Melvin universe as a limit of the C-metric

0.11 MB·

by  Lenka Havrdova

Melvin universe as a limit of the C-metric
Lenka Havrdov´a∗ and Pavel Krtouˇs†
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,
V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech Republic
(Dated: September26, 2006)
It is demonstrated that the Melvin universe representing the spacetime with a strong ‘homoge-
neous’ electric field can by obtained from the spacetime of two accelerated charged black holes by
asuitable limiting procedure. The behaviorof various invariantly definedgeometrical quantitiesin
thislimit is also studied.
PACSnumbers: 04.20.Jb,04.40.Nr
7
0
0
2
I. INTRODUCTION
n
a
J It is a simple exercise to show that the homogeneous electric field in Minkowski spacetime

0 can be obtained from the Born solution representing test fields of two uniformly acceler-
1 ated charged particles by enlarging the distance between these particles accompanied with
increasing the charges.
2
One would thus expect that a similar limit of a ‘homogeneous’field could be obtained also
v
for a fully gravitationally interacting electromagnetic field and strong sources. The non-test
2
analogueof the Bornsolutionis the chargedC-metric representingtwo uniformlyaccelerated
9
0 charged black holes [1, 2] (cf. [3] for review). The gravitationally interacting ‘homogeneous’
1 electric field is described by the Melvin universe [4, 5, 6]. However, because such a field is
1 infinitely extended,the gravitationalinteractionstronglydeformsthe spacetimeandnontriv-
6 ially changesits asymptoticalstructure. The limit ofthe finite sourcesis thus notcompletely
0
straightforward. Most of the na¨ıve limits of the C-metric lead to degenerate results or to
/
c empty Minkowski spacetime [7].
q Let us note that the homogeneous field in Minkowski space can be also obtained by dif-
-
r ferent limiting procedures, some of which have a strong field analogy, see e.g. [8, 9]. Our
g work complements this effort and demonstrates that the limit of the distant strong charged
:
v accelerated sources exists, however it is not a trivial generalization of the test case.
i In the next section we present a suitable way to scale the parameters and coordinates of
X
the C-metric to obtain the desired limit. In Sec. III it is shown that the metric obtained
r indeed corresponds to the Melvin universe, and in Sec. IV the behavior of various physical
a
and geometrical quantities is studied.
II. SCALING THE C-METRIC
Following [10] we write the C-metric in the form
1 1 1
g = Fdt2+ dy2+ dx2+Gdφ2 , (1)
a2(x+y)2(cid:16)− F G (cid:17)
with t R, x ( 1,1), y < x, φ ( Cπ,Cπ), and
∈ ∈ − − ∈ −
F =(y2 1)(1 2may+e2a2y2),
− − (2)
G=(1 x2)(1+2max+e2a2x2).
−
∗Electronicaddress: [email protected]
†Electronicaddress: Pavel.Krtous@mff.cuni.cz
2
The four roots y ,y ,y ,y of the function F are
c a o i
1 1
y = 1, y =1, y = (m m2 e2), y = (m+ m2 e2). (3)
c − a o ae2 −p − i ae2 p −
The metric solves the Maxwell-Einstein equations with electromagnetic field given by
F =edy dt. (4)
∧
The C-metric (1) describes two uniformly acceleratedchargedblack holes moving in oppo-
sitedirectionalongthesymmetryaxis. Theconstantsa,m,e,andC parametrizeacceleration,
mass, charge and conicity of the symmetry axis. The coordinate t represents time of static
observers around the black holes. Near the holes, y is an (inverse) radial coordinate, which
far away from the holes has a bi-spherical character (cf. [11]). The coordinate x is a (co-
sine of) angle measured from the symmetry axis, and φ is the rotational angle around the
axis. The black holes are causally separated by the acceleration horizon y =1, the outer
and inner black hole horizons are given by y =y and y =y, respectively. The conformal
o i
infinity is given by the condition y = x. The properties of C-metric were studied, e.g., in
−
[2,3,12,13,14,15,16],theglobalstructurewasthoroughlydiscussedandvisualisedrecently
in [11].
To obtain the desired limit of a ‘homogeneous’field we have to specify the behavior of the
metricparametersa,m,e,C,andofthe coordinates. Wefixtheaccelerationparameteraand
the conicity parameter C unchanged during the limit. The behaviorof m and e is prescribed
implicitly in terms of the roots (3) as
y =1+y˜ ε, y =1+y˜ε, (5)
o o i i
where y˜ <y˜ are constantsparametrizingthe limiting procedure, andε is a small parameter
o i
which will be sent to zero in the limit. Since the coefficients of the metric functions (2) are
related, the behavior of the roots of F determines also the behavior of the roots of G.
Simultaneously we introduce rescaled coordinates τ,υ,ξ,ϕ
{ }
1 1 1
t= τ , y =1+y˜ y˜ ε2 υ , x=ξ , φ= ϕ. (6)
y˜ y˜ ε2 o i 1+2ma+e2a2
o i
If we redefine the conicity parameter c=C(1+2ma+e2a2), the range of coordinate ϕ is
ϕ ( cπ,cπ). The coordinate ϕ is rescaled in such a way that c gives the actual conicity of
∈ −
the axis between the black holes. The axis is regular if c=1.
According to (6), the coordinate y is rescaled by the factor ε2 around the value y =1.
∼
The limit is thus done in the domain close to the acceleration horizon. The values y and y
o i
are rescaled only as ε (cf. eq. (5)) and thus the corresponding values υ , υ of the rescaled
o i
coordinate υ behave∼as ε−1. Therefore, the horizons υ , υ disappear in the limit ε 0.
o i
∼ →
This corresponds to the intuition gathered in the limit of the Born solution mentioned in
the Introduction: the homogeneous electric field should be obtained in the middle between
very distant sources. The fact that black holes are indeed ‘pushed’ away in the limit is also
confirmed by the evaluation of their physical distance in Eq. (16) below.
The coordinate x is not scaled and the upper bound on υ corresponding to the condition
y < xthusalsobecomestrivial,υ < ,inthelimit. Finally,thethreerootsofthe function
− ∞
G degenerate to the value x= 1 as ε 0.
− →
After the transformations (5) and (6) and sending ε 0 the metric (1) and the electro-
magnetic field (4) become1 →
1 1 1 1
g = 2υdτ2+ dυ2+ dξ2 + (1 ξ2)dϕ2 , (7)
a2(1+ξ)2(cid:16)− 2υ (1 ξ)(1+ξ)3 (cid:17) 16a2 −
−
and
(1+ξ)2
F =edυ dτ =E eυ eτ . (8)
∧ 4 ∧
1 Noticethatm→1/a,e→1/a,andE→4ainthelimit,cf.Eqs.(13)below.
3
Hereeυ andeτ arenormalized1-formsproportionalto dυ,anddτ andE =4ea2 isthe value
of the electromagnetic field at the bifurcation υ =0, ξ =1 of the acceleration horizon.
III. MELVIN UNIVERSE
Themetric (7)stillinheritsthe boost-rotationalstructureofthe C-metricinthe τ-υ plane.
TodemonstratethatitdescribestheMelvinuniverse,wehavetotransformitintocoordinates
which exhibit more properly the symmetries of the Melvin universe. Therefore we perform
an additional coordinate transformations introducing the Melvin coordinates2 t,z,ρ,ϕ :
{ }
t
υ =2a2( t2+z2), tanhsgnυτ = (9)
− z
and
1 Eρ 2 E 2 1 ξ
ξ = − 2 ρ = − . (10)
1+(cid:0)Eρ(cid:1)2 ⇔ (cid:16)2 (cid:17) 1+ξ
2
(cid:0) (cid:1)
Let us observethat the transformation(9) is essentially the transformationfrom Rindler-like
coordinates τ and ζ υ to Minkowski-like coordinates t and z. It is the transformation
∝ | |
fromthe frame ofunifoprmly acceleratedobservers,whichis naturalfor the C-metric,into the
frame of globally static observers more natural for the Melvin universe. The coordinate ρ
could be also given as Eρ=tanϑ, where ξ =cosϑ.
2 2
The transformations (9) and (10) lead to
E2 2 E2 −2
g = 1+ ρ2 dt2+dz2+dρ2 + 1+ ρ2 ρ2dϕ2 , (11)
(cid:16) 4 (cid:17) (cid:16)− (cid:17) (cid:16) 4 (cid:17)
E
F =Edz dt= ez et , (12)
∧ 1+ E2ρ2 2 ∧
4
(cid:16) (cid:17)
with the coordinate ranges t,z R, ρ R+, and ϕ ( cπ,cπ). These are respectively the
∈ ∈ ∈ −
metric and electromagnetic field of the Melvin universe [4, 5, 6] representing the strong ‘ho-
mogenous’ electric field oriented along the symmetry axis. The field is ‘homogeneous’ in the
sense that it does not change under translations along the direction of the field. Indeed, the
metric and the field are invariant under the action of the static, translational, and rotational
Killing vectors ∂ , ∂ , and ∂ , respectively. However, the field changes in the direction or-
t z ϕ
thogonal to the symmetry axis. The constant E measures the electromagnetic field on the
axis and c the conicity of the axis.
IV. GEOMETRICAL QUANTITIES
As a consequence of the expressions (5) we obtain the following behavior of the metric
parameters:
m2a2 =1 (y˜ +y˜)ε+ (ε2), e2a2 =1 (y˜ +y˜)ε+ (ε2),
o i o i
− O − O
1 c (13)
a m2 e2 (y˜ y˜ )ε, E 4a, C .
i o
p − ≈ 2 − ≈ ≈ 4
2 TheMelvincoordinatetisdifferentfromtheC-metriccoordinate tusedin(1).
4
However, it is more interesting to evaluate geometrically invariant quantities. The area and
surface gravity3 of the outer black hole horizon are given by
4πC 1 πc 1
= ,
Ao a2 y 2 1 ≈ 2a2y˜ ε
o − o (14)
1 1 1
κ = a(y 2 1) y˜ (y˜ y˜ )ε2 .
o o o i o
2 − (cid:16)yo − yi(cid:17)≈ −
The surface gravity of the accelerationhorizon is
κ =a(1 2ma+e2a2) a(2y˜2+2y˜2+y˜ y˜)ε2 . (15)
a − ≈ o i o i
Total charge evaluated using the Gauss law is simply Q =Ce c/E. As we already men-
tot
≈
tioned, the conicity of the axis x=1 between the black holes is given by the parameter c. It
transforms to the conicity of the axis of the Melvin universe as ε 0. The conicity of the
→
outeraxes,whichjointheblackholeswithinfinity,vanishesinthelimit. Simultaneously,these
axes are pushedawayto infinity. Finally, the distance of the black hole from the acceleration
horizon measured along the axis (half of the distance between the holes) is
1 yo 1 1 y˜ 1
i
= dy K , (16)
L aZ1 (1+y)√F ≈ a√2y˜i (cid:16)y˜o(cid:17)√ε
where K(w) is the complete elliptic integral of the first kind.
We see that as ε 0 the black holes are pushed far away from each other, the surface of
→
the outer horizons increases to infinity, the surface gravity of both the acceleration and the
outer horizons vanish, and the total charge remains finite.
V. SUMMARY
Wehaveshownthatthefullygravitationallyinteracting‘homogeneous’electricfieldcanbe
obtainedas the limit of the spacetime with localizeduniformly acceleratedsources. Similarly
to the limit of the Born solution (mentioned in the Introduction) the distance between the
sources increases to infinity.
However,unliketheBornsolutionlimit,thetotalchargeofthesourcesremainsfinite. This
is related to the fact that strong ‘homogeneous’ electric field is gravitationally bounded to
the vicinity of the symmetry axis and decays as ρ , cf. Eq. (12). The total electric flux
→∞
through the plane orthogonal to the axis is thus finite.
Amoredelicatequestionisthe behaviorofthemassoftheblackholes.4 Relatingtheblack
hole massto the areaofthe outerblackhole horizon,we haveshownthatit growsto infinity.
Notice also that the resulting field (11), (12) does not depend on details of the limit-
ing procedure—the factors y˜ and y˜ introduced in (5), which parametrize how the limit is
o i
achieved, disappear completely at the end.
A similar limit can also be performed in the presence of a negative cosmological constant
[7]. Rescaling the C-metric with Λ<0 and with an over-critical acceleration (cf. [17]) leads
to a spacetime with an electromagnetic string in the anti-de Sitter universe [18].
3 The surface gravity κ depends on the normalization of the Killingvector. Here we use the Killing vector
a∂t.
4 Itisrelativelyeasytoidentifythe‘netforce’maactingontheacceleratedblackholeintermsofinvariantly
definedquantities(e.g.,takingadifferenceoftensions—proportionaltotheconicities—ofthestringsonthe
bothsideofblackholes). However,itisnotasimpletasktoinvariantlydistinguishthephysical massand
the physical acceleration of the holes. The parameters m and a used in the metric (1) are conventional,
theydependonaparticularchoiceoftheformoftheC-metric.
5
Acknowledgments
This work was supported in part by the grant GACˇR 202/06/0041.
[1] T. Levi-Civita, AttiAccad.Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.26, 307 (1917).
[2] W. Kinnersley and M. Walker,Phys. Rev.D 2, 1359 (1970).
[3] V.Pravda and A. Pravdov´a, Czech. J. Phys. 50, 333 (2000), gr-qc/0003067.
[4] M. A.Melvin, Phys.Lett. 8, 65 (1964).
[5] M. A.Melvin, Phys.Rev. 139, B225 (1965).
[6] K.S. Thorne, Phys. Rev. 139, B244 (1965).
[7] L. Havrdov´a,diploma thesis, Charles University,Prague, Czech Republic(2006), in Czech.
[8] R.Emparan and M. Gutperle, JHEP 0112, 023 (2001), hep-th/0111177.
[9] G.W.GibbonsandC.A.R.Herdeiro,Class.QuantumGrav.18,1677(2001), hep-th/0101229.
[10] K.Hong and E. Teo, Class. QuantumGrav. 20, 3269 (2003), gr-qc/0305089.
[11] J.B.Griffiths,P.Krtouˇs,andJ.Podolsky,Class.QuantumGrav.23,6745(2006),gr-qc/0609056.
[12] A.Ashtekarand T. Dray,Commun. Math. Phys. 79, 581 (1981).
[13] T. Dray,Gen. Rel. Grav. 14, 109 (1982).
[14] H.Farhoosh and R.L. Zimmerman, J. Math. Phys. 20, 2272 (1979).
[15] J. Biˇc´ak, in Galaxies, Axisymmetric Systems and Relativity, edited by M. A. H. MacCaluum
(Cambridge University Press, Cambridge, England, 1985), pp. 91–124.
[16] P.S. Letelier and S. R.Oliveira, Phys.Rev.D 64, 064005 (2001), gr-qc/9809089.
[17] P.Krtouˇs, Phys. Rev.D 72, 124019 (2005), gr-qc/0510101.
[18] O.J. C. Dias and J. P. S.Lemos, Class. QuantumGrav. 19, 2265 (2002), hep-th/0110202.