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

24 J. Stachel(b) General relativity preserves the unique compatibility relation betweenthe special-relativistic chrono-geometric and inertial structures; sothat when the latter becomes dynamical, the former must followsuit.It follows that, in general relativity, there is no kinematics prior to andindependent of dynamics. That is, before a solution to the dynamical fieldequations is specified, there is neither an inertio-gravitational field nor achrono-geometry of space-time. (Of course a similar comment will apply toany generally covariant field equations. For details, see Ref. 17.)Let us look more closely at each of the two assertions above. First ofall, an examination of the Newtonian theory of gravitation from the fourdimensionalpoint of view shows that gravitation always dynamizes theinertial structure of space-time. bb As we have seen, the concept of inertialstructure is based on the behavior of freely falling bodies, i.e., the trajectoriesof bodies (structureless particles cc ) not acted upon by any (net)external force. If one neglects gravitation, force-free motions can be readilyidentified (in principle); but because of the equivalence principle, thepresence of gravitation effectively nullifies the distinction between forcedand force-free motions. How could we realize a force-free motion in principle(that is, ignoring purely practical difficulties)? First of all, the effectof non-gravitational forces (electrical, magnetic, etc.) on a particle can beeither neutralized or shielded from. But gravitation is universal: it cannotbe neutralized or shielded. This still would not constitute a fatal difficulty ifwe could correct for the effect of gravitation on the motion of an otherwiseforce-free body (as we often do for non-gravitational forces when we cannotshield from them). But the effect of gravitation is universal in a second.sense: As Galileo supposedly demonstrated using the leaning tower of Pisa,it has the same effect on the motion of all bodies (this is often called theweak principle of equivalence). This would still not constitute a fatal difficultyif it were possible to single out the class of inertial frames of referencein a way that is independent of the concept of force-free motion. This possibilityis tacitly assumed when gravitation is described, in the traditionof Newton, as a force pulling objects off their inertial paths. But inertialframes cannot be defined independently of inertial motions, which are inturn defined as force-free motions! So our attempt to distinguish betweenbb See, e.g., Ref. 13.cc That is, bodies, for which any internal structure beyond their monopole mass may beneglected.