Gravitational-wave bursts with memory: The Christodoulou effect

Kip S. Thorne

doi:10.1103/PhysRevD.45.520

Access by Xinjiang University

Gravitational-wave bursts with memory: The Christodoulou effect

Kip S. Thorne

Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125

PDF

Phys. Rev. D 45, 520 – Published 15 January, 1992

DOI: https://doi.org/10.1103/PhysRevD.45.520

Abstract

The ‘‘memory’’ of a gravitational-wave burst is the permanent relative displacement that it imposes on free test masses, or more precisely, the permanent change in the burst’s gravitational-wave field $h_{j k}^{TT}$ . This memory, in general, is equal to the change, from before the burst to afterward, in the transverse-traceless (TT) part of the ‘‘1/r, Coulomb-type’’ gravitational field generated by the four-momenta of the source’s various independent pieces. Christodoulou has recently identified a contribution to a burst’s memory that arises from nonlinearities in the vacuum Einstein field equation. This paper shows that the Christodoulou memory is precisely the TT part of the ‘‘1/r, Coulomb-type’’ gravitational field produced by the burst’s gravitons, and it therefore gets built up over the same length of time $τ_{bwm}$ as it takes for the source to emit the gravitons. The sensitivity of broad-band gravitational-wave detectors such as LIGO to the Christodoulou memory is analyzed and discussed.

References (14)

See, e.g., K.S. Thorne, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1987).
Ya.B. Zel'dovich and A.G. Polnarev, Astron. Zh. 51, 30 (1974) [Sov. Astron. \fB18\fR, 17 (1974)]. See also L. Smarr, Phys. Rev. D\fR \fB15\fR, 2069 (1977); R.J. Bontz and R.H. Price, Astrophys. J. \fB228\fR, 560 (1979); S.J. Kovacs and K.S. Thorne, ibid. \fB224\fR, 62 (1978).
V.B. Braginsky and L.P. Grishchuk, Zh. Eksp. Teor. Fiz. \fB89\fR, 744 (1985) [Sov. Phys. JETP \fB62\fR, 427 (1985)].
V.B. Braginsky and K.S. Thorne, Nature (London) \fB327\fR, 123 (1987).
Box 35.1 of C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
D. Christodoulou, Phys. Rev. Lett. \fB67\fR, 1486 (1991).
It has been called to my attention that this Christodoulou memory was discovered independently by Luc Blanchet and Thibault Damour (L. Blanchet, Habilitation Thesis, University of Paris, 1990, pp. 205–213, where is referred to as the ``bundle term'' in the ``hereditary'' part of the metric).
In his expression for the graviton-induced memory, Christodoulou uses dE/d Ω prime - (3/4 π ) $P_{j}$ $ξ^{j}$ prime - E (where $P_{j}$ and E are the total momentum and energy radiated) in place of d E / d Ω prime. (In his notation, he uses F - $F_{[1]}$ in place of F.) However, is not necessary to subtract off the (3/4 π ) $P_{j}$ $ξ^{j}$ prime + E because this quantity gives a vanishing contribution to the TT part of the integral (and correspondingly $F_{[1]}$ gives a vanishing contribution to the full integral in Christodoulou's formula).
R. Epstein, Astrophys. J. \fB223\fR, 1037 (1978); M.S. Turner, Nature (London) \fB274\fR, 565 (1978).
See, e.g., S.J. Kovacs and K.S. Thorne, Astrophys. J. \fB224\fR, 62 (1978), and earlier references cited therein.
For an overview of interferometric detectors see, e.g., Ref. [1] (where they are called ``beam detectors''). For a brief description of the projected capabilities of the Earth-based LIGO and space-based LAGOS, see, e.g., K.S. Thorne, in Recent Advances in General Relativity, edited by A. Janis and J. Porter (Birkhauser, Boston, in press). For overviews of the LIGO Project, see R. Vogt, in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, edited by T. Nakamura and H. Sato (World Scientific, Singapore, in press); and A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker (in preparation).
Equation (46b) of Ref. [1], multiplied by sqrt 2 to correct for an error in its derivation [in Eqs. (42) 2 π ft should have been 2 π int (df/dt)dt], and with (M f $)^{1 / 6}$ apeq 0.5.
Peter Bender (private communication) has pointed out the following possibly important exception: a binary with M << 3000 $M_{cirdot}$ , being studied by LAGOS. Such a binary will spend many years near the frequency $f_{opt}$ apeq 0.003 Hz during its precursor, inspiral stage, so in a reasonable integration time (less than one year), LAGOS will not be able to build up the precursor signal to anywhere near the level of Eq. (8) — and LAGOS might not have even been in operation at the time the precursor passed through $f_{opt}$ . As a result, the memory may be easier for LAGOS to detect than the precursor. However, for such a binary, if the memory is detectable by LAGOS, then the final burst should be detectable by the higher-frequency, Earth-based LIGO.
E.S. Phinney, Astrophys. J. Lett. 380, L17 (1991); R. Narayan, T. Piran and A. Shemi, ibid. 379, L17 (1991).

[c1] See, e.g., K.S. Thorne, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1987).

[c2] Ya.B. Zel'dovich and A.G. Polnarev, Astron. Zh. 51, 30 (1974) [Sov. Astron. \fB18\fR, 17 (1974)]. See also L. Smarr, Phys. Rev. D\fR \fB15\fR, 2069 (1977); R.J. Bontz and R.H. Price, Astrophys. J. \fB228\fR, 560 (1979); S.J. Kovacs and K.S. Thorne, ibid. \fB224\fR, 62 (1978).

[c3] V.B. Braginsky and L.P. Grishchuk, Zh. Eksp. Teor. Fiz. \fB89\fR, 744 (1985) [Sov. Phys. JETP \fB62\fR, 427 (1985)].

[c4] V.B. Braginsky and K.S. Thorne, Nature (London) \fB327\fR, 123 (1987).

[c5] Box 35.1 of C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

[c6] D. Christodoulou, Phys. Rev. Lett. \fB67\fR, 1486 (1991).

[c7] It has been called to my attention that this Christodoulou memory was discovered independently by Luc Blanchet and Thibault Damour (L. Blanchet, Habilitation Thesis, University of Paris, 1990, pp. 205–213, where is referred to as the ``bundle term'' in the ``hereditary'' part of the metric).

[c8] In his expression for the graviton-induced memory, Christodoulou uses dE/d Ω prime - (3/4 π ) $P_{j}$ $ξ^{j}$ prime - E (where $P_{j}$ and E are the total momentum and energy radiated) in place of d E / d Ω prime. (In his notation, he uses F - $F_{[1]}$ in place of F.) However, is not necessary to subtract off the (3/4 π ) $P_{j}$ $ξ^{j}$ prime + E because this quantity gives a vanishing contribution to the TT part of the integral (and correspondingly $F_{[1]}$ gives a vanishing contribution to the full integral in Christodoulou's formula).

[c9] R. Epstein, Astrophys. J. \fB223\fR, 1037 (1978); M.S. Turner, Nature (London) \fB274\fR, 565 (1978).

[c10] See, e.g., S.J. Kovacs and K.S. Thorne, Astrophys. J. \fB224\fR, 62 (1978), and earlier references cited therein.

[c11] For an overview of interferometric detectors see, e.g., Ref. [1] (where they are called ``beam detectors''). For a brief description of the projected capabilities of the Earth-based LIGO and space-based LAGOS, see, e.g., K.S. Thorne, in Recent Advances in General Relativity, edited by A. Janis and J. Porter (Birkhauser, Boston, in press). For overviews of the LIGO Project, see R. Vogt, in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, edited by T. Nakamura and H. Sato (World Scientific, Singapore, in press); and A. Abramovici, W. Althouse, R. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker (in preparation).

[c12] Equation (46b) of Ref. [1], multiplied by sqrt 2 to correct for an error in its derivation [in Eqs. (42) 2 π ft should have been 2 π int (df/dt)dt], and with (M f $)^{1 / 6}$ apeq 0.5.

[c13] Peter Bender (private communication) has pointed out the following possibly important exception: a binary with M << 3000 $M_{cirdot}$ , being studied by LAGOS. Such a binary will spend many years near the frequency $f_{opt}$ apeq 0.003 Hz during its precursor, inspiral stage, so in a reasonable integration time (less than one year), LAGOS will not be able to build up the precursor signal to anywhere near the level of Eq. (8) — and LAGOS might not have even been in operation at the time the precursor passed through $f_{opt}$ . As a result, the memory may be easier for LAGOS to detect than the precursor. However, for such a binary, if the memory is detectable by LAGOS, then the final burst should be detectable by the higher-frequency, Earth-based LIGO.

[c14] E.S. Phinney, Astrophys. J. Lett. 380, L17 (1991); R. Narayan, T. Piran and A. Shemi, ibid. 379, L17 (1991).

Physical Review D

Export citation

Export citation

Gravitational-wave bursts with memory: The Christodoulou effect

Abstract

References (14)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation