Physics Letters B 760 (2016) 759–762
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
On electromagnetic energy in Bardeen and ABG spacetimes
V.S. Manko
a,∗
, E. Ruiz
b
a
Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, A.P. 14-740, 07000 Ciudad de México, Mexico
b
Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca, 37008 Salamanca, Spain
a r t i c l e i n f o a b s t r a c t
Article history:
Received
3 May 2016
Accepted
27 July 2016
Available
online 29 July 2016
Editor:
M. Cveti
ˇ
c
Keywords:
Bardeen
model
Nonlinear
electrodynamics
Regular
black holes
Electromagnetic
energy
We demonstrate that the total energy of electromagnetic field in the Bardeen and Ayón-Beato–García
singularity-free models is equal to the mass parameter M, being therefore independent of the charge
parameter Q . Our result is fully congruent with the original idea of Born and Infeld to use nonlinear
electrodynamics for proving the electromagnetic nature of mass.
© 2016 The Authors. 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 construction of regular black hole models was pioneered by
Bardeen [1] who ingeniously modified the well-known Reissner–
Nordström metric
[2,3] for spherically symmetric mass and charge
to remove the singularity at r = 0. Though the global mathemat-
ical
properties of Bardeen’s spacetime are well understood (see,
e.g., [4]), the corresponding electromagnetic source for this space-
time
was unknown for many years, as the Bardeen model did not
originally arise as a solution to some field equations. The first exact
regular black hole solutions were constructed, within the frame-
work
of Einstein’s gravity coupled to nonlinear electrodynamics, by
Ayón-Beato and García [5,6] who also later reinterpreted Bardeen’s
model as an exact solution for a nonlinear magnetic monopole [7].
Despite a considerable attention these solutions have received in
recent years, it seems that the main physical question concern-
ing
the Bardeen and ABG spacetimes – How can an arbitrarily
small charge remove the physical Schwarzschild singularity of a
collapsed star with enormous mass? – still has not been clarified
so far. Being strongly convinced that the answer to this question
must be closely related to the issue of electromagnetic energy as-
sociated
with the above spacetimes, in the present letter we will
calculate the total electromagnetic energy in the Bardeen and ABG
models to reveal that for all these models it has the same value
that does not actually depend on the charge parameter Q .
*
Corresponding author.
E-mail
address: vsmanko@fis.cinvestav.mx (V.S. Manko).
2. The total energy of electric field in ABG solutions
We start our consideration with the first ABG solution [5] de-
fined
by the metric
ds
2
=−fdt
2
+ f
−1
dr
2
+r
2
d
2
,
f = 1 −
2Mr
2
(r
2
+ Q
2
)
3/2
+
Q
2
r
2
(r
2
+ Q
2
)
2
, (1)
and the associated electric field
E = Qr
4
r
2
−5Q
2
(r
2
+ Q
2
)
4
+
15
2
M
(r
2
+ Q
2
)
7/2
,
(2)
the parameters M and Q standing for the mass and electric charge
of the source, respectively.
For
a static observer, u
α
= (−g
tt
)
−1/2
ξ
α
, ξ
α
being the timelike
Killing vector, the energy density of the electromagnetic field is
defined as
T
αβ
u
α
u
β
= (−g
tt
)
−1
T
tt
=−T
t
t
, (3)
where T
αβ
is the electromagnetic energy–momentum tensor, so
that the quantity −T
t
t
can be calculated either via the construction
of the tensor T
αβ
from the corresponding tensor F
αβ
of nonlinear
electrodynamics or, more directly, from the Einstein equations,
T
αβ
=
1
8π
G
αβ
. (4)
http://dx.doi.org/10.1016/j.physletb.2016.07.066
0370-2693/
© 2016 The Authors. 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
.