The coordinate group symmetries of general relativity

Abstract

The symmetry group of special relativistic theories, the Poincar6 group, was imposed on physical theories to mirror the symmetries of the laws of nature under point mappings of the presumed Minkowskian space-time, thought to be the arena of physics. With the advent of the general theory of relativity the equations of the gravitational field were constructed so as to be invariant under arbitrary curvilinear point transformations of the spacetime, now taken to be a four-dimensional pseudo-Riemannian manifold. Although the dynamical laws of all general relativistic theories are taken to have this enlarged symmetry group, the geometry of any particular spacetime on which all the fields are defined no longer has this symmetry. In fact, in order to facilitate handling &the field equations of general relativity, it is often convenient to exploit the lack of symmetry of generic space-times to impose coordinate conditions upon the field variables, the metric tensor of the space-time. The coordinate transformations leading to the preferred frames of reference in which the coordinate conditions are satisfied, or which preserve those conditions, in so far as they involve specific reference to the metric of the space-time, are best understood, not so much as point mappings within a given four dimensional space-time, but rather as mappings within the function space of the field variables of the theory, guv(x~). (Greek indices are taken to range from 0 to 3, while Latin indices range from 1 to 3.) The general theory of relativity is thus seen to have a much larger natural symmetry group than was initially contemplated, namely transformations of the form 2 ~ = f~(x~, gm(x~))