Closed Timelike Curves in General Relativity

Many solutions of Einstein's field equations contain closed timelike curves (CTC). Some of these solutions refer to ordinary materials in situations which might occur in the laboratory, or in astrophysics. It is argued that, in default of a reasonable interpretation of CTC, general relativity does not give a satisfactory account of all phenomena within its terms of reference. PACS number 0420 In general relativity a timelike curve in spacetime represents a possible path of a physical object or an observer. Normally such a curve will run from past to future, but in some spacetimes timelike curves can intersect themselves, giving a loop, or a closed timelike curve (CTC). CTCs suggest the possibility of time-travel with its well-known paradoxes. The first spacetime in which CTCs were noticed was that of Gödel. [1]. This represents a rotating universe without expansion, and requires a negative cosmological constant. As a model of physical reality it can therefore be dismissed because it is unlike the universe we live in. Another simple space-time containing CTCs is that of van Stockum [2] which represents a cylinder of rigidly rotating dust; however, the cylinder is of infinite length so it could not be realised in practice. Since these early discoveries other spacetimes containing CTCs have been found. Nearly all of these have been regarded as of merely theoretical interest because of some non-physical feature in their composition. 1 Recently, however, there have been published some solutions of Einstein's equations containing CTCs and representing physical situations which in principle could be reproduced in the laboratory, or might occur in astrophysics. One such solution represents two spinning particles of masses m 1 , m 2 and constant angular momenta h 1 , h 2 , their spins both parallel to their line of separation. The particles are fixed on the z-axis at z = ±b. They are not supposed to represent black holes: they could be copper spheres in a laboratory. 1 An exception is the Kerr-Newman solution, which is asymptotically flat and contains CTCs. However, these are unlikely to be realised in astrophysics or in the laboratory.