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

Development of the Concepts of Space, Time and Space-Time 17In the case of Galilei-Newtonian space-time, this group is the inhomogeneousGalilei group. In the special-relativistic case, it is the inhomogeneousLorentz (or Poincaré) group. Seven of the ten generators of the group takethe same form in both groups: four spatio-temporal translations, expressing(respectively) the homogeneity of the relative space of each inertial frameand the uniformity of the time – absolute or relative – of each inertial frame;and three spatial rotations, expressing the isotropy of the relative space ofeach inertial frame. They correspond, respectively to the conservation ofthe linear momentum and energy, and of the angular momentum of thedynamical system.The two groups differ in the form of the three so-called “boosts,” whichrelate the spatial and temporal coordinates of any event with respect totwo inertial frames in motion relative to each other. The Galilei-Newtonianboosts depend on the absolute time, which does not change from frameto frame; so they involve only a transformation of the spatial coordinatesof an event with respect to the two frames. The special-relativistic boostsinvolve a transformation of the temporal as well as the spatial coordinatesof an event. The boosts correspond to the center-of-mass conservation lawfor the dynamical system.The well known spatial (Lorentz) contraction and time dilatation effects,which result from the different breakup of a spatio-temporal process (forthat is what a ruler and a clock are) into spatial and temporal intervalswith respect to two inertial frame in relative motion, can be deduced fromthe relativistic boost transformation formulae relating the global space andtime coordinates of the two frames.9. Fetishism of MathematicsBefore discussing the next topic, the four-dimensional formulation of spacetimetheories, I shall interject a word of caution about a possible pathologyin theoretical physics that I call ‘the fetishism of mathematics’. We inventphysical theories to enable us to better comprehend and cope withthe world, or rather with limited portions of the world. We employ mathematicsas a vital tool in such attempts. Mathematical structures helpus to correctly encode numerous, often extremely complicated, relationsamong physical concepts, relations of both a quantitative and a qualitativenature. If these mathematical structures have been judiciously chosen,they do more than encode the relations that led to their introduction: formalmanipulation of the structures leads to the discovery of new relations