Demetrios Christodoulou

Violation of cosmic censorship in the gravitational collapse of a dust cloud

The behaviour of the outgoing light rays in the gravitational collapse of an inhomogeneous spherically symmetric dust cloud is analyzed. It is shown that, for an open subset of initial density distributions, the first singular event, which occurs at the center of symmetry, is the vertex of an infinity of future null geodesic cones which intersect future null infinity. The frequency of the corresponding light rays is infinitely redshifted.

The problem of a self-gravitating scalar field

In this paper we begin the study of the global initial value problem for Einsteins equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.

The boost problem in general relativity

We show that any asymptotically flat initial data for the Einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface. Furthermore, the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development. We also extend former results on global solutions of the constraint equations. By virtue of this extension, the constraint and evolution parts of the problem fit together exactly. Several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems. This paper contains in addition a systematic exposition of the functional spaces employed.

A mathematical theory of gravitational collapse

We study the asymptotic behaviour, as the retarded timeu tends to infinity, of the solutions of Einsteins equations in the spherically symmetric case with a massless scalar field as the material model. We prove that when the final Bondi massM1 is different from zero, asu → ∞, a black hole forms of massM1 surrounded by vacuum. We find the rate of decay of the metric functions and the behaviour of the scalar field on the horizon.

ON THE GLOBAL INITIAL VALUE PROBLEM AND THE ISSUE OF SINGULARITIES

In the first part of the paper we discuss what is known at present about the global initial value problem for the vacuum Einstein equations with general asymptotically flat initial data. We then give precise formulations of cosmic censorship conjectures. We also point out analogies with fluid dynamics and discuss possibilities suggested by these analogies. In the second part I discuss my work on the spherically symmetric Einstein equations with a real massless scalar field as the material model. I give an outline of the approach which has led to the proof of the conjectures in this context.

Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large

In this paper we study the global initial value problem for the spherically symmetric Einstein-scalar field equations in the large. We introduce the concept of a generalized solution of our problem, and, taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, without any restriction on the size of the initial data, the global, in retarded time, existence of generalized solutions.

The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations

In a previous paper we proved the global existence of generalized solutions of the spherically symmetric Einstein-scalar field equations in the large. In this paper we study the regularity properties of the spacetime and the scalar field corresponding to a generalized solution. We also prove a uniqueness theorem which shows that a generalized solution is an extension of a classical solution.

The Euler Equations of Compressible Fluid Flow

This article is in celebration of the 300th anniversary of the birth of one of the greatest mathematicians and physicists in history, Leonhard Euler. The article is directly concerned with Euler’s work in fluid mechanics, although his work in the calculus of variations and in partial differential equations in general have been instrumental in the developments to be outlined here. Euler did have predecessors in the field of fluid mechanics, who had conceived some of the basic concepts. His immediate predecessor in this regard was his friend D. Bernoulli, whose 1738 work [Be] is likely to have had a great influence on him. However it was Euler who first formulated the general equations describing the motion of a perfect fluid. The general compressible Euler equations first appeared in published form in [Eu2], the second of three Euler articles on fluid mechanics which appeared in the same 1757 volume of the Mémoires de l’Academie des Sciences de Berlin. The third of these articles, [Eu3], is a continuation of the second, while the first, [Eu1], establishes the general validity of the basic concepts and formulates the equations in the static case. However, it seems that the article [Eu4], which formulates the equations of motion in the incompressible case and which was published only in 1761, was actually the first to be composed, as at least a preliminary version of it was presented to the Berlin Academy in 1752. Thus Euler’s fluid equations were among the first partial differential equations to be written down, preceded, it seems, only by D’Alembert’s 1749 formulation [DA] of the one-dimensional wave equation describing the motion of a vibrating string in the linear approximation. Euler was not content to confine himself to the formulation of the basic laws of fluid mechanics, but he proceeded to investigate and explain on the basis of these laws some of the basic observed phenomena. Thus in [Eu5] he made the first, albeit incomplete, study of convection, a phenomenon which depends on compressibility as well as on temperature variation in a gravitational potential. In [Eu7] he studied incompressible flows in pipes in the linear approximation, while in [Eu8] he studied compressible flows in the linear approximation, treating the generation and propagation of sound waves. The contrast to D’Alembert’s equation however could not be greater, for we are still, after the lapse of two and a half centuries, far from having achieved an adequate understanding of the observed phenomena which are supposed to lie within the domain of validity of Euler’s fluid equations. The phenomena displayed in the interior of a fluid fall into two broad classes: the phenomena of sound, the linear theory of which is acoustics, and the phenomena

Convergent and asymptotic iteration methods in General Relativity

We show that the fast motion iteration method in General Relativity gives an asymptotic approximation to exact solutions of the reduced Einstein equations. Rigorous estimates of the error commited at each step of the iteration are derived.

Self-gravitating relativistic fluids : The formation of a free phase boundary in the phase transition from soft to hard

In the 1990s Christodoulou introduced an idealized fluid model intended to capture some of the features of the gravitational collapse of a massive star to form a neutron star or a black hole. This was the two-phase model introduced in ‘Self-gravitating relativistic fluids: a two phase model’ (Demeterios, Arch Ration Mech Anal 130:343–400, 1995). The present work deals with the formation of a free phase boundary in the phase transition from hard to soft in this model. In this case the phase boundary has corners at the null points; the points which separate the timelike and spacelike components of the interface between the two phases. We prove the existence and uniqueness of a free phase boundary. Also the local form of the shock near the null point is established.