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Static solution in three dimensions with naked singularity for gravity with massive scalar field
SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 127 (2004) 148-149
www.elsevierphysics.com
Static solution in three dimensions with naked singularity with massive scalar field
for gravity
G. de Berredo-Peixotoa* “Departamento
de Fisica, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil 36036-330.
We investigate circularly symmetric static solutions in three-dimensional gravity massive scalar field. We integrate numerically the field equations assuming asymptotic do not exist and a naked singularity is present.
1. INTRODUCTION Systems described by gravity and scalar field are extremely interesting and relevant for investigation in the scope of theoretical physics. For instance, scalar-gravity theories are required to study inflationary cosmology, as well as to investigate low energy manifestations of the string theory. Three-dimensional models are extensively explored by physisists as a good theoretical laboratory, since the usual problems of renormalizability and unitarity are less severe than in fourdimensional models. One can find a one-parameter family of charged black holes for the SD-dilaton gravity [l], as well as its uncharged rotating version [2], where the black hole is specified by the mass, angular momentum and the dilaton coupling parameters. A non-trivial black hole solution was found for a minimally coupling scalar field and a class of potentials [3]. The paper in Ref. [4] extends the rotating electrically charged BTZ solution to include Brans-Dicke theory. Solutions for minimally coupled massless scalar field with nontrivial cosmological constant can be found in Ref. [5], without black holes. One can see, for instance, the list of references in [6]. In this paper, we search for static circularly symmetric solutions for gravity minimally coupled with a massive scalar field. The numeric integration can be done if one considers any asymptotically flat solution, however we show that all *e-mail: gbpeixotoOhotmaii.com 0920-5632/$ - see front matter 0 2004 Published by Elsevier B.V. doi: IO. 1016/j.nuclphysbps.2003.12.026
with a minimally coupled flatness, where black holes
these solutions have a naked singularity. 2. ACTION Consider the following action describing gravity minimally coupled with a scalar field, 5’
=
/d3zfi{
-
M2cj2
;(R-2A)-V&V’+ } .
(1)
We are looking for circularly symmetric static solutions of theory (l), thus there are two Killing fields, al& and a/&9 and the metric is given by dt2 + -g(r) dr2 + r2 de2 . f(r)
ds2 = -f(r)
(2)
2.1. Field equations Varying the action (1) with respect to the field variables (gpv, $), one can find the field equations R,, - &vR
= KTPV
(3)
and
qf$-M2c#GO, where the momentum-energy T c1” =
(4) tensor is given by
0,f#n”cfJ - ; ( g~“vcdPvCI~+
+ M2g,d2 ) - $A, and the operator
0 stands for gQfiV,Vp.
(5)
G. de Berredo-Peixoto/Nuclear
Physics B (Pmt. Suppl.) 127 (2004) 148-149
149
has only trivial solution (no massive scalar hair)2. Formally, this solution has a naked singularity. We must remark that ou solution can not be covered by those found in [l] and [5] by changing the appropriate parameters. 4. CONCLUSIONS
Figure 1. assuming curves G, g(T), f(r)
Numerical integration from infinity, the asymptotically flat ansatz. The F and U correspond to the functions and 4(r), respectively.
We have found a numerical solution for gravity minimally coupled to a massive scalar field for vanishing cosmological constant, which exhibits a naked singularity. Although the model is relatively simple, it can not be reached by more complicated solutions by some continuous procedure. Acknowledgment
3. NUMERIC
The author is grateful financial support.
INTEGRATION
Let us consider the case with vanishing cosmological constant. Using the software MAPLE, we first guess suitable initial values for the fields at infinity and then integrate from that point to the origin. At infinity, we assume that the metric is approximately free from the influences of the scalar field, thus the metric at infinity can be regarded as the vacuum solution of the Einstein’s equations, with a conical singularity (see [7]): ds2 = -dt2 + bdr2 + r2de2 ,
(6)
where b = constant. The guess for the scalar field at infinity can be regarded as its solution in the vacuum background, 4(r) = Q Ko(M&r) (modified Bessel function of the second kind). The curves of functions f(r) and g(r) coincide approximately with its constant values corresponding to vacuum solution, deviating by a small amount in the region near to the origin (see more details in [6]). As we increase the parameter M, these deviations are decreased even more. Here, black holes are absent. Figure 1 shows the drawing of the numeric integration, with parameters b = 1.1, M = 0.1 and Q = 2. The scalar field is small along almost all the integration domain. The curve starts to vary substantially near the origin, where it diverges. One can show that the asymptotically flat assumption is incompatible with regularity of the scalar field at the origin or at the horizon, unless the field
to FAPEMIG,
for
REFERENCES
1. K.C.K. Ghan and R.B. Mann, Phys. Rev. D 50 n.10 (1994) 6385-6393. 2. K.C.K. Chan and R.B. Mann, Phys. Lett. B 371 (1996) 199-205. 3. J. Gegenberg, C. Martinez and R. Troncoso, hep-th/0301190. 4. Oscar J.C. Dias and Jo& P.S. Lemos, Phys. Rev. D 64 (2001) 064001. 5. G. ClCment and A. Fabbri, Class. Quantum Grav. 16 (1999) 323-341; G. Clement and A. Fabbri, Class. Quantum Grav. 17 (2000) 2537-2545. 6. G. de Berredo-Peixoto, gr-qc/0208026. 7. S. Giddings, J. Abbott and K. Kuchar, Gen. Rel. Grav. 16 n.8 (1984) 751-775; S. Deser, R. Jackiw and G. t’Hooft, Annals Phys.152 (1984) 220. 8. D. Sudarsky, Class. Quant. Grav. 12 (1995) 579-584.
‘See [8].
www.elsevierphysics.com
Static solution in three dimensions with naked singularity with massive scalar field
for gravity
G. de Berredo-Peixotoa* “Departamento
de Fisica, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil 36036-330.
We investigate circularly symmetric static solutions in three-dimensional gravity massive scalar field. We integrate numerically the field equations assuming asymptotic do not exist and a naked singularity is present.
1. INTRODUCTION Systems described by gravity and scalar field are extremely interesting and relevant for investigation in the scope of theoretical physics. For instance, scalar-gravity theories are required to study inflationary cosmology, as well as to investigate low energy manifestations of the string theory. Three-dimensional models are extensively explored by physisists as a good theoretical laboratory, since the usual problems of renormalizability and unitarity are less severe than in fourdimensional models. One can find a one-parameter family of charged black holes for the SD-dilaton gravity [l], as well as its uncharged rotating version [2], where the black hole is specified by the mass, angular momentum and the dilaton coupling parameters. A non-trivial black hole solution was found for a minimally coupling scalar field and a class of potentials [3]. The paper in Ref. [4] extends the rotating electrically charged BTZ solution to include Brans-Dicke theory. Solutions for minimally coupled massless scalar field with nontrivial cosmological constant can be found in Ref. [5], without black holes. One can see, for instance, the list of references in [6]. In this paper, we search for static circularly symmetric solutions for gravity minimally coupled with a massive scalar field. The numeric integration can be done if one considers any asymptotically flat solution, however we show that all *e-mail: gbpeixotoOhotmaii.com 0920-5632/$ - see front matter 0 2004 Published by Elsevier B.V. doi: IO. 1016/j.nuclphysbps.2003.12.026
with a minimally coupled flatness, where black holes
these solutions have a naked singularity. 2. ACTION Consider the following action describing gravity minimally coupled with a scalar field, 5’
=
/d3zfi{
-
M2cj2
;(R-2A)-V&V’+ } .
(1)
We are looking for circularly symmetric static solutions of theory (l), thus there are two Killing fields, al& and a/&9 and the metric is given by dt2 + -g(r) dr2 + r2 de2 . f(r)
ds2 = -f(r)
(2)
2.1. Field equations Varying the action (1) with respect to the field variables (gpv, $), one can find the field equations R,, - &vR
= KTPV
(3)
and
qf$-M2c#GO, where the momentum-energy T c1” =
(4) tensor is given by
0,f#n”cfJ - ; ( g~“vcdPvCI~+
+ M2g,d2 ) - $A, and the operator
0 stands for gQfiV,Vp.
(5)
G. de Berredo-Peixoto/Nuclear
Physics B (Pmt. Suppl.) 127 (2004) 148-149
149
has only trivial solution (no massive scalar hair)2. Formally, this solution has a naked singularity. We must remark that ou solution can not be covered by those found in [l] and [5] by changing the appropriate parameters. 4. CONCLUSIONS
Figure 1. assuming curves G, g(T), f(r)
Numerical integration from infinity, the asymptotically flat ansatz. The F and U correspond to the functions and 4(r), respectively.
We have found a numerical solution for gravity minimally coupled to a massive scalar field for vanishing cosmological constant, which exhibits a naked singularity. Although the model is relatively simple, it can not be reached by more complicated solutions by some continuous procedure. Acknowledgment
3. NUMERIC
The author is grateful financial support.
INTEGRATION
Let us consider the case with vanishing cosmological constant. Using the software MAPLE, we first guess suitable initial values for the fields at infinity and then integrate from that point to the origin. At infinity, we assume that the metric is approximately free from the influences of the scalar field, thus the metric at infinity can be regarded as the vacuum solution of the Einstein’s equations, with a conical singularity (see [7]): ds2 = -dt2 + bdr2 + r2de2 ,
(6)
where b = constant. The guess for the scalar field at infinity can be regarded as its solution in the vacuum background, 4(r) = Q Ko(M&r) (modified Bessel function of the second kind). The curves of functions f(r) and g(r) coincide approximately with its constant values corresponding to vacuum solution, deviating by a small amount in the region near to the origin (see more details in [6]). As we increase the parameter M, these deviations are decreased even more. Here, black holes are absent. Figure 1 shows the drawing of the numeric integration, with parameters b = 1.1, M = 0.1 and Q = 2. The scalar field is small along almost all the integration domain. The curve starts to vary substantially near the origin, where it diverges. One can show that the asymptotically flat assumption is incompatible with regularity of the scalar field at the origin or at the horizon, unless the field
to FAPEMIG,
for
REFERENCES
1. K.C.K. Ghan and R.B. Mann, Phys. Rev. D 50 n.10 (1994) 6385-6393. 2. K.C.K. Chan and R.B. Mann, Phys. Lett. B 371 (1996) 199-205. 3. J. Gegenberg, C. Martinez and R. Troncoso, hep-th/0301190. 4. Oscar J.C. Dias and Jo& P.S. Lemos, Phys. Rev. D 64 (2001) 064001. 5. G. ClCment and A. Fabbri, Class. Quantum Grav. 16 (1999) 323-341; G. Clement and A. Fabbri, Class. Quantum Grav. 17 (2000) 2537-2545. 6. G. de Berredo-Peixoto, gr-qc/0208026. 7. S. Giddings, J. Abbott and K. Kuchar, Gen. Rel. Grav. 16 n.8 (1984) 751-775; S. Deser, R. Jackiw and G. t’Hooft, Annals Phys.152 (1984) 220. 8. D. Sudarsky, Class. Quant. Grav. 12 (1995) 579-584.
‘See [8].