7.2 Mass formulae

The stationary vacuum Einstein equations describe a two-dimensional σ-model coupled to three-dimensional gravity. The target manifold is the pseudo-sphere SO (2,1 )∕SO (2) ≈ SU (1,1)∕U (1), which is parameterized in terms of the norm and the twist potential of the Killing field (see Section 6.2). The symmetric structure of the target space persists for the stationary EM system, where the four-dimensional coset, SU (2,1)∕S(U (1,1) × U(1)), is represented by a hermitian matrix Φ, comprising the two electro-magnetic scalars, the norm of the Killing field and the generalized twist potential (see Section 6.4).

The coset structure of the stationary field equations is shared by various self-gravitating matter models with massless scalars (moduli) and Abelian vector fields. For scalar mappings into a symmetric target space ¯ ¯ G∕ H, say, Breitenlohner et al. [31Jump To The Next Citation Point] have classified the models admitting a symmetry group, which is sufficiently large to comprise all scalar fields arising on the effective level10 within one coset space, G∕H. A prominent example of this kind is the EM-dilaton-axion system, which is relevant to N = 4 supergravity and to the bosonic sector of four-dimensional heterotic string theory: The pure dilaton-axion system has an SL (2,ℝ ) symmetry, which persists in dilaton-axion gravity with an Abelian gauge field [114]. Like the EM system, the model also possesses an SO (1,2) symmetry, arising from the dimensional reduction with respect to the Abelian isometry group generated by the Killing field. However, Gal’tsov and Kechkin [116, 117] have shown that the full symmetry group is larger than SL (2,ℝ ) × SO (1,2): The target space for dilaton-axion gravity with a U (1 ) vector field is the coset SO (2, 3)∕(SO (2) × SO (1,2)) [113]. Using the fact that SO (2,3) is isomorphic to Sp (4,ℝ), Gal’tsov and Kechkin [118] were also able to give a parametrization of the target space in terms of 4 × 4 (rather than 5 × 5) matrices. The relevant coset space was shown to be Sp (4,ℝ)∕U (1,1); for the generalization to the dilaton-axion system with multiple vector fields we refer to [119, 121].

Common to the black-hole solutions of the above models is the fact that their Komar mass can be expressed in terms of the total charges and the area and surface gravity of the horizon [153Jump To The Next Citation Point]. The reason for this is the following: Like the EM equations (6.26View Equation), the stationary field equations consist of the three-dimensional Einstein equations and the σ-model equations,

1 ¯Rij = -Trace (𝒥i𝒥j ), d¯∗ 𝒥 = 0. (7.4 ) 4
The current one-form 𝒥 := Φ−1d Φ is given in terms of the Hermitian matrix Φ, which comprises all scalar fields arising on the effective level. The σ-model equations, d¯∗ 𝒥 = 0, include dim (G) differential current conservation laws, of which dim (H ) are redundant. Integrating all equations over a space-like hypersurface extending from the horizon to infinity, Stokes’ theorem yields a set of relations between the charges and the horizon-values of the scalar potentials. A very familiar relation of this kind is the Smarr formula [296]; see Eq. (7.8View Equation) below. The crucial observation is that Stokes’ theorem provides dim (G ) independent Smarr relations, rather than only dim (G ∕H ) ones. (This is due to the fact that all σ-model currents are algebraically independent, although there are dim (H ) differential identities, which can be derived from the dim (G ∕H ) field equations.)

The complete set of Smarr type formulae can be used to get rid of the horizon-values of the scalar potentials. In this way one obtains a relation, which involves only the Komar mass, the charges and the horizon quantities. For the EM-dilaton-axion system one finds, for instance [153Jump To The Next Citation Point],

( )2 -1- 2 2 2 2 2 2 4π κ𝒜 = M + N + D + A − Q − P , (7.5 )
where κ and 𝒜 are the surface gravity and the area of the horizon, and the right-hand side comprises the asymptotic flux integrals, that is, the total mass, the NUT charge, the dilaton and axion charges, and the electric and magnetic charges, respectively. The derivation of Eq. (7.5View Equation) is not restricted to static configurations. However, when evaluating the surface terms, one assumes that the horizon is generated by the same Killing field that is also used in the dimensional reduction; the asymptotically time-like Killing field k. A generalization of the method to rotating black holes requires the evaluation of the potentials (defined with respect to k) on a Killing horizon, which is generated by ℓ = k + ΩH m, rather than k.

A very simple illustration of the idea outlined above is the static, purely electric EM system. In this case, the electrovacuum coset SU (2,1)∕S (U (1,1) × U (1)) reduces to G ∕H = SU (1,1)∕ℝ. The matrix Φ is parameterized in terms of the electric potential ϕ and the gravitational potential V := − k kμ μ. The σ-model equations comprise dim (G) = 3 differential conservation laws, of which dim (H ) = 1 is redundant:

( ) ( ) d¯∗ dϕ- = 0, d¯∗ dV--− 2ϕdϕ- = 0, (7.6 ) V V V
( ( ) dϕ dV ) d¯∗ V + ϕ2 ---− ϕ ---- = 0. (7.7 ) V V
[It is immediately verified that Eq. (7.7View Equation) is indeed a consequence of the Maxwell and Einstein Eqs. (7.6View Equation).] Integrating Eqs. (7.6View Equation) over a space-like hypersurface and using Stokes’ theorem yields
Q = Q , M = -κ-𝒜 + ϕ Q , (7.8 ) H 4π H H
which is the well-known Smarr formula; to establish it, one also uses the fact that the electric potential assumes a constant value ϕH on the horizon. Also, the quantity QH is defined by the flux integral of ∗F over the horizon (at time Σ), while the corresponding integral of ∗dk gives κ𝒜 ∕4π (see [153Jump To The Next Citation Point] for details). In a similar way, Eq. (7.7View Equation) provides an additional relation of the Smarr type,
-κ- 2 Q = 2ϕH 4π 𝒜 + ϕHQH , (7.9 )
which can be used to compute the horizon-value of the electric potential, ϕH. Using this in the Smarr formula (7.8View Equation) gives the desired expression for the total mass, 2 2 2 M = (κ𝒜 ∕4π ) + Q.

In the “extreme” case, the Bogomol’nyi–Prasad–Sommerfield (BPS) bound [128] for the static EM-dilaton-axion system, 0 = M 2 + D2 + A2 − Q2 − P 2, was previously obtained by constructing null geodesics of the target space [86]. For spherically-symmetric configurations with non-degenerate horizons (κ ⁄= 0), Eq. (7.5View Equation) was derived by Breitenlohner et al. [31]. In fact, many of the spherically-symmetric black-hole solutions with scalar and vector fields [126, 131, 122] are known to fulfill Eq. (7.5View Equation), where the left-hand side is expressed in terms of the horizon radius (see [120] and references therein). Using the generalized first law of black-hole thermodynamics, Gibbons et al. [130] obtained Eq. (7.5View Equation) for spherically-symmetric solutions with an arbitrary number of vector and moduli fields.

The above derivation of the mass formula (7.5View Equation) is neither restricted to spherically-symmetric configurations, nor are the solutions required to be static. The crucial observation is that the coset structure gives rise to a set of Smarr formulae, which is sufficiently large to derive the desired relation. Although the result (7.5View Equation) was established by using the explicit representations of the EM and EM-dilaton-axion coset spaces [153], similar relations are expected to exist in the general case. More precisely, it should be possible to show that the Hawking temperature of all asymptotically-flat (or asymptotically NUT) non-rotating black holes with massless scalars and Abelian vector fields is given by

2 ∘ ∑-----------∑-------- TH = 𝒜- (QS)2 − (QV )2, (7.10 )
provided that the stationary field equations assume the form (7.4View Equation), where Φ is a map into a symmetric space, G∕H. Here QS and QV denote the charges of the scalars (including the gravitational ones) and the vector fields, respectively.
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