit of the black hole corresponds to |J| → M − Q 2 /2M. In this limit, A → 8π|J|. (We note that rotating black hole solutions in a related theory of dilaton gravity were also found in Ref. [8]. )

A. Unreferenced QLE for Kerr-Sen black hole
Consider a constant stationary-time hypersurface Σ embedded in the Kerr-Sen spacetime with the metric (4.5a). Define the two-surface B to be the surface with some constant value for the Boyer-Lindquist radial coordinate r = r 0 , embedded in Σ. The line element on this surface in the slow-rotation approximation, i.e., a ≪ r 0 , is: where z = r 0 √ κ sin θ 1 + a 2 2r 2 0 κ + ma 2 r 3 0 κ 2 cosh 2 α 2 sin 2 θ , (4.13) and κ = 1 + (2m/r 0 ) sinh 2 α/2. Terms of order O(a 4 /r 4 0 ) and higher have been neglected. We now calculate the unreferenced QLE within the two-surface B, defined by r = r 0 . We assume that r 0 ≥ r + , where r + represents the outer horizon of the Kerr-Sen black hole. Using Eq. (A6) of the appendix, the integral ε for the surface r = r 0 can be written explicitly as where where κ ′ = 1 + (m/r 0 ) sinh 2 α/2. On performing the above integration we get: which is the unreferenced QLE for Kerr-Sen black hole.
In the limit α → 0, the expression (4.15) gives This is the unreferenced QLE of the neutral Kerr black hole [10]. It also has the expected limit when a → 0, in which case which is the unreferenced QLE of the static, charged dilatonic black holes [4]. As r 0 → ∞, we have ε → M − r 0 , which is again divergent. Below, we compute the reference term required to renormalize this QLE.

B. Two-surface isometric to B embedded in flat space
To obtain the reference term in the QLE expression, Eq. (2.1), we first find a 2D surface isometric to (4.12), which is embeddable in a flat 3D slice with the line element (4.18) The equation for the desired 2D surface in the flat slice is denoted by R = g(Θ), where g is a function of the azimuthal angle Θ and the parameters (M, a, r 0 ) of the surface in Σ. Its intrinsic metric is obtained from (4.18). We assume that Θ = Θ(θ) and Φ = φ. Then, on the two-surface R is a function of θ, i.e., R = R(θ), say. Hence, the line-element on it is given by Eq. (3.11). Similar to the case of the charged Kerr black hole discussed in section III B , even for the Kerr-Sen spacetime, requiring the line-element (3.11) to be isometric to (4.12) yields the following first-order ordinary differential equation: This equation is easily solved to give: sin Θ = sin θ 1 + a 2 2κr 2 0 1 + 2m κr 0 cosh 2 α 2 cos 2 θ . (4.20) As in the case of charged Kerr, the above expression can be used to find The two-surface R = g(Θ) once again describes an oblate spheroid.

C. The referenced QLE
In the slow-rotation approximation, Eq. (4.21) implies that the intrinsic metric on B, as embedded in flat space, is The extrinsic curvature k 0 of this surface embedded in flat space can be evaluated using the method detailed in the appendix. This in turn can be used to compute the renormalization integral in the QLE (2.1): The referenced QLE is, therefore, obtained to be: As r 0 → ∞, we have which is indeed the ADM mass (4.7) of the Kerr-Sen solution.
The QLE (4.24) has the correct limit for vanishing charge, namely, for α = 0. In that case, one obtains the expression in Eq. (3.19), which is the QLE for the Kerr black hole. Similarly, for vanishing rotation, i.e., for a = 0, Eq. (4.24) reduces to which is the QLE for charged black holes in string theory [3]. Near the outer horizon r 0 = r + , the energy is (4.28) In the limit of vanishing charge, which is given by α = 0, the above quantity goes over to E(r 0 = r + ) = r + 1 + which are the expected values for the Kerr black hole [10]. Note that in this latter case, to leading order in a/r + . Whether such a relation continues to hold even after the slowrotation approximation is dropped, is not known. For the Kerr-Sen black hole, however, which is not the same as E(r 0 = r + ) given in Eq. (4.27) for such a black hole.

V. DISCUSSION
The Brown-York quasilocal energy of a Kerr black hole, for the type of slice Σ and quasilocal surface B considered in this paper, has not been evaluated yet for the exact case (i.e., beyond the slow-rotation approximation). One of the main hurdles in this computation is the determination of the two-surface R = f (Θ), isometric to B, to be embedded in a flat three-slice. It is nevertheless interesting to explore the slow-rotation regime of such black holes in general relativity and alternative theories of gravity for the results obtained can often tell us about the behavior of certain physical quantities in the exact case. One such quantity is the value of QLE at the (outer) horizon of such black holes, which for the Kerr black hole approaches twice its irreducible mass. In this paper, we find that this result continues to hold even for the charged Kerr black hole. Thus, within general relativity, this identity appears to bear a universal quality as far as its applicability to stationary black hole solutions is concerned.
It is known that the mechanical laws of stationary black hole solutions in general relativity are extendible also to other alternative theories of gravity, especially ones connected with a diffeomorphism invariant Lagrangian. One such leading alternative to general relativity is a string-inspired four-dimensional low-energy effective theory. It is of interest to ask if other identities related to black hole mechanics in general relativity continue to hold for black holes in string theory. This motivated us to study the status of Martinez's conjecture in the context of the Kerr-Sen family of black holes, which arise as solutions in heterotic string theory. We first find the QLE of such black holes for a choice of three-slice Σ quasilocal surface B identical to the ones used to evaluate the QLE of charged Kerr black holes. This expression is found to have the correct limits when the radial coordinate is made to diverge, or when the charge is made to vanish. However, at the outer horizon, the QLE for such black holes does not reduce to twice its irreducible mass. It is important to note that the value of the QLE is influenced by the choice of a reference term. In fact, at the outer horizon it is solely this reference term that contributes to the QLE. It may be, therefore, be possible to motivate an alternative reference term that is concommitant with the applicability of Martinez's conjecture even for such black holes. We hope to return to this issue elsewhere.

ACKNOWLEDGMENTS
We thank Naresh Dadhich and Mohammad Nouri-Zonoz for helpful discussions. One of us (TZN) would like to express his gratitude to IUCAA for the hospitality and financial support, which he received during his stay in IUCAA.

APPENDIX A: QLE ON A TWO-SPHERE EMBEDDED IN Σ
Here we show how to compute the trace the extrinsic curvature, k, corresponding to a closed two-surface B embedded in a 3D axisymmetric Riemannian manifold. The result can then be used in Eq. (2.1) to evaluate the energy integrals appearing in several places in this paper. An analogous calculation is given in Ref. [10]. It is discussed here for the sake of completeness only.
Consider a 3D axisymmetric spacelike hypersurface Σ described by the line element where x i = (r, ϑ, ϕ) denote arbitrary coordinates adapted to the symmetry. The metric coefficients x, y, and z depend only on the "radial" and "azimuthal" coordinates r and ϑ, respectively. An arbitrary 2D axisymmetric surface B having the topology of a two-sphere, and embedded in 3D space Σ is defined by the relation r = R(ϑ), where R is a function of the azimuthal angle and the parameters of the solution. Its two-dimensional line element is σ ab dx a dx b = (x 2 R ′ 2 + y 2 )dϑ 2 + z 2 dϕ 2 , where a prime denotes differentiation with respect to the coordinate ϑ. The functions x, y, and z in (A2) are evaluated at the two-surface r = R(ϑ). Let n i denote the unit outward-pointing spacelike normal to B as embedded in Σ. Its components are (n r , n ϑ , n ϕ ) = 1 √ y 2 + x 2 R ′2 (y/x , −x R ′ /y , 0) .
The extrinsic curvature of the two-surface B as embedded in Σ is denoted by k µν . Its trace k can be written as where h denotes the determinant of the three-metric h µν . Using the coordinate-components of the unit normal (A3) we find that the trace is