New derivation of space-time linear transformation between inertial frames based on the principle of relativity

Abstract

In deriving that the space-time transformation between two inertial frames is linear, one commonly relies on the premises that any inertial frame has physical properties invariant under space-time translations. We present a different derivation of linearity which is based on the principle of relativity. In fact, we show that it is possible to circumvent the discussion of linearity in deriving the space-time transformation between inertial frames. Following the principle of relativity, it is inevitable that the only transformation compatible with the principle of relativity is the Generalized Lorentz Transformation (GLT which is linear). Distinct from other derivations, we prove the existence of a universal speed limit by examining the continuity and monotonicity of the velocity transformation function. In our derivation, we do not explicitly employ the translational invariance of inertial frames as the logical starting point. Neither do we invoke the constant speed of light as a fundamental postulate. The new derivation highlights the pivotal role of the principle of relativity in logically establishing the theory of special relativity.