Volume 98B, number 5 PHYSICS LETTERS 22 January 1981

ON THE STRUCTURE OF GRAVITATIONAL WAVE BURSTS:

IMPLOSION WITH FINITE KINETIC ENERGY

Valeria FERRARI and Remo RUFFINI Istituto di Fisica G. MarconL Universitd di Roma, Rome, Italy

Received 30 October 1980

The structure of a gravitational wave burst emitted by a particle radially imploding into a Schwarzschild black hole with a finite kinetic energy at infinity is analyzed in the framework of the Zerilli-Regge-Wheeler formalism. The three compo- nents of the burst, (a) precursor, (b) main burst, (c) ringing tail, are analyzed and the results compared and contrasted with those obtained for a particle initially at rest at infinity. The ringing tail of the burst is studied in terms of the normal modes of a black hole.

The recent developments in gravitational wave detectors [1 ] point to the necessity o f formulating detailed theo- retical predictions on the structure of gravitational wave bursts emitted by astrophysical sources.

The gravitational radiation burst emitted by a particle, at rest at infinity, imploding into a Schwarzschild black hole has been studied ha refs. [ 2 - 4 ] . The total amount of radiation emitted and the spectral distribution o f the radiation have been given in ref. [3]. The structure o f the burst as a function o f the retarded time has been ana- lyzed in ref. [4] and three different regimes have been identified: a precursor, a main burst and a ringing tail. Al- though all these considerations are obtained for a very idealized process (the infaU of a test particle following a geodesic) there are reasons to believe [5,6] that the above three components of the burst still exist in a general process of gravitational collapse to a black hole, unless the collapsing matter is endowed with such a high degree of symmetry to avoid the emission of radiation (e.g. spherical symmetry). In ref. [7] it was found that the total amount of energy radiated by a particle radially falling into a Schwarzschild black hole greatly increases if the par- ticle is endowed with a finite kinetic energy at infinity. The corresponding spectral distribution of the radiation has also been given in ref. [7].

In this letter we analyze the structure of the burst as a function o f the retarded time. The aim is to prove that, also in this more general case of a particle starting at infinity with finite kinetic energy, the "three components" structure of the burst still persists. We analyze as well the qualitative and quantitative changes introduced in each of the three components by the enhancement o f the total energy found in ref. [7]. Zerilli [8] has shown that a test particle of mass m radially imploding into a Schwarzschild black hole o f mass M determines a perturbation huv in the background geometry g ° v which is fully determined by a Schr6dinger like equation of the form

d2Rl(r, co)/dr .2 + [aft - Vl(r)] Rl(r , co) = St(r, co). (1)

Here R l (r, co) is the Fourier transform of the function describing the radial dependence o f the perturbation in the expansion in tensor harmonics [2]. In eq. (1) and in the following we assume geometrical units with G = c = 1. The coordinate r* in eq. (1) is related to ~he usual Schwarzschild coordinate by

r* = r + 2 M I n ( r / 2 M - 1),

while

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company 381

Volume 98B, number 5 PHYSICS LETTERS 22 January 1981

V/(r) = (1 - 2M/r)[2h2(h + l)r3 + 6A2Mr2 + 18hM2r + 18M3] /r3(hr + 3M),

with X = i<Z - l)(Z + 2), is the effective curvature potential and Sl(r, o) is the Fourier transformed source term generated by the incoming particle expressed in tensor harmonics. If we consider a particle with finite kinetic ener- gy at infinity the source term is given by

where

T(r) --& I($ - l)? t 2r] l/2 - 27 27 -= i - 3

M 0

0 (yi _ 1plnI KY: - lkl ‘I2 + [(T; - lb + 21 1/2I

_ 2 ln 2 (24 - 1)r + 2 - 27, [(y; - l)? t 24 l’2

r-2

is the proper time of the test particle along the geodesic and y. = (1 - u~/c?-~‘~ where u, is the speed of the test particle at infinity.

Eq. (1) has been numerically integrated imposing the asymptotic boundary conditions

RJr, w) = A?*(o) eiwr*, as I* + too; R&r, 0) = Af”(w) eViwr*, as r* + -00,

namely purely outgoing waves at infinity and ingoing waves at the black hole surface. The amplitude of the waves Aout I (a) and At”(w) are given by the Green’s function technique

AFut(w) a jmui(r*, w)S#*, w) dr*, A;“(w) 0: jmu,(r*, o)SJr*, w) dr*, -cc -cc

where u1 (r* , o) and ul(r*, o) are the solutions of the homogeneous equation (1) corresponding, respectively, to a purely outgoing wave at r* = tm and a purely ingoing wave at r* = -=.

Applying further a Fourier transformation we obtain the expression of the radial function as a function of the radial and time coordinate

Fig. 1. The radial functions of the gravitational perturbation for I= 2 and for yo = 1.0 and yo = 1.5 are given as a function of the retarded time (t-r*)/M If ye > 1 the initial and final value of the perturbation are different.

Fig. 2. The second time derivative i7’a of the radial function of the gravitational perturbation for I = 2 is plotted as a function of the retarded time. This quantity is proportional to the tide producing components of the Riemann tensor. If we assume the z axis pointed along the line of propagation of the wave and the principal axis of polarizationSpointed in the x and y directions we have R&,, = -Rtxxo = 3Rl(r*, t) sin*8/2r.

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Volume 98B, number 5 PHYSICS LETTERS 22 January 1981

1 * e_iCotdco. R?utC, t)= f ,

Both for 70 = 1 and 70 > 1 the main contribution to the gravitational radiation burst comes from the quadrupole term (l = 2) and we, therefore, report in the following uniquely the l = 2 contributions.

The relative importance o f the higher multipoles, however, increases with 70 > 1. The complete treatment in- cluding the terms with l > 2 is given in ref. [6].

In fig. 1 the results of the numerical integration o f the radial function for 3`0 = 1 and )0 = 1.5 are given as a ]'unction of the retarded time (t-r*)/M. If 3`0 > 1 the initial value of the perturbation differs from the final value and this in turn leads to a possible characteristic "signature" in the gravitational wave signal [6]. In fig. 2 the sec- ond time derivative of the radial function, proportional to the tide producing components of the gravitational field, is plotted versus the retarded time, again for 3'0 = 1.5. In both cases we can still isolate three main components in the burst structure [4] :

(a) A precursor (_oo < (t-r*)/M < - 1 0 ) emitted when the test particle is far from the black hole (the Schwarzschild radial coordinate o f the particle varies between 7.8 ~< rim < +0%

(b) A main burst ( - 1 0 < (t-r*)/M < +5) emitted when the test particle approaches the black hole horizon (4.5 <~r/M<~ 7.8).

(c) A ringing tail (5 < (t-r*)/M <: +~) emitted when the particle coalesces into the black hole (2 + 2 × 10 -7 ~ r / M ~ 4.5).

Finally the analysis o f the ringing tail both for 3'0 = 1 and 3'o = 1.5 has been made using the gravitational nor- mal modes analysis first introduced by Chandrasekar [8] ~:1

The best fit has been obtained using a linear combination o f the two principal normal modes varying both their amplitude and their phase. The results for 3`0 = 1 and 70 = 1.5 are given in table 1 and fig. 3.

The normal modes excited are the same both for 70 = 1 and 3`0 = 1.5 although their amplitude and phase change significantly.

:~ 1 An earlier analysis pointing out the relevance of the Chandrasekhar normal modes to the structure of the gravitational wave signal, found in ref. [4], is given by Thorne [9].

!,, , m ~ 0.4L

, A C Z - x ?'~:to

' 5 10 I~ 20 \ \ 25 30 ~ 40

/ " - -

-17.3

410 P I-r*

Fig. 3. The "ringing tail" of the gravitational wave signal corresponding to a particle with 70 = 1 and one with ~'o = 1.5 are plotted as a function of the retarded time. The doubled line corresponds to the best fit obtained using a linear combination of two princip- al normal modes of the Schwarzschild black hole.

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Volume 98B, number 5

Table 1 Parameters of the ringing tail's fit.

PHYSICS LETTERS 22 January 1981

°91,2 A1,2 tPl,2

"to = 1 70 = 1.5 70 = 1 "to = 1.5

0.37367 + i0.08896 0.77 1.26 0.5166 1.9271 0.34844 + i0.27469 2.00 1.83 0.2373 3.4877

It was purported by Price [10] that the late stages of gravitational collapse would be determined by a perturba- t ion decreasing with time as

t -(21÷2), (2)

l being the order o f the multipole. From the previous fit of the ringing tail we see that the agreement of the ob- tained results with a combinat ion of normal modes is excellent and no contributions of the kind (2) are found.

Since the amplitude of the normal modes decreases exponential ly we have to infer that the above component , if it at all exists, has a very small amplitude with respect to the other components of the burst.

In conclusion the existence of a finite kinetic energy in the imploding matter greatly increases the total amount of energy radiated and modifies both qualitatively and quantitatively the structure o f the burst, as shown in the pre- vious figures. The three components structure of the burst is strongly confirmed, although differences in amplitude and phase in the three components are encountered.

It is a pleasure to acknowledge discussions on all the aspects of this problem with Professor E. Amaldi.

References

[1] E. Amaldi and G. PizzeUa, Relativity, quanta and cosmology, ed. F. de Finis, Vol. 1 (Johnson Reprint Corporation, 1979) pp. 9-136.

[2] F. Zerilli, Phys. Rev. D2 (1970) 2141. [3] M. Davis, R. Ruffini and R. Price, Phys. Rev. Lett. 27 (1971) 1466. [4] M. Davis, R. Ruffini and J. Tiommo, Phys.Rev. D5 (1972) 2932. [5 ] R. Ruffini, Physics and astrophysics of neutron stars and black holes, Rendiconti della Scuola Internazionale di Fisica E.

Fermi (Luglio, 1975) (Societ~ Italiana di Fisica, 1978) pp. 287-342. [6] E. Amaldi, V. Ferrari, R. Ruffini and M. Sasaki, to be published. [7] R. Ruffini, Phys. Rev. 7 (1973) 972. [8] S. Chandrasekhar, Proc. Roy. Soc. London A344 (1975) 441. [9] K.S. Thorne, in: Theoretical principles in astrophysics and relativity, eds. R. Lebovitz, W. Reid and O. Vandervoort (Chicago

U.P., 1978) pp. 141-216. [10] R. Price, Phys. Rev. D5 (1972) 2419.

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