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

32 J. Stachelwhich merely defines the chronogeometrical relations between events givenpriori. This is incorrect for at least two reasons. As noted earlier, withoutthe specification of a particular metric tensor field, we cannot specify theglobal topology of the base space differentiable manifold. But, even locally,there is no means of physically individuating a priori the points of a regionof a bare manifold. If particles or other, non-gravitational fields are presentin the region, then their properties may suffice to individuate the points ofthe region. But even then, without a metric, it is generally impossible tofully interpret the physical properties of such non-gravitational entities.If we confine ourselves to otherwise empty regions, in which only thechronogeometrical (and corresponding inertio-gravitational) field is present,then the points of such a region cannot be physically individuated by anythingbut the properties of this field. oo16. The Second Relativization of TimeAmong the properties of any physical event are its position in space-time:the ‘here’ and ‘now’ of the event. One cannot give meaning to the concepts‘here’ and ‘now’ of an event in otherwise empty regions of space-timewithout use of the chronogeometry (metric tensor). This leads to a secondrelativization of the concept of time in general relativity that is even moredrastic than that required by the special theory of relativity.To recall what has been said earlier: In Galilei-Newtonian kinematics,the absolute time function as a global time, the same for all rigid referenceframes, regardless of their states of motion, even when gravitation is takeninto account; and it also serves as a local time, the time elapsed betweentwo events is the same for all time-like paths between them.In special-relativistic kinematics, this unique concept of absolute timesplits into two different concepts, both of which are relativized. Time inthe global sense (the time used to compare events at different places) isrelative to the inertial frame chosen, i.e., to one member of a preferredclass of reference frames. Time in the local sense (the proper time thatelapses between events along a time-like path) is relative to the path anddiffers for different paths between the same two events.In both pre-relativistic and special-relativistic kinematics, the use ofdifferent frames of reference leads to different descriptions of some processundergone by a dynamical system: The expression for the initial state ofoo For a more detailed discussion of this point, see Ref. 10.