Is there a coordinate independent method to compute particle production in curved spacetime for some scalar field? In all methods, they fix the quantum state of the field (they usually take the 'vacuum') by specifying a complete set of solutions of the classical wave equation of the field. Those solutions are written in a specific coordinate system. Another coordinate system will evoke different complete set of solutions (like say plane waves in cartesian coordinates and spherical waves in spherical coordinates). The results would be different if one chooses different complete sets of solutions. Moreover, no matter in what coordinate system one writes the wave equation, any complete set of solutions can be expressed in that system only some sets would look 'natural' (like a set of plane waves would look natural in cartesian coordinates but nobody can stop you to get the spherical waves set in the 'unnatural' for them cartesian coordinates). I can't understand why the scalar field wold 'prefer' a specific complet set of solutions or specific coordinate system in which those solutions look 'natural'. Isn't that a total violation of the GR principle that all coordinate systems are equivalent ???