First quantization light ray in Schwarzschild's black hole

[Andre' Quanta]

The dinamic of a light ray in a Schwarzschild' s metric is governed by a lagrangian where the potential is V(x)= a*(1/r)^2-b*(1/r)^3 with a and b positive costants.
The presence of a Lagrangian it means that is possible to apply a first quantization of this sistem; if so which are the eigenvalues of the Hamiltonian of this sistem?

 

[eljose79]

I would like to clarify that the dynamic of a light ray in a Schwarzschild's metric is governed by the geodesic equation, not a Lagrangian. The presence of a potential in the Lagrangian may suggest the possibility of a particle with mass moving in this metric, but it is not applicable to a massless particle such as light.

In terms of first quantization, it is not possible to apply it to a light ray in a Schwarzschild's metric as it violates the principle of equivalence in general relativity. However, if we consider a particle with mass moving in this metric, then we can apply first quantization and obtain the eigenvalues of the Hamiltonian.

The eigenvalues of the Hamiltonian for this system would depend on the specific values of the constants a and b. Without knowing their values, it is not possible to determine the exact eigenvalues. However, we can make some general observations.

Since both a and b are positive constants, the potential V(x) would always be positive. This suggests that the eigenvalues of the Hamiltonian would also be positive. The specific values of the eigenvalues would depend on the specific values of a and b, and would require further analysis and calculations.

In conclusion, it is not possible to apply first quantization to a light ray in a Schwarzschild's metric. However, if we consider a particle with mass moving in this metric, we can apply first quantization and determine the eigenvalues of the Hamiltonian. The specific values of the eigenvalues would depend on the specific values of the constants a and b in the potential.