Rotated charged black holes in Einstein-Born-Infeld theories
In this work the solution of the Einstein equations for slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties, singularities, horizons of this solution are analyzed. There are considered the conditions when the black hole modifies its mass (like in the non-linear mon...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Rotated charged black holes in Einstein-Born-Infeld theories / Diego Julio Cirilo Lombardo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 71-73. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859620421060526080 |
|---|---|
| author | Diego Julio Cirilo Lombardo |
| author_facet | Diego Julio Cirilo Lombardo |
| citation_txt | Rotated charged black holes in Einstein-Born-Infeld theories / Diego Julio Cirilo Lombardo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 71-73. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | In this work the solution of the Einstein equations for slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties, singularities, horizons of this solution are analyzed. There are considered the conditions when the black hole modifies its mass (like in the non-linear monopole cases) and angular momentum for the same non-linear electromagnetic field what produces the black hole.
|
| first_indexed | 2025-11-29T03:08:31Z |
| format | Article |
| fulltext |
ROTATED CHARGED BLACK HOLES
IN EINSTEIN-BORN-INFELD THEORIES
Diego Julio Cirilo Lombardo
Buenos Aires University, Buenos Aires, Argentina
e-mail: diegoc@iafe.uba.ar
In this work the solution of the Einstein equations for slowly rotating black hole with Born-Infeld charge is
obtained. Geometrical properties, singularities, horizons of this solution are analyzed. There are considered the
conditions when the black hole modifies its mass (like in the non-linear monopole cases) and angular momentum for
the same non-linear electromagnetic field what produces the black hole.
PACS 04.20.Ha 04.70.Bw
INTRODUCTION
The four dimensional solutions with spherical
symmetry of the Einstein equations coupled to Born-
Infeld fields have been well studied in the literature [1].
In particular the Born-Infeld monopole, in contrast to
Maxwell, contributes to the mass of the compact object
source of the field. B. Hoffmann was the first who
studied such static solutions in the context of the general
relativity with the idea to obtain a consistent particle-
like model. Unfortunately, these static Einstein-Born-
Infeld models generate conical singularities at the origin
[1,9]. This type of singularities cannot be removed like
global monopoles or other non-localized topological
defects of the space-time [6,7]. In this report the
solution for slowly rotating black hole with Born-Infeld
charge is obtained. This solution presents coupled terms
of charge and angular momentum what modifies
asymptotically the mass and the angular momentum of
the rotating source. Families of solutions are obtained
varying b (Born-Infeld parameter). The fundamental
feature of this metric is the lack of conical singularities
at the origin (the rotation produces a repulsive potential
what protects the singular zone), and for particular
values of b there are solutions with M a Q2 2 2< + without
violation of cosmic censorship hypothesis.
ROTATED EINSTEIN-BORN-INFELD MODEL
In this case, one expects found a metric with an
asymptotic behaviour like the well-known Kerr-
Newman’s metric for the rotating and charged black
hole. But there exist few difficulties: the metric is non-
diagonal (rotating frame) and the energy-momentum
tensor of Born-Infeld includes the invariant
pseudoscalar [2,3], because of the rotation of the
compact object (it is appear magnetic field). The used
convention is the spatial of Landau and Lifshitz (1974),
with signatures of the metric, Riemann and Einstein all
positives (+++) [4,5].
STATEMENT OF THE PROBLEM
One proposes a line element like Kerr’s geometry in
the Boyer and Lindquist generalization [8,9], with the
expected asymptotic behaviour:
θρ+∆
ρ+−φ+
ρ
θ+
+φθ−
ρ
∆−=
ddradtdarSin
daSindtds
222222
2
2
22
2
2
where the functions ∆ and ρ are in principle depending
of r and θ and are to be determined. For obtaining the
Einstein equations, the most powerful is the Cartan’s
method [5,8,9]. This method applies differential forms
and is based on two fundamental geometric equations
(structure equations). In the 1-forms orthonormal basis
the line element is written:
( ) ( ) ( ) ( ) 2
3
2
2
2
1
2
02
ω+ω+ω+ω−=ds ,
where the association between coordinate and
orthonormal frame is not trivial in the case of an axially
rotating symmetry and requires to solve an equation
system. Because of the geometrical symmetries of the
Riemann tensor, ρ immediately can be determined:
θ+=ρ
2222
Cosar .
One can see, the ρ function is the same what the ρ of
Boyer and Lindquist and does not depend of the axially
symmetric source considered. With the function ρ
founded, only ∆ left to be find.
The calculation of the energy-momentum tensor
components in the rotating system will give all
information to determine the function ∆. For this we
shall use the metric symmetrized expression of Tab :
b
l
Fl
a
FP
BIL
b
l
Fl
a
FS
BIL
BILb
a
b
a
T ~
∂
∂
−∂
∂
−δ=
where:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 71-73. 71
( )4
2
2
2114
2
b
P
b
Sb
BIL −−−π=
and
( )cdFabcdabF
ab
FabFP
MaxwellLabFabFS
ε≡−≡
=−≡
2
1~~
4
1
;4
1
For the electromagnetic tensor F, one proposes the
form similar to the Boyer and Lindquist generalization
for a Kerr-Newman problem [8,9]:
13
31
02
20
22
31
2
20
ω∧ω+ω∧ω=
=−φ+∧θθ+
φθ−∧=
FF
adtdardSinF
daSindtdrFF
where F20 and F31are to be determined. One can see,
F20 and F31are the only field components in the tetrad
and the energy-momentum tensor takes the diagonal
form:
( ) ,14
2
2200 u
b
TT −π==−
( )11
4
2
3311
−−
π
== u
b
TT ,
where:
( )
( )
( )
b
abF
abF
F
F
u ≡
−
+
≡ 2
201
1
2
31
.
The fields are to be obtained from the dynamical
(Eulerian) equations in the tetrad form. Let us solve
these equations with the following boundary condition:
the fields asymptotically have the same behavior like
the electromagnetic fields of the Kerr-Newman model.
Then one obtains )( 24
0 bQr = :
( ) ( ) ( )[ ]4
0
224
22222
44
0
8
24
02
20 rCosaCosarr
rr
rr
F θ−θ−
+ρ
=
,
( ) ( ) ( )[ ]
( )[ ] 84
0
224
22222
22228
0
2224
24
0
44
02
31
ρθ−θ−
θ−θ+
=
rCosaCosarr
CosarrarCos
rr
rr
F .
Putting all the ingredients in the Einstein equations,
one obtains the next expression:
.2
22244
0
44
0
22422
222
222
44
0
22422
44
0222224
∆∂∂=+
θ−
+
+θ−ρ
ρ−
−
θ−
+θ−ρ
+
ρ−ρ
rr
Cosar
r
rr
rrCosar
b
r
Cosar
rrCosar
rr
bb
This expression, although exact, is not integrable by
transcendental functions like in the static cases. One
must to make an expansion in power series for a r/
small (slowly rotating). Looking the last equation, one
can see that ∆ depends of the radial coordinate r and
the angular coordinate θ. The integrals are calculated in
indefinite form and the values of the two constants A(θ)
and B(θ) of the problem are selected according to the
asymptotical behavior of ∆ and the metric. The
obtained solution takes the form:
( ) ( )
( )
( ) ( ) ,1,11
11
17cos2
385
21
385
313
770
16312cos2
2,
2
1
5
9
10
1
5
4cos2
8576.022
8653.2222525.12,
45
92,
0
4
1
0
4
1
24
4
0
2
10
10
0
6
6
0
2
2
0
4
0
4
2
0
2
24
4
0
2
2
0
2
0
44
04
4
044
0
22
2
4
0
2
4
4
0
2
422
424
5
0
2
2
3
0
2
22
−
−−θ−
−
−−−++
−θ+
+
−+−++θ+
+θ−θ++
+
θ−θ+−
θ+
π++=θ∆
r
riArcSinhF
r
ra
r
Q
r
r
r
r
r
r
r
r
r
rArcSinha
r
Q
r
riArcCosEirrrr
r
rrrr
r
Qa
a
r
QSinSina
rSinSina
r
QMSinEi
r
QaPrr
KN
ST
where the constants have been selected to obtain
asymptotically Kerr-Newman metric and PST is
identical to the similar quantity in the static case:
( ) ( )
≡≡−−−+++=
0
;4
0
221,4
3
14/11214242
2
3
r
rr
r
QbrArcSinFrrrrQSTP
72
One can see that this solution contains new terms
that do not appear in the Kerr-Newman model, like
Reissner-Nordstrom and the static Born-Infeld model.
There are coupled terms of charge and angular
momentum. The expansion is for
a
r
< < 1 and r r0 ≤ .
ANALYSIS OF THE METRIC
IN THE BORN-INFELD ROTATING CASE
The general behavior of the metric is similar to the
Kerr-Newman’s model. How one can see from the last
expression for the ∆, the metric has two horizons and
depends strongly of r0 (related with Born-Infeld
parameter: b≡Q2/(r0)4) and its quotient to a2 . The
asymptotical behavior of the ∆(r) is:
( )
+++
−−+−≅∆
4
0
22
8576.0222
8653.24
0
42
2
0
22525.1854.1
3
4
0
2
22
r
QaaKNaQ
r
r
a
r
a
r
QMrr
what corresponds to have asymptotically an effective
mass:
[ ]MEff M Q
r
a
r
a
r
= + − −
2
0
2
3
1 854 1 525
2
04 2 8653
4
04. . .
and an effective angular momentum:
aEff aKN
Q
r
a= +2 2
2
04
4 0 8576. ,
where M is Schwarzschild mass and aKN ≡ Kerr-
Newman’s model angular momentum (this asymptotic
term is due only to the rotation of the spherical body
and is characteristic of the Kerr’s model [8])
CONCLUSIONS
In this report a solution of the Einstein equation for
slowly rotating black hole is presented. The general
behavior of the geometry is strongly modified according
to the value what take 0r (Born-Infeld radius [1,2])
relative to a value. This metric has not the problem of
the conical singularities at the origin of the static Born-
Infeld models and permits solutions with M a Q2 2 2< +
without violation of the cosmic censorship hypothesis.
Solutions with M = 0 and a = 0 are possible, too; the
Born-Infeld-rotating model gives mass and angular
momentum to the source of the non-linear fields. In next
papers the particle-like models with Born-Infeld rotating
black holes will be studied and analyzed, and also the
possibility of supersymmetrical extensions (see recent
papers [10,11,12]) for such model will be considered.
ACKNOWLEDGMENTS
I am very grateful to Professor Nikolaj F. Shul’ga
for the opportunity of participate of the International
Conference, dedicated to the 90th anniversary of
Alexandver Il’ich Akhiezer (great man and great
scientist) and giving the possibility to visit the Kharkov
Institute of Physics and Technology. I appreciate deeply
the people at the Kharkov Institute of Physics and
Technology for their great hospitality. I am very
thankful to Professor Yurii Stepanovsky for his guide in
my scientific formation and for his help in the
preparation of this text.
REFERENCES
1. B. Hoffmann. Gravitational and electromagnetic
mass in the Born-Infeld electrodynamics // Phys.
Rev. 1935, v. 47, p 877-880.
2. M. Born and L. Infeld. Foundations of the new field
theory // Proc. Roy. Soc. (London). 1934, v. 144,
p. 425-451.
3. M. Born. On the quantum theory of the
electromagnetic field // Proc. Roy. Soc. (London).
1934, v. 143, p. 411-437.
4. L.D. Landau and E.M. Lifshitz, Teoria Clasica de
los Campos. Buenos Aires “Reverte”, 1974, 504 p.
5. C. Misner, K. Thorne and J.A. Weeler, Gravitation.
San Francisko “Freeman”, 1973, 474 p.
6. D. Harari and C. Lousto. Repulsive gravitational
effects of global monopoles // Phys. Rev. D. 1990,
v. 42, p. 2626-2631.
7. A. Borde. Regular Black Holes and topology
change. gr-qc/9612057.
8. S. Chandrasekhar. The Mathematical theory of
Black Holes. Oxford “Oxford University Press”,
1992, 632 p.
9. D.J. Cirilo Lombardo. The axially symmetric
geometry with Born-Infeld fields: Rotated EBI
model, Tesis de la Licenciatura en Ciencias Fisicas
(undergraduate tesis), Universidad de Buenos Aires,
September of 2001, 61 p.
10.V.V. Dyadichev, D.V. Gal’tsov, A.G. Zorin, and
M.Yu. Zotov. Non Abelian Born-Infeld Cosmology,
hep-th/0111099.
11.E.A. Ivanov, B.M. Zupnik, N=3 Supersymmetric
Born-Infeld theory, hep-th/011074; Balaz et al.
Dyons in Non Abelian Born-Infeld Theory, hep-
th/0110245.
73
ROTATED CHARGED BLACK HOLES
Diego Julio Cirilo Lombardo
Buenos Aires University, Buenos Aires, Argentina
INTRODUCTION
ROTATED EINSTEIN-BORN-INFELD MODEL
STATEMENT OF THE PROBLEM
ANALYSIS OF THE METRIC
IN THE BORN-INFELD ROTATING CASE
CONCLUSIONS
ACKNOWLEDGMENTS
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79426 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-29T03:08:31Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Diego Julio Cirilo Lombardo 2015-04-01T19:32:43Z 2015-04-01T19:32:43Z 2001 Rotated charged black holes in Einstein-Born-Infeld theories / Diego Julio Cirilo Lombardo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 71-73. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS 04.20.Ha 04.70.Bw https://nasplib.isofts.kiev.ua/handle/123456789/79426 In this work the solution of the Einstein equations for slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties, singularities, horizons of this solution are analyzed. There are considered the conditions when the black hole modifies its mass (like in the non-linear monopole cases) and angular momentum for the same non-linear electromagnetic field what produces the black hole. I am very grateful to Professor Nikolaj F. Shul’ga for the opportunity of participate of the International Conference, dedicated to the 90th anniversary of Alexandver Il’ich Akhiezer (great man and great scientist) and giving the possibility to visit the Kharkov Institute of Physics and Technology. I appreciate deeply the people at the Kharkov Institute of Physics and Technology for their great hospitality. I am very thankful to Professor Yurii Stepanovsky for his guide in my scientific formation and for his help in the preparation of this text. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Rotated charged black holes in Einstein-Born-Infeld theories Вращающиеся заряженные черные дыры в теориях Эйнштейна-Борна-Инфельда Article published earlier |
| spellingShingle | Rotated charged black holes in Einstein-Born-Infeld theories Diego Julio Cirilo Lombardo Quantum field theory |
| title | Rotated charged black holes in Einstein-Born-Infeld theories |
| title_alt | Вращающиеся заряженные черные дыры в теориях Эйнштейна-Борна-Инфельда |
| title_full | Rotated charged black holes in Einstein-Born-Infeld theories |
| title_fullStr | Rotated charged black holes in Einstein-Born-Infeld theories |
| title_full_unstemmed | Rotated charged black holes in Einstein-Born-Infeld theories |
| title_short | Rotated charged black holes in Einstein-Born-Infeld theories |
| title_sort | rotated charged black holes in einstein-born-infeld theories |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79426 |
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