SpaceTime Continuum in Special and General Relativity
Section Twenty-Six and Twenty-Seven
These two sections are rather hard to understand, without suitable preparations; you have to review Einstein's Hole Argument, and his solution of that. First, Einstein illustrates the special character of the (Lorentz) metric in Minkowski space. This is rather straightforward, but this metric should be understood with the general Gaussian coordinates in view. When Einstein speaks of the "Euclidean" character of Minkowski space-time, he means that it is easy to give a physical meaning to coordinate values.
However, the same does not seem to hold in the space-time continuum of general relativity, since coordinates there do not have any direct physical meaning, unlike the case of the classical and special-relativistic picture. Thus Einstein wants to assert that we've got to reconsider the manner how we give a physical meaning to a point in a mathematical (abstract) continuum characterized by a set of coordinate values in the Gaussian manner. Recall that Gaussian coordinates may be arbitrarily chosen, and the space-time continuum is considered relative to that choice.
In section 27, Einstein is trying to say this (judging from his statements in other papers, from his solution of the Hole Argument): If you want to talk about physical space-time point, you need more than abstract coordinate values for that point. Any physical space-time points must be identified by encounters (coincidences) of two physical events, e.g., an encounter of two trajectories of material objects. Mere points in a space-time (4-dimensional) continuum do not signify physical events, unless they are matched with such encounters (coincidences). Here, he comes close to Leibniz's poisition that two physically indiscernible points are identical and that any points without such physical means of identification are merely "ideal", not "physically real". And if we realize this, Einstein is saying, it is now easy to see that the physical significance of Euclidean coordinates comes from the same factor, i.e., physical encounters or coincidences.
(This continuum is considered relative to an arbitrarily chosen coordinate system)
Although a coordinate system may be arbitrarily chosen, there are certain invariant quantities independent of that choice. The condition of general covariance (often confused with general principle of relativity by Einstein) assures that such invariants are preserved in any system.
Last modified, May 23, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
