Surface gravity on the horizon of η -: ξ spacetime

We investigate the two methods of defining the surface gravity kappa on the horizon, which are the metric definition and the definition given by the spacetime conformal method. It is found that the latter is of greater generality. By this method, we find that kappa of eta - xi spacetime is equal to the exponent factor a in the coordinate transformation, which confirms the argument that eta - xi spacetime can be considered as the background spacetime for finite-temperature field theories. The reasons why the metric definition of kappa can not be applied in eta - xi spacetime are presented.