arXiv subjects

Richard H. Price

Publications and source records attributed to Richard H. Price.

44 records · Page 3Linked to original sources

arXiv1997

Understanding initial data for black hole collisions

Numerical relativity, applied to collisions of black holes, starts with initial data for black holes already in each other's strong field. The initial hypersurface data typically used for computation is based on mathematical simplifying prescriptions, such as conformal flatness of the 3-geometry and longitudinality of the extrinsic curvature. In the case of head on collisions of equal mass holes, there is evidence that such prescriptions work reasonably well, but it is not clear why, or whether this success is more generally valid. Here we study these questions by considering the ``particle limit'' for head on collisions of nonspinning holes. Einstein's equations are linearized in the mass of the small hole, and described by a single gauge invariant spacetime function psi, for each multipole. The resulting equations have been solved by numerical evolution for collisions starting from various initial separations, and the evolution is studied on a sequence of hypersurfaces. In particular, we extract hypersurface data, that is psi and its time derivative, on surfaces of constant background Schwarzschild time. These evolved data can then be compared with ``prescribed'' data, evolved data can be replaced by prescribed data on any hypersurface, and evolved further forward in time, a gauge invariant measure of deviation from conformal flatness can be evaluated, etc. The main findings of this study are: (i) For holes of unequal mass the use of prescribed data on late hypersurfaces is not successful. (ii) The failure is likely due to the inability of the prescribed data to represent the near field of the smaller hole. (iii) The discrepancy in the extrinsic curvature is more important than in the 3-geometry. (iv) The use of the more general conformally flat longitudinal data does not notably improve this picture.

gr-qc↗

arXiv1996

Head-on collisions of black holes: the particle limit

We compute gravitational radiation waveforms, spectra and energies for a point particle of mass $m_0$ falling from rest at radius $r_0$ into a Schwarzschild hole of mass $M$. This radiation is found to lowest order in $(m_0/M)$ with the use of a Laplace transform. In contrast with numerical relativity results for head-on collisions of equal-mass holes, the radiated energy is found not to be a monotonically increasing function of initial separation; there is a local radiated-energy maximum at $r_0\approx4.5M$. The present results, along with results for infall from infinity, provide a complete catalog of waveforms and spectra for particle infall. We give a representative sample from that catalog and an interesting observation: Unlike the simple spectra for other head-on collisions (either of particle and hole, or of equal mass holes) the spectra for $\infty>r_0>\sim5M$ show a series of evenly spaced bumps. A simple explanation is given for this. Lastly, our energy vs. $r_0$ results are compared with approximation methods used elsewhere, for small and for large initial separation.

gr-qc↗

arXiv1996

Analytic approximations to the spacetime of a critical gravitational collapse

We present analytic expressions that approximate the behavior of the spacetime of a collapsing spherically symmetric scalar field in the critical regime first discovered by Choptuik. We find that the critical region of spacetime can usefully be divided into a ``quiescent'' region and an ``oscillatory'' region, and a moving ``transition edge'' that separates the two regions. We find that in each region the critical solution can be well approximated by a flat spacetime scalar field solution. A qualitative nonlinear matching of the solutions across the edge yields the right order of magnitude for the oscillations of the discretely self-similar critical solution found by Choptuik.

gr-qc↗

arXiv1995

Head-on collision of two black holes: comparison of different approaches

A benchmark problem for numerical relativity has been the head-on collision of two black holes starting from the ``Misner initial data,'' a closed form momentarily stationary solution to the constraint equations with an adjustable closeness parameter $μ_0$. We show here how an eclectic mixture of approximation methods can provide both an efficient means of determining the time development of the initial data and a good understanding of the physics of the problem. When the Misner data is chosen to correspond to holes initially very close together, a common horizon surrounds both holes and the geometry exterior to the horizon can be treated as a non-spherical perturbation of a single Schwarzschild hole. When the holes are initially well separated the problem can be treated with a different approximation scheme, ``the particle-membrane method.'' For all initial separations, numerical relativity is in principle applicable, but is costly and of uncertain accuracy. We present here a comparison of the different approaches. We compare waveforms, for $\ell=2$ and $\ell=4$ radiation, for different values of $μ_0$, from the three different approaches to the problem.

gr-qc↗

arXiv1994

Embedding initial data for black hole collisions

We discuss isometric embedding diagrams for the visualization of initial data for the problem of the head-on collision of two black holes. The problem of constructing the embedding diagrams is explicitly presented for the best studied initial data, the Misner geometry. We present a partial solution of the embedding diagrams and discuss issues related to completing the solution.

gr-qc↗

arXiv1994

Colliding Black Holes: The Close Limit

The problem of the mutual attraction and joining of two black holes is of importance as both a source of gravitational waves and as a testbed of numerical relativity. If the holes start out close enough that they are initially surrounded by a common horizon, the problem can be viewed as a perturbation of a single black hole. We take initial data due to Misner for close black holes, apply perturbation theory and evolve the data with the Zerilli equation. The computed gravitational radiation agrees with and extends the results of full numerical computations.

gr-qc↗

arXiv1992

The Escape of Gravitational Radiation from the Field of Massive Bodies

We consider a compact source of gravitational waves of frequency $ω$, in or near a massive spherically symmetric distribution of matter or a black hole. Recent calculations have led to apparently contradictory results for the influence of the massive body on the propagation of the waves. We show here that the results are in fact consistent and in agreement with the ``standard'' viewpoint in which the high frequency compact source produces the radiation as if in a flat background, and the background curvature affects the propagation of these waves.

astro-ph↗

arXiv1992

The Emergence of Radiation from Gravitational Potential Wells: The Absence of $ωM$ Effects

We consider a source of gravitational waves of frequency $ω$, located near the center of a massive galaxy of mass $M$ and radius $R$, with $ω\gg R^{-1}$. In the case of a perfect fluid galaxy, and of odd-parity waves, there is no direct matter-wave interaction and the propagation of the waves is affected by the galaxy only through the curvature of the spacetime background through which the waves propagate. We find that, in addition to the expected redshift of the radiation emerging from the galaxy, there is a small amount of backscatter, of order $M/ω^2R^3$. We show that there is no suppression of radiative power by the factor $1+ω^2M^2/4$ as has been recently predicted by Kundu. The origin of Kundu's suppression lies in the interpretation of a term in the expansion of the exterior field of the galaxy in inverse powers of radius. It is shown why that term is not related to the source strength or to the strength of the emerging radiation.

hep-th↗