  The Klein-Gordon and Dirac equations in a semi-infinite lab ($x > 0$), in the
background metric $\ds^2 = u^2(x) (-\dt^2 + \dx^2) + \dy^2 + \dz^2$, 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 $mg\hbar c$. 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.
