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Dynamical Spacetime in Gravitational Theory: Insights and Solutions from Quantum Mechanics

This paper explores the concept of dynamical spacetime within gravitational theory, questioning the traditional view where spacetime is merely described by coordinates. We analyze the implications of this view in the context of quantum mechanics, particularly the absence of a "time" operator conjugate to energy. By examining variations in gravitational fields and developing equations reflecting these dynamics, we derive solutions applicable to flat and arbitrary spacetimes. We also propose extensions to string gas cosmology and discuss potential ramifications for quantum cosmology, including supersymmetric solutions and addressing the cosmological constant.

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Dynamical Spacetime in Gravitational Theory: Insights and Solutions from Quantum Mechanics

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  1. GRAVITATIONAL THEORY WITH A DYNAMICAL SPACETIME Eduardo Guendelman, arXiv:0911.0178 [gr-qc] Ben Gurion University, Israel, MIAMI 2009, December 19, 2009

  2. 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?

  3. We start with a simple example

  4. The Equacion obtained from variation of a is:

  5. The integration of this eq. w/r to t leads us to the conservation of energy

  6. ordinary equations are reproduced

  7. 4D theory with invariance under transformation of coordinates

  8. Symmetries of Killing vectors

  9. Then the following is a symmetry

  10. Models with point particles

  11. Symmetries of Killing tensors

  12. Equations of the Gravitational Field

  13. Variation w/r to the metric gives

  14. The gravitacional energy momentum tensor

  15. We could add a conventional term

  16. Solutions for flat space-time

  17. 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 .

  18. Contribution to the mass: study small perturbations of Flat Space.

  19. The constant c only redefines G .We study the string gas cosmology

  20. Milne coordinates

  21. Other Cosmological solutions

  22. 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.

  23. 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 .

  24. 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.

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