Paper preprint:

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Keywords:

general relativity, quantum gravity, quantum mechanics, de Sitter spacetime, Reissner-Nordstroem extremal spacetime, Israel junction conditions, relativistic shells, WKB (semiclassical) approximation, Bohr-Sommerfeld quantization, stationary states.

Project goals:

to describe a reasonably simple general relativistic system in a regime where quantum effects might be important.

Project results:

numerical determination of the "spectrum" of the system, i.e. of the values of the parameters describing the system, when its action is of the order of the quantum of action; parameter values are restricted by the "quantization condition" a la Bohr-Sommerfeld.

Short Description.

The system under consideration is constituted by a relativistic shell (described using Israel's junction conditions) which separates an interior region with the de Sitter geometry from an exterior region with an extremal Reissner-Nordstroem geometry. An analytical Hamiltonian description is given by using an already developed formalism, which has been tested on other similar systems and which gives an expression for the momentum associated to the dynamics of the shell. This expression of the momentum is used to (numerically) calculate the action 'S' along classically allowed trajectories. Applying the Bohr-Sommerfeld quantization condition:

S =nh

,

n = 1,2,...,

the quantum admissible states of the system are obtained. Since the parameters of the system are the de Sitter cosmological constant, the external charge and the rest mass of the dust composing the shell, in the quantum regime the quantization condition above gives a relationship among these parameters, which classically are all free. From the point of view of quantum gravity, the quantization condition constrains the possible geometries of spacetimes, i.e., in our case, the possible values of the cosmological constant in the inside region, of the charge in the outside region and of the mass of the shell. Moreover, from the point of view of an observer moving across the shell from the inside to the outside region, a cosmological constant is converted into charge and mass.

Numerical computations have been performated with Mathematica(R) and some numerical problems require to go beyond machine precision to obtain significative results.

Additional material:

Summary of files with solutions

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