Gravitomagnetism and spinor quantum mechanics

We give a systematic treatment of a spin $1/2$ particle in a combined electromagnetic field and a weak gravitational field that is produced by a slowly moving matter source. This paper continues previous work on a spin zero particle, but it is largely self-contained and may serve as an introduction to spinors in a Riemann space. The analysis is based on the Dirac equation expressed in generally covariant form and coupled minimally to the electromagnetic field. The restriction to a slowly moving matter source, such as the earth, allows us to describe the gravitational field by a gravitoelectric (Newtonian) potential and a gravitomagnetic (frame-dragging) vector potential, the existence of which has recently been experimentally verified. Our main interest is the coupling of the orbital and spin angular momenta of the particle to the gravitomagnetic field. Specifically, we calculate the gravitational g-factor to be ${\mathrm{g}}_{\mathrm{g}}=1$; this is to be compared with the electromagnetic g-factor of ${\mathrm{g}}_e=2$ for a Dirac electron. Lastly, we discuss a number of possible experimental approaches to observing gravitomagnetic effects in atomic and macroscopic systems.

Figure 1

The electron in the wave function scatters from the field and back into the wave function.