Inertial Constraints in a Curved Spacetime

Abstract

A precise kinematical definition of the deformation of an arbitrary space or spacetime is considered. This approach introduces which, combined with the vacuum field equations of general relativity, lead to a unique class of solutions involving plane gravitational waves. A general expression which relates accelerations in a smoothly deformed back­ ground has been proposed recently.l) This particular formula has also been used for the definition of the inertial backgrounds (i.e., the global inertial frames of reference) which can be associated, in principle, with an arbitrary curved space or spacetime. An approach of this particular kind is useful for the reexamination of the various covariant formulations of mechanics in n-dimensions (n~4) in relation or not to the general theory of relativity.2) However, the physical interpretation of the terms involved in the aforementioned expression is rather obscure and in a certain sense misleading. In particular, the analysis is restricted in the subcase of affine motions of the background, the general case of a deformation not being considered. In this sense it is worth reconsidering the whole subject from a different point of view. It turns out that a precise kinematical definition of the deformation of an arbitrary space is always possible. In particular, we may use the deformation of a test particle's acceleration in order to establish criteria which give us infomation about the structure of the underlying space. Clearly, there is a hierarchy of such possibilities. How­ ever, even the simplest physically interesting one, where the test particle's accelera­ tion does not feel the deformation of the background, gives non-trivial results. In fact, a combination of the aforementioned case with the vacuum field equations of general relativity prescribes uniquely, a special class of solutions which involve plane gravitational waves. In an arbitrary space M, where space means any connected paracompact mani­ fold, the acceleration vector (in contradistinction to the velocity one) cannot be defined without an explicit reference to the geometry of the space under consideration. This means that a (symmetric) affine connection or a Finslerian one is at least required so that the acceleration vector is properly and consistently defined in some local neighborhood of the point mEM. If the space is continuouslydeformedl) then we have