  We derive the solutions of gravitational waves in the future (F) expanding
and the past (P) shrinking Kanser spacetimes as well as in the left (L) and
right (R) Rindler wedges in the Regge-Wheeler gauge. The solutions for all
metric components are obtained in an analytic form in each region. We identify
the master variables, which are equivalent to massless scalar fields, to
describe the gravitational degrees of freedom for the odd parity and even
parity modes under the transformation in the two-dimensional plane symmetric
space. Then, the master variables are quantized, and we develop the quantum
field theory of the gravitational waves in the F, P, L, and R regions. We
demonstrate that the mode functions of the quantized gravitational waves in the
left and right Rindler wedges are obtained by an analytic continuation of the
left-moving and right-moving wave modes in Kasner spacetime. On the basis of
these analyses, we discuss the Unruh effect of quantized gravitational waves
for an observer in a uniformly accelerated motion in Minkowski spacetime.
