GRAVITATIONAL THEORY WITH A DYNAMICAL SPACETIME

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GRAVITATIONAL THEORY WITH A DYNAMICAL SPACETIME

• Eduardo Guendelman, arXiv0911.0178 gr-qc
• Ben Gurion University, Israel,
• MIAMI 2009, December 19, 2009

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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?

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We start with a simple example
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The Equacion obtained from variation of a is
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The integration of this eq. w/r to t leads us to
the conservation of energy
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ordinary equations are reproduced
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4D theory with invariance under transformation of
coordinates
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Symmetries of Killing vectors
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Then the following is a symmetry
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Models with point particles
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Symmetries of Killing tensors
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Equations of the Gravitational Field
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Variation w/r to the metric gives
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The gravitacional energy momentum tensor
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We could add a conventional term
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Solutions for flat space-time
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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 .

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Contribution to the mass study small
perturbations of Flat Space.
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The constant c only redefines G .We study the
string gas cosmology
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Milne coordinates
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Other Cosmological solutions
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For strings the same relation for the Grav. EM
tensor is valid

• we can reformulate the particle so as to have
  invariance under world line reparametrizacion,
  the expression for the EM gravitacional tensor
  is not changed.
• the solution for the vector that implements the
  conservacion of the original EM tensor is
  another indicacion that its interpretacion is
  that of a dynamical space-time.

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Conclusions

• we have formulated a theory with a dyna-
• mical 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 ten-
• sors killing are found, as shifts of that vector.
• 2 EM tensors original and gravitacional
• Cosmological solutions of string gas that do not
  curve space time are found, the Milne universe
  with non trivial matter .

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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.