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.