summary of the afternoon session i—thursday, april 25, 1985

2
Summary of the afternoon session I-Thursday, April 25,1985 A. DAS Sitnon Fraser University, Burt~aby, B.C., Canada V5A IS6 Can. J. Phys. 64. 131 (1986) The first speaker of this session was Professor F. I. Cooper- stock, University of Victoria. He spoke on The correct formula for graiitational-radiation loss and the axially symmetric two- body problem. He first gave the history of the problem and Einstein's prediction of radiation loss. Then he derived the correct quadrupole formula by using linearized field equations of Einstein. Then he applied the formula to two axially symmetric bodies falling towards each other. After the talk, several questions were asked. Will: Why is the size of the region related to the number of orbits? Cooperstock: Integration is performed in the region where orbits are situated. Glass: How do you establish that you are not adding radiation to the system when you cut the strut? Cooperstock: At that stage two bodies do not have free fall but approach each other in a different way. The masses can be viewed as a perfect fluid enclosed in a thin membrane that starts decaying as the strut starts breaking. Professor Israel asked about the derivation of the formula treating a strong gravitational field on a flat background. Professor Cooperstock mentioned the difficulties of treating strong fields by perturbation techniques. The second speaker of the afternoon was Professor C. M. Will, Washington University, St. Louis, and he spoke on the Approximation methods for grailitational-radiation reaction. He said that there is no exact solution of Einstein's equations that describes a dynamical system with realistic material sources (like binary pulsars) and the emission of gravitational radiation to "infinity;" therefore, a variety of approximation methods must be brought to bear on the problem. Since the quadrupole formula describes gravitational-radiation flux and reaction for systems with slow motions and weak dynamical gravitational fields, the so called "quadrupole formalism" comprises a variety of approximation methods: the post-Newtonian iteration schemes of Chandrasekhar, Anderson, and Carmelli; the asymptotic matching techniques of Burke, Thorne, and Damour; the statis- tical initial-value approach of Futamase and Schutz; and the characteristic initial-value approach of Isaacson and Winicour. All these approximation methods contribute to the quadrupole formalism. He discussed the relative merits of these various approaches. Then he mentioned the statistical initial-value approach he has utilized to compute electromagnetic reaction to a freely falling charge. Finally, he discussed application of the Wentzell-Kramers-Brillouin approximation to the black- hole perturbation theory, which involves the "master equation" The third speaker was Professor E. Pechlaner, Simon Fraser University. He spoke on Self-similar spinning rods. Mathe- matically conformal killing motion or homothetic motion is given by the existence of a vector field 5' in a differentiable manifold, which satisfies Lggij = Agii There is a conjecture that the end points of a spinning rod in general relativity might describe world lines that are integral curves of conformal killing motion. Such spinning rods are called self-similar. Exact solu- tions of Einstein's equations due to self-similar spinning rods are difficult to discover. Professor Pechlaner indicated that using the approximation method of Das-Florides-Synge (DFS), he has managed to find such an approximate solution. The first- order mass turns out to be zero, as also shown by Madore. In the higher approximation, positive mass is obtained. His model corresponds to a spinning rod whose mass and length are proportional to time t and the angular velocity is propor- tional to t - I. The following questions were asked. Cooperstock: Are you aware of Synge's later (1970) approxi- mation method which is different from the DFS method? Pechlaner: Yes, but I do not understand it fully. Henriksen: What are the physical constants in the solution? Pechlaner: Besides c and G, there is k, a measure of the mass density; L, the speed with which the rod shrinks; and e, the angular velocity of the rod multiplied by times t. The next speaker was Professor L. A. Nelson, Massachu- setts Institute of Technology, who spoke On the evolution of ultracompact binary systems driven by gravitational-radiation losses. Many of the important characteristics of low-mass X-ray binaries and cataclysmic variables are explained by models in which orbital-angular-momentum losses due to gravitational- radiation drive mass transfer from a low-mass hydrogen-rich secondary to a collapsed degenerate dwarf. However, the two highly compact X-ray binaries 4U- 1626-67 and 4U- 19 16-05 and cataclysmic variable G61-29 have orbital periods (41-50 min) that are too short to be explained by models with hydrogen-rich secondaries. One resolution of the difficulties may be found by invoking binary stellar systems whose sec- ondary components are severely hydrogen depleted. Evolu- tionary models in which the only loss of orbital angular mo- mentum is that due to gravitational radiation (as described by Einstein's quadrupole formula) are constructed that are in good agreement with observations of the three ultracompact binaries. The relevant formula for the rate of angular-momentum loss for a circular binary is The next speaker was Dr. M. Fichett, Canadian Institute for Theoretical Astrophysics (CITA). He spoke on Gravita- tional radiation from perturbed wide binaries. For gravitational radiation from a binary system, people usually consider a cir- cular system. However, there exists the possibility that one of the members of a binary system will collapse to a black hole. Can. J. Phys. Downloaded from www.nrcresearchpress.com by McMaster University on 11/14/14 For personal use only.

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Page 1: Summary of the afternoon session I—Thursday, April 25, 1985

Summary of the afternoon session I-Thursday, April 25,1985

A. DAS Sitnon Fraser University, Burt~aby, B.C., Canada V5A IS6

Can. J . Phys. 64. 131 (1986)

The first speaker of this session was Professor F. I. Cooper- stock, University of Victoria. He spoke on The correct formula for graiitational-radiation loss and the axially symmetric two- body problem. He first gave the history of the problem and Einstein's prediction of radiation loss. Then he derived the correct quadrupole formula

by using linearized field equations of Einstein. Then he applied the formula to two axially symmetric bodies falling towards each other. After the talk, several questions were asked.

Will: Why is the size of the region related to the number of orbits?

Cooperstock: Integration is performed in the region where orbits are situated.

Glass: How do you establish that you are not adding radiation to the system when you cut the strut?

Cooperstock: At that stage two bodies do not have free fall but approach each other in a different way. The masses can be viewed as a perfect fluid enclosed in a thin membrane that starts decaying as the strut starts breaking.

Professor Israel asked about the derivation of the formula treating a strong gravitational field on a flat background. Professor Cooperstock mentioned the difficulties of treating strong fields by perturbation techniques.

The second speaker of the afternoon was Professor C. M. Will, Washington University, St. Louis, and he spoke on the Approximation methods for grailitational-radiation reaction. He said that there is no exact solution of Einstein's equations that describes a dynamical system with realistic material sources (like binary pulsars) and the emission of gravitational radiation to "infinity;" therefore, a variety of approximation methods must be brought to bear on the problem. Since the quadrupole formula describes gravitational-radiation flux and reaction for systems with slow motions and weak dynamical gravitational fields, the so called "quadrupole formalism" comprises a variety of approximation methods: the post-Newtonian iteration schemes of Chandrasekhar, Anderson, and Carmelli; the asymptotic matching techniques of Burke, Thorne, and Damour; the statis- tical initial-value approach of Futamase and Schutz; and the characteristic initial-value approach of Isaacson and Winicour. All these approximation methods contribute to the quadrupole formalism. He discussed the relative merits of these various approaches. Then he mentioned the statistical initial-value approach he has utilized to compute electromagnetic reaction to a freely falling charge. Finally, he discussed application of the Wentzell-Kramers-Brillouin approximation to the black- hole perturbation theory, which involves the "master equation"

The third speaker was Professor E. Pechlaner, Simon Fraser

University. He spoke on Self-similar spinning rods. Mathe- matically conformal killing motion or homothetic motion is given by the existence of a vector field 5' in a differentiable manifold, which satisfies Lggij = Agii There is a conjecture that the end points of a spinning rod in general relativity might describe world lines that are integral curves of conformal killing motion. Such spinning rods are called self-similar. Exact solu- tions of Einstein's equations due to self-similar spinning rods are difficult to discover. Professor Pechlaner indicated that using the approximation method of Das-Florides-Synge (DFS), he has managed to find such an approximate solution. The first- order mass turns out to be zero, as also shown by Madore. In the higher approximation, positive mass is obtained. His model corresponds to a spinning rod whose mass and length are proportional to time t and the angular velocity is propor- tional to t - I .

The following questions were asked.

Cooperstock: Are you aware of Synge's later (1970) approxi- mation method which is different from the DFS method?

Pechlaner: Yes, but I do not understand it fully.

Henriksen: What are the physical constants in the solution?

Pechlaner: Besides c and G , there is k , a measure of the mass density; L , the speed with which the rod shrinks; and e , the angular velocity of the rod multiplied by times t .

The next speaker was Professor L. A. Nelson, Massachu- setts Institute of Technology, who spoke On the evolution of ultracompact binary systems driven by gravitational-radiation losses. Many of the important characteristics of low-mass X-ray binaries and cataclysmic variables are explained by models in which orbital-angular-momentum losses due to gravitational- radiation drive mass transfer from a low-mass hydrogen-rich secondary to a collapsed degenerate dwarf. However, the two highly compact X-ray binaries 4U- 1626-67 and 4U- 19 16-05 and cataclysmic variable G61-29 have orbital periods (41-50 min) that are too short to be explained by models with hydrogen-rich secondaries. One resolution of the difficulties may be found by invoking binary stellar systems whose sec- ondary components are severely hydrogen depleted. Evolu- tionary models in which the only loss of orbital angular mo- mentum is that due to gravitational radiation (as described by Einstein's quadrupole formula) are constructed that are in good agreement with observations of the three ultracompact binaries. The relevant formula for the rate of angular-momentum loss for a circular binary is

The next speaker was Dr. M. Fichett, Canadian Institute for Theoretical Astrophysics (CITA). He spoke on Gravita- tional radiation from perturbed wide binaries. For gravitational radiation from a binary system, people usually consider a cir- cular system. However, there exists the possibility that one of the members of a binary system will collapse to a black hole.

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Page 2: Summary of the afternoon session I—Thursday, April 25, 1985

132 CAN. I . PHYS. VOL. 64, 1986

In such a process, it will recoil and give rise to a binary system the decay time of a binary orbit is with a highly eccentric orbit. These systems can compete with the closed systems, both in terms of their gravitational-radiation 768

Tdec(a, e) = - Tdec(a)(l - e2)7/2 luminosity and their duration of emission. The characteristics 425 of the gravitational radiation emitted are, however, quite dif- where a is the radius of a circular orbit and is the eccentricity ferent. This difference will likely be significant in the study of of a wide binary. very massive object binary systems. An interesting formula for

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For

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