XXII MG - ParisXXII MG - Paris

An invariant approach to define repulsive gravity An invariant approach to define repulsive gravity

Roy Kerr, Orlando Luongo, Hernando Quevedo and Remo RuffiniRoy Kerr, Orlando Luongo, Hernando Quevedo and Remo Ruffini

AbstractAbstract

A remarkable property of naked singularities in general relativity is A remarkable property of naked singularities in general relativity is their repulsive nature. The effects generated by repulsive gravity are their repulsive nature. The effects generated by repulsive gravity are

usually investigated by analyzing the trajectories of test particles usually investigated by analyzing the trajectories of test particles which move in the effective potential of a naked singularity. This which move in the effective potential of a naked singularity. This

method is, however, coordinate and observer dependent. We propose method is, however, coordinate and observer dependent. We propose to use the properties of the Riemann tensor in order to establish in an to use the properties of the Riemann tensor in order to establish in an invariant manner the regions where repulsive gravity plays a dominant invariant manner the regions where repulsive gravity plays a dominant role. In particular, we show that in the case of the Reissner-Nordstrom role. In particular, we show that in the case of the Reissner-Nordstrom

and Kerr naked singularities the method delivers plausible results. and Kerr naked singularities the method delivers plausible results.

OutlinesOutlines

Properties of curvature invariants in the Properties of curvature invariants in the case of naked singularities.case of naked singularities.

Properties of complex eigenvalues in the Properties of complex eigenvalues in the case of naked singularities.case of naked singularities.

Possibility to infer a definition for repulsive Possibility to infer a definition for repulsive effects in general relativity.effects in general relativity.

Conclusions and perspectives.Conclusions and perspectives.

Naked singularitiesNaked singularities

What happens to gravity What happens to gravity near a naked singularity? near a naked singularity? Effects of repulsive Effects of repulsive gravity?gravity?

6

248

r

MRRK

Curvature invariant

Example of repulsive gravity: Schwarschild spacetime with negative mass

drdrr

GMdt

r

GMds 22

122 2

12

1

SIMMETRY M -> -M

)()( MKMK

Complex eigenvaluesComplex eigenvalues

An alternative class of invariants is requested in the case of An alternative class of invariants is requested in the case of complex eigenvalues in the SO(3, C) representation. They deal with complex eigenvalues in the SO(3, C) representation. They deal with an invariant class of solutions, always found for all the metrics.an invariant class of solutions, always found for all the metrics.

Decomposition of the Riemann tensor in Weyl tensor, traceless Ricci tensorand scalar curvature

Little indeces are tetrad indices in a orthonormal frame.

Big indices are called bivector indices. (6 x 6 representation)

SO(3,C) representationSO(3,C) representation

Definition of SO(3,C) representation

Definition of complex eigenvalues 0det 3 IW Here we find three different (in principle)complex eigenvalues.

Complex eigenvaluesComplex eigenvalues

The form of eigenvalues for Schwarzschild spacetime is 3r

M

The form of thecomplex invariantchanges its form with the change

of the sign of the mass.

On the left is plottedwith +M,

on the right with-M.

The use of complex eigenvalues The use of complex eigenvalues VS the curvature invariantsVS the curvature invariants

nnn iba

We compare the two classes of invariants and we find which values of radius arecommon 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

sign. its changes )(:.0

,0jj

n

n rKrb

a

KN spacetimeKN spacetime

Kerr Newman metricsKerr Newman metrics

wherewhere 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?

For Kerr – Newman invariants only numerical values of r have been found on the axial plane; one of these, in the quoted case is

Mr 8.3

Behavior of the invariant with a=1,Q=1

In the case of Reissner – Nordstrom there isno real solution for the radius, while in the caseof Kerr spacetime, a solution is found

.cos32 ar

This suggests that the use of complex eigenvalues or effective potential is strongly required!!!

Another example: Zipoy-Voorhees Another example: Zipoy-Voorhees with eigenvalueswith 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 wehave 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)

The effective potentialThe effective potentialEffective potential in Kerr-Newman metric: Effective potential in Kerr-Newman metric:

Its form is not invariant under Its form is not invariant under coordinate transformations. 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).

Interpretation of effective mass and Interpretation of effective mass and repulsive effectsrepulsive effects

An effective mass is An effective mass is required?required?

MM eff

The study of potential suggests the use of an effective negative mass to better understandthe 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

r

QMM eff 2

Kerr

),,( raLMM eff

Vanishing of eigenvalues and their Vanishing of eigenvalues and their first derivativesfirst derivatives

.3

cos3 22242

M

MaQQr

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-Nordstromsolution the radius is less than the classical radius. Then a possible explanation ofrepulsive gravity could deal with the first derivative of complex eigenvalues equal to zero, i.e. respectively for Reissner – Nordstrom solution and Kerr solution

.cos21

,22

ar

M

Qr

This method is completely equivalent tothe matching method proposed byQuevedo.

Conclusion and perspectivesConclusion and perspectives

The idea to explain the role of repulsive effects The idea to explain the role of repulsive effects in general relativity has been investigated by in general relativity has been investigated by studying naked singularities. studying naked singularities. A definite interpretation of repulsive gravity is A definite interpretation of repulsive gravity is possible by using the first order derivative of the possible by using the first order derivative of the eigenvalues of the curvature tensor. eigenvalues of the curvature tensor. Matching condition between interior solution and Matching condition between interior solution and exterior one must be Cexterior one must be C33..

Work in progress: Matching with interior Work in progress: Matching with interior solutions in more general cases.solutions in more general cases.