Causal Anomalies in Kaluza–klein Gravity Theories

Abstract

Causal anomalies in two Kaluza-Klein gravity theories are examined, particularly as to whether these theories permit solutions in which the causality principle is violated. It is found that similarly to general relativity the field equations of the space-time-mass Kaluza-Klein (STM-KK) gravity theory do not exclude violation of causality of Gödel type, whereas the induced matter Kaluza-Klein (IM-KK) gravity rules out noncausal Gödel-type models. The induced matter version of general relativity is shown to be an efficient therapy for causal anomalies that occurs in a wide class of noncausal geometries. Perfect fluid and dust Gödel-type solutions of the STM-KK field equations are studied. It is shown that every Gödel-type perfect fluid solution is isometric to the unique dust solution of the STM-KK field equations. The question as to whether 5-D Gödel-type non-causal geometries induce any physically acceptable 4-D energy-momentum tensor is also addressed.Kaluza-Klein-type theories in five or more dimensions have been of interest in a few contexts. In the framework of gauge theories they have been used in the quest for unification of the fundamental interactions in physics. The idea that the various interactions in nature might be unified by enlarging the dimensionality of the space-time has a long history that goes back to the works of Kaluza and Klein [1] - [2]. They showed how one could unify Einstein's theory of gravitation and Maxwell's theory of electromagnetism in a five-dimensional framework. The higher-dimensional Kaluza-Klein approach to unification was later used by the unified-field theorists, especially among those investigating the eleven-dimensional supergravity and ten-dimensional superstrings.In 1983, Wesson [3,4] has given an impetus to the study of Kaluza-Klein type theories by investigating a five-dimensional Kaluza-Klein theory of gravitation with a variable rest mass. In this theory we have five-dimensional (5-D) space-time-mass manifolds, and the fifth dimension is a convenient mathematical way of geometrizing the rest mass and of allowing one to study the possibility that it may be variable. The four-dimensional (4-D) general relativity theory is recovered when the rate of change of the rest mass is zero.We shall refer to this theory as space-time-mass Kaluza-Klein gravity theory (STM-KK gravity theory, for short). Ever since the foundations of the STM-KK gravity theory were laid, there have been investigations on its potentialities, particularly as concerns to its consequences for astrophysics and cosmology, the confrontation between theory and observation, its consistency with Mach's principle, the solutions to its field equations, and the like [4] - [18]. For fair list of references on the STM-KK theory see Overduin andWesson [19].More recently, Wesson and co-workers [5,20] have introduced a new approach to general relativity, conceptually in line with an old idea due to Einstein [21], in which the matter and its role in the determination of the space-time geometry is given from a purely f...

Gödel-type universes in string-inspired charged gravity

“…Recently, geometrical aspects of Gödel-type solutions have also been studied, either determining solutions that could be interpreted as the exterior of Gödel spacetimes [14] or connecting the 3+1 Gödel geometry with a 2+1 anti-de Sitter subspace [15]. Five-dimensional models that admit generalized Gödel-type solutions have also been constructed [16] and the same type of solutions was proved to arise also in Riemann-Cartan spacetimes with non-zero torsion [17].…”

Gödel-type space–times in induced matter gravity theory

“…As no use of any particular field equations was made in this first paper, its results hold for any 5D Gödel-type manifolds regardless of the underlying 5D Kaluza-Klein gravity theory. In the second article [65] the classes of 5D Gödel-type spacetimes discussed in [64] were investigated from a more physical viewpoint. Particularly, it was examined the question as to whether the induced matter theory of gravitation permits the family of noncausal solutions of Gödel-type metrics studied in [64].…”

“…In both articles [64,65] the 5D Gödel-type family of metrics discussed is the simplest 5D class of geometries for which the section u = const (u is the extra coordinate) is the 4D Gödel-type metric of general relativity. Actually the 5D Gödel-type line element of both papers does not depend on the fifth coordinate u, and therefore as regards to the IM theory a radiation-like equation of state is an underlying assumption of both articles.…”

Violation of causality in f(T) gravity

“…[68,69,70,71,72,73,74,75]) and in the framework of other gravity theories (see, for example, Refs. [76,77,78,79,80,81,82,83,84,85,86,87,88,89,90]). 4 The local violation causality is also related to ill-defined Cauchy problem in f(T) gravity.…”