Table Of ContentOn Magnetic solution to 2+1 Einstein–Maxwell gravity
Mauricio Cataldo,1,∗ Juan Cris´ostomo,2,† Sergio del Campo,2,‡ and Patricio Salgado3,§
1Departamento de F´ısica, Facultad de Ciencias, Universidad del B´ıo–B´ıo,
Avenida Collao 1202, Casilla 5-C, Concepci´on, Chile.
2Instituto de F´ısica, Facultad de Ciencias B´asicas y Matem´aticas,
Pontificia Universidad Cat´olica de Valpara´ıso, Avenida Brasil 2950, Valpara´ıso, Chile.
3Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas,
Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.
(Dated: February 1, 2008)
Thethree–dimensionalmagneticsolution totheEinstein–Maxwell fieldequationshavebeencon-
sideredbysomeauthors. Severalinterpretationshavebeenformulated forthismagneticspacetime.
Uptonowthissolution hasbeenconsidered asatwo–parameter self–consistent field. Wepointout
that the parameter related to the mass of this solution is just a pure gauge and can be rescaled
4 to minus one. This implies that the magnetic metric has really a simple form and it is effectively
0 one-parameter solution, which describes a distribution of a radial magnetic field in a 2+1 anti–de
0 Sitterbackgroundspace–time. WeconsideranalternativeinterpretationtotheDias–Lemosonefor
2 themagnetic field source.
n
a PACSnumbers: 04.20.Jb
J
5
The 2+1 magnetic solution to the Einstein–Maxwell the component g becomes zero. This occurs to be for
2 φφ
field equations has been studied by some authors. The some value of r =r, which satisfies the constraint
1 staticsolutionhasbeenfoundbyClement[1],Pelda´n[2],
v Hirschmann and Welch [3] and Cataldo and Salgado [4], r2+Q2 ln r2 M =0. (2)
9 using different procedures. The generalizationto the ro- m (cid:12)l2 − (cid:12)
8 tatingcasewasdonebyDiasandLemos[5]. Thissolution (cid:12)(cid:12) (cid:12)(cid:12)
1 may be written in the form Thisequationimpliestha(cid:12)triscons(cid:12)trainedtobebetween
1
0
r2 l√M <r l√M +1. (3)
4 ds2 = M dt2+ ≤
0 −(cid:18)l2 − (cid:19)
h/ r2dr2 The metric (1)appearsto changethe signatureatr =r.
+ Thisindicatesusthatweareusinganincorrectextension.
p-t rl22 −M r2+Q2mln rl22 −M The correct one can be found setting
he (cid:0) (cid:18)r(cid:1)2(cid:0)+Q2mln(cid:12)(cid:12)rl22(cid:12)(cid:12) −M(cid:12)(cid:12)(cid:19)(cid:12)(cid:12)d(cid:1)φ2, (1) x2 =r2−r2, (4)
: (cid:12) (cid:12)
v (cid:12) (cid:12) since the physical space–time has sense only for r r
i where l is the radius of a pseudo–sphere related to ≥
X the cosmological constant via l = 1/√Λ, Q and M and we have 0 x < . Taking into account the con-
m ≤ ∞
− straint (2), the metric (1) becomes
r are self–consistent integrationconstants of the Einstein–
a
Maxwell field equations. The vector potential 1–form of
x2 α2 l2x2dx2
this gravitationalfield is given by ds2 = + dt2+ +F2(x)dφ2,
−(cid:18)l2 l2 (cid:19) (x2+α2)F2(x)
Q r2 (5)
m
A= 2 ln(cid:12)l2 −M(cid:12) dφ. where α2 = r2 l2M and the function F2(x) is defined
(cid:12) (cid:12) as −
(cid:12) (cid:12)
(cid:12) (cid:12)
When Q = 0 the metric (1) reduces to the nonrotat-
m x2
ingthree–dimensionalBan˜ados–Teitelboim–Zanelliblack F2(x)=x2+Q2 ln 1+ (6)
m (cid:18) α2(cid:19)
hole [6], where M is the mass of this uncharged met-
ric, which has an event horizon at r = √Ml. Let us
This metric is horizonless, without curvature singulari-
study the behavior of this Einstein–Maxwell field. We
ties andin particular,there is no a magneticallycharged
shall consider the values of the r–coordinate for which
three–dimensionally black hole [3]. The presented mag-
netic solution shows a strange behavior. As the param-
eter Q , related to the strength of the magnetic field,
m
goes to zero we should recoverthe Ban˜ados–Teitelboim–
∗[email protected]
Zanelli black hole, but it does not occur. Since, in this
†[email protected]
‡[email protected] case“thelimitofatheoryisnotthetheoryofthelimit”.
§[email protected] Surprisingly, this strange behavior can be eliminated by
2
introducinganewsetofcoordinates. Effectively,making the electric charges is at rest and the other is spinning
the following rescaling transformations around it. In view of the symmetry of the space–time
this configuration is located at the origin of the coordi-
t′2 = r2−Ml2 t2, r′2 = l2 x2, nate system.
l2 r2 Ml2 We propose here another interpretation based on the
−
r2 Ml2 similaritiesofstaticEinstein–Maxwelltheoryfor2+1di-
φ′2 = − φ2, (7)
l2 mensional rotationally symmetric spacetimes and 3+1
dimensional axially symmetric spacetimes [7]. Let us
and introducing them into Eq.(5), we obtain the follow- refer to the static magnetic fields in four dimensional
ing metric general relativity. Bonnor [8], studying this topic, have
consideredaxiallysymmetricmagnetostaticgravitational
r′2 r′2dr′2
ds2 = +1 dt′2+ +F′2(r′)dφ′2, fieldsinemptyspacesgeneratedfromknownelectrostatic
−(cid:18) l2 (cid:19) (rl′22 +1)F′2(r′) solutions. Bonnor noted here that when we generate
(8) magnetostatic solutions from electrostatic ones “there is
where not an equivalence between sources of the static elec-
r′2 tric and magnetic fields; by this is meant that whereas
F′2(r′)=r′2+q˜2 ln +1 , (9) the electrostatic field in empty space may be consid-
m (cid:18) l2 (cid:19)
ered to arise from point–charges,the magnetostatic field
and q˜2 = Q2 er2/Q2m. In the present form, the constant must arise from dipoles, or from stationary electric cur-
m m rents”[8]. Theaboveremarkmayhaveprofoundimplica-
M has been eliminated and the metric (8) has one in-
tionsforthe natureofthe studied2+1dimensionalmag-
tegration constant q . This parameter is well behaved
m netic spacetime. Effectively, Bonnor generates a mag-
for r = 0 and the magnetic field can be switched off
netostatic solution from a set of electrostatic ones, for
without any problem. When q˜ = 0, the anti–de Sitter
m electric fields containing no matter or charge except at
space is obtained. Clearly the metric (8) is a particu-
singularities (see Eqs. (3.4) and (3.5) of the Ref. [8]).
lar solution of the line element (1), where we need to
Thissolutionhastwoconstantsofintegration,represent-
put M = 1. This implies that the parameter related
− ing the mass and the electric field strength. The gener-
to the mass of this solution is just a pure gauge and it
ated 3+1 dimensional magnetostatic solution has physi-
can be rescaled to the value 1. This agrees with the
− calsense only if we take zerothe parameterrepresenting
Dias–Lemos result [5], who have shown that the mass
the mass; then the solution is regarded as referring sim-
of the magnetic solution (5) is negative. However, the
ply to a uniform magnetic field produced by a solenoid
examined by authors three–dimensional static magnetic
without mass [8]. The similarity that happen between
field is still a two–parameter solution, since the mass is
2+1 and 3+1 dimensions is clear: the three–dimensional
considered a free parameter (see their Eq. (3.24) with
magnetic solution may be generated from the electro-
Ω = J = Q = 0). Thus, the metric (5) is really a
e static Ban˜ados–Teitelboim–Zanelli black hole with the
one–parameter solution with a distribution of a radial
help of a duality mapping [4, 7]. In this case the electric
magneticfieldina2+1anti–deSitterbackground,which
field arises from a charged point mass (excluding inte-
takes the form of Eq.(8). This metric can be considered
riorsolutionsfromconsideration). Aswehaveshownthe
as the general “physical solution” to the self–consistent
three–dimensionalmagnetostaticgravitationalfieldisre-
problemforasuperpositionofaradialmagneticfieldand
ally a one–parameter solution, where the free parameter
a2+1Einsteinstaticgravitationalfield. Clearlythemet-
is only the integration constant related to the magnetic
ric(8)isnotamagneticallychargedthree–dimensionally
field strength. Thus, the source of the magnetic field
black hole. This metric is horizonless (in this sense this
may be considereda two dimensional solenoid,i.e. a cir-
is a particle–like solution), without curvature singular-
cular current. We prefer to locate this current at spatial
ities and it has no signature change. The solution (8)
infinity, since the curvature is regular everywhere.
does have a conical singularity at r′ = 0 which can be
We should note that the Bonnor solution with an uni-
removed by identifying the φ′–coordinate with the pe-
form magnetic field is valid for the case in which the
riod Tφ′ = 2π/(1+q˜m2 /l2) [3]. It is well behaved, since cosmological constant is vanished [8]. In our case the
ifq˜ approachesinfinity,thisperiodbecomeszero,while
m magnetic field is given by
if q˜ approaches zero, this period goes to 2π, since the
m
anti–de Sitter space has no angle deficit. Finally, let us
1
consider an alternative interpretation of this magnetic B(r) , (10)
solution. In the reference quoted above [5], the authors ∼ rl22 +1
q
haveshownthatthe magneticfieldsourcecanbeneither
a Nielson–Oleson vortex solution nor a Dirac monopole. and is regular everywhere. From this expression we see
Thusthey attemptedtoprovideaninterpretationofthis that the magnetic field at the origin has a maximum
magneticsolution. DiasandLemosinterpretedthestatic value, and at infinity approaches to zero. This magnetic
magnetic field source as being composed by a system of field is not a constant since the cosmological constant is
two symmetric and superposed electric charges. One of negative and then it acts as an attractive gravitational
3
force. This implies that the magnetic lines held together I. ACKNOWLEDGEMENTS
near the origin.
Note added. Inarecentlyappearedworkthethinshell
collapse, leading to the formation of charged rotating
black holes in 2+1 dimensions, is considered [9]. In this
context, from physical considerations, the author singles This work was partially supported by CONICYT
outfromthe solution(1)the caseM = 1,sinceforthis FONDECYT N0 1010485 (MC, SdC and PS), N0
−
choiceoftheparameterM themagneticsolutiondoesnot 1030469(SdC and MC) and by Ministerio de Educaci´on
exhibitapathologicalbehavior. Inthiscaseachargedro- through MECESUP grant FSM 9901 (JC). It also was
tatingthinshellisinterpretedastheanalogtoasolenoid supported by the Direcci´on de Investigaci´on de la Uni-
carrying a steady current, and then inside the thin shell versidad del B´ıo–B´ıo (MC) and by grants 123.764-2003
the three dimensional M = 1 magnetic static solution of Vicerrector´ıa de Investigaci´on y Estudios Avanzados
−
is valid, and the magnetic field just vanishes outside the of Pontificia Universidad Cat´olica de Valpara´ıso (SdC)
rotating thin shell. and UdC/DI 202.011.031-1.0(PS and MC).
[1] G. Clement, Class. Quant.Grav. 10 (1993) L49. 69 (1992) 1849.
[2] P. Peld´an, Nucl. Phys. B395 (1993) 239. [7] M. Cataldo, Phys.Lett. B 529 (2002) 143.
[3] E. Hirschmann and D. Welch, Phys. Rev. D 53 (1996) [8] W.B. Bonnor, Proc. Phys. Soc. Lond.A 67 (1954) 225.
5579. [9] R. Olea, Charged Rotating Black Hole Formation
[4] M.CataldoandP.Salgado,Phys.Rev.D54(1996)2971. from Thin Shell Collapse in Three Dimensions,
[5] O.J.C. Dias and J.P.S. Lemos, JHEP 01 (2002) 006. hep–th/0401109.
[6] M.Ban˜ados,C.TeitelboimandJ.Zanelli,Phys.Rev.Lett.
