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JHEP07(2016)004

Published for SISSA by Springer

Received: June 15, 2016

Accepted: June 15, 2016

Published: July 1, 2016

Dyonic (A)dS black holes in Einstein-Born-Infeld

theory in diverse dimensions

Shoulong Li,

H. L¨u

and Hao Wei

School of Physics, Beijing Institute of Technology,

5 South Zhongguancun Street, Beijing 100081, China

Center for Advanced Quantum Studies,

Department of Physics, Beijing Normal University,

19 Xinjiekouwai Street, Beijing 100875, China

E-mail:

sllee phys@bit.edu.cn, mrhonglu@gmail.com, haowei@bit.edu.cn

Abstract: We study Einstein-Born-Infeld gravity and construct the dyonic (A)dS planar

black holes in general even dimens i ons, that carry both the electric charge and magnetic

ﬂuxes along the planar space. In four dimensions, the solution can be constructe d with

also spherical and hyperbolic topologies. We study the black hole thermodynamics and

obtain the ﬁrst law. We also classify the singularity structure.

Keywords: Black Holes, Classical Theorie s of Gravity, Spacetime Singularities

ArXiv ePrint: 1606.02733

Open Access,

 The Authors.

Article funded by SCOAP

doi:

10.1007/JHEP07(2016)004

JHEP07(2016)004

BI theory has enjoyed further attentions since the invention of string theory. It turns

out that the BI action can arise from string theory [

2], describing the low energy dynamics

of D-branes [

3]. We refer to e.g. [4, 5] for some comprehensive reviews on the BI theory

in s tr i ng theory. The special Born-Infeld-like nonlinear form is al s o very useful to con-

struct analogous new theori e s, such as Dirac-Born-Infeld (DBI) inﬂation theory [

6, 7] and

Eddington-inspired Born-Infeld (EiBI) cosmologies [

8]. BI theory can als o be adopted to

explore issues of dark energy [

9, 10].

In this paper, we focus on the study of black holes in Einstein -B orn -Inf el d (EBI)

theory. The most general static type-D metric of the BI theory in four dimensions was

constructed in [

11]. (See also [12] and [13].) The sp he ri c al l y -s y mme t ri c solution was

generalized to arbitrary D dimensions in [

14] wher e the black hole thermodynamics was

studied. The black hole solutions was also generalized to include diﬀerent topologies [

15].

The Born-Infel d black hole solutions were also studied in Einstein theory with a dilaton

ﬁeld [

16] and in the modiﬁed gravity theories such as Gauss-Bonnet theory [17], Lovelock

theory [

18], Brans-Dicke theory [19], f(T ) theory [20], massive gravity [21] and so on. The

extended thermodynamics [

14–20, 22–24], geodesics [25, 26] , and AdS/CFT correspondence

properties [

27–29] were studied too. Other Born-Infeld solutions are also studied, for

example, thin-shell wormhol e s [

30].

(A)dS black hole solutions in BI theory considered in literature typically involves only

either the electric or magnetic charges. Although the dyonic black hole in EBI theory was

constructed in [

11], it is written in the general (static) type-D form. The global structure

in the spherically symmetric form was analysed in [

25] for the asymptotically-ﬂat case. In

this paper, we shall ﬁrst study the dyonic (A)dS black holes in the EBI theory in four

dimensions with general topologie s, focus on analysi ng the black hole thermodynamics and

singularity structure. We then construct dyonic AdS planar black holes in arbitrary even

dimensions, where the solutions carry both the elec t r i c ﬂux as well as the magnetic 2-form

ﬂux along the planar space.

Interest i ngl y in almost all the prev i ous works on constructing black holes, the equiv-

alent action in four dimensions was us ed , rather than the original one. In D = 4, the

Lagrangian can be equivalently expressed as [1]

L = b

− b

1 + I

+ I

, (1.3)

where

µν

− E

, I

= −

16b



µν



= −

(E · B)

, (1.4)

in which E and B are electric and magnetic ﬁelds, and

µν

µνρσ

ρσ

−det(η

)

µνρσ

ρσ

, (1.5)

where ε

µνρσ

is a tensor density with ε

0123

= 1.

The equivalence of (1.3) and (1.1) is only true in four dimensions; it is no longer valid in

higher dimensions. However, if one considers only static solutions carrying electric charges,

– 2 –

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