Canonical quantization of modified non-gauge invariant Einstein-Maxwell gravity and stability of spherically symmetric electrostatic stars

Abstract

We consider a non-minimally coupled Einstein-Maxwell gravity with
no $U(1)$ symmetry property to study stability of an electrostatic
star via canonical quantization approach and obtain that the
stability is free of gauge field effects. By calculating the
Hamiltonian density of the stellar system we show that the
corresponding Wheeler-DeWitt wave functional is similar to a
simple harmonic quantum Oscillator for which a non zero ADM mass
of the system causes a quantization condition on the metric
fields. Probability wave packets are described by the Hermit
polynomials. Our mathematical calculations show that in this
approach of quantum gravity the metric fields are regular for all
values of the electric potential and so the quantized spacetime
has not both of event and apparent horizons. The most probability
of the quantized line element is for ground state of the system.
To check validation of the model we use Bohr`s correspondence
principal and generate directly semi classical approach of the
quantized metric states at large quantum numbers where they reach
to Schwarzschild like metric according to the Birkhoff's theorem.
Also we check that the generated semi classical solutions are
satisfied exact classical metric solutions which are obtained from
Euler Lagrange equations. We show that `charge to mass ratio` of
the electrostatic star is a constant defined by the coupling
constant of the model and it is in accord to other alternative
approaches.