Mass-Energy and Momentum Extraction by Gravitational Wave Emission in the Merger of Two Colliding Black Holes: The Non-Head-On Case

0.66 MB·English

by  R. F. Aranha

Mass-Energy and Momentum Extraction by Gravitational Wave Emission in the
Merger of Two Colliding Black Holes: The Non-Head-On Case
R. F. Aranha1,2, I. Damia˜o Soares1 and E. V. Tonini3
1Centro Brasileiro de Pesquisas F´ısicas, Rio de Janeiro 22290-180, Brazil,
2Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA,
3Instituto Federal do Esp´ırito Santo, Vito´ria 29040-780, Brazil.∗
Weexaminenumericallythepost-mergerregimeoftwoSchwarzschild blackholesinnonhead-on
collision. Ourtreatment ismade in therealm ofnon-axisymmetric Robinson-Trautmanspacetimes
which are appropriate for the description of the system. Characteristic initial data for the system
areconstructedandtheRobinson-Trautmanequationisintegratedforthesedatausinganumerical
code based on the Galerkin spectral method which is accurate and sufficiently stable to reach the
final configuration of the remnant black hole, when the gravitational wave emission ceases. The
2 initial data contains three independent parameters, the ratio mass α of the individual colliding
1 black holes, the boost parameter (that characterizes the initial pre-merger infalling velocity of the
0 two black holes) and the incidence angle of collision 0 ρ 90o. The remnant black hole is
0
≤ ≤
2 characterized by its final boost parameter, the final rest mass and scattering angle. The motion
of the remnant black hole is restricted to the plane determined by the directions of the two initial
n
collidingblackholes, characterizingaplanarcollision. Theenergy-momentumfluxescarried outby
a
J gravitationalwavesareconfinedtothisplane. Weevaluatetheefficiencyofmass-energyextraction,
thetotalenergyandmomentumcarriedoutbygravitationalwavesandthemomentumdistribution
1
of the remnant black hole for a large domain of initial data parameters. Our analysis is based on
3
theBondi-Sachsfourmomentumconservationlaws. Theprocessofmass-energyextractionisshown
] to be less efficient as the initial data departs from the head-on configuration. Head-on collisions
c (ρ =0)andorthogonalcollisions(ρ =90o)constitute,respectively,upperandlowerboundstothe
0 0
q poweremission and totheefficiency of mass-energy extraction. On thecontrary,head-on collisions
-
and orthogonal collisions constitute, respectively, lower and upper bounds for the momentum of
r
g the remnant. The momentum extraction and the pattern of the momentum fluxes, as a function
[ of the incidence angle, are examined. The momentum extraction characterizes a regime of strong

deceleration ofthesystem. Theangularpattern ofgravitational wavesignals arealso examined for
4
small mass ratios α and early times u. We show that, in the plane of collision (x,z), the pattern
v
is typically bremsstrahlung, corresponding to a strong deceleration regime at early times, with two
3
dominant lobes in the forward direction of motion. Gravitational waves are also emitted outside
2
theplaneofcollision, theangulardistribution ofwhich issymmetricwith respect tothey-axisand
2
1 thereforewithazeronetmomentum,consistent with theplanarnatureofthecollision. Finally the
. relationbetweentheincidenceangleandthescatteringanglecloselyapproximatesarelationforthe
1 inelastic collision of classical particles in Newtonian dynamics.
1
1
PACSnumbers:
1
:
v
I. INTRODUCTION previous recent papers[2, 3] where we examined the case
i
X of head-on collisions. The collision and merger of two
r black holes are considered to be important astrophysi-
a
It is by now theoretically well established that, in cal sources of strong gravitational emission (cf.[4] and
the nonlinear regime of General Relativity, gravitational references therein) and the relevance of the processes of
waves extract mass, momentum and angular momen- generation and emission of gravitational waves in these
tum of the source and that the radiative transfer in- configurations lies in the fact that the associated wave
volved in these processes may turn out to be funda- patterns will be crucial for the present efforts towards a
mental for the astrophysics of the collapse of stars, the direct detection of gravitational waves. In spite of the
formation of black holes and the collision and merger enormous progress achieved until now using approxima-
of two or more black holes. In the present paper we tion methods and numerical techniques, the information
will examine the radiative transfer processes by gravi- onwaveformpatternsandradiativetransferprocessesin
tational wave emission in the post-merger phase of two the dynamics of gravitational wave emission is far from
colliding black holes, with focus on the case of non- being complete[5].
head-on collisions, described in the realm of Robinson- The object of this paper is the numerical treatment
Trautman (RT) spacetimes[1]. This work extends two of the gravitational wave production and the associated
processes of radiative transfer in the non-head-on colli-
sion of two boosted Schwarzschild black holes, modeled
in the context of Robinson-Trautman (RT) spacetimes.
∗Electronicaddress: [email protected];[email protected];[email protected] initial data for the RT dynamics are con-
2
structed that represent instantaneously two black holes be fundamental in the treatment of the energy and lin-
in non-head-oncollision. RT spacetimes presenta global earmomentumextractedfromthesourcebythegravita-
apparenthorizon[6] so that the dynamics corresponds to tionalwavesemitted. SectionsVIItoXcontainthemain
a regime where the merger has already set in. During results of the paper, as the numerical evaluation of the
the merger part of the rest mass and kinetic energy of efficiency of mass-energy transfer by gravitational wave
the two initial individual black holes are radiated away emission, the conservation of energy in the processes via
by the gravitationalwaves,and part will be absorbed to the Bondi mass formula and linear momentum extrac-
constitute the mass-energy of the remnant. The linear tion by gravitational waves. Distinct patterns of gravi-
momentum of the initial system is also carried out by tationalwaveemissionarediscussedrelatedtothe range
the gravitational waves emitted. We also examine the of the physical parametersof the initial data. In Section
two regimes of momentum extractionin the post-merger XI we summarize our results and discuss their relevance
phasebycontemplatingthetimebehaviorofthemomen- and limitations as compared to previous results in the
tum flux of the gravitational waves emitted. We use the literature. Throughoutthe paper we use units such that
momentum-energy conservation law in the Bondi-Sachs 8πG=c=1.
formulation of gravitational wave emission by bounded
sources[7, 8]. Also the analysis of the energy flux car-
ried out by gravitational waves will allow us to charac- II. DYNAMICS OF ROBINSON-TRAUTMAN
terizetwodistinctregimesofbremsstrahlungemissionfor SPACETIMES. WAVE ZONE CURVATURE
distinct domains of the initial data parameters. Several
characteristics of a non-head-on collision are drastically RT spacetimes[1] are asymptotically flat solutions of
distinct from a head-on collision, mainly for large values Einstein’s vacuum equations that describe the exterior
oftheinitialmassratioparameter. Thegeneraloutcome gravitational field of a bounded system radiating gravi-
will be a remnant boosted black hole with a rest mass tational waves. The RT metric is expressed as
larger than the sum of the rest masses of the two indi-
ds2=α2(u,r,θ,φ)du2+2dudr r2K2(u,θ,φ)
vidual initial black holes. −
(dθ2+sin2θdϕ2). (1)
As compared to previous estimates of the literature, ×
our approach differs basically in that we have adopted where r is an affine parameter defined along the shear-
the characteristic surface initial data formalism, which free null geodesics determined by the vector field ∂/∂r.
hasseveraladvantagesforthedescriptionofgravitational Einstein equations imply that, in a suitable coordinate
radiation and the construction of algorithms[9]. In this system,
direction,anaccuratecodebasedontheGalerkinmethod
2m K
was constructed to integrate the RT non-axisymmetric α2(u,r,θ,φ)=λ(u,θ,φ) 0 +2r u, (2)
− r K
field equations. The code is accurate and highly sta-
ble for long time runs in the nonlinear regime so that where m0 is a positive constant, and λ(u,θ,φ) is the
we are able to reach numerically the final configuration Gaussian curvature of the surfaces (u=const, r=const)
of the system, when the gravitational emission ceases. defined by
This will allow us to describe processes of mass-energy,
1 (K sinθ/K) 1 K2 K
andmomentumextractionduetogravitationalradiation λ= θ θ + φ φφ . (3)
K2 − K2sinθ sin2θ K4 − K3
emission, by using physically meaningful quantities con- (cid:18) (cid:19)
nected to initial and final configurations of the source.
The remaining Einstein equations yield
We organize the paper as follows. In Sec. II we re-
K 1 (λ sinθ) λ
viewsomebasicaspectsofthegeneralnon-axisymmetric 6m u + θ θ + φφ =0. (4)
Robinson-Trautman spacetimes necessary for our anal- − 0 K 2K2(cid:18) sinθ sin2θ(cid:19)
ysis of gravitational radiation emission in the case of a
In the above, subscripts u, θ and φ denote derivatives
non-head-on collision. In Section III we construct char-
with respect to u, θ and φ, respectively. Eq. (4), de-
acteristic initial data for the non-axisymmetric RT dy-
noted RT equation, governs the dynamics of the grav-
namicscorrespondingto twoboostedSchwarzchildblack
itational field which is totally contained in the metric
holesinanon-head-oncollision. InSectionIVwepresent
function K(u,θ,φ). Chrusciel and Singleton[10] estab-
a basic descriptionof the numericalcodes used to evolve
lishedthatRTspacetimesexistgloballyforallpositiveu
the characteristic initial data via the RT dynamics. The
andconvergeasymptoticallytothe Schwarzschildmetric
codes are based on Galerkin spectral methods and the
as u – this global time extension being realized for
dynamical evolution of the initial data is performed and →∞
arbitrary smooth initial data.
discussed, including its accuracy and stability. In Sec-
An important feature of RT spacetimes, that estab-
tion V we discuss the planar nature of the collision and lishes its radiative character, arises from the expression
show how this property allows us to save a lot of com- of its curvature tensor that in the semi-null tetrad basis
putational effort. In Section VI the Bondi-Sachs four
momentum for the non-axisymmetric RT spacetimes are Θ0 = du, Θ1 =(α2/2) du+dr
introduced together with its conservation laws that will Θ2 = rKdθ, Θ3=rKsinθdφ (5)
3
assumes the form The K(θ,φ) function (11), which depends on three pa-
rameters,isaK-transformationofthegeneralizedBondi-
N III II
RABCD = ABrCD + Ar2BCD + ArB3CD, (6) Metznergroup[7]discussedbySachs[13](theBMSgroup)
and represents the general form of Lorentz boosts con-
wherethescalarquantitiesNABCD,IIIABCDandIIABCD tainedinthe homogeneousorthochronousLorentztrans-
are of the algebraic type N, III and II, respectively,
formations of the BMS.
in the Petrov classification of the curvature tensor[11],
The Bondi mass function of this solution is given by
and r is the parameter distance along the principal null m(θ,φ) = m K3(θ,φ). The total mass-energy of this
direction∂/∂r. Eq. (6)displaysthe peelingproperty[12] 0
gravitationalconfiguration is given by the Bondi mass
of the curvature tensor, showing that indeed RT is the
exterior gravitational field of a bounded source emitting
2π π
gravitationalwaves. For large r we have M = (1/4π) dφ dθ m(θ,φ)sinθ
Z0 Z0
RABCD ∼ NABrCD, (7) = m0K03coshγ =m0K03/ 1−v2. (12)
p
so that at large r the gravitational field looks like a The interpretation of (11) as a boosted black hole is
gravitational wave with propagation vector ∂/∂r. The relative to the asymptotic Lorentz frame which is the
nonvanishing of the scalars N is an invariant cri- rest frame of the black hole when γ =0.
ABCD
terion for the presence of gravitational waves, and the
asymptotic regionwhere (1/r)-terms are dominant de-
O
finedasthewavezone. Thecurvaturetensorcomponents
III. CHARACTERISTIC INITIAL DATA
in the above basis that contribute to N , namely, to
ABCD
thegravitationaldegreesoffreedomtransversaltothedi-
rection of propagationof the wave,are R0303 = R0202 = As well known the initial data problem for RT space-
D(u,θ,φ)/r+ (1/r2)andR0203 = B(u,θ,φ)/r+− (1/r2) times is within the class of characteristic initial value
− O − O
where
formulation as opposed to the 1+3 formulation, accord-
1 K K 2 K2 ing to the classification of York[14]. For RT spacetimes
D(u,θ,φ)= 2K2 ∂u Kθθ − Kθ cotθ− K2θ the function K(u0,θ,φ) given in a characteristic surface
(cid:18) (cid:19) u = u corresponds to the initial data to be evolved via
1 K 2K2 0
+ ∂ φφ + φ , (8) the RT equation (4). Our task is now to exhibit an ini-
2K2sin2θ u − K K2
(cid:18) (cid:19) tial K(u0,θ,φ) that represents instantaneously the gen-
and eral collision of two Schwarzschild black holes (head-on
collision or not), by extending a procedure outlined in
1 K K K K
B(u,θ,φ)= ∂ θφ 2 θ φ cotθ φ , (9) Refs. [2,15]to constructinitialdatafor RT dynamics in
K2sinθ u(cid:18) K − K2 − K (cid:19) the non-axial case.
In analogy to bispherical coordinates[16] in the 3-
From(7)wecanseethatthefunctionsD andB contain
dim Cartesian plane Σ, let us introduce the following
all the information of the angular, and time dependence
parametrizationfor Cartesian coordinates
of the gravitational wave amplitudes in the wave zone
onceK(u,θ,φ)isgiven. D andB actuallycorrespondto
a sinθ sinhη
the two independent polarization modes of the gravita- x = cosφ,
coshη+cosθ sinhη
tional wave,transverse to its direction of propagationat
a sinθ sinhη
the wave zone. y = sinφ, (13)
coshη+cosθ sinhη
The field equations present two stationary solutions,
a
which will play a crucial role in our future discussions. z = ,
±coshη+cosθ sinhη
The first is the Schwarzschild solution corresponding to
for z > 0 and z < 0 respectively. In the above 0 η
K =K0 =const, λ=1/K02 (10) , 0 θ π, 0 φ 2π. In this parametriz≤ation≤,
∞ ≤ ≤ ≤ ≤
and mass MSchw =m0K03. The second is η =η0 corresponds to two spheres, one at z >0 and the
other at z < 0, centered at (x = y = 0,z = acoshη )
0
±
K respectively, with radius asinhη . The Cartesian vector
K(θ,φ)= 0 , (11) 0
coshγ+(n xˆ)sinhγ from a point P :(x,y,z) of Σ has length
·
where xˆ = (sinθcosφ,sinθsinφ,cosθ) is the unit vector coshη cosθsinhη
along an arbitrary direction x and n = (n1,n2,n3) is a r(η,θ)=ascoshη−+cosθsinhη (14)
constantunitvector(satisfyingn2+n2+n2 =1);alsoK
1 2 3 0
andγ areconstants. We notethat(11)yieldsλ=1/K2, For η = the spheres degenerate into the planes z =0
0 ∞
resulting in its stationary character. This solution can andz = . Theusefulnessofthisparametrizationwill
±∞
be interpreted[7] as a boosted black hole along the axis become clear in what follows. We note that the Carte-
determined by the unit vector n with boost parameter sian coordinates are continuous functions, with continu-
γ, or equivalently, with velocity parameter v = tanhγ. ous derivatives, of (η,θ,ϕ). Singularities occurring are
4
the usual singularities of a spherical coordinate system. (dx)2+(dy)2+(dz)2 is expressed as
For future reference let us introduce the functions
a2
S(±)(η,θ,φ,n)=pcoshη±(n·xˆ)sinhη . (15) ds2flat = S(4+)(η,θ,φ,n)(cid:20)dη2+sinh2η (dθ2+sin2θdφ2)(cid:21).(16)
where xˆ = (sinθcosφ,sinθsinφ,cosθ) and n =
(n ,n ,n ), with n2 + n2 + n2 = 1. In the above We now take Σ as a spacelike surface of initial data,
1 2 3 1 2 3
parametrization (13), the flat space line element ds2 = with geometry defined by the line element
ds2 =a2K2(η+γ ,θ,φ) dη2+sinh2(η+γ )(dθ2+sin2θdφ2) (17)
0 0
(cid:20) (cid:21)
where γ is an arbitrary parameter. By assuming time- namely, when (η ,θ π). In this asymptotic limit,
0
→∞ ≃
symmetric data (namely, Σ a maximal slice with zero returning to Cartesian coordinates, the 3-geometry (22)
extrinsiccurvature)weobtainthattheHamiltoniancon- can be given in the approximate form
straints reduce to (3)R = 0. With the substitution
2 M
K Φ2, the constraint equation reduces to the Laplace g 1+ (1) δ , (23)
≡ ij ≃ r(η,θ) ij
equation (cid:26) (cid:27)
where we fixed the scale of bispherical-type coordinates
sin1θ Φθsinθ + Φ′sinh2(η+γ0) ′ by taking 2√2a = m0(α1 + α2)√1+n3 α2/α1. The
(cid:18) (cid:19)θ (cid:18) (cid:19) Schwarzschildmass M(1) =m0(α1+α2).
+ 1 Φ + 3sinh2(η+γ )Φ=0, (18) From the above construction we can now extract ini-
sin2θ φφ 4 0 tialdatafor the RT dynamics,whichhas its initialvalue
problem on null cones. Based on the initial data formu-
where a prime denotes derivative with respect to η. It
lation on characteristic surfaces proposed by D’Inverno
is not difficult to verify that the functions
and Stachel[17, 18] – in which the degrees of freedom of
1 the vacuum gravitational field are contained in the con-
Φ= (η+γ ,θ,φ,n) (19) formal structure of 2-spheres embedded in a 3-spacelike
(cid:18)S(±) 0 (cid:19) surface – we are then led to adopt the conformal struc-
satisfy Eq. (18) and, with respect to metric (17), cor- ture given by the conformal factor (20) defined on the
respond to flat space solutions (zero curvature). It then surface η =0,
follows that
α α 2
K(u ,θ,φ)= 1 + 2 , (24)
Φ= (η+αγ1,θ,φ,n) + (η+αγ2,θ,φ,n˜) (20) 0 (cid:18)S(−)(γ0,θ,φ,n) S(+)(γ0,θ,φ,n˜)(cid:19)
(cid:18)S(−) 0 (cid:19) (cid:18)S(+) 0 (cid:19) with n˜ = (0,0,1), as initial data for the RT dynamics.
is a nonflat solution of (18), where α and α are ar- This conformal structure is to be extended along null
1 2
bitrary positive constants. The nonflat 3-dim geometry bicharacteristics and propagated along a timelike con-
defined by (20), gruence of the spacetime via RT dynamics. A restricted
spacetime may then be constructed locally as the prod-
ds2 =a2 Φ4 dη2+sinh2(η+γ0)(dθ2+sin2θdφ2) , (21) uct of the two-sphere geometry times a timelike plane
(cid:20) (cid:21) (u,r˜)generatedbyanullvector∂/∂r˜andatimelikevec-
tor ∂/∂u with geometry dσ2 =α2(u,r˜,θ,φ)du2+2dudr˜.
is asymptotically flat with a form analogous to that of
The four geometry is then taken as
the 3-dim spatial section of the Schwarzschild geometry
in isotropic coordinates, as we proceed to show. ds2 = α2(u,r˜,θ,φ)du2+2dudr˜
Withoutlossofgeneralitywetakein(20)n˜ =(0,0,1),
that corresponds to choose the z-axis along n˜. In this r˜2K2(u,θ,φ)(cid:16)dθ2+sin2θdφ2(cid:17). (25)
−
instance a straightforward manipulation shows that, for
η >>γ , the metric (21) can be rewritten as Eq. (25) is the RT metric, the dynamics of which (ruled
0
byEinstein’svacuumfieldequations)propagatestheini-
4
aα √coshη cosθsinhη tialdata(24)forwardintimefromthe characteristicini-
ds2 = α2+ r(η,1θ) coshη −(n xˆ)sinhη! ds2flat. (22) tial surface u = u0. We note that Einstein’s vacuum
− · equationsdemandthatthefunctionα2(u,r˜,θ,φ)hasthe
Now to probe thepasymptotic structure of the metric form given in (2).
(22) let us consider η very large and, for this η, points The interpretation of the asymptotically flat ini-
(x,y,z)whosedistancefromthe originisalsoverylarge, tial data (24) as two instantaneously interacting
5
Schwarzschild black holes boosted along the z-axis is Schwarzschild black hole. In this sense we associate the
now discussed, based on perturbations of the RT met- perturbationwithablackholeofrelativesmallrestmass,
ric (25) constructed with such data. As shown in Sec- boostedalongthedirectionn(orn˜)innonhead-oncolli-
tion II, for α = 0 the data correspond in (25) to a sion with a larger black hole boosted along the direction
1
static Schwarzschild black hole (with the total Bondi n˜ (or n). The initial infalling velocity of each black hole
mass m (α )6coshγ ) boosted along the direction de- considered individually is given by v =tanhγ .
0 2 0 0
finedbythe unitvectorn˜,withv =tanhγ . Forα =0, Without loss of generality, in the remaining of the pa-
0 1
6
with α <<α , the configurationis no longer static and per we fix α =1 and drop subscripts 1 and 0 of the pa-
1 2 2
cannot therefore be a black hole, but can still be inter- rameters α and γ , respectively, in order to avoid over-
1 0
preted as an initially perturbed boosted Schwarzschild cluttering in formulae and Figures. In this instance, the
black hole. Conversely the same consideration holds for initial data (24) will in principle contain four indepen-
(α = 0,α = 0) and α = 0 with α << α , the latter dentparameters,namely,(α,γ,n)withn2+n2+n2 =1,
1 6 2 1 6 2 1 1 2 3
casecorrespondingalsotoaninitiallyperturbedboosted and assumes the form
1 α 2
K(u ,θ,φ)= + . (26)
0 √coshγ+cosθsinhγ coshγ (n xˆ)sinhγ
(cid:18) − · (cid:19)
p
We mustcommentthat,sinceonlytwoblackholesare IV. NUMERICAL EVOLUTION OF THE DATA
involved in the collision, the data (26) should actually
depend (besides α and γ) only on the incidence angle We proceed now to discuss the numerical evolution of
ρ0 betweenthetwoinitialblackholes,determinedbythe the initial data (26) via the non-axisymmetric RT equa-
scalarproductnz n. Thereforeweshouldexpectthat(i) tion (4). Throughout the present Section the variable
·
theresultingdynamicsofthesystemwouldnotbealtered θ will be expressed in terms of the variable x = cosθ.
underarigidrotationofthetwoinitialblackholesabout The numerical integration of the non-axisymmetric RT
the z-axis, which could be used to locate n of the initial equation is performed using a Galerkin spectral method
which is now described in detail. In the present pro-
data in the x z plane; and consequently (ii) the initial
− cedure we rewrite the equations (3)-(4) using the vari-
data will result in a planar dynamics of the collision, a
able P(u,x,φ) 1/K(u,x,φ) (instead of the former
fact that would save a lot of computational effort. The ≡
K(u,x,φ)), an approach already adopted in Ref. [20].
above statements (i)-(ii) will indeed be confirmed by the
We obtain that
numericalresultsobtainedfromtheevolutionofthedata
(26), as shown in section VI, and establish the planar λ(u,x,φ)=(1 x2) PP P2 2xPP +P2+
− xx− x − x
nature of a general non head-on collision of two black 1
(cid:2) + (cid:3) PP P2 (27)
holes. (1 x2) φφ− φ
−
(cid:2) (cid:3)
and
Finallyweshouldremarkthat, inthe fullBondi-Sachs
problem, the analysis of field equations in the 2 + 2 P˙(u,x,φ)= P3 (1 x2)λ 2xλ + λφφ . (28)
formulation[7, 17] shows that the two news functions −12m0(cid:20) − xx− x (1−x2)(cid:21)
c(1)(u,θ,φ)andc(2)(u,θ,φ)arepartoftheinitialdatato
u u where a dot and the the subscripts x and φ denote,
be prescribed for the evolution of the system[19]. How-
respectively, derivatives with respect to u, x and φ.
ever for the RT dynamics the news are already speci-
The Galerkin method establishes that P(u,x,φ) can
fied once the initial data for the RT equation, namely
be expanded in a convenient set of basis functions of a
K(u ,θ,φ), is given and consequently K(u,θ,φ) is given
0 projection space by which we can reduce the RT partial
for all u>u from the numerical evolution of the data.
0 differentialequation(28)intoafinitesetofnonlinearcou-
pledordinarydifferentialequations. Thissetofequations
The initial data (26) will be evolvednumerically – via constitutesanautonomousdynamicalsystem,thedimen-
RT dynamics – up to a final configuration that corre- sion of which depends directly on the truncation in the
sponds to a remnant Schwarzschild black hole boosted Galerkin method to approximate the RT dynamics. As
along the axis determined by the unit vector n and theconformalfunctionP(u,x,φ)definedonthe2-sphere
f
having the form of the solution (11), as we discuss in is assumedto be sufficiently smooth, we can use the real
the next Section. A numerical code using the Galerkin Spherical Harmonics (SHs) [21] as the appropriate ba-
spectralmethodiswasimplementedtointegratethenon- sisthatbetter approximatesourdesiredsolution. Before
axisymmetric RT equation (4), the basis of which is de- startingwiththenumericalscheme,wefirstpresentsome
scribed in the next Section. definitions and properties of these functions.
6
UnlikethecomplexSHs,therealSHsconsistofabreak as our approximatedsolutionfor P(u,x,φ). Here N is
P
of the complex SHs into their sine and cosine parts as apositiveintegerthatdefinesthetruncationorderofthe
follows, Galerkin method. By using the orthogonality relations
(32)and(33),themodalcoefficientsA (u)andB (u)
hx,φ|l,mi(+) ≡ (+)Ylm(x,φ)≡Wl,mPlm(x)cos(mφ) are given by l,m l,m
= (+) l,mx,φ ,
h | i 2π 1
hx,φ|l,mi(−) ≡ (−)Ylm(x,φ)≡Wl,mPlm(x)sin(mφ) Al,m(u) = Z0 Z−1(+)Ylm(x,φ)P(a)(u,x,φ)dxdφ
= (−)hl,m|x,φi, (29) ≡ (+) l,m|P(a) , (36)
where Pm(x) are the associated Legendre functions de- 2π 1
fined by l Bl,m(u) = (cid:10) (−)Y(cid:11)lm(x,φ)P(a)(u,x,φ)dxdφ
Z0 Z−1
Plm(x)≡(1−x2)|m|/2ddmxPml(x), (30) ≡ (−) l,m|P(a) . (37)
and the normalization factors W are given by Inserting the expan(cid:10)ded solu(cid:11)tion (35) in the RT partial
l,m
differential equation (28) we obtain an autonomous dy-
1(2l+1)(l m)! namical system of dimension (NP + 1)2 for the modal
Wl,m ≡s2 π(l+m−)! . (31) coefficients (Al,m(u),Blm(u)), namely,
From these formulas we see that the case m = 0 gives A˙ (u)= A (u),B (u) ,
l,m l,m(cid:16) l,m l,m (cid:17)
us the basis functions for the axial case, the Legendre A
polynomials Pl(x) xl [21]. B˙ (u)= A (u),B (u) , (38)
≡h | i l,m l,m(cid:16) l,m l,m (cid:17)
The real SH orthogonality relations are given by B
(±) l,ml′,m′ (±) where l,mand l,marepolynomialsoforder(NP+1)2in
| ≡ A B
2π 1 Al,m(u)andBl,m(u)andadotdenoteshereu-derivative.
(cid:10) (±) l(cid:11),mx,φ x,φl′,m′ (±)dx dφ The Galerkin scheme guarantees that the projections
≡ h | i |
Z0 Z−1 of (P(u,x,φ) P (u,x,φ)) onto each basis function,
= 2π 1 (±)Ylm(x,φ)(±(cid:10))Ylm′ ′(x,φ)(cid:11)dx dφ namely, < (P−(u,x(a,)φ) − P(a)(u,x,φ)),Ylm(x,φ) > ap-
Z0 Z−1 proach zero when NP so that (35) approaches
→ ∞
= δl,l′δm,m′, (32) the exact solution P(u,x,φ), in the sense of the norm
of the projection basis space of the SHs. The initial
and
conditions (A (u ),B (u )) to be used to integrate
l,m 0 lm 0
(±) l,ml′,m′ (∓) the dynamical system (38) are provided by the initial
| ≡
2π 1 data K(u ,x,φ) constructed in last Section (Eqs. (26)).
= (cid:10) (±) l(cid:11),mx,φ x,φl′,m′ (∓)dx dφ 0
h | i | These initial values are obtained from the Galerkin de-
Z0 Z−1
2π 1 (cid:10) ′ (cid:11) composition of P(u0,x,φ) (cf. Eq. (35)) evaluated ac-
= (±)Ylm(x,φ)(∓)Ylm′ (x,φ)dxdφ=0, (33) cording to (36).
Z0 Z−1 Asamatteroffact,theprojectionsthattransformthe
and are fundamental in the treatment of the Galerkin RTequationintothedynamicalsystem(38)demandsac-
method. Here the following completeness relationis also tually that we replace P(u,x,φ) by P(a)(u,x,φ) in (27)
used, and (28). But in doing so we excessively increase the
computational demand, overloading the computer data
2π 1
x,φ x,φ dx dφ=1. (34) storage. To circumvent this problem, we divide the RT
Z0 Z−1| ih | equation into two parts that we called constraints: the
λ-constraintandtheP3-constraint. Theprocesswillcon-
Now we are able to construct our numerical Galerkin sist then in also expanding – in the same way as in (35)
scheme,thatwillallowus toobtainanaccurateapproxi- – the expressions for λ and P3,
matednumericalsolutionfor(28)correspondingtogiven
initial data. As every twice continuously differentiable, Nλ 1
suitably periodic real function defined on the surface of λ(a)(u,x,φ)= 2El,0(u)(+)Yl0(x,φ)+
a sphere admits an absolutely convergent expansion in Xl=0
terms of the SHs, let us consider the expansion Nλ l
+ E (u)(+)Ym(x,φ)+
P (u,x,φ)= NP 1A (u)(+)Y0(x,φ)+ Xl=1mX=1 l,m l
(a) 2 l,0 l Nλ l
Xl=0 + F (u)(−)Ym(x,φ), (39)
NP l l,m l
+ A (u)(+)Ym(x,φ)+ Xl=1mX=1
l,m l
Xl=1(cid:20)mX=1
l NP3 1
+ B (u)(−)Ym(x,φ) , (35) P3 (u,x,φ)= C (u)(+)Y0(x,φ)+
l,m l (a) 2 l,0 l
mX=1 (cid:21) Xl=0
7
NP3 l A (u ),B (u )
l,m 0 l,m 0
+ C (u)(+)Ym(x,φ)+ { }
l,m l
+Xl=N1PmX3=1l Dl,m(u)(−)Ylm(x,φ). (40) BAll,,mm((uu00)) ≡≡ ((+−))(cid:10)ll,,mm||PP00((aa))(cid:11),. (44)
Xl=1 mX=1 The effect of the truncation(cid:10) N on(cid:11)the initial data
P
Here, Nλ and NP3 are integers that define the trunca- may be evaluated by the relative error R(u0) between
tion orders for the respective expansions. Throughout theapproximateandexactexpressionsgivenby(35)and
the paper, we take NP = Nλ = NP3, but these orders P(u0,x,phi) 1/K(u0,x,phi) (cf. (26)),
are totally independent and can be taken with diferent ≡
values. Now,wehaveanew twosetofmodalcoefficients P(u ,x,φ) P (u ,x,φ)
0 (a) 0
t({erCml,imn,eDdlb,my}thaendfir{sEtl,emm,pFllo,mye}d)stehtatofwmillobdealuncoiqeuffieclyiednets- R(u0,x)= | P(u−0,x,φ) |. (45)
A ,B by
{ l,m l,m} In all the numerical experiments of the paper we have
adopted N = 7; with this truncation the relative error
Cl,m(u) ≡ (+) l,m|P3(a) aboveisofPtheorderof,orsmallerthan10−8,forall 1
− ≤
= C ( A (u),B (u) ), x 1 and 0 φ 2π. The RT dynamics furnishes us
l,m(cid:10) { l,m (cid:11) l,m } ≤ ≤ ≤
D (u) (−) l,mP3 withafurthertesttochecktheaccuracyandreliabilityof
l,m ≡ | (a) our numerical codes. In fact, for any sufficiently smooth
= Dl,m(cid:10)({Al,m(u)(cid:11),Bl,m(u)}), K(u,x) we have that the quantity
E (u) (+) l,mλ
l,m ≡ | (a) 2π 1
= El,m(cid:10)({Al,m(u(cid:11)),Bl,m(u)}), ζ(u)= dφ P−2(u,x,φ) dx, (46)
F (u) (−) l,mλ Z0 Z−1
l,m ≡ | (a)
= Fl,m(cid:10)({Al,m(u(cid:11)),Bl,m(u)}). (41) is conserved along the dynamics, namely, ∂uζ(u) = 0.
Evaluating its exact value from the initial data (26) and
The RT dynamical equation will have then a new form: at distinct steps of the computation we that ζ(u )
0
the left side will be written by the u-derivative of P(a) ζ(u) 10−10 for all our numerical experiments|and fo−r
| ≤
allthe sampledvalues ofu>u , forthe adoptedtrunca-
0
NP 1 tion N =7.
P˙ (u,x,φ) = A˙ (u)(+)Y0(x,φ)+ P
(a) left 2 l,0 l Ournumericalexperimentsarerealizedwiththeinitial
Xl=0 data (26) – which corresponds to one initial black hole
NP l
+ A˙ (u)(+)Ym(x,φ)+ boosted alongthe z-axisand the secondblack hole mov-
l,m l ing along the direction determined by n = (n ,n ,n )
Xl=1mX=1 with n2+n2+n2 =1– having in principle four1inde2pen3-
NP l 1 2 3
+ B˙ (u)(−)Ym(x,φ), (42) dent parameters.
l,m l
Xl=1mX=1 We vary α in the interval (0,1.5) for γ = 0.2 and
several values of n . Exhaustive numerical experiments
1
and, its right side, will be given by the replacement of
show that for a sufficiently large computation time u
λ and P3 in (28) by the constraint expressions (39) and f
all modal coefficients become constant, namely, at u
(40). f
the emission of gravitationalwaves is considered to have
P˙ (u,x,φ) = P(3ap) λ 2xλ + λ(a)φφ . cAease(du. )For u10f−w10e, aBctua(lluy h+avhe)thaBt |A(l,um()uf +1h0)−1−2
(a) right −12m0(cid:20) (a)xx− (a)x (1−x2)(cid:21) folr,mall lf=| ≤0...7, wher|e hl,mis thfe integ−ratilo,mn stfep|.≤
From these modal coefficients A (u ) and
l,m f
Finally we project both sides to get the relation be- B (u ) we reconstruct the final configuration
l,m f
ttwheeennewthme oud-vaalrciaoteiffioncieonfttshe set {Al,m(u),Bl,m(u)} and Pbe(uefx,pxr,eφss)ed≃aPs(a)(uf,x,φ) that, in all cases, can
A˙l,m(u) ≡ (+) l,m|P˙(a) P(u ,x,φ) = A0,0(uf) + A1,0(uf)cosθ
= A˙ D( C (uE),D (u),E (u),F (u) ) f 2 2
l,m { l,m l,m l,m l,m } + A1,1(uf)sinθcosφ
B˙l,m(u) ≡ (−) l,m|P˙(a) + B1,1(uf)sinθcosφ+O(10−10)
= B˙l,mD({Cl,m(uE),Dl,m(u),El,m(u),Fl,m(u)})(.43) ≃ K1 (coshγf +nf ·xˆ sinhγf). (47)
f
This completesourschemeto obtainthe dynamicalsys-
tem version of the non-axisymmetric RT equation (28). The rms error of the second equality in (47) is of the
To solve it, we use a fourth-order Runge-Kutta recur- order of, or smaller that 10−12. The expression in the
sive method adapted to our constraints. So we have second equality of (47) corresponds to a boosted black
to get the initial values for the modal coefficients, say hole along the direction n = (n ,n ,n ) with boost
f 1f 2f 3f
8
parameterγ andrestmassm K3,aconfirmationofthe V. THE PLANAR NATURE OF THE GENERAL
f 0 f
NON-HEAD-ON COLIISION
Chrusciel-Singleton theorem[10].
From (47) we can identify
Fromthenumericalexperimentsdonetocheckthelong
time evolution of the initial data (26) and to obtain the
2coshγ
A (u ) = f, basic parameters of the remnant black hole, we observe
0,0 f K
f that the final direction of remnant velocity is contained
A (u ) = 2 n3f sinhγf, (48) in the plane determined by the the two directions of the
1,0 f Kf initial individual colliding black holes. For illustration
n sinhγ letusconsidertheevolutionoftheinitialdata(26),with
A1,1(uf) = 1f Kf f, nz =(0,0,1)correspondingtothe directionofthe initial
colliding black hole (boosted along the positive z-axis),
n sinhγ
B1,1(uf) = 2f K f, with n = (0.26,0.07,0.963068) corresponding to the di-
f rection of the second initial colliding black hole and the
upto (10−12),fromwhichwecanreadthefinalparam- severalvaluesofα=0.1, 0.1, 0.20, 0.3, 0.50. Theresult-
eters oOf the remnant stationary black hole (K ,γ ,n ). ing final direction of the remnant is given by (cf. (49))
f f f
It results
n (α=0.1) (1.363345 10−2,3.670544 10−3,9.999003 10−1),
f
≃
n1f = 2AA11,,01((uuff)) n3f, n2f = 2AB11,,01((uuff)) n3f nnf((αα==00..23)) ≃ ((30..006040807070 1100−−22,,80..205010509020 1100−−33,,99..909040906010 1100−−11)),,
f
n3f = 1+(2AA1,1((uuf)))2+(2AB1,1((uuf)))2 −1/2 (49) nf(α=0.5) ≃≃ (1.197374 10−1,3.223699 10−2,9.922820 10−1).
(cid:18) 1,0 f 1,0 f (cid:19)
In all cases we obtain
1 A
γ = tanh−1( 1,0)
f n3f A0,0 n n n 0. (52)
z f
2 · ∧ ≃
K = coshγ . (50)
f A0,0 f Within machine precision, (52) is valid up to (10−10),
O
and it also holds for all numerical experiments done in
An alternative to evaluate the parameter K is the use
f the paper considering a large domain of the initial data
oftheinitialdataintheconservedquantity(46),namely,
parameters,implyingthatthedynamicsisindeedplanar,
1 2π 1 1/2 as discussed at the end of Section III. The result (52)
Kf = 4π dφ K2(u0,x,φ) dx . (51) could also be considered as an additional test for the
(cid:18) Z0 Z−1 (cid:19) accuracy of our numerical code.
The agreement is within 10 significant decimal digits. This fundamental result – which will be used to save
This is also a test of the accuracy of (n )2 +(n )2 + computational efforts in the dynamical system evolution
1f 2f
(n )2 =1. of the RT non-axisymmetric dynamics – has an impor-
3f
One of the basic results to be extracted from our nu- tant physical counterpart connected to the fact that the
merical experiments are the values of (K ,γ ,n ), for linearmomentumfluxcarriedoutbygravitationalwaves
f f 3f
eachoftheindependentparameters(α,γ,n)oftheinitial is confined to the plane of the initial collision. In other
data. Theformerarethebasicparametersoftheremnant words,thelinearmomentumofthemergedsysteminthe
that – together with the initial data of the system and directionnz nisconserved,asdiscussedinSectionVIII.
∧
the function K(u,θ,φ) for all u u u – allow us to Let us then consider a rigid rotation of the two initial
0 f
≤ ≤
evaluate quantities which are characteristic of the radia- black hole system about the direction of motion of the
tive transfer processesinvolvedinthe gravitationalwave first initial black, namely, the z-axis. This rotation will
emission. Finally we remark that the integration of the be chosen so that the plane of the initial collision, deter-
dynamicalsystem(43))usedafourth-orderRunge-Kutta minedbythetwodirectionsn =(0,0,1)andn,coincide
z
recursive method (adapted to our constraints) together withthe x z plane. Inthiscasethe directionofmotion
−
with a C++ integrator. Unless otherwise stated, all our ofthe secondinitialcollidingblackholeis nowexpressed
numerical results are restricted to the choice γ = 0.2, by the unit vector n = (n ,0,n ), with n2+n2 = 1. In
1 2 1 3
corresponding to the initial infalling collision velocity thisinstancetheinitialdata(26)canassumethesimpler
v 0.1973. form
≃
1 α 2
K(u ,θ,φ)= + . (53)
0 √coshγ+cosθsinhγ coshγ (cosρ cosθ+sinρ sinθcosφ)sinhγ
(cid:18) − 0 0 (cid:19)
p
This initial data contains now three independent pa- rameters(α,γ,ρ )only,whereρ istheangleformedbyn
0 0
9
andthez-axisandisusedtoparametrizen =cosρ and satisfy the Bondi-Sachs boundary conditions. It should
1 0
n =sinρ . Again, as alreadyestablished, the motionof be noticed that in the RT coordinate system the pres-
2 0
the resulting remnant black hole will then be restricted enceoftheterm2rK /K doesnotfulfillthe appropriate
u
to the x z plane for any values of the initial data pa- boundary conditions. Although the coordinate transfor-
−
rameters. From the preceding discussions of the present mationsfromRTcoordinatestoBondi-Sachscoordinates
Sectionwehavethattheinitialdata(53)describesagen- cannotbe expressibleinaclosedform(they aregivenby
eralnonhead-oncollisionoftwoblackholesinthe realm an infinite series in powers of r−1) [22], their asymptotic
of RT dynamics. expansion allows us to obtain the form of the required
Now, for exhaustive numerical tests with the initial physical quantities. In this section we will restrict our-
data (53) and a large range of parameters (α,ρ ) we selves to the Bondi-Sachs energy-momentum for the RT
0
verify that: (i) the initial modal coefficients B (u ) . spacetimes as well as its conservation laws. Our deriva-
l,m 0
10−17,and(ii)itsevolutionviatheRTdynamicalsystem tionforthenon-axisymmetriccase[19]followscloselythe
(43) maintains B (u).10−17, forall(l,m)in their al- workofG¨onaandKramer[23]doneforthe axisymmetric
l,m
lowedrange,andforallu <u u . Thiscorresponds– case.
0 f
≤
withintheprecisionofourcomputation–toB (u)=0 From the supplementary vacuum Einstein equations
l,m
for all u. Obviously, from the second equation (49), we RUU = 0, RUΘ=0, and RUΦ=0 in the 2+2 Bondi-Sachs
formulation[7,8](where(U,R,Θ,Φ)aretheBondi-Sachs
have that the unit vector determining the direction of
the remnant will always have the form n=(n ,0,n ). coordinates), we obtain in RT coordinates
1f 3f
Therefore our computational task can be simplified by
taking all the Bl,m coefficients equal to zero, which is ∂m(∂uu,θ,φ) =−K c(u1)2+c(u2)2 + 21∂∂u 3c(θ1)cotθ
equivalent to restrict all our expansions in the Galerkin (cid:18) (cid:19) (cid:20)
cosθ 2 1
method decomposition to the cosine series only. In the + 4c(2) 2c(1)+c(1)+ c(2) c(1) (54)
φ sin2θ − θθ sinθ θφ − sin2θ φφ
remaining of the paper we have adopted this procedure (cid:21)
inthenumericalevaluationsofRTdynamicsforthedata
where m(u,θ,φ) is the Bondi mass function and
(53). We have however made sample tests by verifying
c(1)(u,θ,φ) and c(2)(u,θ,φ) are the two news functions
the absolute differences in the physical results originat- u u
for the non-axisymmetric case[19], corresponding to the
ingfromdataeitherfromthe completedecompositionor
twomodesofpolarizationofthegravitationalwaves. The
the decomposition using the cosine series only. The dif-
extra factor K in the first term of the second-hand-side
ference remains always of the order of, or smaller than of Eq. (54) comes from the transformation
10−17. The angle ρ (0 ρ π/2) will be denoted the
0 0
≤ ≤
collision angle, the limiting cases ρ = 0 corresponding ∂U 1
0 lim = , (55)
to a head-on collision and ρ = π/2 corresponding to a r→∞∂u K
0
right angle collision. U beingtheBonditimecoordinate. Forthenewssatisfy-
It will be possible to follow the full evolution of (53)
ing the appropriate boundary conditions c(1) = c(2) = 0
by considering from very small α up to those values for
and c(1) = c(2) = 0 at θ = 0,π , we obtain the Bondi-
which the nonlinearities start to play an important role θ θ
Sachs four-momentum conservation
in the dynamics. We will restrict ourselves to the range
0<α 1.2Therangeoftheinitialincidenceangleinthe dPµ(u)
numer≤ical experiments will be 0 ρ0 90o, the limits du =PWµ, (56)
≤ ≤
corresponding respectively to a head-on collision and an
wheretheBondi-Sachsfour-momentumPµ(u)isdefined
orthogonalcollision.
as
1 2π π
Pµ(u)= dφ m(u,θ,φ) lµsinθ dθ, (57)
VI. THE BONDI-SACHS FOUR MOMENTUM 4π
Z0 Z0
AND CONSERVATION LAWS
and
RTspacetimesdescribetheasymptoticallyflatexterior Pµ(u)= 1 2πdφ πK lµ c(1)2+c(2)2 sinθ dθ, (58)
gravitationalfieldofaboundedsystemradiatinggravita- W −4π u u
Z0 Z0 (cid:18) (cid:19)
tionalwavesandinthissensetheyareintherealmofthe
correspondstothenetfluxofenergy-momentumcarried
2+2 Bondi-Sachs formulation of gravitational waves in
out by the gravitational waves emitted. In the above
GeneralRelativity[7,8]. FurthermoreinitialdataforRT
lµ =(1, sinθcosφ, sinθsinφ, cosθ) is a null vector
dynamics are prescribed on null characteristic surfaces.
− − −
relative to an asymptotic Lorentz frame at infinity. We
Thereforesuitableexpressionsforthephysicalquantities
notethatthelasttermin(54)vanishesintheintegrations
to be used in the description of gravitationalwave emis-
due to the boundary conditions of the news.
sionprocessesanditsconservationlawsmustbederived.
For µ=0, Eq. (56) yields the Bondi mass formula
To exhibit such expressions it is necessary to perform a
coordinate transformation from RT coordinates used in
dM (u)
(1)toacoordinatesysteminwhichthemetriccoefficients dBu =−PW(u) (59)
10
where M (u) is the Bondi mass at a time u and
B
PW(u)= 41π 2πdφ πK c(u1)2+c(u2)2 sinθdθ. (60) ΡΡ0==4950oo
Z0 Z0 (cid:18) (cid:19) 0.00004 Ρ0=21o
0
is the power extracted from the system by the gravita- Ρ=15o
0
tional wave emission in a time u. The total energy E Ρ=0o
W 0
carried out of the system by the gravitational waves is 0.00003
given by the time integral of (60) up to u , and cor-
f W
responds to the total Bondi mass extracted from the P
system, M (u ) M (u ) = E , where M (u ) = 0.00002
B 0 B f W B f
m K3coshγ is t−he Bondi mass of the remnant.
0 f f
For µ =x,y,z, Eqs. (56) determine the liner momen-
tum conservation of the system, 0.00001
d P(u)
=P (u) (61)
du W 0
0.01 0.1 1
where the vector
u(cid:144)m
0
PW(u)= 41π 2πdφ πKxˆ c(u1)2+c(u2)2 sinθdθ. (62) FIG.1: Linear-logplotoftheenergyflux(power)carriedout
Z0 Z0 (cid:18) (cid:19) by gravitational waves as a function of u/m0, for α = 0.1
and γ = 0.2 and several values of the collision angle ρ . We
is the net momentum flux carried out by the 0
gravitational waves emitted. In the above xˆ = see that, for this relatively small value of α, the gw emis-
sion corresponds to a pulse of short duration, ∆u/m 3.5,
(sinθcosφ,sinθsinφ,cosθ). 0 ∼
and its initial intensity decreases as ρ increases. The case
0
Wearenowabletoanalyzetheradiativeprocessesthat
ρ =0 (head-on collision) constitutes an upper bound, while
0
lead the merged system from its initial configuration to theorthogonal collision (ρ =90o)constitutesalower bound
0
the final configuration of the remnant black hole. for the total energy emitted (the area below the curve), in
accordance with the efficiency behavior.
VII. THE POWER EXTRACTED BY THE
GRAVITATIONAL WAVE EMISSION AND THE
EFFICIENCY OF THE MASS-ENERGY
RADIATIVE TRANSFER: UPPER AND LOWER
BOUNDS gravitationalwaves,aswewilldiscusslaterinthepresent
Section.
Wenowexaminetheenergyextractedfromthemerged
Second,althoughthecurvespresentthesamebehavior
system by the gravitational waves. From our discussion
for distinct α’s, we can see that for α small (cf. Fig. 1
in Section VI on the Bondi-Sachs conservationlaws, and
for α=0.1) the emission correspondsto a short pulse of
specifically from Eqs. (59) and (60) we have that the
gravitational waves. We indeed obtain that, for any of
power emitted by the system in a time u is given by
the incident angles ρ (here including the head-on colli-
0
P (u) dE (u)/du sion),the initialpoweremitted decreasesby threeorders
W W
=≡ 41π 2πdφ πK c(u1)2+c(u2)2 sinθdθ. (63) othfemcaognntirtaurdye, fionraαn=int0e.r6va(clfo.fFtiimg.e2∆)ut/hme 0in∼itia3l.5p.owOenr
Z0 Z0 (cid:18) (cid:19) emitted (which is one order of magnitude smaller than
In Figs. 1 and 2 we plot the power emitted P (u) as a in the case α = 0.1) decreases by three orders of mag-
W
function of u/m0 for α = 0.1 and α = 0.6, respectively, nitude in an much larger interval ∆u/m0 ∼ 72 for all
and γ =0.2 fixed, and for severalvalues of the incidence the incidence angles ρ0 considered. In fact, analogousto
angleρ . ThetotalenergyemissionE (u )ineachcase the case of head-on collision analyzed in Ref. [2], there
0 W f
is measured by the area below the respective curve – is a threshold value of α 0.67 that separates regimes
∼
in accordance with (60). Two important features are of short bursts of gravitational waves from a regime of
to be noted in the Figures. First, we can see that a quiescent long time emission.
head-on collision (ρ = 0) constitutes an upper-bound
0 Forαintheinterval(0,1.5]andforseveralvaluesofρ
for the total energy emitted, as well as the orthogonal 0
in the interval (0,π/2) we have determined (K ,γ ,ρ )
collision (ρ = 90o) constitutes a lower-bound for this f f f
0 which characterize the final boosted black hole configu-
energy. This patternis typicalfor any 0<α<1 andfor
ration. Fromtheseparametersoftheblackholeremnant
all γ. Actually in our numerical experiments we verified and the total radiated energy E (u ), the efficiency ∆
W f
this important result for 0 α 1, consistent with ofmass-energyextractionbygravitationalwaveemission
≤ ≤
the curves of the efficiency of mass-energy extraction by canbeevaluated. AccordingtoEardley[24]theefficiency