Comments on tails in Schwarzschild spacetimes

0.18 MB·English

by  Janusz Karkowski

Comments on tails in Schwarzschild spacetimes.
Janusz Karkowski∗, Zdobys law S´wierczyn´ski+ and Edward Malec∗
∗ Institute of Physics, Jagiellonian University, 30-059 Cracow, Reymonta 4, Poland.
+ Pedagogical University, Cracow, Podchora¸z˙ych 1, Poland.
the otherhanda detaileddiscussionin[4]showsthatfor
initially nonstatic, ∂tΨ6=0 analytic initial data satisfy-
We performed a careful numerical analysis of the late tail
ingcertainboundaryconditions(attheeventhorizonofa
behaviour of waves propagating in the Schwarzschild space-
blackhole andat spatialinfinity) the decayof a solution
time. Specifically the scalar monopole, the electromagnetic
agreeswith that of the Price’s initially static l-pole field.
dipoleandthegravitationalaxialquadrupolewaveshavebeen
investigated. Theobtainedresultsagreewithafalloff1/t2l+3 This might suggest in turn that one can understand the
for thegeneral initial data and 1/t2l+4 for theinitially static ”staticl-polefield”asaspeciallypreparedgeneralinitial
4 data. waveprofile. Correspondinginitialdata(bothΨand∂tΨ
0 arenonzero)haveanoncompactsupport(thespatialsup-
0 port has infinite extension, as measured in terms of the
2 I. INTRODUCTION tortoise variable r∗ – see below). The work of Gundlach

n et al. [5] numerically confirmed the analytically derived
a Waves propagating in a curved spacetime usually un- 1/t2l+2 decay of late tails. Let us remark an internal in-
J
dergo backscatter; that leads to the emergence of two consistency–thecontentionmustbenowthatthefalloff

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classes of effects, the so-called quasinormal modes (dis- actuallydependsontheprofileofinitialdata,incontrast

covered by Vishveshvara [1]) and the tails. Price, who to the primary expectation that the ”tail” is a universal
2
v investigatedthe temporal behaviour of tails in 1972,has phenomenonduetothebackscatterofthewavesignaloff
1 shown some kind of universality. The late tail behaviour the curvature of the geometry.
0 of scalar, electromagnetic and gravitational waves hap- Leaving aside the area of initially static fields, let us
1 penedtodependonthe orderofthe multipoleexpansion notice that in the case of data of compact support the
3
[2]. Theexpositionof[2]issomewhatconfusingandthat predictionsof[2]areclear–solutionsshouldhaveafalloff
0
3 forcesus to saymore thanwouldnormallybe neededfor t−(2l+3). That estimate applies to solutions generated
0 the purpose of a paper reporting only numerical results. by generic initial data (Ψ 6= 0, ∂tΨ 6= 0) [4]. From a
c/ PricestatesIf a staticl-pole field is present outside the naive inspection of the formal solution Ψ =∂tG∗Ψ(t =
q star,prior totheonsetofcollapse, thefieldwillfall offas 0)+G∗∂tΨ(t=0)(where G is the Greenfunction), one
- t−(2l+2). If there is no field initially outside the star, but expectsthatinthecaseofmomentarilystaticinitialdata
r
g an l-pole perturbation develops during the collapse pro- the fall off is faster (by 1/t) than that corresponding to
: cess, it will fall off as t−(2l+3) at large time [2]. Thatre- general initial data. It is interesting that no numerical
v
i quires a comment. Since the evolution equation (2) (see investigation has been performed in the case of moment
X below)islinearandoftheform−∂2Ψ+LΨ=0,whereL of time symmetry initial data of compact support. It
t
r isalineartime-independentoperator,itssolutionscanbe should be noticed also that the status of the analytic in-
a
superposed. Thatmeansthatifthereisastaticsolution, vestigationis still not quite satisfactory, since it consists
then it cannot influence any evolving perturbation. Irre- either of heuristic analysis [2] or rather formal investiga-
spective of whether there are static solutions or not, the tion of the Green function formulae ( [4], [6]) [8]. It is
perturbation will decay in the same way. Therefore the onlyrecently thatthe (generic case)falloff1/t3 has been
quotedstatementcannotbetakenatthefacevalue. The rigorously proven by Dafermos and Rodnianski for the
natural understanding would be the following: A static monopolemodeoftherealscalarfield[9]. Thesamecon-
field means that there is initially a field (Ψ(t=0)6=0), clusionisbeing derived,inadifferent(spectral)method,
but ∂tΨ = 0 (hence data are momentarily static). The by Machedon and Stalker [10].
initial absence of field in turn means that Ψ = 0 but The main focus of our numericalwork will be on find-
∂tΨ 6= 0. This agrees with the standard way of posing ingthefalloffoftailsgeneratedbyinitialdataofcompact
the initial valueprobleminmathematicalliterature,and support in either of the aforementioned two fundamen-
withthisinmindweinspectthelaterpapersonthissub- talcases. We study the evolutionof the scalarmonopole
ject. The notion of initially static l-pole has been clearly (l =0), the electromagnetic dipole (l =1) and the grav-
specifiedintwopublicationsthatappearedinlateseven- itational axial quadrupole (l = 2). The obtained results
ties;itbecomesaninitiallystationarymultipoleoforderl reveal a falloff 1/t2l+3 for the data ∂tΨl 6= 0,Ψl = 0,
(that falls off like 1/t2l+2) [3]. Therein the term initially in agreement with [2], and 1/t2l+4 for the initially static
stationary means momentarily static as defined above, initial data [11]. This latter result on the behaviour of
but with Ψ satisfying special asymptotic condition. On late tails generated by initial data ∂tΨl = 0,Ψl 6= 0 is
1
new in the numerical literature. Thus the generic initial The expected asymptotic tail falloff should be the same
data have the falloff 1/t2l+3, which agrees with Leaver as for generic initial data.
[4], Ching et al. [6] and Poisson[7]. ii) Symmetric initial data with Ψ(R,t = 0) =
Our results support the view that the l-th moment sin4 πr∗−r∗(a) for r∗ ∈ (r∗(a) = r∗(3m),r∗(a)+4m)
(cid:16) 4m (cid:17)
of any of the waves (scalar, electromagnetic or gravita- ∗ ∗ ∗ ∗
and Ψ(r,0) = 0 for r < r (a) or r > r (a)+4m, and
tional) will decay like 1/t2l+3 for general initial data (in
accordance with [6], [4] and [7]) and 1/t2l+4 for the l-th ∂tΨt=0 = 0 everywhere. It is shown below that the cor-
responding tail decays faster than that of i).
momentsevolvingfrominitiallystaticdata(inagreement
with the formal analysis of [4] and [7]). These results iii) ∂tΨ = sin4(cid:16)πr∗−4rm∗(a)(cid:17) for r∗ < r∗(a) = r∗(3m)
∗ ∗ ∗ ∗
should be of practical significance for numerical relativ- or r > r (a)+4m and ∂tΨ(r,0) = 0 for r < r (a) or
∗ ∗
ity. The determination of the late tail behaviour is a r > r (a)+4m, while Ψ|t=0 = 0 everywhere; the tail
nontrivial but at the same time feasible numerical task behaviour happens to be like in i).
that can serve as a useful test for the accuracy and sta- One finds convenient to split the wave equation (2),
bility of numerical codes. into the pair of first order differential equations
(∂t+∂r∗)Ψ=Φ (6)
II. DEFINITIONS AND EQUATIONS
∗
(∂t−∂r∗)Φ=−V(r )Ψ. (7)
The spacetime geometryisdefined by the line element
We use the following mesh scheme (see Fig. 1)
dR2
ds2 =−ηRdt2+ +R2dΩ2, (1)
ηR
wheretisatime coordinate,Risthe radialarealcoordi-
nate, ηR =1− 2Rm and dΩ2 =dθ2+sin2θdφ2 is the line ab
element on the unit sphere, 0 ≤ φ < 2π and 0 ≤ θ ≤ π. a1 b1
dt
Throughout this paper the Newtonian constant G and
a2 b2
the velocity of light c are put equal to 1.
dr = 2 dt
The propagation of the scalar, the dipole electromag-
FIG.1. This figure shows the mesh used in the numerical
netic and the axial (quadrupole) gravitational waves is computations. Intheone-step schemethefunctions Φab,Ψab
given by arecomputedfromEqs(9–11)usingvaluesatpointsa1and
b1. Thesecondalgorithmneedsvaluesatfourpointsa1,a2,b1
(−∂t2+∂r2∗)Ψ=VΨ. (2) and b2 in orderto obtain Φab,Ψab from Eqs(14).
∗ R
Here r (R)=R+2mln(cid:16)2m −1(cid:17) is the tortoise coordi- andtwodifferencingschemes: the one-stepimplicital-
nate while the potential term reads: for the l = 0 mode ghoritm and the multistep Adams-like algorithm. In the
of the scalar field first approach we approximate the equations (7) by
ηR dt
V(R)=2m , (3)
R3 Ψab−Ψa1 =(Φab+Φa1) (8)
2
for the l =1 (dipole) electromagnetic mode
dt
V(R)=2ηR (4) Φab−Φb1 =−(VabΨab+Vb1Ψb1) 2 . (9)
R2
In the next step these equations are solved for Ψab,Φab,
and for the quadrupole axial mode of GW
η2 m Ψab =Ψab(Ψa1,Ψb1,Φa1,Φb1) (10)
R
V(R)=6 (1− ). (5)
R2 R
Φab =Φab(Ψa1,Ψb1,Φa1,Φb1). (11)
The second alghoritm is based on the Adams-Moulton
III. NUMERICAL SCHEMES
corrector formula for ordinary differential equations [12]
Wi)eΨw(Rill,tcho=ose0)fol=lowsiinng4(cid:16)cπlars∗s−e4srm∗o(af)i(cid:17)nitaianldda∂ttaΨ:|t=0 = yn+1 =yn+ 1d2t(5yn′+1+8yn′ −yn′−1)+O(dt4) (12)
∗ ∗ ∗ ∗
−∂r∗Ψt=0 for r ∈ (r (a) = r (3m),r (a) +4m), and Applyingthisformulatoeachoftheequations(7)weget
∗ ∗ ∗ ∗
Ψ(r,0) = ∂tΨ = 0 for r < r (a) or r > r (a)+4m. approximatelly
2
dt
Thenumericaltime rangedfrom5000m(forthe grav-
Ψab =Ψa1+ (5Φab+8Φa1−Φa2) (13)
12 itational and electromagnetic waves) up to 40000 m (for
thescalarwaves). Itturnedoutthatthepreciseshapeof
and
this function, withoutnumericalnoise,canbe calculated
dt using numbers with 64 significant digits (gravitational
Φab =Φb1− (5VabΨab+8Vb1Ψb1−Vb2Ψb2) (14)
12 and electromagnetic cases) and the ordinary double pre-
cision numbers (scalar case). In order to achieve this
Again this linear set of equations can be solved for
aim we have used the freely distributable ”qd” (quad
Ψab,Φab
precision) and ”arprec” (arbitrary precision) numerical
libraries [13].
Ψab =Ψab(Ψa1,Ψb1,Ψa2,Ψb2,Φa1,Φb1,Φa2,Φb2)
Φab =Φab(Ψa1,Ψb1,Ψa2,Ψb2,Φa1,Φb1,Φa2,Φb2).
(15) 5
4.98
Let us stress that these two alghoritms give almost the
4.96
same results, hinting at the numerical stability of our
4.94
methods.
4.92
4.9
4.88
IV. NUMERICAL RESULTS
4.86
1000 2000 3000 4000 5000
∗ ∗
We expect the field Ψl(r ,t) to behave (for a fixed r ) FIG. 3. The solid (general initial data) and broken line
like (datawith φ=0), thebehaviourof f(t) for theselectromag-
netic waves.
∗ −α
lim Ψ(r ,t)=Ct (16)
t→∞
Ourgoalistogetavalueofαwithareasonableaccuracy.
7
Therefore we calculate the function 6.99
∗ ∗ 6.98
f(t)=−dlog(Ψl(r ,t)) =−tdlog(Ψl(r ,t)) (17) 6.97
dlog(t) dt 6.96
6.95
which asymptotically should be equal to α. 6.94
In the first instance we analysed the case with waves, 6.93
whichbelongs to the classi) ofSec. 3, andforthe initial 6.92
data of class iii) of the preceding section. The results 6.91
1000 2000 3000 4000 5000
are presented in Figs. 2 – 4 which show the temporal
∗ ∗ FIG. 4. The solid (general initial data) and broken line
behaviourofthefuntionf(t)asseenatr =r (3m)+4m.
(datawith φ=0),thebehaviouroff(t)forthegravitational
In all figures (2 – 7) below the time is put on the x-axis,
waves.
which is scaled in units of m.
The high accuracy calculations are time consuming
3.02 and therefore the achieved calculational time for the
3 scalar case exceeds by a factor of 8 the evolution time
2.98
fortheremainingcases. Thenumericallyachievedvalues
2.96
of the exponent , for the general initial data i), range
2.94
from 2.999 for the scalar field, through 4.99 for the elec-
2.92
2.9 tromagnetic field up to 6.99 for the gravitational waves
2.88 (figures 2 – 4, solid line). Both sets of exponents is in a
2.86 good agreement with 3, 5 and 7, respectively, obtained
2.84
by Ching et al. [6]. Similar results (broken line Fig. 3)
10000 20000 30000 40000
have been obtained for the initial data of class iii) of the
FIG. 2. The solid (general initial data) and broken line
preceding section, in agreement with [2].
(datawithφ=0),thebehaviouroff(t)forthescalar waves.
The momentarilystatic initialdata(classii)fromSec.
3) produced profiles f(t) shown in Figs. 5 – 7. In this
case the numerical exponents have been determined at
∗ ∗
r (b) = r (3m) + 4m and also at another obervation
3
∗ ∗
point,r =100m+r (b). Inthefirstcaseweobtained4,
6and8(uptothefourthdigitnumber)whileinthelatter 8
casewearrivedat3.99,5.88and7.84,forthescalar,elec- 7.8
tromagneticandgravitationalwaves,respectively. These
7.6
exponentsseemtobestabilizeat4,6and8forthescalar,
7.4
electromagneticandgravitationalfields,respectively;no-
∗ ∗ 7.2
tice that the results detected at r = r (3m)+4m con-
vergequickertotheprospectivelimitingvaluethanthose 7
taken at the point located farther. The asymptotic tail 6.8
regime is evidently achievedquicker at a regioncloser to 1000 2000 3000 4000 5000
thehorizon,whichisanewfeature,unknownintheexist- FIG. 7. Time-symmetric initial data. The solid and bro-
ken lines show the behaviour of f(t) for gravitational waves,
ing literature. But in both detection points, the asymp-
∗ ∗
as seen at the observation points r (b) = r (3m)+4m and
totic values of α are similar and very close in the case
r∗ =r∗(b)+100m, respectively.
of the scalar waves, see Fig. 5. The calculation of these
exponents constitutes the main result of this paper.
Acknowledgments. This work has been supported in
part by the KBN grant2 PO3B006 23. One of us (EM)
4.1 acknowledges a discussion with Eric Poisson.
4
3.9
3.8
3.7
3.6
3.5
3.4
3.3 [1] C. V.Vishveshwara, Nature 227, 936(1970).
3.2 [2] R. Price, Phys. Rev. D5, 2419(1972).
10000 20000 30000 40000 [3] C. T. Cunningham, R. Price and V. Moncrief, Astr. J.
FIG. 5. Time-symmetric initial data. The solid and bro- 224, 643(1978); ibid, 230, 870(1979).
ken lines show the behaviour of f(t) for scalar waves, as [4] E. W.Leaver, Phys. Rev. D34, 384(1986).
seen at the observation points r∗(b) = r∗(3m) + 4m and [5] C. Gundlach, R. Price and J. Pullin, Phys. Rev. D49
r∗ =r∗(b)+100m, respectively. 883(1994); ibid,890(1994).
[6] E.S.C.ChingP.T.Leung,W.M.Suen,K.Young,Phys.
Rev. D52, 2118(1995).
[7] E. Poisson, Phys.Rev. D66, 044008(2002).
6.2
[8] J. Bicak, Gen. Rel. Grav. 3, 331(1972); ibid, 12,
6 195(1980); S.Hod, Phys. Rev. D58, (1998) 024019.
5.8 [9] M. Dafermos and I. Rodnianski, A proof of Price’s
law for the collapse of a self-gravitating scalar field gr-
5.6
qc/0309115.
5.4
[10] John Stalker, privatecommunication.
5.2 [11] From now on we do not make distinction between data
5 thatareinitiallystatic,momentarilystaticormomentof
4.8 time symmetry initially.
1000 2000 3000 4000 5000 [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B.
FIG.6. Time-symmetricinitialdata. Thesolidandbroken P. Flannery, Numerical Recipes in C, Oxford Press.
lines show the behaviour of f(t) for electromagnetic waves, [13] Quad: Y. Hida, X. S. Li and D. H. Bailey; arprec: Y.
as seen at the observation points r∗(b) = r∗(3m)+4m and Hida X.S.Li. H.Bailey and B. Thompson. Available at
r∗ =r∗(b)+100m, respectively. http:// users.bigpond.net.au/amiller.
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