  As long as we neglect backreaction, the Hawking temperature of a given black
hole would not depend upon the parameters of the particle species we are
considering. In the semiclassical complex path analysis approach of Hawking
radiation, this has been verified by taking scalar and Dirac spinors separately
for different stationary spacetime metrics. Here we show, in a coordinate
independent way that, for an arbitrary spacetime with any number of dimensions,
the equations of motion for a Dirac spinor, a vector, spin-$2$ and
spin-$\frac{3}{2}$ fields reduce to Klein-Gordon equations in the WKB
semiclassical limit. We then obtain, under some suitable assumptions, the
complex solutions of those resulting scalar equations across the Killing
horizon of a stationary spacetime to get a coordinate independent expression
for the emission probability identical for all particle species. Finally we
consider some explicit examples to demonstrate the validity of that expression.
