Universality and scaling in gravitational collapse of a massless scalar field

Abstract

I summarize results from a numerical study of spherically symmetric collapse of a massless scalar Field. I consider families of solutions, 8[p], with the property that a critical parameter value, p, separates solutions containing black holes from those which do not. I present evidence in support of conjectures that (1) the strong-field evolution in the p~p * limit is universal and generates structure on arbitrarily small spatiotemporal scales and (2) the masses of black holes which form satisfy a power law MsH Ix ]p -p*~~, where p 0.37 is a universal exponent. ds = -n (r, t)dt + a (r, t)dr + r dA, where the radial coordinate r, which directly measures proper surface area, is a geometric quantity. A geometric time variable is provided by the proper time of a central observer,Introducing the auxiliary scalar field variables 4 = P' and II = aP/n, where an overdot denotes 0/Bt and a prime denotes 8/Br, a sufficient set of equations for the model is i=( -rr)',(4) (5)The above system is invariant under the trivial rescaling r -+ kr, t -+ kt, where k is an arbitrary positive constant -such transformations amount to a choice of units, and refiect the absence of a mass/length scale in the model. I will discuss the dynamics of the scalar Geld in terms of variables, X and Y, which are form-invariant under these transformations: 4'(r, t)-:+2~-4=Q2~-r rBQ (6) a a Br Y (r, t)-: v 2' -II = g2m-r r0$ (7) a nBt PACS numbers: 04.20.JbThe model problem [1 -5] of a single, massless scalar field, P, minimally coupled to the gravitational field, g», provides one of the simplest arenas in which to investigate the nature and consequences of nonlinearity in general relativity [6]. With large-scale computation and advanced numerical techniques, it is now possible to make detailed surveys of the phenomenology of the model. Here I summarize such a survey which reveals several intriguing nonlinear phenomena arising in the regime where black holes form, or "almost form. " The general time-dependent, spherically symmetric metric can be written In terms of these new variables, the total (conserved) mass, M, of the spacetime is dm dr = X+Ydr, where the mass aspect function m is related to the metric function a in (1) by a = (1 -2m/r) I solve Eqs. (3) -(5) using finite-difference techniquesand an adaptive mesh-refinement algorithm [4,7], where the basic scale of discretization, 6, is allowed to vary locally (in both space and time) in response to the development of solution features. This adaptivity has been instrumental in the discovery and study of the phenomena described here which can unfold on arbitrarily small spatiotemporal scales. The large dynamic range required to resolve the phenomena and the extreme sensitivity of the solutions to initial conditions complicate the task of ensuring that the numerics faithfully represent the underlying physics. I have carefully examined this issue and will discuss it in detail elsewhere; here I assert that there is very strong evidence that the observed phenomena are not numerical artifacts.Let 8 denot...

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