General analysis of self-dual solutions for the Einstein-Maxwell-Chern-Simons theory in (1+2) dimensions

3 Pages·0.09 MB·English

by  T. Dereli

General analysis of self-dual solutions for the Einstein-Maxwell-Chern-Simons theory
in (1+2) dimensions
∗
T. Dereli and Yu.N. Obukhov
Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
Here f,g,h,a and E,B are the functions of the radial
The solutions of the Einstein-Maxwell-Chern-Simons the-
coordinate r.
oryarestudiedin(1+2)dimensionswiththeself-dualitycon-
Without any loss of generality it will be convenient to
ditionimposedontheMaxwellfield. Wegiveaclosedformof
absorbthemetricfunctiong(r)bythesimpleredefinition
thegeneral solution which is determined bya single function
of the radial coordinate:
having the physical meaning of the quasilocal angular mo-
mentumofthesolution. Thisfunctioncompletelydetermines
1
the geometry of spacetime, also providing the direct compu- ρ= g(r)dr (hence ϑ =dρ). (6)
Z
0 tationoftheconservedtotalmassandangularmomentumof
0 theconfigurations. ¿From now on, the derivatives w.r.t. new coordinate ρ
0
will be denoted by prime.
2
After all these preliminaries, the Einstein field equa-
n
tions read explicitly
a
J The(1+2)-dimensionalgeneralrelativityhasattracted
1 ′
7 considerable attention recently (see, e.g., [1] and refer- β βγ = EB, (7)
− 2 − −
ences therein). This is explained by two main reasons.
1 1
1 Firstly, since the discovery of the BTZ black hole solu- γ′+γ2+ β2+λ= (E2+B2), (8)
v 4 −2
tions [2], the three-dimensionalgravity became a helpful
7 3 1
′ 2 2 2 2
1 laboratory for the study of geometrical, statistical and α +α β +λ= (E +B ), (9)
− 4 2
0 thermodynamics properties of black holes. Secondly, the
1 1
1 quantization of these models may give new insights into αγ β2 λ= (E2 B2), (10)
0 the general quantum gravity problem. − − 4 − 2 −
0
A number of generalizations of BTZ solution to the and this system is supplemented by the (modified)
0
/ case of nontrivial electromagnetic field source were de- Maxwell equations:
c
veloped previously [4–7]. The aim of our present paper
q ′
istogiveanewgeneralanalysisoftheself-dualEinstein- B αB+βE+µE =0, (11)
- − −
gr Maxwell solutions in three dimensions. E′ γE+µB =0. (12)
The Lagrangian3-form, − −
:
v
Here we introduced the functions
i 1 1 µ
X L= 2R ∗1−λ∗1− 2F ∧∗F − 2A∧F, (1) f′ a′h h′
r α= , β = , γ = . (13)
a f f h
containstheEinstein-Hilbertterm,thecosmologicalcon-
stant λ, and the standard Maxwell field F = dA La- which actually describe the Levi-Civita connection co-
grangian along with the Chern-Simons term with the efficients. The remarkable feature is that the com-
coupling constant µ, [3]. Variation of L with respect to plete Einstein-Maxwell system (7)-(12) involves no met-
thecoframefieldϑα andtheelectromagneticpotentialA ricfunctions(i.e.,f,g,h,a),butonlytheconnectioncom-
yields the system of field equations: binations α,β,γ.
Let us assume “self-duality” of the electromagnetic
G ϑβ +λ ϑ =Σ , (2)
αβ α α field:
∗ ∗
d F +µF =0. (3)
∗ E =kB, with k2 =1. (14)
1
Here Σ = [(e F) F F e F] is the Maxwell
α 2 α⌋ ∧∗ − ∧ α⌋∗
field energy-momentum 2-form, and G is the Einstein Substituting this into (11)-(12) and (10), we find that
αβ
tensor. the twounknownfunctions areexpressedinterms of the
In the study of the “spherically”-symmetric solutions, third:
we choose the local coordinates (t,r,φ) and make the
k k
general ansatz for the coframe 1-form, α= β+ℓ, γ = β+ℓ. (15)
2 − 2
0 1 2
ϑ =fdt, ϑ =gdr, ϑ =h(dφ+adt), (4) Here we denote ℓ:= √ λ.
± −
Taking into account the algebraic relations (14) and
and for the Maxwell field
(15), we are left with two essential equations for deter-
0 1 1 2 mining the functions β and B. Explicitly, the equations
F =Eϑ ϑ +Bϑ ϑ , (5)
∧ ∧ (7) and (12) are reduced to
′ 2 2
β kβ +2ℓβ =2kB , (16) of the Einstein-Maxwell (with or without Chern-Simons
−
(B2)′ kβB2+2ℓB2 =2kµB2. (17) term)fieldequationsisalwaysrepresentedsolelyinterms
− the function ϕ.
This system of nonlinear coupled equations is simplified Becauseofsuchanimportantroleplayedbyϕ,itwould
with the help of the substitution be interesting to find out its physical meaning. The lat-
ter is revealedin the analysis of the quasilocal mass and
1 2 k ϕ′ angular momentum which characterize our general solu-
β = , B = , (18)
ω 2ω ϕ tion.
We refer the reader to [8] for a comprehensive discus-
which yields for the new functions ϕ and ω the linear
sion of the conserved quantities for gravitating systems
equations:
withintheframeworkofHamiltonianformulationofgen-
′′ ′ eralrelativity theory. As a first step, let us use the coor-
ϕ =2kµϕ. (19)
dinate freedom and replaceρ by a new radialcoordinate
ϕ′
ω′+ 2ℓ ω+k =0. (20) defined by
(cid:18)ϕ − (cid:19)
r =h(ρ). (29)
Multiplying (20)byϕe−2ℓρ, weeasily obtainthe general
solution Then a nontrivial metric function g will reappear in the
coframe (4) [and hence in the metric (28)]. Using (25)
ρ
ω = ϕek−Ω2ℓρ, with Ω:=c0−Z dρ˜ϕ(ρ˜)e−2ℓρ˜. (21) we find explicitly
dρ h2 −1
0
g = = ℓr ϕ . (30)
Note that in fact it is not necessary to know the ex- dr (cid:18) − 2r (cid:19)
plicit form of ϕ when solving (20). At the same time, of
course,the equation(19)is straightforwardlyintegrated. Now we can write the quasilocal angular momentum at
Depending on µ, it admits two solutions: a distance r, which reads
ϕ=ρ+ρ0, when µ=0, (22) j(r)= g−1r3 da, (31)
ϕ=1+u0e2kµρ, when µ=0. (23) f dr
6
in our notations. Using (24)-(26) and (29)-(30), we find
Herec0,ρ0,u0 areintegrationconstants. Itisworthwhile
to note that an overallconstant factor is irrelevantfor ϕ 2
j(r)=kh ϕ. (32)
0
because this function appears everywhere only through
the ratio (18). Clearly, one should invert (29) and use ρ=ρ(r) in (32),
Quite remarkably, however, we will not need the ex- oralternatively,onecanconsidertheangularmomentum
plicit form of ϕ till the very end of our analysis. Such j as a function of ρ.
a formulation is extremely convenient since it makes it Thequasilocalenergyisgiven,inournotations,bythe
possible to treat the cases of standard Maxwell theory difference
with µ = 0, and the Maxwell-Chern-Simons with µ = 0
simultaneously. 6 E(r)=g−1 g−1, (33)
0
−
It remains to integrate the equations for the metric
functions(13). Thisisstraightforward,andusing(21)in where the first term describes the contribution of the
(15), we find: background “empty” spacetime. The latter, as usually,
−1
is given by g = ℓr. Making use of (30), we obtain
0
f =f0eℓρΩ−21, (24) explicitly
h=h0eℓρΩ12, (25) h2 k
0
E(r)= ϕ= j(r). (34)
kf0 −1 2r 2r
a= Ω a0. (26)
h0 −
Finally, the quasilocalmassis determinedby the expres-
For completeness, the magnetic field reads: sion
B2 = ϕ′ e−2ℓρΩ−1. (27) m(r)=2fE(r)−ja. (35)
2
Substituting (24), (26), (32) and (34), we arrive at the
Here f0,h0,a0 are integration constants. result:
The first main result which we learned in our study,
is that a general “spherically”-symmetric (rotating, for m(r)=a0j(r). (36)
nontrivial a) solution
Wethushavedemonstratedthatthefunctionϕ,which
ds2 = ϑ0 2+ ϑ1 2+ ϑ2 2 (28) determines the spacetime geometry via (21) and (24)-
− (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (26), is also determining all the quasilocal quantities of
thegravitatingsystem: itsenergy,massandangularmo- [1] S.Carlip,Class.QuantumGrav.12(1995)2853;S.Carlip,
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| |
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large classes of such configurations: (A) kµ < 0,ℓ = 0,
then J = kh2 and M = 0, and (B) kµ < 0,ℓ > 0,
0
then J = kh2 and M = f0h0. Imposing the standard
0 c0
asymptotic condition f2g2 r→∞ = 1, one finds a0 = kℓ,
|
and thus the solutions of the class (B) are all charac-
terized (irrespectively of the value of the Chern-Simons
coupling constant µ) by M2 = ℓ2J2. This class also
contains the extremal BTZ solution, as a particular case
when the electromagnetic field is absent. [The general
non-extremalBTZ solution cannot be recoveredbecause
of the algebraic relations (15) which necessarily hold for
the self-dual electromagnetic field].
Summarizing, we have obtained a general solution of
the Einstein-Maxwell-Chern-Simonstheoryin (1+2)di-
mensions which covers all the particular cases studied
previously. The form of the solution (24)-(27), (21) is
transparent and easy to analyse: everything is deter-
mined by a single function ϕ(ρ) which has a clear physi-
cal meaning as the quasilocal angular momentum of the
gravitationalfieldconfiguration. Thecomputationofthe
totalmassandangularmomentumisstraightforwardand
it involves only the analysis of the asymptotic behaviour
of ϕ.
TheauthorsaregratefultoTUBITAKforthe support
of this research. Y.N.O. is also grateful to the Depart-
ment of Physics, Middle East Technical University, for
the warm hospitality.