GRAVITATIONAL THEORY WITH A DYNAMICAL SPACETIME Eduardo Guendelman

GRAVITATIONAL THEORY WITH A DYNAMICAL SPACETIME Eduardo Guendelman, ar. Xiv: 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

4 D 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 spacetime.

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.