Stable characteristic evolution of generic 3-dimensional single-black-hole spacetimes

4 Pages·0.11 MB·English

by  The Binary Black Hole Grand Challenge Alliance

Stable characteristic evolution of generic 3-dimensional single-black-hole spacetimes
R. G´omeza, L. Lehnera, R. L. Marsaa,b, J. Winicoura
A. M. Abrahamsc, A. Andersond, P. Anninosc, T. W. Baumgartec, N. T. Bishope, S. R. Brandtc, J. C. Browneb,
K. Camardaf, M. W. Choptuikb, R. R. Correlb, G. B. Cookg, C. R. Evansd, L. S. Finnh, G. C. Foxi, T. Haupti,
M. F. Huqb, L. E. Kidderh, S. A. Klaskyi, P. Lagunaf, W. Landryg, J. Lenaghand, J. Massoc, R. A. Matznerb,
S. Mitrab, P. Papadopoulosf, M. Parasharb, L. Rezzollac, M. E. Ruprightd, F. Saiedc, P. E. Saylorc, M. A. Scheelg,
E. Seidelc, S. L. Shapiroc, D. Shoemakerb, L. Smarrc, B. Szil´agyia, S. A. Teukolskyg, M. H. P. M. van Putteng,
P. Walkerc, J. W. York Jrd.
aUniversity of Pittsburgh, Pittsburgh, Pennsylvania 15260
bThe University of Texas at Austin, Austin, Texas 78712
cUniversity of Illinoisat Urbana-Champaign, Urbana, Illinois 61801
dUniversity of North Carolina, Chapel Hill, North Carolina 27599
eUniversity of South Africa, P.O. Box 392, Pretoria 0001, South Africa
8 fPenn State University, University Park, Pennsylvania 16802
9 gCornell University, Ithaca, New York 14853
9 hNorthwestern University, Evanston, Illinois 60208
1 iSyracuse University, Syracuse, New York 13244-4100

n
a
J ingevenastrictlyspherical(onespatialdimension)single

0 Wereportnewresultswhichestablishthattheaccurate3- black hole led to the formulation of “the Holy Grail of
2 dimensional numericalsimulation ofgenericsingle-black-hole
numerical relativity”, a list of requirements for “ a code
spacetimeshasbeenachievedbycharacteristicevolutionwith
that simultaneously
1 unlimited long term stability. Our results cover a selection
v of distorted, moving and spinning single black holes, with • Avoids singularities
9 evolution times up to 60,000M. Handles black holes
•
6 Maintains high accuracy
0 •
04.25.Dm,04.30.Db Runs forever.” [6]
1 •
Theresultsreportedheredefinitelyachievethisgoalin
0
8 the 3-dimensional, single black hole case. The challenge
Accurate numerical simulation of black holes is neces-
9 for the 1990sandbeyond is the Binary Black Hole prob-
/ sary to calculate gravitational waveforms in the nonlin- lem. The results here may become directly applicable to
c
earregimethatcannotbeapproximatedbyperturbation
q that multiple black hole stage.
- theory. Theimportanceofsuchwaveformstothesuccess The characteristic algorithm is a new computational
r of the LIGO gravitywave detector was a prime factor in
g treatment of hyperbolic systems. The theoretical frame-
: organizing the Binary Black Hole Grand Challenge Al- workisthecharacteristicinitialvalueproblem,pioneered
v
liance,whosegoalistoprovidethecapabilityofobtaining
i byBondi[7]andPenrose[8]inthe1960s. Almostallnu-
X waveforms from the inspiral and merger of binary black
merical modeling of hyperbolic systems has been based
r holes [1]. The Alliance is developing a code consisting of upon the Cauchy initial value problem, which evolves
a
aCauchymodulematchedtoanouterboundarymodule
fieldsonspacelikehypersurfacesalongadiscretesequence
usingeitheracharacteristicorperturbativemethod. For
of time steps. The major new idea in the characteristic
reports on the Cauchy and perturbative modules see [2]
approachistoevolvefieldsonoutgoing(oringoing)light
and [3], respectively. Here we report two new tests of a
cones along a sequence of retarded (or advanced) time
3-dimensional characteristic evolution module [4,5] that
steps. Figure 1 shows the schematic setup for the out-
establish its unlimited capability to accurately simulate
going case. A world tube Γ has been placed as an inner
a generic single black hole spacetime and that establish
boundary on the light cones (characteristics) to excise
a calibrated tool to attack the binary problem: (i) We
caustics from the evolution domain. Boundary data on
have evolved a black hole of mass M moving with peri-
Γanddataontheinitiallightcone determineaunique
0
odictimedependenceinducedbyacoordinatewobblefor N
exterior evolution.
a time of 60,000M;and (ii) we have evolved an initially
Characteristic evolution has severaladvantageous fea-
distorted,spinning blackhole upto the finalequilibrium
tures [9]: The evolution variables reduce to one com-
state,whichremainsstationarytowithinmachineround-
plex function related to the two gravitational polariza-
offerrorandisadiscretizedversionoftheKerrblackhole
tion modes; the Einstein equations reduce to propaga-
spacetime. (We use units with Newton’s constant G=1
tionequationsalongthelightrays;andtherearenocon-
and the speed of light c = 1. Thus M GM/c3 is a
straints on the initial data. Furthermore, because the
≡
time. Also, M GM/c2 is a length.)
light cones are the spacetime hypersurfaces along which
≡
Inthe1970sand1980s,thedifficultyofstablysimulat-
1
waves propagate, such propagating disturbances appear tially proposedby Unruh (see [17])for Cauchy evolution
fairlysmoothalongthem. Thisfeatureallowsimplemen- of black holes.
tation of Penrose’s spacetime compactification [8] to in- This strategy is based upon the properties of trapped
clude points atfuture lightlikeinfinity (inthe casebased surfaces[18]. Normally,thelightraysemittedintheout-
on outgoing light cones), where the waveform is calcu- ward normal direction to a (topologically) spherical sur-
lated in the numerical grid. The major disadvantage is face form an expanding beam. But strong gravitational
the difficulty in treating caustics. One early strategy for lensing can make such an outgoing spherical beam ev-
a characteristicalgorithmproposedtackling the caustics erywhereconvergent. Suchasurfacewhoseoutgoingand
head-onaspartoftheevolution[10]. Buttodatethishas ingoing raysall convergeis called trapped. A marginally
only been accomplishedfor point causticsin axisymmet- trapped surface (MTS) is the borderline case in which
ricspacetimes[11];andtheextensionto3Dwouldbepro- the outward light cone neither expands nor converges.
hibitive on present-day machines. Cauchy-characteristic Underreasonableassumptions,aMTScannotlieoutside
matching is a strategyforcombining the complementary a black hole (see [19]). Consequently, if the worldtube Γ
strengths of Cauchy and characteristic evolution. in Fig. 2 is outside the black hole then the ingoing light
The implementation and calibrationof the 3D charac- cone mustextend some finite distance inwardfromΓ
0
N
teristic module has been described elsewhere [4,12]. For beforereachingaMTS( inFig.2). Inallknownexam-
S
a grid of discretization size ∆, the numerical solutions ples of black holes the singularities are located inside a
converge in the continuum limit to exact analytic values MTS. Excision of the interior of the MTS thus protects
in a wide variety of test beds, with O(∆2) error. The the evolution from encountering a singularity.
longtermstabilityoftheoutgoingproblemhasalsobeen In order to implement this strategy (i) the evolution
established [4]. In these studies, the inner world tube module must be equipped with an MTS finder and (ii)
Γ was chosen to be the ingoing branch of the r = 2M the singularregioninside the MTSmustbe excisedfrom
horizon in a Schwarzschild spacetime. The initial data the computational grid without influencing the exterior
consisted of a pulse of ingoing radiation on . This evolution. In a characteristic evolution, item (i) is facili-
0
N
set the data for the scattering of a pulse of radiation tatedbylocatingtheMTSinanaturalwaybydeforming
by a Schwarzschild black hole, the classic problem first aninitialguessalongtheingoinglightrays[5]. Similarly,
studied perturbatively by Price [13]. The angular mo- item(ii) is facilitatedbecause the excisionofthe interior
mentum of the ingoing pulse leads to a final black hole of the MTS reduces to a 1-dimensionalproblem with re-
with spin. The pulse is partially transmitted into the spect to a radial grid variable. There is no need for any
black hole and partially scattered to (compactified) in- further boundary condition on the MTS: by construc-
finity along outgoing light cones, where its waveform is tion, waves emitted from its surface cannot expand into
obtained. The evolution handles highly distorted black theexteriorregion. Thistheoreticalpropertyisbuiltinto
holeswithbackscatteredradiationathousandtimesmore the characteristic algorithm.
massivethantheinitialblackholeandwithapeakpower Detailsandcalibrationoftheingoingmodulearegiven
105 in dimensionless units (equivalentto conversionof in[5]. Ininitialsimulationsofanon-spinningblackhole,
≈
our galaxy’s mass into gravitationalwaves in 1 second). data on Γ was induced from the exterior geometry of a
The calculation of the waveform at infinity for the bi- Schwarzschildspacetimeandinitialdataon consisted
0
N
naryproblemcanbe posedina similarwayby matching of a Schwarzschild black hole of mass M distorted by a
a Cauchy interior module matched at a worldtube Γ to pulse of radiation. The worldtube Γ was also placed in
a characteristicouter module. (See [3] for an alternative motion relative to the static symmetry of the exterior
perturbative matching scheme). In model 3D nonlinear Schwarzschild spacetime to produce a time dependent
problems, Cauchy-characteristic matching dramatically location of the black hole in the numerical grid. The
outperformsotherexistingouterboundaryconditionsfor dynamics was monitored by tracking the surface area
Cauchy evolution [14]. It has been successful in 1D gen- of the MTS. For a non-spinning black hole in equi-
A
eral relativity [15,16] but its efficacy in 3D general rela- librium, this surface area equals its Schwarzschild value
tivity isyettobe determined, becausea stablematching S =16πM2. Calculationof isanespeciallydemand-
A A
scheme has not yet been found. ing test when the world tube is offset from the spherical
A simple transformation switches an outgoing charac- symmetry of the Schwarzschild exterior and then placed
teristic evolution module into an ingoing module [5,16]. in a periodic circular orbit. This periodic wobble of the
In this case, to uniquely define a black hole spacetime, coordinates leads to a periodic time dependence of both
boundary data is prescribed on an outer worldtube and the metric and the location of the MTS, even in the fi-
on an incoming light cone (Γ and in Fig. 2); and in nalstateofintrinsicallystaticequilibrium. Theareaofa
0
N
order to excise the singular region interior to the black MTSdeterminesitsHawkingmass[20]andgivesauseful
hole, an inner boundary is constructed at a world tube measureof the energy inside it. Initially, < S due to
A A
traced out by a marginally trapped surface ( in Fig. 2). the energy content of the initial pulse on . The MTS
0
T N
This extends to characteristic evolution the strategy ini- grows as this energy falls into it. For a non-spinning
2
blackhole, S asthe MTSsettlesintoequilibrium, worldtubedatainBondicoordinates,whicharespherical
A→A
eventhoughthemetricandthelocationoftheMTSvary coordinatesbaseduponthelightconesemanatinginward
periodically (see Fig. 3). from Γ (advanced time coordinates). The transforma-
It is important to establish that the ingoing charac- tion from Cartesian Kerr-Schild coordinates to spheri-
teristic module has no long term instabilities and that it cal Bondi coordinates is carried out numerically in the
canhandlespinningblackholes. Wenowpresenttwonew neighborhood of the worldtube by an extraction module
tests which demonstrate that it can essentially evolve a which forms part of the Alliance’s Cauchy-characteristic
generic black hole “forever”. matching procedure [12,23]. The initial data for a Kerr
Since “forever” cannot be rigorously attained in any black hole on the ingoing light cone is complicated
0
N
finite simulation, we appeal to a characteristic time nec- to specify analytically(the geodesicequationleadsto el-
essary to obtain accurate waveforms for the inspiral and liptic integrals). Instead, we choose initial data which
merger of two black holes. If one of the holes is small approximatesKerrdata but distortsthe blackhole. The
then the test particle approximation can be used. Con- amountofdistortioncanbemeasuredintermsoftheini-
sider a test particle in a quasi-circular orbit about a tial surface area of the MTS as compared with the Kerr
Schwarzschildblackhole,wherethefinalstableorbitisat value = 8πM(M +√M2 a2). In Fig. 3, we plot
K
A −
r =6M. From the quadrupole approximation (see [19]), vs time for both the initially distorted Kerr and wob-
the radiation rate per orbit is 10−2 of the binding en- Abling Schwarzschild cases. Tests show that at late times
≈
ergy,suggestinghundredsoforbitsforthetransitionfrom converges to the exact (Kerr or Schwarzschild) value
A
inspiral to merger. The period measured by an observer as the discretization size ∆ 0.
→
at infinity is 12π√6M 90M for this orbit so that the The Kerr evolution was run for a time of 15,000M,
≈
decaytimewouldbe 10,000M. Forblackholesofcom- at which the only changes were at machine round-off.
≈
parable mass, perturbation theory cannot reliably treat As apparent from Fig. 3, the final Kerr equilibrium is
the regime intermediate between an orbital separation effectively reached at 20M.
of 12M and merger. However, as estimated in [21] the The success of the ingoing characteristic module sug-
decaytimeforthisstageis 1500M andtojointheevo- gests a possible strategy for excising the singularities in
≈
lutionsmoothlytoapost-Newtonianorbitat20M would the binary case (see Fig. 4). Two disjoint characteristic
require an evolution time of 10,000M. evolutions based upon ingoing light cones are matched
≈
In our first test, we have successfully evolved acrossworldtubesΓ andΓ toaCauchyevolutionofthe
1 2
an initially distorted, moving (but non-spinning) shadedregionbetweenthem. Theingoinglightconesare
Schwarzschild black hole for a time of 60,000M, clearly each truncated at a MTS surrounding the singularities.
as long as needed for a smooth transition from the post- TheouterboundaryΓoftheCauchyregionismatchedto
Newtonianregime to merger,if this successcould be du- an exterior characteristicevolution based upon outgoing
plicatedintheultimatebinarycode. Therunwastermi- light cones extending to infinity, where the waveform is
nated because it had achieved a steady state, with no calculated. This global strategy has been successfully
sign of instability, and could be extended further. It implemented for spherically symmetric self-gravitating
was carried out in a wobbling coordinate system (see scalarwavesevolvinginasingleblackholespacetime[16].
Fig. 3) which induces an “artificial” time dependence. Justasseveralcoordinatepatchesarenecessarytode-
(The wobble of the outer worldtube Γ in the vicinity of scribe a spacetime with nontrivial topology, an effective
r = 7M is the only time dependence seen at late times attackonthebinaryblackholeproblemcouldbetopatch
intheevolution). Thiscapabilityisimportantbecauseit together regions of spacetime handled by different algo-
may not be possible to simulate a binary in coordinates rithms. The Cauchy-characteristic modules are in place
which become exactly stationary after the merger and andcalibratedforaccuracy. TheAlliance’sCauchymod-
ring-down to final equilibrium. ule [2] has evolved a Schwarzschild black hole for a time
Our second test establishes that the ingoing charac- of 475M. Its performance and stability are now being
teristic module handles spinning black holes. The outer studied in a wide variety of tests. (For Cauchy evolution
worldtubedataisinducedfromtheexteriorgeometryof of a black hole with another 3D code, see [24]). The key
a Kerr spacetime with mass M and angular momentum missingingredientisthe longtermstabilityofmatching,
parametera=M/5(spinequalto M2/5). The metric is which is a major current project.
written in the Cartesian Kerr-Schild form [22] This work was supported by the Binary Black Hole
Grand Challenge Alliance, NSF PHY/ASC 9318152
ds2 = dt2+dx2+dy2+dz2+2Hkµkνdxµdxν, (1) (ARPA supplemented). Computer time wasprovidedby
−
the Pittsburgh Supercomputing Center.
where k is tangentto aningoing congruenceof twisting
µ
light raysandH is a potential(which equals M/r in the
a = 0 nonspinning case). The outer worldtube Γ is lo-
cated at x2+y2+z2 =49M2. The module requires this
3
[1] ForinformationaboutthegoalsandstatusoftheAlliance
Γ visit: http://www.npac.syr.edu/projects/bh/.
[2] The Binary Black Hole Grand Challenge Alliance,
Boosted 3-dimensional black hole evolutions with singu-
larity excision, (submitted for publication).
N [3] The Binary Black Hole Grand Challenge Alliance,
0
Schwarzschild-perturbative gravitational wave extraction
FIG. 1. The outgoing formulation: The exterior of Γ is and outer boundary conditions, Phys. Rev. Lett. (to ap-
covered by a sequenceof outgoing light cones. pear).
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Winicour, Phys. Rev. D 56, 6298 (1997).
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T gr-qc/9710138.
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S and Numerical Relativity, ed. J. Centrella (Cambridge
N
0 UP, Cambridge, 1986) p.74.
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Proc. R. Soc. Lond. A269, 21, 1962.
FIG. 2. The ingoing formulation: The interior of Γ is
[8] R. Penrose, Phys. Rev. Lett., 10, 66 (1963).
covered by a sequence of ingoing light cones. The interior of
T is excised from the evolution. [9] R. G´omez and J. Winicour, in Approaches to Numerical
Relativity,ed.R.d’Inverno(CambridgeUniversityPress,
Cambridge, 1992) p. 143.
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A 12.38
P. Papadopoulos, and J. Winicour, J. Comput. Phys.
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1 10 100 1000 10000
t/M [18] R. Penrose, Phys. Rev. Lett., 14, 57 (1965).
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rotationfrequency0.1,offset0.1andmass0.5)andaninitially Press, Chicago, 1984).
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Γ
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Γ Γ
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(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) Anninos, J. Mass´o, E. Seidel and W.-M. Suen, Physics
FIG. 4. A matching scheme for two orbiting black holes World 9, 43 (1996).
(in a co-rotating frame which eliminates the major source of
time dependence).
4