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.