Spin 0 and spin 1/2 quantum relativistic particles in a constant gravitational field

The Klein-Gordon and Dirac equations in a semi-infinite lab ($x>0$), in the background metric ${\text{ds}}^2=u^2(x)(-{\text{dt}}^2+{\text{dx}}^2)+{\text{dy}}^2+{\text{dz}}^2$, are investigated. The resulting equations are studied for the special case $ u(x) = 1 + g x$. It is shown that in the case of zero transverse-momentum, the square of the energy eigenvalues of the spin-1/2 particles are less than the squares of the corresponding eigenvalues of spin-0 particles with same masses, by an amount of $mg\mathrm{\hslash }c$. Finally, for nonzero transverse-momentum, the energy eigenvalues corresponding to large quantum numbers are obtained, and the results for spin-0 and spin-1/2 particles are compared to each other.