JHEP10(2015)161
two-form fields, for a generic choice of boundary conditions, the mass and also the angular
momentum become automatically finite, and acquire explicit contributions from the matter
field. In section 2.1, it is shown that requiring compatibility of the boundary conditions
with the scaling symmetry of the class of stationary solutions, singles out a precise set of
“holographic boundary conditions” that is described by an arbitrary function of a single
variable. It can also be seen that if one further requires the holographic boundary conditions
to be compatible with Lorentz symmetry, selects a very special subset that is parametrized
just by a single arbitrary fixed constant. In section 3 we compare the global charges that
are obtained in the case of an electrically charged rotating black hole solution for different
choices of boundary conditions. It is shown that the standard result in [2] corresponds
to the simplest choice of Lorentz invariant boundary conditions. Noteworthy, it is found
that if the boundary conditions are compatible with both Lorentz and scaling symmetries,
the energy spectrum of an electrically charged (rotating) black hole is nonnegative, and
also, for a fixed value of the mass, the electric charge becomes bounded from above. We
conclude with further remarks in section 4.
2 Reduced action principle: stationary solutions, integrability conditions
and conserved charges
Let us consider the electromagnetic field minimally coupled to General Relativity with
negative cosmological constant in three spacetime dimensions. The action reads
I =
ˆ
d
3
x
√
−g
1
2κ
(R − 2Λ) −
0
4
F
αβ
F
αβ
, (2.1)
where the Newton constant G and the AdS radius l are defined through κ = 8πG and
Λ = −l
−2
, respectively. The “vacuum permeability”
0
has units of length, and is assumed
to be fixed as
0
= 1. A wide family of exact stationary circularly symmetric solutions has
already been found in the literature [1, 2, 7–22]. Hereafter we deal with generic stationary
circularly symmetric configurations, so that the spacetime metric can be written as
ds
2
= −N (r)
2
F (r)
2
dt
2
+
dr
2
F (r)
2
+ R(r)
2
N
φ
(r) dt + dφ
2
, (2.2)
and the gauge field can be chosen to be given by
1
A = A
t
(r) dt + A
φ
(r) dφ . (2.3)
It is then clear that the form of (2.2) and (2.3) is mapped into itself under the action of
a Lorentz boost in the “t − φ cylinder”. It is also worth pointing out that the metric and
the gauge field are invariant under scalings of the form
r → λr , t → λ
−1
t , φ → λ
−1
φ , (2.4)
1
Here the gauge field has been assumed to be stationary and circularly symmetric. Had we made the
same assumption for the field strength, one would obtain a wider class of configurations, so that (2.3)
corresponds to one of the two possible branches (see e.g., [22, 23]).
– 2 –
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