By discussing on the dynamical equations of sorts of particles in the non-stationary Rindler and the Vaidya space-time,it is shown that after introducing new coordinates into a time-dependent metric,the method to define the tortoise coordinate transformation in a stationary space-time can be generalized to a non-stationary case.The character of the obtained tortoise transformation is that the temperature function is obviously dependent on time.It is also shown that the Hawking processes are independent of the source terms in the dynamical equations.Particularly,we find a picture where particle-scattering must be twice in the Hawking radiation described by the general time t such as in the non-stationary Rindler case.
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