Physics Letters B 773 (2017) 290–294
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Physics Letters B
www.elsevier.com/locate/physletb
On the Smarr formula for rotating dyonic black holes
Gérard Clément
a,∗
, Dmitri Gal’tsov
b,c
a
LAPTh, Université Savoie Mont Blanc, CNRS, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France
b
Department of Theoretical Physics, Faculty of Physics, Moscow State University, 119899, Moscow, Russia
c
Kazan Federal University, 420008 Kazan, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
11 July 2017
Accepted
17 August 2017
Available
online 24 August 2017
Editor:
M. Cveti
ˇ
c
We revisit the derivation by Tomimatsu of the generalized Komar integrals giving the mass and angular
momentum of rotating Einstein–Maxwell black holes. We show that, contrary to Tomimatsu’s claim, the
usual Smarr formula relating the horizon mass and angular momentum still holds in the presence of
both electric and magnetic charges. The simplest case is that of dyonic Kerr–Newman black holes, for
which we recover the modified Smarr formula relating the asymptotic mass and angular momentum,
the difference between asymptotic and horizon masses being equal to the sum of the two Dirac
string masses. Our results apply in particular to the case of dyonic dihole solutions which have been
investigated recently.
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The Smarr formula [1] relating the mass, angular momentum,
entropy and electric charge of black holes was originally designed
for the electrically charged Kerr–Newman solution. Its possible
generalizations were extensively analyzed by Carter [2] on the
basis of Komar integrals [3]. More general solutions describing
axisymmetric configurations of multiple rotating black holes (pos-
sibly
joined by strings) endowed with electric and also magnetic
charges were discussed recently. These solutions usually have a
simple description in terms of Ernst potentials, while the met-
ric
and the electromagnetic potentials are rather complicated. For
such situations Tomimatsu designed his original formulas [4,5] ex-
pressing
black hole parameters in terms of both the metric vari-
ables
and the Ernst potentials taken on the symmetry axis. These
formulas suggested in 1984 were successfully applied for multiple
electrically charged rotating black holes [6].
On
the other hand, when the Tomimatsu formulas (completed
by an analogous expression for magnetic charge) were applied to
multi-dyons [5,7–10], it was observed that the resulting values
for the black hole parameters failed to obey the standard Smarr
relation, but obeyed a generalized Smarr relation with both elec-
tric
and magnetic contributions. However, the derivation by Tomi-
matsu [5] gives
little details of the underlying calculations, so to
*
Corresponding author.
E-mail
addresses: gclement@lapth.cnrs.fr (G. Clément), galtsov@phys.msu.ru
(D. Gal’tsov).
clarify the situation a new derivation is necessary. Here such a
derivation is presented, showing that in the original Tomimatsu
formulas an important term is missing. Correcting the Tomimatsu
formulas, we obtain a new version which, when applied to dyons,
leads to the standard Smarr relation between the local horizon
mass, angular momentum and electric charge.
In
passing we establish the crucial role played by the Dirac
strings associated with magnetic monopoles in the mass and
angular momentum balance equations. We show that for the
Kerr–Newman solution with both electric and magnetic charges
the Dirac strings are endowed with non-zero generalized Komar
masses which should be taken into account in the Smarr formula
for the total mass. We also find that the symmetric choice of
gauge for the vector potential (with both North and South pole
Dirac strings present with equal weights) for dyons is essential to
achieve the total angular momentum balance of the configuration.
2. Generalized Komar mass and angular momentum
We first review the generalized Komar formulas [2] giving the
masses and angular momenta of extended sources of Einstein–
Maxwell
fields. The Komar mass and angular momentum for
an asymptotically flat, stationary, axisymmetric configuration are
given by the integrals over a spacelike 2-surface at infinity [3]
1
:
1
We use the metric signature (−+++) and the convention d
μν
=
1/2
√
|g|
μνλτ
dx
λ
dx
τ
with
tρzϕ
= 1in Weyl coordinates. We will label t, ϕ by
an index a, and the remaining coordinates ρ, z by i, j. In Sect. 4 we will also use
http://dx.doi.org/10.1016/j.physletb.2017.08.041
0370-2693/
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.