The Relativistic Generalization of the Gravitational Force for Arbitrary Spacetimes

It has been suggested that re-expressing relativity in terms of forces could provide fresh insights. The formalism developed for this purpose only applied to static, or conformally static, space-times. Here we extend it to arbitrary space-times. It is hoped that this formalism may lead to a workable definition of mass and energy in relativity.Comment: 16 page

Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non-trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is $r$-dependent. This rescaling factor is compared with that for the Reissner-Nordstr\"{o}m metric.Comment: 17 Page

General Formula for the Momentum Imparted to Test Particles in Arbitrary Spacetimes

Ehlers and Kundt have provided an approximate procedure to demonstrate that gravitational waves impart momentum to test particles. This was extended to cylindrical gravitational waves by Weber and Wheeler. Here a general, exact, formula for the momentum imparted to test particles in arbitrary spacetimes is presented.Comment: 6 page