100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Development of the Concepts of Space, Time and Space-Time 33the system at some time, and for the changes in that state as it evolvesin accord with the dynamical laws governing the system, depend on (arerelative to) the choice of inertial frame. These different descriptions are allconcordant with one another, leading, for example, to the same predictionfor the outcome of any experiment performed on the system. pp Mathematically,this is because different descriptions amount to no more thandifferent slicings (foliations) and fibrations of the background space-time,in which a unique four-dimensional description of the dynamical process canbe given.The Galilei-Newtonian absolute time and the special-relativistic global(inertial-frame relative) time have this in common: They are defined kinematicallyin a way that is independent of any dynamical process. qqIn general relativity, no time – either global (frame-dependent) or local(path-dependent) – can be defined in a way that is independent of thedynamics of the inertio-gravitational field. In general relativity, that is,time is relative in another sense of the word: It is relative to dynamics; inparticular, it is relative to the choice of a solution to the gravitational fieldequations.The proper (local) time is of course relative to a path in the manifold,but also depends on the particular solution in a new way: Whether a pathin the base space manifold is time-like or not cannot be specified a priori.Similarly, introduction of a frame-relative global time depends upon aslicing (foliation) of the space-time into three-dimensional space-like slices,and whether such a slicing of the base manifold is even space-like cannot bespecified a priori. Indeed, while a space-like slicing is always possible locally,a global space-like foliation may not even exist in a particular space-time.Even if such global space-like slicings are possible for a particular solution,there is generally no preferred family of slicings (such as the spacelikehyperplanes of Minkowski space) because in general the metric tensor of aspace-time has no symmetries. The symmetries of a particular metric tensorfield may entail preferred slicings. For example, there is a class of preferredslicings of a static metric such that the spatial geometry of the slices doesnot change from slice to slice. Hence, there is a preferred global frame relativeto any static solution. As in the pre-general-relativistic case, differentchoices of slicings (frames) may lead to different descriptions of the samepp This is the case in classical physics. Quantum mechanically, the different descriptionsmust all lead to predictions of the same probability for a given process.qq Of course, the measurement of such time intervals depends on certain dynamical processes,but that is a different question.