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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. - PowerPoint PPT Presentation
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GRAVITATIONAL THEORY WITH A DYNAMICAL
SPACETIMEEduardo 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 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 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.