chapter  Chapter 13
24 Pages

Tensors on manifolds

ByJames H. Luscombe

A coordinate grid covering Minkowski space can be constructed from the worldlines of free particles—straight lines at constant speed, always and everywhere. Real gravitational fields are inhomogeneous and lead to relative accelerations between neighboring observers. The effective interaction brought about by inhomogeneous gravitational fields implies a metrical relation between neighboring points in spacetime involving a metric tensor that varies in spacetime, a tensor field, such as occurs in accelerated systems. The treatment of tensors implicitly presumes a flat geometry, that spacetime can be covered by a single coordinate system. A relativistic theory of gravitation requires a spacetime more general than Minkowski space, one that can be flat locally, approximating local inertial frames, but which is not flat globally. This need is met by the mathematical structure of a manifold. This need is met by the mathematical structure of a manifold.