Physics Letters B 798 (2019) 135019
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Smarr mass formulas for BPS multicenter black holes
E. Torrente-Lujan
a,b,∗
a
Dept. de Física, U. de Murcia, Campus de Espinardo, 30100 Murcia, Spain
b
TH division, CERN, CH-1211 Geneve 23, Switzerland
a r t i c l e i n f o a b s t r a c t
Article history:
Received
4 September 2019
Received
in revised form 5 October 2019
Accepted
8 October 2019
Available
online 11 October 2019
Editor:
M. Cveti
ˇ
c
Mass formulas for multicenter BPS 4D black holes are presented. In the case of two center BPS solutions,
the ADM mass can be related to the intercenter distance r, the angular momentum J
2
, the dyonic charge
vectors q
i
and the value of the scalar moduli at infinity (z
∞
)by the Smarr-like expression
M
2
ADM
= A
1 + α J
2
1 + 2M
ADM
/r + A/r
2
where A(Q ), α(q
i
) are symplectic invariant quantities (Q , the total charge vector) depending on the
special geometry prepotential defining the theory. The formula predicts the existence of a continuos
class, for fixed value of the charges, of BH’s with interdistances r ∈ (0, ∞) and M
ADM
∈ (∞, M
∞
). First
Law expressions incorporating the intercenter distance are obtained from it:
dM ≡ dJ+
i
dq
i
+ Fdr,
in addition to an effective angular velocity and electromagnetic potentials
i
, the equation allows
to define an effective “force”, F , acting between the centers. This effective force is always negative: at
infinity and at short distances we recover the familiar Newton law F ∼ 1/r
2
at the leading order. Similar
results can be easily obtained for more general models and number of centers.
© 2019 The Author. 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
Extended theories of gravity as Supergravity and its solutions,
in particular supersymmetric extremal black hole solutions, con-
tinues
to be central for M-theory, string theory phenomenology,
quantum properties of black holes, and the AdS/CFT correspon-
dence.
Applications can be found from extensions of the particle
physics SM to supersymmetric black hole solutions and strongly
coupled systems.
Black
hole physics, in general relativity or in extended theories
as supergravity, is of interest in different backgrounds: from astro-
physics
to classical general relativity, quantum field theory, particle
physics, string and supergravity. Nowadays black holes in Super-
gravity
theories are used to answer to condensed matter questions
in particular in strongly correlated fermionic systems including
high T superconductors [1–7].
*
Correspondence to: Dept. de Física, U. de Murcia, Campus de Espinardo, 30100
Murcia, Spain.
E-mail
address: torrente@cern.ch.
It is a rather trivial problem the existence and construction of
extremal BH solutions in a wide number of well known theories.
The two-parameters Reissner-Nordstrom (RN) metric, for example,
describes black holes of (ADM) mass M and charge Q only when
the ratio Q /M is sufficiently small. In the extremal case, the bor-
derline
between naked singularities and black hole solutions, the
mass and electromagnetic charges P , Q are related by
M
2
= P
2
+ Q
2
.
This can be considered a (necessary and sufficient) condition on
the macroscopic parameters for the existence of a extremal RN BH.
Solutions saturating this bound can be considered as the stable fi-
nal
state of Hawking evaporation [8]. For the Majumdar-Papatreou
solution with
H = 1 +
i
M
i
/|x − x
i
|,
the conditions are M
i
> 0with M
2
i
= q
2
i
, M
ADM
=
M
i
. For the
three parameter Kerr-Newman extremal case, the mass, charge and
https://doi.org/10.1016/j.physletb.2019.135019
0370-2693/
© 2019 The Author. 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
.
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