Second order perturbations of a Schwarzschild black hole: inclusion of odd parity perturbations

12 Pages·0.15 MB·English

by  Carlos O. Nicasio

Second order perturbations of a Schwarzschild black hole: inclusion of odd parity
perturbations
Carlos O. Nicasio1, Reinaldo Gleiser1, Jorge Pullin2
1. Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba,
Ciudad Universitaria, 5000 C´ordoba, Argentina.
2. Center for Gravitational Physics and Geometry, Department of Physics,
The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802
WeconsiderperturbationsofaSchwarzschildblackholethatcanbeofbothevenandoddparity,
keeping terms up to second order in perturbation theory, for the ℓ = 2 axisymmetric case. We
developexplicitformulaefortheevolutionequationsandradiatedenergiesandwaveformsusingthe
Regge–Wheeler–Zerilli approach. This formulation is useful, for instance, for the treatment in the
“close limit approximation” of thecollision of counterrotating black holes.
1
0
I. INTRODUCTION
0
2
There has been a recent surge of interest in black hole perturbation theory, stemming from the successful applica-
n
a bility of such a formalism in the case of black hole collisions (see [1] for a review). In particular, the use of second
J order perturbative computations [2] has proven quite useful to endow the perturbative formalism with “error bars”

0 and also to increase its accuracy.
2 Thecomputationsassociatedwiththe secondorderformulationarequite laborious. Inparticular,the secondorder
perturbativeequationsarelineardifferentialequationsthatcontain“source”termsthatarequadraticinthefirstorder

2 perturbations. This forbids us from doing a “generic” second order computation, since it would involve an infinite
v
number of source terms. Computations have to be restricted to certain multipolar orders and parities to achieve a
1
computable expression. The expressions we have presented in the past [2] are all for even parity axisymmetric ℓ=2
2
0 first order perturbations. This is a significant case, since for first order perturbations such modes are the ones that
1 dominate the initial data for black hole collisions (at least in the head-on case, in the non-head on case one also
0 has non-axisymmetric m = 2 modes). If the black holes individually have spin, however, odd parity terms appear
0 as well. If one wishes to st±udy such collisions with the second order formulation one needs to lay out the second
0
order equations for odd and even parity perturbations. Since the “source” terms that appear in the second order
/
c equations are quadratic, one has quadratic contributions from odd parity first order modes in the even parity second
q order equations. The intention of this paper is to present a comprehensive account of the perturbative equations
- involving odd terms, up to second order. In a separate publication we will apply the formalism to the collision of
r
g counterrotating black holes.
:
v
i
X
II. PERTURBATIVE FORMALISM: REVIEW OF SOME FIRST ORDER RESULTS
r
a
A. The Regge–Wheeler decomposition and the Regge–Wheeler gauge
In order to fix notation and to set the computationalphilosophy for the rest of the paper, we recollect here several
results of first order perturbation theory that are known, but that it is useful to have laid out in a compact fashion
for even and odd perturbations together.
We consider the axially symmetric perturbations of a spherically symmetric background, up to and including
second order, using the Regge–Wheeler framework. In first order perturbations there is no loss of generality in
the consideration of axisymmetric perturbations since modes with different m value decouple. This is not the case,
however, in second order theory and our treatment is therefore restricted to the axisymmetric case. In terms of
the usual Schwarzschild-like coordinates t,r,θ,φ, any ℓ = 2 axially symmetric perturbation may be written in the
“Regge–Wheeler form” [3]
g = (1 2M/r) 1 ǫH(1)(r,t)+ǫ2H(2)(r,t) P (θ) (1)
tt − − − 0 0 2
gerr =(1−2M/r)−1n1+hǫeH2(1)(r,t)+ǫ2eH2(2)(r,t)i P2(θ)o (2)
n h i o
egrt = ǫH1(1)(r,t)+ǫ2H1(2)(r,et) P2(θ) e (3)
h i
e e e
1
g = ǫh(1)(r,t)+ǫ2h(2)(r,t) ∂P (θ)/∂θ) (4)
tθ 0 0 2
h i
getφ = ǫeh(01o)(r,t)+ǫ2eh(02o)(r,t) sin(θ)∂P2(θ)/∂θ) (5)
h i
gerφ = ǫeh(11o)(r,t)+ǫ2eh(12o)(r,t) sin(θ)∂P2(θ)/∂θ) (6)
egrθ =hǫeh(11)(r,t)+ǫ2eh(12)(r,t)i∂P2(θ)/∂θ) (7)
h i
geθθ =r2e 1+[ǫK(1)(re,t)+ǫ2K(2)(r,t)]P2(θ)
n
e +[ǫG(1)(r,et)+ǫ2G(2)(r,et)]∂2P2(θ)/∂θ2) (8)
o
gφφ =r2 sein2θ+sin2θ[eǫK(1)(r,t)+ǫ2K(2)(r,t)]P2(θ)
n
e +sin(θ)cos(θ)[ǫG(1)e(r,t)+ǫ2G(2)e(r,t)]∂P2(θ)/∂θ) (9)
1 o
gθφ = 2 ǫh(21o)(r,t)+ǫ2eh(22o)(r,t) coes(θ)∂P2(θ)/∂θ−sin(θ)∂2P2(θ)/∂θ2 (10)
where P2(θ)=3cose2(θ)/2 h1/e2. We introdeuce theiex(cid:2)pansionparameterǫ, which defines th(cid:3)e perturbationorder,and
−
identify the corresponding metric coefficients with the superscripts (1), (2) and use the subscript “o” to refer to the
odd parity quantities.
As we stated above, any axisymmetric perturbation can be put into the Regge–Wheeler form. However, for
calculationalsimplicity, it is usually convenientto takeadvantageof the coordinate freedomto restrictsomewhatthe
formofthe metric. Anexample ofsuchchoiceis thesocalledRegge - Wheeler gauge(RWG),wherethe nonvanishing
metric coefficients are
g = (1 2M/r) 1 [ǫH(1)(r,t)+ǫ2H(2)(r,t)]P (θ) (11)
tt − − − 0 0 2
g =(1 2M/r)−1n1+[ǫH(1)(r,t)+ǫ2H(2)(r,t)]P (θ)o (12)
rr − 2 2 2
g =[ǫH(1)(r,t)+ǫn2H(2)(r,t)]P (θ) o (13)
rt 1 1 2
g =[ǫh(1)(r,t)+ǫ2h(2)(r,t)]sin(θ)∂P (θ)/∂θ (14)
tθ 0o 0o 2
g =[ǫh(1)(r,t)+ǫ2h(2)(r,t)]sin(θ)∂P (θ)/∂θ (15)
rθ 1o 1o 2
g =r2 1+[ǫK(1)(r,t)+ǫ2K(2)(r,t)]P (θ) (16)
θθ 2
n o
g =r2 sin2θ+sin2θ[ǫK(1)(r,t)+ǫ2K(2)(r,t)]P (θ) , (17)
φφ 2
n o
that is, we are demanding that h(i) =h(i) =G(i) =h(i) =0, i=1,2.
0 1 2o
Aremarkableaspectofthe Regge–Wheelergauge(RWG)isthatitisunique,inthesensethatthemetricperturba-
tion coefficients in the RWG can be uniquely recoveredfrom those in an arbitrary gauge. That is, given an arbitrary
metric, in an arbitrary gauge, there is a well defined procedure that brings it to the RWG.
Letus exhibitthis propertyforfirstordergaugetransformations. One canuniquely choosea gaugetransformation
vector that brings the metric to the RWG form [2]. The resulting transformation formulae are,
2
K(1) =K(1)+(r 2M) G(1), h(1) (18)
− r−r2 1
(cid:18) (cid:19)
2 2
H(1) =He(1)+(2r 3M) eG(1), eh(1) +r(r 2M) G(1), h(1) (19)
2 2 − r−r2 1 − r−r2 1
(cid:18) (cid:19) (cid:18) (cid:19),r
H(1) =He(1)+r2G(1), h(1e), 2Me h(1)+h(1), +re(r−3M)Ge(1), (20)
1 1 tr− 1 t−r(r 2M) 0 0 r r 2M t
− −
H(1) =He(1) MeG(1), e 2 h(1) 2re h(1),e+ r3 G(1), e (21)
0 0 − r−r2 1 − r 2M 0 t (r 2M) tt
(cid:18) (cid:19) − −
1
h(1) = eh(1) +h(1)e e e e (22)
0o 2 2o,t 0o
1
h(1) = e rh(1)e 2h(1) +h(1). (23)
1o 2r 2o,r− 2o 1o
(cid:16) (cid:17)
e e e
2
Something that should be stressed is that given the uniqueness of the RWG, quantities computed in such a gauge
(like,forinstance,metriccomponents),aregauge invariant. Thatis,substitutingtheaboveformulaeinthedefinition
of any quantity in the RWG one obtains expressions for the quantity in any gauge.
The procedure can in principle be repeated at every order in perturbation theory. To achieve this, one first
performs a first order transformation that yields the metric (at first order) in the RWG. This transformation will
induce transformations at all higher orders as well. One then makes a purely second order gauge transformationthat
brings the second order portion of the metric to the RWG. This will leave the first order piece in RWG, and will
modify the third and higher orders as well. One can continue this procedure up to an arbitrarily high order and the
metric will be, up to that order,in the RWG. Unfortunately, the procedure is not quite unique at higher orders. The
reason for this is that if before performing the second order gauge transformation, one performs an ℓ = 0 first order
gaugetransformation,the procedurewe outlined yields atthe endofthe day a different secondordermetric, butstill
in the RWG.That is, the metric one obtains at the end of the procedure is not unique, in spite of being in the RWG.
Fortunately, one can isolate second order quantities that are invariant under these ℓ=0 transformations. This point
has been discussed in reference [4] in detail, so we will not repeat it here. From the expressions for waveforms given
in this paper, one can apply the techniques of reference [4] in order to compare with numerical results without the
ℓ=0 ambiguity.
From now on, we will do computations in the RWG. Because of what we just discussed, there is no issue of gauge
invariance in our formalism, since all quantities in the RWG (appropriately modified to take into account the ℓ = 0
issue) will be “gauge invariant” in the sense described above.
B. The first order Zerilli equation
As shown by Zerilli, we may introduce a function ψ(1)(r,t), such that if we write for the metric components in the
RWG,
r2+rM +M2 M ∂ψ(1)(r,t)
K(1)(r,t)=6 ψ(1)(r,t)+ 1 2 (24)
r2(2r+3M) − r ∂r
(cid:18) (cid:19)
∂ 2r2 6rM 3M2 ∂ψ(1)(r,t)
H(1)(r,t)= − − ψ(1)(r,t)+(r 2M) K(1)(r,t) (25)
0 ∂r r(2r+3M) − ∂r −
(cid:20) (cid:21)
2r2 6rM 3M2 ∂ψ(1)(r,t) ∂2ψ(1)(r,t)
H(1)(r,t)= − − +r (26)
1 (r 2M)(2r+3M) ∂t ∂r∂t
−
H(1)(r,t)=H(1)(r,t), (27)
2 0
then the linearized Einstein equations are satisfied, provided only that ψ(1)(r,t) is a solution of the Zerilli equation
(28).
∂2ψ(1)(r,t) ∂2ψ(1)(r,t)
V (r∗)ψ(1)(r,t)=0 (28)
∂r∗2 − ∂t2 − Z
where
∗
r =r+2Mln[r/(2M) 1] (29)
−
and
M 4r3+4r2M +6rM2+3M3
V (r)=6 1 2 (30)
Z − r r3(2r+3M)2
(cid:18) (cid:19)
It is straightforwardto obtain “inversion”formulas for ψ(1)(r,t) in terms of the metric coefficients. From (24) and
(26), we have
∂ψ(1)(r,t) r ∂K(1) M
(1)
= r 1 2 H (31)
∂t 2r+3M ∂t − − r 1
(cid:20) (cid:18) (cid:19) (cid:21)
while from (24) and (25) we find
3
r(r 2M) ∂K(1) r
ψ(1)(r,t)= − H(1) r + K(1) (32)
3(2r+3M) 0 − ∂r 3
(cid:18) (cid:19)
Equations(31)and(32)areequivalent,asfaras firstorderperturbationsareconcerned,upto anadditive function
of r. This ambiguity is of no concern since the relevant physical quantities, like radiated energies and waveforms are
allcomputedintermsofthe timederivativeofψ(1),soitisconventionallyignored,wewillassumewesetthe additive
function to zero. Equation (32) can be used to obtain a gauge invariant form for ψ(1)(r,t), i.e., an expression for
ψ(1)(r,t), valid in an arbitrary gauge. This is an especially important result, because it allows us to obtain the initial
data for the Zerilli equation, directly in an appropriate gauge. It should be noted, however, that although ψ(1)(r,t)
is uniquely defined in a general gauge by (32), we do not have, in a general gauge, unique expressions for the metric
perturbation functions in terms of ψ(1)(r,t). The general statement of Zerilli’s results is, in this case, that if the
metric perturbations satisfy the first order Einstein equations, then ψ(1)(r,t) satisfies the Zerilli equation (28).
The situation, as we shall see, is somewhat different for the second order perturbations, because of the presence of
“source terms” in the Zerilli equation.
C. The Odd parity equation
One can also write a “Zerilli”-like equation (known as the Regge–Wheeler equation, historically it was discovered
earlierthantheevenparityone[3])fortheoddpartoffirstordermetricperturbationsandthe equationstheysatisfy.
If we define
2M ∂
h(1)(r,t)= 1 r Q(1)(r,t)+Q(1)(r,t) (33)
0o − r ∂r
(cid:18) (cid:19)(cid:18) (cid:19)
r ∂
h(1)(r,t)= Q(1)(r,t) (34)
1o 1 2M/r∂t
−
the Einstein equations are satisfied, provided that Q(1)(r,t) is a solution of the equation
∂2Q(1)(r,t) ∂2Q(1)(r,t)
V (r∗)Q(1)(r,t)=0 (35)
∂r∗2 − ∂t2 − Q
where
∗
r =r+2Mln[r/(2M) 1] (36)
−
and
(r M)(r 2M)
V (r)=6 − − . (37)
Q r4
It is easier to obtain inversionformulas in this case that in the even parity case. We can write,
1 ∂h(1)(r,t) ∂h(1)(r,t)
Q(1)(r,t)= r 1o r 0o +2h(1)(r,t) (38)
−4 ∂t − ∂r 0o !
∂Q(1)(r,t) (1 2M/r)
= − h(1)(r,t) (39)
∂t r 1o
We can make the same comments as in the even case. Equation (38) can be used to give a gauge invariant form for
the odd parity Zerilli function. Bothequations allow us to give initial conditions for Q(1)(r,t) in anany gauge. As in
the even case, there is no unique way to write the metric perturbations in terms of Q(1)(r,t) in an arbitrary gauge.
III. SECOND ORDER PERTURBATIONS
We define the second order Regge–Wheeler gauge by imposing that h(i) =h(i) =G(i) =h(i) =0, i=1,2. It is easy
0 1 2o
to check thatthis gaugeis uniquely defined, as longas we restrictto ℓ=2 axisymmetric firstandsecondordergauge
transformations, but the general expressions for the second order gauge transformations are rather lengthy.
4
It should be clear from the perturbation expansion, that the second order Einstein equations imply, in turn, that
the second order perturbation functions satisfy the same linear set of equations as the first order perturbations, but
with additionalterms quadraticin the firstorderperturbations,that maybe thoughtofasrepresenting“sources”for
these equations.
Therefore if motivated by the corresponding expression for the time derivatives ∂ψ(1)/∂t, ∂Q(1)/∂t, of the Zerilli
functions in first order perturbation theory, we introduce gauge invariant functions, given by
r ∂K(2)(r,t) 2M
χ(2)(r,t)= r 1 H(2)(r,t) (40)
2r+3M ∂t − − r 1
(cid:20) (cid:18) (cid:19) (cid:21)
(1 2M/r)
Θ(2)(r,t)= − h(2)(r,t) (41)
r 1o
it follows that if the second order Einstein equations are satisfied, then χ(2)(r,t), Θ(2) satisfy equations of the form
∂2χ(2)(r,t) ∂2χ(2)(r,t)
V (r∗)χ(2)(r,t)+ =0 (42)
∂r∗2 − ∂t2 − Z SZ
∂2Θ(2)(r,t) ∂2Θ(2)(r,t)
V (r∗)Θ(2)(r,t)+ =0 (43)
∂r∗2 − ∂t2 − Q SQ
where , are quadratic polynomials in the first order functions and their t and r derivatives, with r dependent
Z Q
S S
coefficients. , may be considered as a kind of ”source terms” for the otherwise homogeneous equations satisfied
Z Q
S S
by χ(2), Θ(2).
But, precisely because of the presence of these “sourceterms”, if we redefine χ(2)(r,t), Θ(2)(r,t) by the addition of
a quadratic polynomial in the first order functions and their derivatives, the new, redefined χ(2)(r,t), Θ(2)(r,t) will
stillsatisfyanequationoftheform(42),providedonlythatthefirstandsecondorderEinsteinequationsaresatisfied.
ThisarbitrarinessinthedefinitionoftheZerillifunctions associatedtothesecondorderperturbationsmaybeused
tosimplifytheform,andasymptoticbehaviorofthe“sourceterms” , . Aswasdiscussedin[2],thesourceterms
Z Q
S S
are delicate in the sense that they can be divergentfor largevalues of t and r. The divergence of the source does not
imply a physicaldivergencein the problem,but it poses seriousdifficulties for numeric computations. If we make the
following choice, basedon the study of the solutions of the Zerilli equations for large values of r, the source simplifies
and also is well behaved at spatial infinity,
r ∂K(2) 2M 2 r2 ∂K(1)
χ(2)(r,t)= r 1 H(2) K(1) +(K(1))2 (44)
2r+3M ∂t − − r 1 − 7 2r+3M ∂t
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21)
as our definition for χ(2)(r,t), while Θ(2)(r,t) is defined by (41). Making the appropriate replacements, we then find
for the sources,
12 (r 2M)3 24r5 108Mr4 72M2r3+24M3r2+180M4r+180M5 (ψ)2
= − − − −
SZ − 7 r3(2r+3M)"(cid:0) r5(2r+3M)(r 2M)2 (cid:1)
−
64r5+176Mr4+592M2r3+1020M3r2+1122M4r+540M5 ψ ψ
r
+
r4(2r+3M)2(r 2M)
(cid:0) (cid:1)
−
112r5+480Mr4+692M2r3+762M3r2+441M4r+144M5 ψ ψ
t
− r2(2r+3M)3(r 2M)2
(cid:0) (cid:1)
−
8r3+16r2M +36rM2+24M3 ψrrψ 8r2+12rM +7M2 ψrtψ Mψψrrt rψtψrrr
+ + + +
r3(2r+3M) r(2r+3M)( r+2M) 2r+3M 3
(cid:0) (cid:1) (cid:0) − (cid:1)
12r3+36r2M +59rM2+90M3 (ψ )2 18r3+4r2M +33rM2+48M3 ψ ψ
r r t
+ −
− 3r3(r 2M) 3(2r+3M)r(r 2M)2
(cid:0) − (cid:1) (cid:0) − (cid:1)
12r2+20rM +24M2 ψrψrr ( 3r+7M)ψrψrt rψrψrrt
−
− 3r2 − 3r+6M − 3
(cid:0) (cid:1) −
12 r2+rM +M2 2(ψ )2 2r2+M2 ψ ψ 4r2+4rM +4M2 ψ ψ
t rr t t rt
+ − +
(2r+3M)r(r 2M)3 − (2r+3M)(r 2M) (r 2M)2
(cid:0) −(cid:1) (cid:0) (cid:1)− (cid:0) − (cid:1)
(2r+3M)(r 2M)(ψ )2 (2r+3M)r(ψ )2 ( 2r+10M)(3r+8M)Q Q
rr rt r t
− −
− 3r − 3r+6M − r(r 2M)2
− −
5
22r2+70rM +66M2 Q Q 136r3 462r2M +50rM2+456M3 QQ
r rt t
+ − −
(2r+3M)(r 2M) − r2(2r+3M)(r 2M)2
(cid:0) − (cid:1) (cid:0) − (cid:1)
14r2+58rM +66M2 QQrt (22r+54M)(r+M)QtQrr
+ (45)
(cid:0) r(2r+3M)(r−2M(cid:1) ) − (2r+3M)(r−2M) #
12 (r 2M)3 40r4 150Mr3 138r2M2+162M3r+81M4 Qψt
= − − −
SQ − 7 r3(2r+3M)"(cid:0) 3r3(2r+3M)(r 2M)2 (cid:1)
−
+ (2r+3M)(−r+M)Qψrrt + 4r3−17r2M −33rM2+81M3 Qψrt
r(r 2M) 3r2(r 2M)2
− (cid:0) − (cid:1)
160M2r3+216M5+102M3r2 160Mr4+387M4r 128r5 Q ψ
t
+ − − −
2r3(2r+3M)2(r 2M)2
(cid:0) (cid:1)
−
+ 75M3r2−12r5+81M5−10M2r3+14Mr4+150M4r ψQrt (2r+3M)rψrrQrrt
r2(2r+3M)2(r 2M)2 − 3
(cid:0) (cid:1)
−
4r3+4r2M +6rM2+3M3 ψQ 8r4+56Mr3 40r2M2 39M3r+72M4 Q ψ
rrt t r
+ − − −
r(2r+3M)(r 2M) − 2r2(2r+3M)( r+2M)2
(cid:0) − (cid:1) (cid:0) − (cid:1)
18r4+19Mr3 13r2M2+78M3r+81M4 ψ Q 2r2 2rM +3M2 ψ Q
r rt r rrt
− − + −
− 3r(2r+3M)(r 2M)2 3r+6M
(cid:0) − (cid:1) (cid:0) − (cid:1)
28r4 72Mr3 96r2M2+114M3r+81M4 ψ Q 2r2+6rM +3M2 ψ Q
t r t rrr
− − + −
− 3r2(2r+3M)(r 2M)2 3r+6M
(cid:0) − (cid:1) (cid:0) − (cid:1)
2r4 9Mr3+9r2M2+22M3r+9M4 ψ Q 4r3 11r2M 15rM2+63M3 Q ψ
t rr t rr
− + − −
− r(2r+3M)(r 2M)2 3r(r 2M)2
(cid:0) (cid:1) (cid:0) (cid:1)
− −
6r2+7rM +18M2 ψ Q 10r3 10r2M 37rM2+27M3 ψ Q
rr rt rt r
+ − −
6r+12M − 3r(r 2M)2
(cid:0) − (cid:1) (cid:0) − (cid:1)
6r2+7rM +18M2 ψrtQrr (2r+3M)rQrrrψrt (2r+3M)( r+3M)Qtψrrr
+ −
− 6r+12M 3 − 3r+6M
(cid:0) − (cid:1) −
(2r+3M)rψ Q (2r+3M)( r+M)ψ Q (2r+3M)rQ ψ
rrr rt rrt r rr rrt
+ − + (46)
− 3 3r+6M 3
− (cid:21)
where ψ =ψ(1), Q=Q(1) and the subindices indicate partial derivatives.
IV. GRAVITATIONAL WAVE AMPLITUDE
A straightforwardcomputational way to define a gravitationalwave amplitude, is to consider a gauge in which the
space-time is manifestly asymptotically flat. For instance, consider the typical gauge discussed in reference [5]. In
this gauge the metric has components that behave in the following way,
H O(1/r3)
2
≃
h O(1/r)
1
≃
K O(1/r)
≃
G O(1/r)
≃
h O(1/r)
1o
≃
h O(r) (47)
2o
≃
and, of course, H = H = h = h = 0. The gravitational wave amplitude can then simply be read off from the
0 1 0 0o
components K, G and h . We therefore need to relate the behavior of these components of the metric to the Zerilli
2o
function in an asymptotic region far away from the source.
6
In orderto study the asymptotic behavior,one considersa solutionof the Zerilliequationin terms of anexpansion
∗
in inverse powers of r with coefficients that are functions of the retarded time t r , as discussed in detail in [2].
−
Substituting in the Zerilli equations we obtain,
3 3
ψ(1) =F(iii)(t r∗)+ F(ii)(t r∗) 7F(ii)(t r∗)M +4F(i)(t r∗) +O(1/r3)
ψ − r ψ − − 4r2 − ψ − ψ −
3 h i
Q(1) =F(iii)(t r∗)+ F(ii)(t r∗)
Q − r Q −
1 3
+ MF(ii)(t r∗)+3F(i)(t r∗) +O(1/r3) (48)
r2 −2 Q − Q −
(cid:20) (cid:21)
For this behavior of first order Zerilli functions, the sources S , S for second order Zerilli functions χ(2), Θ(2)
Z Q
decay as O(1/r2) asymptotically. Therefore, χ(2),Θ(2) will behave in a similar way to ψ(1), Q(1) for large r:
1
χ(2) =F(20)(t r∗)+ F(21)(t r∗)+O(1/r2)
χ − r χ −
1
Θ(2) =F(20)(t r∗)+ F(21)(t r∗)+O(1/r2) (49)
Q − r Q −
We can now compute the asymptotic form for the metric perturbations, to first order and second order, in Regge–
Wheeler gauge, using (48) and (49) in (25) and (27). Unfortunately, in the Regge–Wheeler gauge the metric is not
in the manifestly asymptotically flat form (47). We therefore need to perform a gauge transformation to bring it to
such a form, to first and second order.
In order to do this, we start by recalling that for any ℓ = 2 axisymmetric gauge transformation, we can write the
gauge transformation vectors as,
ξ(1)t = (1)(r,t)P (cos(θ)) ξ(1)r = (1)(r,t)P (cos(θ))
(even) M0 2 (even) M1 2 (50)
ξ(1)θ = (1)(r,t)∂P (cos(θ))/∂θ ξ(1)φ = 0
(even) M 2 (even)
for even parity, and
ξ(1)t = 0 ξ(1)r = 0
(odd) (odd) (51)
ξ(1)θ = 0 ξ(1)φ = (1)(r,t)sin(θ)−1∂P (cos(θ))/∂θ
(odd) (odd) M2 2
for the odd parity case.
To achieve the asymptotically flat gauge (47) we demand that H(1)(r,t)= H(1)(r,t)=h(1)(r,t) =h(1) =0. From
0 1 0 0o
the general form of the gauge transformation equations we have,
2M ∂ (1)
0=H(1)+ (1)+2 M0
0 r(r 2M)M1 ∂t
−
0=He(1) r ∂M(11) + r−2M ∂M(01)
1 − r 2M ∂t r ∂r
−
∂ (1) r 2M
0= er2 M + − (1)
− ∂t r M0
2M ∂ (1)
H(1) =H(1)+ (1) 2 M1 (52)
2 2 r(r 2M)M1 − ∂r
−
h(1) = e r (1) r2∂M(1)
1 −r 2MM1 − ∂r
−
G(1) = 2 (1)
− M
2
K(1) =K(1) (1)
− rM1
∂ (1)(r,t)
0=he(1)+r2 M2
0o ∂t
∂ (1)(r,t)
h(1) =eh(1)+r2 M2
1o 1o ∂r
h(1) = 2r2 (1)(r,t)
2o e− M2
7
where the quantities with tildes are given in the Regge–Wheeler gauge, (1), (1), (1) , are the components of
{M M0 M1 }
the even parity gauge transformation vector, and (1) that of the odd parity gauge transformation vector.
M2
We can see that, given H(1) and H(1), the first and second equation above may be used to solve for (1) and
0 1 M0
(1). Then,thethirdequationleadstoapartialdifferentialequationfor (1) whiletheequationforh(1) fixes (1).
M1 M 0o M2
Therefore, the set of equatieons, and theerefore the gauge choice, is consistent. The solution, assuming only that H(1),
0
H(1), H(1), and K(1) are given, is non unique since one has to integrate partial differential equations, so the solution
1 2
ingeneralinvolvesfree functions. One maychoosesuchfunctions ina judicious wayto ensurethat the non-vanisehing
meetricecomponenets behave in a manner consistent with asymptotic flatness.
Similarly,wedefinethesecondorderasymptoticallyflatgaugebyimposingthefirstorderasymptoticallyflatgauge
conditions and H(2)(r,t) = 0, H(2)(r,t) = 0, h(2)(r,t) = 0 and h(2)(r,t) = 0. Clearly, the same reasoning as above,
0 1 0 0o
applied to the second order gauge transformations, shows that this is a consistent choice.
The end result of these calculations is that one may write the first order gauge transformationvectors in the form,
1 1 1 1 3
(1)(r,t)= F(iii)(t r∗)+ F(ii)(t r∗)+ MF(ii)(t r∗)+ F(i)(t r∗) +O(1/r4)
M 2r ψ − r2 ψ − r3 4 ψ − 2 ψ −
(cid:20) (cid:21)
1
(1)(r,t)= rF(iv)(t r∗)+ F(iii)(t r∗)+MF(iv)(t r∗)
M0 2 ψ − ψ − ψ −
1 9 h 3 i
+ MF(iii)(t r∗)+ F(ii)(t r∗)+2M2F(iv)(t r∗) +O(1/r2)
r 4 ψ − 2 ψ − ψ −
(cid:20) (cid:21)
1 3
(1)(r,t)= rF(iv)(t r∗)+ F(iii)(t r∗)
M1 2 ψ − 2 ψ −
1 3 3
+ F(ii)(t r∗) MF(iii)(t r∗) +O(1/r2)
r 2 ψ − − 4 ψ −
(cid:20) (cid:21)
3 2
(1)(r,t)= F(iii)(t r∗)+ F(ii)(t r∗)
M2 r Q − r2 Q −
1 1
+ 3F(i)(t r∗)+ MF(ii)(t r∗) +O(1/r4) (53)
r3 Q − 2 Q −
(cid:20) (cid:21)
The second order gauge equations are more complex then the first order ones. It can be seen that the equations can
alsobesolvediterativelyforallordersinr, butforsimplicityweonlycomputethetermsthataregoingtoberelevant
in the gravitationalwaveforms. The meaningful terms are given by
1
(2) = (20)(t r∗)+O(1/r2)
M rM −
(2) =r (20)(t r∗)+O(r0)
M0 M0 −
(2) =r (20)(t r∗)+ (21)(t r∗)+O(1/r)
M1 M1 − M1 −
1
(2) = (20)(t r∗)+O(1/r3) (54)
M2 rM2 −
and if we compute them in terms of F ,F , we finally get
ψ Q
∂ (20)(t r∗) 1 1 1 3
M − = F(20)(t r∗) F(v)(t r∗)F(iii)(t r∗) F(iv)(t r∗)2 F(iv)(t r∗)2
∂t 2 χ − − 28 ψ − ψ − − 28 ψ − − 14 Q −
1 1 3
(20)(t r∗)= F(20)(t r∗) F(v)(t r∗)F(iii)(t r∗) F(iv)(t r∗)2
M0 − 2 χ − − 14 ψ − ψ − − 14 Q −
1 1 3
(20)(t r∗)= F(20)(t r∗) F(v)(t r∗)F(iii)(t r∗) F(iv)(t r∗)2
M1 − 2 χ − − 14 ψ − ψ − − 14 Q −
∂2 (21)(t r∗) 1 3 12
M1 − = F(21)(ii)(t r∗)+ MF(v)(t r∗)2 F(iv)(t r∗)F(v)(t r∗)
∂t2 2 χ − 14 ψ − − 7 ψ − ψ −
3 9
F(ii)(t r∗)F(vii)(t r∗) F(vi)(t r∗)F(iii)(t r∗)
−14 ψ − ψ − − 14 ψ − ψ −
69 9
(v) ∗ (iv) ∗ (vi) ∗ (iv) ∗
F (t r )F (t r )+ F (t r )F (t r )
−14 Q − Q − 14 Q − Q −
3 3
+ MF(v)(t r∗)F(v)(t r∗) MF(iii)(t r∗)F(vii)(t r∗)
28 ψ − ψ − − 28 ψ − ψ −
8
15 9
F(vi)(t r∗)F(iii)(t r∗)+ MF(v)(t r∗)2
−14 Q − Q − 14 Q −
∂ (20)(t r∗) 1 1
M2 − =F(20)(t r∗) F(v)(t r∗)F(iii)(t r∗) F(iii)(t r∗)F(v)(t r∗) (55)
∂t Q − − 14 Q − ψ − − 14 Q − ψ −
Wecannowcomputethegravitationalwaveformsbyreadingofftheappropriatemetriccomponents. Forfirstorder
we find,
∂G(1)(r,t) 1
= F(iv)(t r∗)+O(1/r2)
∂t r ψ −
∂K(1)(r,t) 3
= F(iv)(t r∗)+O(1/r2)
∂t r ψ −
∂h(1)(r,t)
2o = 2rF(iv)(t r∗)+O(r0) (56)
∂t − Q −
and the second order components are
∂G(2)(r,t) 1 ∂ 3 1
= F20(t r∗)+ F(iv)(t r∗)F(iii)(t r∗) F(iv)(t r∗)2 +O(1/r2)
∂t χ − 7∂t ψ − ψ − − 7 Q − r
(cid:26) h i (cid:27)
∂K(2)(r,t) ∂G(2)(r,t)
=3 +O(1/r2)
∂t ∂t
∂h(2)(r,t) 1 ∂2
2o = 2r F20(t r∗)+ F(iii)(t r∗)F(iii)(t r∗)
∂t − Q − 7∂t2 Q − ψ −
(cid:26) h i
1
+ F(iv)(t r∗)F(iv)(t r∗) +O(r0) (57)
7 Q − ψ −
(cid:27)
and in terms of first and second order Zerilli functions, we get
∂G(1)(r,t) 1∂ψ(1)
= +O(1/r2)
∂t r ∂t
∂K(1)(r,t) ∂G(1)(r,t)
=3
∂t ∂t
∂h(1)(r,t) ∂Q(1)(r,t)
2o = 2r +O(r0) (58)
∂t − ∂t
and
∂G(2)(r,t) 1 ∂ ∂ψ(1) 3 ∂Q(1) 2 1
= χ(2)+ ψ(1) +O(1/r2)
∂t ( 7∂t(cid:20) ∂t (cid:21)− 7(cid:20) ∂t (cid:21) ) r
∂K(2)(r,t) ∂G(2)(r,t)
=3 +O(1/r2)
∂t ∂t
∂h(2)(r,t) 1 ∂2 1 ∂ ∂
2o = 2r Θ(2)+ ψ(1)Q(1) + ψ(1) Q(1) +O(r0) (59)
∂t − 7∂t2 7∂t ∂t
(cid:26) h i (cid:27)
It is straightforward to compute the radiated energy, since we are in an explicitly asymptotically flat gauge. One
can therefore simply apply the formulae stemming from Landau and Lifshitz [6,7],
2 2
dPower 1 ∂ 1 ∂ 1 ∂
= h + h h . (60)
dΩ 16πr2 "(cid:18)∂t θφ(cid:19) 4(cid:18)∂t θθ− sin2θ∂t φφ(cid:19) #
For the perturbations we are considering,
h =r2 [ǫK(1)(r,t)+ǫ2K(2)(r,t)]P (θ)
θθ 2
n
+[ǫG(e1)(r,t)+ǫ2G(2e)(r,t)]∂2P2(θ)/∂θ2) (61)
o
e e
9
h =r2 sin2θ[ǫK(1)(r,t)+ǫ2K(2)(r,t)]P (θ)
φφ 2
n
+sin(θ)cose(θ)[ǫG(1)(r,t)e+ǫ2G(2)(r,t)]∂P2(θ)/∂θ) (62)
1 o
hθφ = 2 ǫh(21o)(r,t)+ǫ2eh(22o)(r,t) coes(θ)∂P2(θ)/∂θ−sin(θ)∂2P2(θ)/∂θ2 (63)
h i(cid:2) (cid:3)
and substituting, we get, e e
3 ∂ψ(1) 1 ∂ ∂ψ(1) 3 ∂Q(1) 2 2
Power= ǫ +ǫ2 χ(2)+ ψ(1)
10" ∂t 7∂t(cid:18) ∂t (cid:19)− 7(cid:18) ∂t (cid:19) !#
36 ∂Q(1) 1 ∂2Q(1)ψ(1) ∂Q(1)∂ψ(1) 2
+ ǫ +ǫ2 Θ(2)+ + . (64)
35 ∂t 7 ∂t2 ∂t ∂t
(cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21)
Ascanbeseen,theenergyhastwoquadraticcontributions,onecomingfromtheh componentofthemetricandone
θφ
from the diagonal elements of it. They correspond to the “ ” and “+” polarization modes of the gravitational field
×
and therefore we can consider the expressions in the square brackets as the“waveforms” of the gravitational waves
emitted. Contrary to the first order case, in which the waveform is directly given by the Zerilli (or Regge–Wheeler)
function, for second order corrections the relationship is more involved.
V. INITIAL DATA
Initial data for solutions of Einstein equations is usually given in terms of the initial value of the three-metric and
the initial value of the extrinsic curvature of the initial Cauchysurface. For the case of initial data for colliding black
holes, the popular families of initial data of Bowen and York [8] and punctures [9] naturally come cast in a gauge in
which1 H =0, H =0, h =0 and h =0. We will therefore present the appropriate formulaethat wouldallow us,
0 1 0 0o
given a metric and extrinsic curvature cast in such a gauge, to provide the initial data for the first and second order
Zerilli functions and their time derivatives.
The extrinsic curvature K of a spatial slice is defined by
ab
K = n (δc +ncn ) δd +ndn (65)
ab (c;d) a a b b
wheren istheunitnormaltotheCauchysurfacewherethedat(cid:0)aarespecifi(cid:1)ed. Onecanrelatetheextrinsiccurvature
a
to the time derivative of the three-metric through,
ds2 =g dxidxj N2(dt)2
ij
−
∂g
ij
= 2NK . (66)
ij
∂t −
That is, if through some procedure we are given the extrinsic curvature and the three metric, we can compute the
time derivative of the three metric. Re-expressing this in terms of components of the tensor spherical harmonic
decomposition due to Regge and Wheeler we get,
∂h(1)/∂t = 2 1 2M/rK(1) ∂H(1)/∂t = 2 1 2M/rK(1)
1 − − (h1) 2 − − (H2)
∂K(1)/∂t = −2p1−2M/rK((K1)) ∂G(1)/∂t = −2p1−2M/rK((G1)) (67)
∂h(11o)/∂t = −2p1−2M/rK((h1)1o) ∂h(21o)/∂t = −2p1−2M/rK((h1)2o)
for first order components. K(1) arpe spherical tensor harmonic componpents of the first order extrinsic curvature
(xx)
tensor, written in the Regge–Wheeler notation that we used in formulae (1-10). For second order we get,
∂h(2)/∂t = 2 1 2M/rK(2) ∂H(2)/∂t = 2 1 2M/rK(2)
1 − − (h1) 2 − − (H2)
∂K(2)/∂t = −2p1−2M/rK((K2)) ∂G(2)/∂t = −2p1−2M/rK((G2)) (68)
∂h(12o)/∂t = −2p1−2M/rK((h2)1o) ∂h(22o)/∂t = −2p1−2M/rK((h2)2o)
p p
1
This happensto coincide exactly with thesame gauge condition of theasymptotically flat gauge.
10