5.3 The Israel-Wilson Class5 Applications of the Coset 5.1 The Mazur Identity

5.2 Mass Formulae 

The stationary vacuum Einstein equations describe a two-dimensional tex2html_wrap_inline3625 -model which is effectively coupled to three-dimensional gravity. The target manifold is the pseudo-sphere tex2html_wrap_inline4309, which is parametrized in terms of the norm and the twist potential of the Killing field (see Sect. 4.3). The symmetric structure of the target space persists for the stationary EM system, where the four-dimensional coset, tex2html_wrap_inline4295, is represented by a hermitian matrix tex2html_wrap_inline4215, comprising the two electro-magnetic scalars, the norm of the Killing field and the generalized twist potential (see Sect. 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 tex2html_wrap_inline4315, say, Breitenlohner et al. [14Jump To The Next Citation Point In The Article] have classified the models admitting a symmetry group which is sufficiently large to comprise all scalar fields arising on the effective level Popup Footnote within one coset space, G / H . A prominent example of this kind is the EM-dilaton-axion system, which is relevant to tex2html_wrap_inline4319 supergravity and to the bosonic sector of four-dimensional heterotic string theory: The pure dilaton-axion system has an tex2html_wrap_inline4321 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 [63], [64] have shown that the full symmetry group is, however, larger than tex2html_wrap_inline4325 : The target space for dilaton-axion gravity with an U (1) vector field is the coset tex2html_wrap_inline4329 [62]. Using the fact that SO (2,3) is isomorphic to tex2html_wrap_inline4333, Gal'tsov and Kechkin [65] were also able to give a parametrization of the target space in terms of tex2html_wrap_inline4335 (rather than tex2html_wrap_inline4337) matrices. The relevant coset was shown to be tex2html_wrap_inline4339 . Popup Footnote

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 In The Article]. The reason for this is the following: Like the EM equations (32Popup Equation), the stationary field equations consist of the three-dimensional Einstein equations and the tex2html_wrap_inline3625 -model equations,

  equation991

The current one-form tex2html_wrap_inline4343 is given in terms of the hermitian matrix tex2html_wrap_inline4215, which comprises all scalar fields arising on the effective level. The tex2html_wrap_inline3625 -model equations, tex2html_wrap_inline4349, include tex2html_wrap_inline4351 differential current conservation laws, of which tex2html_wrap_inline4353 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. Popup Footnote The crucial observation is that Stokes' theorem provides tex2html_wrap_inline4351 independent Smarr relations, rather than only tex2html_wrap_inline4357 ones. (This is due to the fact that all tex2html_wrap_inline3625 -model currents are algebraically independent, although there are tex2html_wrap_inline4353 differential identities which can be derived from the tex2html_wrap_inline4357 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 In The Article],

  equation1019

where tex2html_wrap_inline3689 and tex2html_wrap_inline4367 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. Popup Footnote

A very simple illustration of the idea outlined above is the static, purely electric EM system. In this case, the electrovac coset tex2html_wrap_inline4295 reduces to tex2html_wrap_inline4379 . The matrix tex2html_wrap_inline4215 is parametrized in terms of the electric potential tex2html_wrap_inline4045 and the gravitational potential tex2html_wrap_inline4385 . The tex2html_wrap_inline3625 -model equations comprise tex2html_wrap_inline4389 differential conservation laws, of which tex2html_wrap_inline4391 is redundant:

  equation1038

  equation1049

[It is immediately verified that Eq. (39Popup Equation) is indeed a consequence of the Maxwell and Einstein Eqs. (38Popup Equation).] Integrating Eqs. (38Popup Equation) over a space-like hyper-surface and using Stokes' theorem yields Popup Footnote

  equation1062

which is the well-known Smarr formula. In a similar way, Eq. (39Popup Equation) provides an additional relation of the Smarr type,

  equation1070

which can be used to compute the horizon-value of the electric potential, tex2html_wrap_inline4403 . Using this in the Smarr formula (40Popup Equation) gives the desired expression for the total mass, tex2html_wrap_inline4405 .

In the ``extreme'' case, the BPS bound [75] for the static EM-dilaton-axion system, tex2html_wrap_inline4407, was previously obtained by constructing the null geodesics of the target space [45]. For spherically symmetric configurations with non-degenerate horizons (tex2html_wrap_inline4409), Eq. (37Popup Equation) was derived by Breitenlohner et al. [14]. In fact, many of the spherically symmetric black hole solutions with scalar and vector fields [73], [76], [69] are known to fulfill Eq. (37Popup 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. [72] recently obtained Eq. (37Popup Equation) for spherically symmetric solutions with an arbitrary number of vector and moduli fields.

The above derivation of the mass formula (37Popup 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 (37Popup 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

  equation1100

provided that the stationary field equations assume the form (36Popup Equation), where tex2html_wrap_inline4215 is a map into a symmetric space, G / H . Here tex2html_wrap_inline4415 and tex2html_wrap_inline4417 denote the charges of the scalars (including the gravitational ones) and the vector fields, respectively.



5.3 The Israel-Wilson Class5 Applications of the Coset 5.1 The Mazur Identity

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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