Intersections of self-gravitating charged shells in a Reissner-Nordstrom field

0.24 MB·English

by  Marco Pizzi

January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
International JournalofModernPhysicsD
(cid:13)c WorldScientificPublishingCompany
9
0
0
2
Intersections of self-gravitating charged shells in a Reissner-Nordstrom
n
field
a
J

6
2
M.PIZZI

Physics Department, Rome University “LaSapienza”,
]
c Piazzale A. Moro, 00185 Rome, Italy, and
q ICRANet, Pescara, Italy 65122.
- [email protected]
r
g
[ A.PAOLINO

Physics Department, Rome University “LaSapienza”,
3
Piazzale A. Moro, 00185 Rome, Italy.
v
6
0 ReceivedDayMonthYear
4 RevisedDayMonthYear
2 CommunicatedbyManagingEditor
.
2
1
8 We describe the equation of motion of two charged spherical shells with tangential
0 pressureinthefieldofacentralReissner-Nordstrom(RN)source.Wesolvetheproblem
: ofdeterminingthemotionofthetwoshellsafter theintersectionbysolvingtherelated
v
Einstein-Maxwellequationsandbyrequiringaphysicalcontinuityconditionontheshells
i
X velocities.
Weconsideralsofourapplications:post-Newtonianandultra-relativisticapproxima-
r
a tions,atest-shellcase,andtheejectionmechanismofoneshell.
ThisworkisadirectgeneralizationofBarkov-Belinski-Bisnovati-Koganpaper.
Keywords: Classicalgravity;Exactsolutions.
1. Introduction
The mathematical model that we analyze in this paper describes the dynamic evo-
lutionoftwosphericalshellsofchargedmatterwhichfreelymoveoutsidethefieldof
acentralReissner-Nordstrom(RN)source.Microscopicallytheseshellsareassumed
tobecomposedbychargedparticleswhichmoveonellipticalorbitswithacollective
variableradius.The angularmotion,distributeduniformly andisotropicallyonthe
shell surfaces,is mathematically described by a tangential-pressureterm in the en-
ergymomentumtensorofthe Einsteinequations.Thedefinitionoftheshellimplies
that all the particles have the same following three ratios: energy/mass, angular
momentum/mass, and charge/mass. Indeed, since the equations of motion for any
1
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
2 Pizzi and Paolino
singled-out particle “a” are
dt 1
a
= (E +e A (r )) (1)
ds m c2g (r ) a a 0 a
a tt a
−
dr 2 1 1 l2 1 1
a = (E +e A (r ))2 a +1
ds m2c4 a a 0 a g (r )g (r ) − m2c2r2 g (r )
(cid:18) (cid:19) a (cid:18)− tt a rr a (cid:19) (cid:18) a (cid:19) rr a
(2)
dθ 2 l2 1 k2 1
a = a a (3)
ds m2c2r4 − m2c2r4sin2θ
(cid:18) (cid:19) a a a
dϕ k 1
a a
= (4)
ds macr2sin2θa
(g and g are the components of a spherical symmetric metric and A is the
tt rr 0
electricpotential;k andl arearbitraryconstants),itiseasytoseethattheradial
a a
motion for all particles is the same if
E e l
a a a
=const, =const, | | =const, a, (5)
m m m ∀
a a a
whereeachconst.doesnotdependontheindexa.Therefore,ifatthebeginningthe
particles are on the same radius r = R , then the shell will evolve “coherently”,
a 0
i.e. all particles will evolve with the same radius.
Now the problemwe are interested in is to find the exchange of energy between
the two shells after the intersection. Indeed the motion of the shells before and
after the crossing can be easily deduced from the equation of motion for just one
1
shell,whichequationhasbeenfoundmanyyearsagobyChase witha geometrical
2
method first used by Israel . Instead, the intersection problem was consideredfirst
5
by Langer-Eid : they applied it to the particular case of electrically neutral and
6
pressureless shells (neutral dust), then Barkov et al. considered the more general
caseofshellsintersectionswhenthe shellshavealsotangentialpressure(soLanger-
Eid’sresultsfollowsfromBarkovetal.resultsasparticularcase).Nowwegeneralize
this problem for the shells with tangential pressure and also electric charge.
What we achieve in the present paper is the determination of the constant pa-
rametersafterthe intersectionknowingjust the parametersbeforethe intersection.
Actually the unknownparameterisonly one,m ,whichis the Schwarzschildmass
21
parameter measured by an observer between the shells after the intersection. This
parameterisstrictlyrelatedtotheenergytransferwhichtakesplaceinthecrossing,
and it is found imposing a proper continuity condition on the shells velocities.
In the model we assume that there are no other interactions between the two
shells apartthe gravitationalandelectrostatic ones.In particularthe shells, during
the intersection, are assumed to be “transparent” each other (i.e. no scattering
processes).
The paper is divided as follows: in Sec.2 we preliminarily discuss the one-shell
case;in Sec.3,which is the centralpart ofthis article, we find the unknownparam-
eter m ; then, Secs.4-7 are devoted to some applications: post-Newtonian approx-
21
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
Intersections of self-gravitating charged shells 3
imation, zero effective masses case (i.e. ultra-relativistic case), test-shell case, and
finally the ejection mechanism.
In this paper we deal only with the mathematical aspects of the problem; some
astrophysicalapplicationsofchargedshellsinthefieldofaRNblackholehavebeen
considered in Ref.[3].
2. A gravitating charged shell with tangential pressure
The motion of a thin charged dust-shell with a central RN singularity was firstly
4
studied by De La Cruz and Israel , while the case with tangential pressure was
1
achieved by Chase in 1970. All these authors used the extrinsic curvature tensor
and the Gauss-Codazzi equations. However we followed a different way, indeed the
same solution can be found also by using δ and θ distributions and then by di-
rect integrationof the Einstein-Maxwellequations (see Ref.[6] and the appendix in
Ref.[7]). This method has the advantage of a clearer physical interpretation, and
it is also straightforward in the calculations; however in the following we will give
only the main passages.
Let there be a central body of mass m and charge e and let a spherical
in in
massive charged shell with charge e move outside this body. It is clear in advance
that the field internalto the shell will be RN, while externally we will have againa
RN metric but with different mass and charge parametersm and e =e +e.
out out in
Using the coordinates x0 = ct and r, which are continuous when passing through
the shell, we can write the intervals inside, outside, and on the shell as
(ds)2 = eT(t)f (r)c2dt2+f−1(r)dr2+r2dΩ2 (6)
− in − in in
(ds)2 = f (r)c2dt2+f−1(r)dr2+r2dΩ2 (7)
− out − out out
(ds)2 = c2dτ2+r (τ)2dΩ2 (8)
− on − 0
where we denoted
dΩ2 =dθ2+sin2θdφ2
and
Gm Ge2 Gm G(e +e)2
f =1 2 in + in , f =1 2 out + in . (9)
in − c2r c4r2 out − c2r c4r2
In the interval (8), τ is the proper time of the shell. The “dilaton” factor eT(t) in
(6) is required to ensure the continuity of the time coordinate t through the shell.
If the equation of motion for the shell is
r =R (t), (10)
0
then joining the angular part of the three intervals (6)-(8), one has
r (τ)=R [t(τ)], (11)
0 0
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
4 Pizzi and Paolino
where the function t(τ) describes the relationship between the global time and the
propertime oftheshell.Joiningthe radial-timepartsofthe intervals(6)-(7)onthe
shell requires that the following relations hold:
2 2
dt dr
f (r ) eT(t) f−1(r ) 0 =1, (12)
in 0 dτ − in 0 cdτ
(cid:18) (cid:19) (cid:18) (cid:19)
dt 2 dr 2
f (r ) f−1(r ) 0 =1. (13)
out 0 dτ − out 0 cdτ
(cid:18) (cid:19) (cid:18) (cid:19)
If the equation of motion for the shell —i.e. the function r (τ)— is known, then
0
the function t(τ) follows from (13) and consequently T(t) can be deduced by (12).
Thus the problem consist only in determining r (τ), which can be done by direct
0
integration of the Einstein-Maxwell equations
Rk 1Rgk = 8πGTk
i − 2 i c4 i
(14)

(√−gFik),k =√−g4cπρui
with the energy-momentum tensor given by:

Tk =ǫu uk+(δ2δk+δ3δk)p+T(el)k (15)
i i i 2 i 3 i
1 1
T(el)k = (F Fkl δkF Flm) . (16)
i 4π il − 4 i lm
Here on we employ the following notations:
ds2 =g dxidxk, g has signature ( ,+,+,+)
ik ik
− −
xk =(ct,r,θ,ϕ) i,j,k...=0,1,2,3
p p(R )=p =p =tangential pressure (p =0)
0 θ ϕ r
≡
F =A A
ik k,i i,k
−
The above equations are to be solved for the metric
ds2 =g (t,r)c2dt2+g (t,r)dr2+r2dΩ2, (17)
00 11
−
and for the potential
A =A (t,r), A =A =A =0, (18)
0 0 1 2 3
As followsfrom the Landau-Lifshitz approach[8] (see Ref.[6]) the energy distri-
bution of the shell is
M(t)c2δ[r R (t)]
0
ǫ= − , (19)
4πr2u0√ g g
00 11
−
while its charge density is
ceδ[r R (t)]
0
ρ= − , (20)
4πr2u0√ g g
00 11
−
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
Intersections of self-gravitating charged shells 5
where δ is the standard δ-function. In the absence of tangential pressure p, the
quantity M in Eqn.(19) would be a constant, but in presence of pressure, Mc2
includes the rest energy along with the energy (in the radially comoving frame) of
the tangential motions of the particles that produce this pressure.
It can be checked that the Einstein part of (14) actually lead to the solution
(6)-(8) with, in addition, the “joint condition”
dr 2 dr 2 (m m ) e2+2ee
0 0 out in in
f (r )+ + f (r )+ =2 − , (21)
s in 0 (cid:18)cdτ(cid:19) s out 0 (cid:18)cdτ(cid:19) µ(τ) − µ(τ)c2r0
where we denoted
µ(τ)=M[t(τ)], (22)
while
m m =E/c2 (23)
out in
−
is a constant which can be interpreted as the total amount of energy of the shell.
Then,fromtheMaxwellsideof(14)theonlynon-vanishingcomponentoftheelectric
field is
√ g g
00 11
F = − e +eθ[r R (t)] (24)
01 − r2 { in − 0 }
(θ(x) is the standard step function). Finally, the equationsTk =0 can be reduced
i;k
to the only one relation:
dM c2δ[r R (t)]
0
p= − (25)
− dt 8πru1√ g g
00 11
−
We will not treat here the steady case (i.e. r =const) which should be treated
0
separately; thus in the following we will assume always r =const. .
0
6
The joint condition (21) can be written in several different forms: two of them,
which will be useful in the following, are
dr 2 (m m ) Gµ2(τ) e2 2ee
0 out in in
f (r )+ = − + − − (26)
s in 0 (cid:18)cdτ(cid:19) µ(τ) 2µ(τ)c2r0
and
dr 2 (m m ) Gµ2(τ)+e2+2ee
0 out in in
f (r )+ = − . (27)
s out 0 (cid:18)cdτ(cid:19) µ(τ) − 2µ(τ)c2r0
As in Ref. 6, all the radicals encountered here are taken positive, since for astro-
physicalconsiderationsonlythesecasesaremeaningful.Toproceedfurther,wemust
specifytheequationofstate,i.e.thefunctionµ(τ).Hereweconsideraparticle-made
shell, therefore the sum of kinetic and rest energy of all the particles is
p2
Mc2 = m c2 1+ a , (28)
a a s m2ac2!
X
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
6 Pizzi and Paolino
where p is the tangential momentum of each particle (the electric interaction be-
a
tween the particles is already taken into account by the self-energy term of, e.g.,
(26), thus one has not to include it in M too). From the definition of the shell (see
Introduction) it follows:
p2 l2 const
a = a = , (29)
m2 m2R2 R2
a a 0 0
the square root in (28) does not depend on the index a; then defining
m c2 =mc2, l =L,
a a
| |
a a
X X
formula (28) can be re-written (remembering definition (22) too) as
L2
µ(τ)= m2+ . (30)
s c2r02(τ)
Thus,now,onecandeterminethefunctionr (τ)fromequation(21)[orfromoneof
0
the equivalentforms(26)-(27)]if the initialradius oftheshell andthe six arbitrary
constants m , m , m, e , e and L are specified. Accordingly with (19), (25),
in out in
(22) and (30), the equation of state that relates the shell energy density ǫ to the
tangential pressure p is
ǫ L2 L2 −1
p= 1+ (31)
2m2c2R2 m2c2R2
0 (cid:18) 0(cid:19)
asinthe unchargedcase,i.e.the presenceofthe chargesdo notmodify the relation
between energy density and pressure (indeed the presence of the charge is hidden
in the equationof motion).Note that when the shellexpands to infinity (R )
0
→∞
the angular momentum becomes irrelevant and the equation of state tends to the
dust case p<<ǫ.
3. The shells intersection
Letusnowconsiderthecaseoftwoshellswhichmoveinthefieldofacentralcharged
mass. The generalization from the previous (single-shell) case is straightforward if
theshellsdonotintersect:indeedtheoutershelldonotaffectthemotionoftheinner
one, while the inner one appears from outside just as a RN metric. Therefore the
principalaimofthissectionistoconsidertheintersectioneventualityandtopredict
themotionofthetwoshellsafter thecrossing,havingspecifiedtheinitialconditions
before the crossing. After the intersection one has a new unknown constant that
has to be found by imposing opportune joining conditions as now we are going to
explain (the analysis follows step by step the Ref.6’s one).
Let us previously analyze the space-time in the (t,r) coordinates (which are
continuous through the shells). We define the point O (t ,r ) as the intersection
∗ ∗
≡
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
Intersections of self-gravitating charged shells 7
r
C B
2
O
A
1
D
t
Fig.1. Thefourregioninwhichitisdividedthespacetime;thetwolinesrepresentthetrajectories
ofshell-1andshell-2.
point; then the space-time is divided in four regions (see Fig.1):
COB (r >R ,r >R ),
1 2
COA (R <r <R ),
1 2 (32)
AOD (r <R ,r <R ),
1 2
BOD (R <r <R ).
2 1
Correspondinglytotheseregionswehavethemetricinform(13)butwithdifferent
coefficients g and g :
00 11
g(COB) = f (r), g(COB) =f−1(r) (33)
00 − out 11 out
g(COA) = eT1(t)f (r), g(COA) =f−1(r) (34)
00 − 12 11 12
g(AOD) = eT0(t)f (r), g(AOD) =f−1(r) (35)
00 − in 11 in
g(BOD) = eT2(t)f (r), g(BOD) =f−1(r) (36)
00 − 21 11 21
ThedilatonfactorT(t)allowstocoverallthespace-timewithonlyonet-coordinate;
here,f andf arethesameasthosein(9)whilef andf aregivenbysimilar
in out 12 21
expressions:
2Gm G(e +e )2
12 in 1
f =1 + (37)
12 − c2r c4r2
2Gm G(e +e )2
21 in 2
f =1 + (38)
21 − c2r c4r2
As we said, the parameters m , m , m , e , e , e are assumed to be specified
in 12 out in 1 2
at the beginning, while m is the actual unknown constant which has yet to be
21
determined from the joining conditions on (t ,r ).
∗ ∗
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
8 Pizzi and Paolino
Before the intersection
Letuswritetheequationofmotionforthetwoshellsbeforetheintersection(shell-1
inner andshell-2 outer). This can be made easily adapting the (27) and (26) to the
present case:
dr 2 (m m ) GM2+e2+2e e
f (r )+ 1 = 12− in 1 1 in 1 (39)
s 12 1 (cid:18)cdτ1(cid:19) M1 − 2M1c2r1
for shell 1, while for shell 2
dr 2 (m m ) GM2 e2 2(e +e )e
f (r )+ 2 = 12− in + 2 − 2− in 1 2 (40)
s 12 2 (cid:18)cdτ2(cid:19) M2 2M2c2r2
with
L2 L2
M = m2+ 1 , M = m2+ 2 . (41)
1 s 1 c2r12 2 s 2 c2r22
Here, τ and τ are the proper times of the first and second shells respectively,
1 2
while r (τ )=R [t(τ )] and r (τ )=R [t(τ )]. Now we have to impose the joining
1 1 1 1 2 2 2 2
conditions for the intervals onboth the shells.For the firstshell (oncurve AO)one
has:
2 2
dt dr
eT1(t)f (r ) f−1(r ) 1 =1 (42)
12 1 dτ − 12 1 cdτ
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
2 2
dt dr
eT0(t)f (r ) f−1(r ) 1 =1; (43)
in 1 dτ − in 1 cdτ
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
while for the second shell:
2 2
dt dr
f (r ) f−1(r ) 2 =1 (44)
out 2 dτ − out 2 cdτ
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
2 2
dt dr
eT1(t)f (r ) f−1(r ) 2 =1. (45)
12 2 dτ − 12 2 cdτ
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
If all free parameters and initial data to Eqs.(39)-(41) were specified and if the
functions r (τ ) and r (τ ) were derived, then their substitution in (42)-(45) gives
1 1 2 2
the functions τ (t), τ (t) and T (t), T (t), which is enough for determining the
1 2 1 0
motionofthe shellsbefore the intersection.Thereforethe intersectionpoint(t ,r )
∗ ∗
can be found by solving the system
r =r (τ (t ))
∗ 1 1 ∗ (46)
r =r (τ (t )) ,
(cid:26) ∗ 2 2 ∗
which we assume that has a solution.
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
Intersections of self-gravitating charged shells 9
After the intersection
The equation of motion for the shells after the intersection time t can be con-
∗
structed in the same way again by turning to Eqns.(26) and (27), and introducing
the new parameterm whichcharacterizethe “Schwarschildmass” seenby an ob-
21
server in the region BOD. We use Eq.(26) for (now outer) shell 1 and Eq.(27) for
(now inner) shell 2:
dr 2 (m m ) GM2 e2 2e (e +e )
f (r )+ 1 = out− 21 + 1 − 1− 1 in 2 , (47)
s 21 1 (cid:18)cdτ1(cid:19) M1 2M1c2r1
dr 2 (m m ) GM2+e2+2e e
f (r )+ 2 = 21− in 2 2 2 in . (48)
s 21 2 (cid:18)cdτ2(cid:19) M2 − 2M2c2r2
Naturally, M (r ) and M (r ) are given by the same expression of (41) but now
1 1 2 2
they have to be calculated on r (τ ) and r (τ ) after the intersection.
1 1 2 2
Joining the intervals on the first shell (on curve OB) yields
2 2
dt dr
f (r ) f−1(r ) 1 =1 (49)
out 1 dτ − out 1 cdτ
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
2 2
dt dr
eT2(t)f (r ) f−1(r ) 1 =1. (50)
21 1 dτ − 21 1 cdτ
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
Then, joining the second shell (on curve OB) we obtain:
2 2
dt dr
eT2(t)f (r ) f−1(r ) 2 =1 (51)
21 2 dτ − 21 2 cdτ
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
2 2
dt dr
eT0(t)f (r ) f−1(r ) 2 =1. (52)
in 2 dτ − in 2 cdτ
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
Since the initial data to Eqs.(47) and (48) have already been specified (from the
previous evolution), then the evolution of the shells after the intersection would
be determined from Eqs.(47)-(52) if parameter m were known. Thus we need an
21
additional physical condition from which we could determine m .
21
This condition follows from the fact that the Christoffel symbols (i.e. the accel-
erations) of the shells have only finite discontinuities (finite jumps), therefore the
relative velocity of the shells must remain continuous through the crossing point.
In the presence of two shells, we can construct one more invariant than in the
singleshellcase(whereonlyu ui = 1waspossible):thescalarproductbetweenthe
i
−
two4-velocitiesoftheshells.Wecanalsoavoidtoapplytheparalleltransportifwe
evaluate the 4-velocities on the intersectionpoint (t ,r ). The continuity condition
∗ ∗
can be found imposing that the scalar product has to have the same value when
evaluated in both the two limits t t− and t t+.
→ ∗ → ∗
January 26, 2009 13:19 WSPC/INSTRUCTION FILE
Charged˙Shell˙Intersections
10 Pizzi and Paolino
Determination of Q.
Let us start determining the quantity
Q≡{g0(C0OA)u0AOu0CO+g1(C1OA)u1AOu1CO}t=t∗,r=r1=r2=r∗, (53)
whichisthescalarproductofthetwo4-velocitiesevaluatedintheintersectionpoint
from the region AOC (along the curves AO and CO). Written explicitly, the unit
tangent vector to trajectory AO is
ui =(u0 ,u1 ,u2 ,u3 )
AO AO AO AO AO
dt dr
1
= , ,0,0 , (54)
dτ cdτ
(cid:18) 1 1 (cid:19)t≤t∗
while for the trajectory CO we have
ui =(u0 ,u1 ,u2 ,u3 )
CO CO CO CO CO
dt dr
2
= , ,0,0 . (55)
dτ cdτ
(cid:18) 2 2 (cid:19)t≤t∗
The fact that these are actually unit vectors follows from the joining equations
(42) and (45).
The components of the vector (54) can be easily expressed from Eqs.(39) and
(42) as
dt e−T1(t)/2 GM2(r )+e2+2e e
= m m 1 1 1 1 in (56)
dτ M (r )f (r ) 12− in− 2c2r
(cid:18) 1(cid:19)t≤t∗ 1 1 12 1 (cid:18) 1 (cid:19)
dr
1
=
cdτ
(cid:18) 1(cid:19)t≤t∗
δ GM2(r )+e2+2e e 2
= 1 m m 1 1 1 1 in M2(r )f (r )
M1(r1)f12(r1)s(cid:18) 12− in− 2c2r1 (cid:19) − 1 1 12 1
(57)
where
dr
δ =sgn 1 . (58)
1
cdτ
(cid:18) 1(cid:19)t≤t∗
Analogously,forthecomponentsofvector(55),weobtainthe followingexpressions
from Eqs.(40) and (45):
dt e−T1(t)/2 GM2(r ) e2 2e (e +e )
= m m + 2 2 − 2− 2 in 1
dτ M (r )f (r ) out− 12 2c2r
(cid:18) 2(cid:19)t≤t∗ 2 2 12 2 (cid:18) 2 (cid:19)
(59)