Gravitational radiation in black-hole collisions at the speed of light. III. Results and conclusions

Abstract

This paper summarizes results following from the two preceding papers, I and 11, on the gravitational radiation emitted in the head-on collision of two black holes, each with energy p, at or near the speed of light. The radiation (in the speed-of-light case) near the forward and backward directions 8=0, a, where 8 is the angle from the symmetry axis in the center-of-mass frame, is given by the series A co( ?, 0 )=2,7=0a2n ( r^/p )sin2"8 for the news function co of retarded time ? and angle 8; successive terms can in principle be found from a perturbation treatment. Here the form of a,(r^/p) is presented. Knowledge of a, allows the new mass-loss formula of paper I to be applied, giving a calculation of the mass of the (assumed) final Schwarzschild black hole. Since the "final mass" resulting from the calculation exceeds 2p, the assumptions of the new mass-loss formula must not all hold. The most likely explanation is that there is a "second burst" of radiation present in the space-time, centered for small angles 6 on retarded times roughly 18p ln6( later than the "first burst" described above. A more realistic crude estimate of the energy emitted in gravitational waves is given by the Bondi expression, taking only the first two terms a . and a2 in co; this gives an efficiency of 16.4% for gravitational wave generation.PACS number(s): 04.30. + x, 97.60.Lf

Classical black hole production in high-energy collisions

“…In particular, understanding the case of non-zero impact parameter is crucial to improving the estimate (1). We will use the methods of Penrose [12] and D'Eath and Payne [13,14,15], where each incoming particle is modelled as a point particle accompanied by a plane-fronted gravitational shock wave, this wave being the Lorentz-contracted longitudinal gravitational field of the particle. At the instant of collision the two shock waves pass through one another, and nonlinearly interact by shearing and focusing.…”

“…At the instant of collision the two shock waves pass through one another, and nonlinearly interact by shearing and focusing. Penrose [12] and D'Eath and Payne [13,14,15] studied the case of zero impact parameter b, and by finding a closed trapped surface, derived a rigorous lower bound of M > s/2 and improved estimate M ≈ .84 √ s for the mass of the resulting black hole. This paper extends this analysis to b = 0.…”

“…Penrose [12] and D'Eath and Payne [13,14,15] (for a comprehensive treatment see [16]) have studied the problem of a classical collision with zero impact parameter and shown that a closed trapped surface forms, [11] argues that such collisions will not form black holes at non-zero impact parameter. Finally, collisions of cosmic rays with our atmosphere have energy reach beyond that of LHC, and reasonable conjectures about neutrino fluxes suggest the possibility that their black hole products might be seen with present or future cosmic ray observatories [17,18,19,20][9] or absence of their observation could place improved 1 For a review, see [9].…”

How black holes form in high energy collisions

“…We will describe a very simple and intuitively clear physical picture whereby a black hole forms in a classical capture process. This is why the estimate of black hole formation cross section by the classical horizon area is the correct thing to do, as argued by [7][8][9][10][11], and why the nonlinear corrections should not change it qualitatively. To see this, it is enough to use the original, simple escape velocities argument, that led John Michell [15] in 1784 and Pierre Simon, Marquis de Laplace [16] in 1796 to deduce the existence of black holes, some 130 years before the inception of General Relativity.…”

“…As a result, the analyses of black hole formation processes [2][3][4][5][6] estimate the black hole production cross section by the horizon area of a black hole whose horizon radius r h is set by the center-of-mass collision energy √ s. When the impact parameter b is smaller than r h it is assumed that the efficiency of formation of a black hole is close to 100%. The strongest support for this comes from examining the gravitational shock wave fields of zero rest mass particles [7][8][9][10][11], and seeing when a trapped surface forms as the shock waves cross. These analyses, in particular the work of Eardley and Giddings [11] which covers off-center collisions with b > 0, find that trapped surfaces form when b < ∼ r h , and have the area scaling as the horizon area ∼ r 2 h ∼ G 2 N s, where r h is the horizon radius and G N is Newton's constant.…”