Frank J. Tipler

Singularities and Causality Violation

Abstract A number of important theorems in General Relativity have required a causality assumption; for example, the Geroch topology change theorem, and most of the Hawking-Penrose-Geroch singularity theorems. It is shown in this paper that the causality condition can be replaced by weaker causality conditions, and in some cases removed altogether. In particular, (a) it is shown that if the Einstein equations (and the weak energy condition) hold on the “topology-changing” space-time considered by Geroch, then topology change cannot occur. No causality assumption is needed in the proof. Furthermore, it is shown that if topology change occurs within a finite region, then this change of topology must be accompanied by singularities. (b) It is shown that causality violation cannot prevent the Hawking-Penrose-Geroch singularities unless the causality violation begins “at infinity”—a region which is free of matter and gravitational radiation—and this seems very unlikely.

Singularities in conformally flat spacetimes

Abstract It is shown that the energy density must diverge at least as fast as 1/t along certain null geodesics near a strong curvature singularity in any conformally flat spacetime.

Analysis of the generic singularity studies by Belinskii, Khalatnikov, and Lifschitz

Abstract We apply some modern mathematical methods of global analysis to a series of studies undertaken by Belinskii, Khalatnikov and Lifschitz (BKL) to elucidate the structure of space-time near a general cosmological singularity. A brief summary of BKLs large body of work on inhomogeneous cosmological models is given (their work on homogeneous models is not under discussion here). Various theorems are proven and analyses of a mathematical and physical nature are made to show that the constructions of BKL cannot be general and in some cases do not give Lorentz manifolds. We conclude that although the work of BKL has led to very significant advances in our understanding of the dynamics of homogeneous cosmological models, the local techniques they employ do not extend to give us reliable information about the global structure of generic space-times. A detailed discussion of stability, generality, function counting, linearization stability, physical singularities and fictitious singularities is given together with an outline of various physical considerations which might be useful in future studies of the structure of generic space-times.

The growth of curvature near a space-time singularity

The causal structure of space-time constrains the rate of growth of Ricci curvature near a singularity. Roughly speaking, Ro0 cannot grow faster than 1/t 2 near a singularity in any physically realistic space-time, and Roo must grow at least as fast as 1/t near a curvature singularity in a conformally flat spacetime. The following theorems make this statement precise. Theorem 1. Let 7(t) be a null geodesic generator of the boundary of a TIF. Then there is no affine parameter interval (s l , s2) of 7(s) satisfying

What is a black hole

Until recently, the concept of black hole has been rigorously defmed only in asymptotically tlat space-times: a black hole has been defined to be the set of events not in J-(~+). However, Penrose has proposed a different definition of a black hole: a black hole is the set of events that are not past endpoints of timelike curves of infinite length. Although the Penrose definition does not depend on asymptotic flatness, it assumes that every observer falling into a black hole always hits a singularity and that every observer not falling into a black hole never hits a singularity. Thus the Penrose definition cannot be used in closed universes. I propose a definition of black hole that I believe will work in any stably causal space-time. Definition. A black hole is the closure of the smallest future set that has the following properties.

Mach's principle in spatially homogeneous spacetimes☆

Abstract On the basis of Machs Principle we should like to conclude that the only singularity-free solution to the empty space Einstein equations is flat space. It is shown that the only singularity-free solution to the empty space Einstein equations which is spatially homogeneous and globally hyperbolic is in fact suitably identified Minkowski space.