Role of Space Time in Jainas Syadavada and Quantum Theory

Author(s): Filita Bharucha
Publisher: Shri Satguru Publications

Page 63

General Theories of Space-Time
Clearing up the discussion about the classical Spacetime theories we come to the perspective on Einsten's geometrical interpretation of 1908. On this interpretation of the Special Theory of relativity describes a four-dimensional semi-Euclidean manifold, with the line element given by. ds2 = dx2 - dx2 - dx2 - dx2
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The inertial frames are just the Cartesian co-ordinates systems for this line element and the Lorentz transformations are analogous to the Euclidean orthogonal transformations. The trajectories or world-lines of free particles are straight lines or geodesics of the metric (1) they are four-dimensional curves of extreme in fact maximal-Length according to ds and which they satisfy the linear equation x = au+b, in all inertial frames.
There is indeed no three-dimensional Euclidean embedding space but there is a four-dimensional semiEuclidean space-time in which all physical events, are embedding. Within this space-time manifold, a privileged class of straight lines or inertial trajectiones represents the world-lines of free particles. Once again we start the flat Minkowski manifold with the line element given as above earlier.
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ds2 = 2 g1, dx, dx, where g's are not constant.
4,j = 0
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(1) ds2 = dx2 - dx2 - dx2 - dx2
for which we impart a variable curvature to this manifold, where the degree of curvature in a given region depends on the distribution of mass and energy, since our new manifold is not flat or semi-Euclidean we can no longer use the simple for (1) for the line element, but we need the general form.