The two faces of the coin of Special Relativity

TL;DR

This paper clarifies the interdependence of space and time in Special Relativity, resolves common misconceptions, and demonstrates how different definitions lead to equivalent physical laws like length contraction and time dilation.

Contribution

It presents a detailed analysis of how space and time interdependence underpins Special Relativity and shows the equivalence of different mathematical formulations from a physical perspective.

Findings

Interdependence of space and time is crucial for understanding Special Relativity.

Different assumptions about measurement tools lead to different but equivalent equations.

Both formulations produce the same physical effects like length contraction and time dilation.

Abstract

It is rarely emphasized in modern physics textbooks that our definitions of space and time have to reflect their complete interdependence. Our intuitive methods of always picturing one-dimensional space as a sum of unit-length rods and of picturing one-dimensional time as consecutive ticks of a clock suggest subconsciously that changes in space never alter time and vice versa. In this paper, we present a compiled list of popular arguments against Special Relativity, and we show how our erroneous intuitive conviction that the definition of space is always independent of the definition of time and vice versa is indeed the driving force behind these arguments. Once interdependence between space and time is established within their respective definitions, it is shown how such arguments are quickly resolved. Interdependence between space and time is described in two different ways, which are mathematically different but physically equivalent. We show how by assuming a time-independent measurement tool for space, time should be defined in terms of space to reflect their interdependence and this yields Lorentz Transformation. If we assume a space-independent measurement tool for time, space should be defined in terms of time, and this yields a different set of equations. Next, it is shown how the two different faces of the coin of Special Relativity, while mathematically different, are essentially equivalent from a physical point of view, and that both yield the same laws of addition of velocity, length contraction and time dilation effects. The apparent differences in physical interpretations between the two sets of equations are attributed to the differences in the definitions of space and time used in both cases.