  The vacuum of an accelerating observer is defined on the local inertial
frame, which is called a ``moving frame''. However, in the discussion of the
Unruh-deWitt detector, many papers define the vacuum on a fixed frame. This
paper discusses the Unruh-deWitt detector by defining the vacuum directly on
the local inertial frame, showing that the problem of the Stokes phenomenon can
be solved by using the exact WKB.
