5.2 Mass formulae

The stationary vacuum Einstein equations describe a two-dimensional σ-model which is effectively coupled to three-dimensional gravity. The target manifold is the pseudo-sphere SO (2, 1)∕SO (2) ≈ SU (1,1)∕U (1), which is parametrized in terms of the norm and the twist potential of the Killing field (see Section 4.3). 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 4.5).

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. [15Jump 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 level48 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,IR ) symmetry which persists in dilaton-axion gravity with an Abelian gauge field [61]. 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. Gal’tsov and Kechkin [6364] have shown that the full symmetry group is, however, larger than SL (2,IR ) × SO (1,2): The target space for dilaton-axion gravity with an U (1) vector field is the coset SO (2,3)∕(SO (2) × SO (1,2)) [60]. Using the fact that SO (2,3) is isomorphic to Sp (4,IR), Gal’tsov and Kechkin [65] were also able to give a parametrization of the target space in terms of 4 × 4 (rather than 5 × 5) matrices. The relevant coset was shown to be Sp (4,IR)∕U (1,1).49

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 [89Jump To The Next Citation Point]. The reason for this is the following: Like the EM equations (32View Equation), the stationary field equations consist of the three-dimensional Einstein equations and the σ-model equations,

1 ¯Rij = --Trace {𝒥i𝒥j } , d¯∗𝒥 = 0 . (36 ) 4
The current one-form 𝒥 ≡ Φ −1dΦ 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 hyper-surface extending from the horizon to infinity, Stokes’ theorem yields a set of relations between the charges and the horizon-values of the scalar potentials.50 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 formulas 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 [89Jump To The Next Citation Point],

( )2 -1- 2 2 2 2 2 2 4π κ𝒜 = M + N + D + A − Q − P , (37 )
where κ and 𝒜 are the surface gravity and the area of the horizon, and the RHS 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.51

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

( dϕ ) (d σ dϕ ) d¯∗ --- = 0, d¯∗ --- − 2ϕ --- = 0, (38 ) σ σ σ
(( ) dϕ dσ ) d¯∗ σ + ϕ2 ---− ϕ --- = 0. (39 ) σ σ
[It is immediately verified that Eq. (39View Equation) is indeed a consequence of the Maxwell and Einstein Eqs. (38View Equation).] Integrating Eqs. (38View Equation) over a space-like hyper-surface and using Stokes’ theorem yields52
Q = Q , M = -κ-𝒜 + ϕ Q , (40 ) H 4 π H H
which is the well-known Smarr formula. In a similar way, Eq. (39View Equation) provides an additional relation of the Smarr type,
Q = 2ϕH -κ-𝒜 + ϕ2HQH , (41 ) 4π
which can be used to compute the horizon-value of the electric potential, ϕH. Using this in the Smarr formula (40View Equation) gives the desired expression for the total mass, M 2 = (κ 𝒜∕4 π)2 + Q2.

In the “extreme” case, the BPS bound [74] for the static EM-dilaton-axion system, 0 = M 2 + D2 + A2 − Q2 − P 2, was previously obtained by constructing the null geodesics of the target space [45]. For spherically symmetric configurations with non-degenerate horizons (κ ⁄= 0), Eq. (37View Equation) was derived by Breitenlohner et al. [15]. In fact, many of the spherically symmetric black hole solutions with scalar and vector fields [727669] are known to fulfill Eq. (37View Equation), where the LHS is expressed in terms of the horizon radius (see [67] and references therein). Using the generalized first law of black hole thermodynamics, Gibbons et al. [75] recently obtained Eq. (37View Equation) for spherically symmetric solutions with an arbitrary number of vector and moduli fields.

The above derivation of the mass formula (37View 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 formulas which is sufficiently large to derive the desired relation. Although the result (37View Equation) was established by using the explicit representations of the EM and EM-dilaton-axion coset spaces [89], 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 = -- (QS )2 − (QV )2 , (42 ) 𝒜
provided that the stationary field equations assume the form (36View 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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