Smarr formula for black holes endowed with both electric and magnetic charges

Abstract

The present paper clarifies how to consistently take into account the contribution of magnetic charges in the generalized Smarr mass formula by using properly the Tomimatsu's representation of Komar integrals. It is shown in particular that in all three examples of the dyonic solutions considered by us, the sum of the two electromagnetic terms in Smarr's formula can be cast into the formQΦ H ext , where Q is the complex charge and Φ H ext the complex extension of the electric potential.

Equatorially symmetric configurations of two Kerr-Newman black holes

“…At the same time, while the potentials Φ e and Φ m are symmetric with respect to the change q → β, β → q in the solutions where the magnetic charges are introduced by means of the duality rotation of the Ernst potential Φ [19], in our nontrivial dyonic configuration these Φ e and Φ m are defined by different, nonsymmetric expressions. As a consequence, the generalized Smarr formula (40) in our case cannot be cast into a more elegant form (by introducing a complex charge q + iβ) like this was done in the paper [19].…”

“…2). Therefore, we have a nontrivial binary system of dyonic KN black holes in which the magnetic charges are not introduced via the duality rotation of the potential Φ, in contrast to all the dyonic solutions studied for example in the paper [19]. In what follows we shall see that thermodynamics of the black holes in our system is subject to the generalized Smarr mass formula which takes into account the contribution of the magnetic field.…”

“…The other physical characteristics that might be of interest to us are the horizon's area A, the surface gravity κ, horizon's angular velocity Ω and the magnetic potential Φ m , which all can be calculated by means of the formulas of the paper [21], taking into account the relation of Φ m to the electric potential Φ e of the associated problem [19]. Assuming the validity of the Bekenstein-Hawking formula S = A/4 between the entropy S and horizon's area A [22,23], and also recalling that the Hawking temperature T is related to the surface gravity as T = κ/(2π), we give below the formulas for S, T , Φ m and Ω calculated for the lower black hole of our particular binary configuration:…”

“…It is worth noting that the case of corotating KN black holes with opposite electric charges is trivially obtainable from the above configuration of magnetically charged KN black holes by formally changing in (19) the electromagnetic Ernst potential Φ to iΦ, in which case b becomes an electric dipole parameter, while β becomes the electric charge. Moreover, the 19), ( 43) and ( 22) leads, after the analogous complex extension of the magnetic charge parameter β → β − iq, β 2 → β 2 + q 2 to the case of two dyonic KN black holes endowed with opposite electric and magnetic charges, and then the Smarr mass relation takes the form discussed in [19]. In the paper [30] it was clarified that in order to treat correctly the solutions involving both electric and magnetic charges it is best to identify first the particular case in which only the electric charges are present and then apply the extension parameter procedure.…”

“…Though our main results which will be discussed in the present paper were obtained more than a year ago, their publication was postponed due to the paper [7] criticizing the extension of the well-known Smarr mass formula [8] to the case of dyonic black holes [19],…”