Galilean Covariance in Curved Space
Galilean covariance, curved space, non-relativistic black hole, covariant Schrödinger equation.
Galilean covariance allows for the analysis of non-relativistic phenomena without abandoning the consequences of the covariance principle. Thus, the formalism is used to construct a covariant Schrödinger equation, which is employed to describe a field of non-relativistic spinless particles. Galilean covariance can also be extended to curved spaces, enabling the construction of a non-relativistic black hole, Schwarzschild-like. In this way, the interaction between spinless particles and a non-relativistic black hole was studied in the vicinity of the event horizon. The solution of the Schrödinger equation in this curved space presents analytical solutions, in terms of confluent Heun functions. As a result, it was possible to observe that particles do not escape to infinity, exhibiting quantized energy levels. Additionally, by examining the transmission and reflection coefficients, it was also possible to demonstrate that there is no non-relativistic equivalent of Hawking radiation.