1. XXII MG - Paris An invariant approach to define repulsive gravity Roy Kerr, Orlando Luongo, Hernando Quevedo and Remo Ruffini Abstract A remarkable property of naked singularities in general relativity is their repulsive nature. The effects generated by repulsive gravity are usually investigated by analyzing the trajectories of test particles which move in the effective potential of a naked singularity. This method is, however, coordinate and observer dependent. We propose to use the properties of the Riemann tensor in order to establish in an invariant manner the regions where repulsive gravity plays a dominant role. In particular, we show that in the case of the Reissner-Nordstrom and Kerr naked singularities the method delivers plausible results. A remarkable property of naked singularities in general relativity is their repulsive nature. The effects generated by repulsive gravity are usually investigated by analyzing the trajectories of test particles which move in the effective potential of a naked singularity. This method is, however, coordinate and observer dependent. We propose to use the properties of the Riemann tensor in order to establish in an invariant manner the regions where repulsive gravity plays a dominant role. In particular, we show that in the case of the Reissner-Nordstrom and Kerr naked singularities the method delivers plausible results.

2. Outlines Properties of curvature invariants in the case of naked singularities. Properties of complex eigenvalues in the case of naked singularities. Possibility to infer a definition for repulsive effects in general relativity. Conclusions and perspectives.

3. Naked singularities What happens to gravity near a naked singularity? Effects of repulsive gravity? Curvature invariant Example of repulsive gravity: Schwarschild spacetime with negative mass SIMMETRY M -> -M

4. Complex eigenvalues An alternative class of invariants is requested in the case of complex eigenvalues in the SO(3, C) representation. They deal with an invariant class of solutions, always found for all the metrics. Decomposition of the Riemann tensor in Weyl tensor, traceless Ricci tensor and scalar curvature Little indeces are tetrad indices in a orthonormal frame. Big indices are called bivector indices. (6 x 6 representation)

5. SO(3,C) representation Definition of SO(3,C) representation Definition of complex eigenvalues Here we find three different (in principle) complex eigenvalues.

6. Complex eigenvalues The form of eigenvalues  for Schwarzschild spacetime is The form of the complex invariant changes its form with the change of the sign of the mass. On the left is plotted with +M, on the right with -M.

7. The use of complex eigenvalues VS the curvature invariants We compare the two classes of invariants and we find which values of radius are common for the two classes. The idea is to find a set of complex eigenvalues and to find from them the values of r in which they vanish

8. KN spacetime Kerr Newman metrics where Is it possible to find a radius at which gravity changes its sign? What properties has this radius? From this radius is it always possibleto have a definition of effective mass and repulsive gravity?

9. For Kerr – Newman invariants only numerical values of r have been found on the axial plane; one of these, in the quoted case is Behavior of the invariant with a=1,Q=1 In the case of Reissner – Nordstrom there is no real solution for the radius, while in the case of Kerr spacetime, a solution is found This suggests that the use of complex eigenvalues or effective potential is strongly required!!!

10. Another example: Zipoy-Voorhees with eigenvalues The different behavior of the naked singularity case could be interpreteted as the result of repulsive effects. In particular near 0 and in the first maximum we have repulsivity but the strange behavior of the curve, which goes to repulsive to Minkowski, does not allow us to understand which maximum is correct. In order to understand it, we can match the interior solution with the smooth one maximum (see Quevedo talk)

11. The effective potential Effective potential in Kerr-Newman metric: Its form is not invariant under coordinate transformations. We expect that some of the results from the effective potential will be in contraddiction with the study of invariants. For example in the case Q=0, Kerr case, the radius found with the effective potential method are in contradiction with the K changing of sign. It is expected, because for an invariant there is no dependence from the properties of the particle (L).

12. Interpretation of effective mass and repulsive effects An effective mass is required? The study of potential suggests the use of an effective negative mass to better understand the behavior of gravity in the case of naked singularity (see right figure) but not for all the metrics is possible to have a negative mass; this suggests again to use invariants quantities into account. Schwartchild & Z. V. Reisser-Nordstrom Kerr

13. Vanishing of eigenvalues and their first derivatives We look for the solution to the equation  n =0 in order to find the values of radius. We found This appears to be a non physical result, because in the limit of Reissner-Nordstrom solution the radius is less than the classical radius. Then a possible explanation of repulsive gravity could deal with the first derivative of complex eigenvalues equal to zero, i.e. respectively for Reissner – Nordstrom solution and Kerr solution This method is completely equivalent to the matching method proposed by Quevedo.

14. Conclusion and perspectives The idea to explain the role of repulsive effects in general relativity has been investigated by studying naked singularities. A definite interpretation of repulsive gravity is possible by using the first order derivative of the eigenvalues of the curvature tensor. Matching condition between interior solution and exterior one must be C 3. Work in progress: Matching with interior solutions in more general cases.