HOW BLACK HOLES GROW

Abstract

A summary of on how black holes grow in full, non-linear general relativity is presented. Specically , a notion of dynamical horizons is introduced and expressions of uxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy ux is positive. Change in the horizon area is related to these uxes. The ux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null innit y and provide generalizations of the rst and second laws of black hole mechanics. Black holes are perhaps the most fascinating manifestations of the curvature of space and time predicted by general relativity. Properties of isolated black holes in equilibrium have been well-understood for quite some time. However, in Nature, black holes are rarely in equilibrium. They grow by swallowing stars and galactic debris as well as electromagnetic and gravitational radiation. For such dynamical black holes, the only known major result in exact general relativity has been a celebrated area theorem, proved by Stephen Hawking in the early seventies: if matter satises the dominant energy condition, the area of the black hole event horizon can never decrease. This theorem has been extremely inuen tial because of its similarity with the second law of thermodynamics. However, it is a ‘qualitative’ result; it does not provide an explicit formula for the amount by which the area increases in any given physical situation. One might hope that the change in area is related, in a direct manner, to the ux of matter elds and gravitational radiation falling in to the black hole. Is this in fact the case? If so, the formula describing this dynamical evolution of the black hole would give us a ‘nite’ generalization of the rst law of black hole mechanics: The standard rst law, E = (= 8 G) a + J, relates the innitesimal change a in the black hole area due to the innitesimal inux E of energy and angular momentum J as the black hole makes a transition from one equilibrium state to a nearby one, while the exact evolution law would provide its ‘integral version’, relating equilibrium congurations which are far removed from one another. From a general, physical viewpoint, these expectations seem quite reasonable. Why, then, had this question remained unresolved for three decades? The reason is that when one starts thinking of possible strategies to carry out these generalizations, one immediately encounters severe diculties. To begin with, to carry out this program one would need a precise notion of the gravitational energy ux falling in to the black hole. Now, as is well known, in full