The Condition for the Invariance of the Speed of Light in Special and General Relativity

Abstract

Hyoungseok Koh*

This study aims to investigate the conditions for the the law of invariance of the instantaneous speed of light in both inertial and non-inertial systems. By applying inertial forces, it argues that the invariance of the instantaneous speed of light applies not only to inertial systems but also to non-inertial systems, since inertial forces transform them into instantaneous inertial systems. In particular, while acceleration acts from the objects, inertial force acts from the rest field, thus presupposing the existence of such a rest field. Furthermore, the Lorentz transformation and its inverse, derived solely from the invariance of light speed, represent the relativity of observers. However, acceleration and gravity, both of which belong to non-inertial systems, are not equivalent, because the conditions under which the invariance of instantaneous light speed holds differ. In accelerating coordinate systems, the equivalence principle between acceleration and inertial force allows transformation into an instantaneous uniformly moving system. In contrast, a gravitational field involves not only the equivalence between gravity and inertial force but also the equivalence between gravitational and inertial mass, thereby forming an instantaneous inertial system. Meanwhile, in a gravitational field, applying the Lorentz transformation to the momentarily comoving rest system of a freely falling object yields the Schwarzschild metric, which corresponds to an instantaneous inertial system. Within the gravitational field of a star, by substituting the Schwarzschild metric into the geodesic equation, one can derive the spacetime paths of matter and light. A more detailed discussion of this will be reserved for future work.

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