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

Development of the Concepts of Space, Time and Space-Time 23four-dimensional pseudo-metric of signature two. A metric is a quadraticform used to compute the ‘length-squared’ of vectors, which is always apositive quantity. If the tensor does not assign a positive ‘length-squared’to all vectors, i.e., if some of them have zero or negative ‘length squared’, thetensor is called a pseudo-metric. The Minkowski metric is such a pseudometricof signature two. This means that, when diagonalized, it has threeplus terms and one minus term (or the opposite – at any rate three terms ofone sign and one of the other resulting in a signature of two for the pseudometrictensor), which represent space and time respectively. If the ‘lengthsquared’ of a vector computed with this pseudo-metric is positive, the vectoris called space-like; if negative, the vector is called time-like; if zero, thevector is called null or light-like. Signature two for the four-dimensionalMinkowski pseudo-metric implies that the null vectors at each point forma three-dimensional cone with vertex at that point, called the null cone. Atime-like and a space-like vector are said to be pseudo-orthogonal if theirscalar product vanishes when computed with the pseudo-metric. A fibrationof parallel lines is time-like if it has time-like tangent vectors and such afibration represents an inertial frame. A hyperplane pseudo-orthogonal tothe fibration (that is, with all space-like vectors in the hyperplane pseudoorthogonalto the tangents to the fibration) represents the set of all events ofequal global (Poincaré-Einstein) time relative to that inertial frame; and theslicing (foliation) consisting of all hyperplanes parallel to this one representsthe sequence of such global times relative to that inertial frame.The compatibility conditions between pseudo-metric and flat affinestructure are now expressed by the vanishing of the covariant derivative ofthe former with respect to the latter. In both the Newtonian and specialrelativisticcases, the compatibility conditions require that the ‘lengthsquared’of equal parallel vectors (space-like, time-like, or null) be thesame.11. Enter Gravitation: General Non-RelativityAs indicated above, in general relativity, both elements of the kinematicstructure, the chronogeometrical and the inertial, lose their fixed, a prioricharacter and become dynamical structures. There are two basic reasonsfor this:(a) General relativity is a theory of gravitation and, even at the Newtonianlevel, gravitation transforms the fixed inertial structure into thedynamic inertio-gravitational structure.