XXII MG - Paris

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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.

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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.

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Naked singularities

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

Example of repulsive gravity Schwarschild
spacetime with negative mass
Curvature invariant
SIMMETRY M -gt -M
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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)
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SO(3,C) representation
Definition of SO(3,C) representation
Definition of complex eigenvalues
Here we find three different (in
principle) complex eigenvalues.
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Complex eigenvalues
The form of eigenvalues x 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.
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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
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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?
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Behavior of the invariant with a1,Q1
For Kerr Newman invariants only numerical
values of r have been found on the axial plane
one of these, in the quoted case is
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!!!
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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)
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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 Q0, 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).
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Interpretation of effective mass and repulsive
effects

• An effective mass is required?

Schwartchild Z. V.
Reisser-Nordstrom
Kerr
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.
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Vanishing of eigenvalues and their first
derivatives
We look for the solution to the equation xn0 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.
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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 C3.
• Work in progress Matching with interior
  solutions in more general cases.