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GRAVITATIONAL THEORY
WITH A DYNAMICAL
SPACETIME
Eduardo Guendelman,
arXiv:0911.0178 [gr-qc]
Ben Gurion University, Israel,
MIAMI 2009, December 19, 2009
DYNAMICAL SPACE TIME?
• IN Gen. Rel., THE SPACE- TIME IS
REPRESENTED BY COORDINATES,
THESE ARE PARAMETERS, NOT
DYNAMICAL VARIABLES.
• In Quantum mechanics there is no “time”
operator that would be conjugate to the
energy. Is there a possibility to construct a
variable of this type?
We start with a simple example
The Equacion obtained from
variation of a is:
The integration of this eq. w/r to t
leads us to the conservation of
energy
ordinary equations are reproduced
4D theory with invariance under
transformation of coordinates
Symmetries of Killing vectors
Then the following is a symmetry
Models with point particles
Symmetries of Killing tensors
Equations of the Gravitational Field
Variation w/r to the metric gives
The gravitacional energy
momentum tensor
We could add a conventional term
Solutions for flat space-time
Solution for an arbitrary space time
in a locally inertial frame
• The solution found before for flat space
time is in fact (up to a constant) the
solution for the vector field for ANY space
time in a locally inertial frame (LIF).
• This shows a way to construct the
solution for the vector field in general: it is
proportional to the local Minkowski
coordinate in the LIF, then transform back
to the Lab. frame .
Contribution to the mass: study
small perturbations of Flat Space.
The constant c only redefines G .
We study the string gas cosmology
Milne coordinates
Other Cosmological solutions
For strings the same relation for the
Grav. E&M tensor is valid
• we can reformulate the particle so as to
have invariance under world line
reparametrizacion, the expression for the
E&M gravitacional tensor is not changed.
• the solution for the vector that implements
the conservacion of the original E&M
tensor is another indicacion that its
interpretacion is that of a dynamical
space-time.
Conclusions
• we have formulated a theory with a dynamical spacetime. In flat space time and in a locally
inertial frame the vector field is proportional to
the Minkowski coordinates.
• Symmetries associated to vector and tensors killing are found, as shifts of that vector.
• 2 E&M tensors: original and gravitacional
• Cosmological solutions of string gas that do not
curve space time are found, the Milne universe
with non trivial matter .
Perspectives
• Extension to fields, no only particles and
strings, relation with the problem of the
cosmological constant, as it has been
done in the special case of the TMT,
• Possibles supersymmetric extensions,
• Exploration of more general cosmological
solutions.
• Possible aplication to the problem of time
in quantum cosmology.