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The traditional approach to special relativity is based on the constancy of the speed of light. In a recent work, we aim to offer a different conceptual perspective where relativity does not depend on light, but on causality and the geometry of space, reinforcing its fundamental and universal nature. How can we understand it from the realm of theoretical physics?

There are many ways to see that the symmetry group that characterizes special relativity must contain the Poincaré group. This leads to the idea that Poincaré symmetry is inevitable for any reasonable theory and invites the search for the most fundamental and minimal set of hypotheses. 

In this regard, in an article published in Foundations of Physics, we propose a derivation of special relativity based exclusively on the principle of causality, avoiding the traditional postulates about the constant speed of light. It starts from the idea that events must be unique for each observer and that communication between observers requires a finite time. This excludes any instantaneous interaction, since if it existed, the observers would not be spatially different. This is a natural requirement that avoids possible paradoxes and leads to the existence of a maximum speed for the transfer of information between different observers. In other words, all messengers between observers move at speeds less than or equal to this maximum speed and need a finite and non-zero time to reach an observer located at a different position in space. This result is valid in any reference system, inertial or not. Causality also follows from this condition, again in any reference system. 

A geodesic is usually defined as the shortest line or segment. When you think about it, you realize that we don’t really know what a line is. In our article, we give an operational definition of geodesic: minimum information transfer time. This definition provides an experimental construction of geodesic coordinates. It is worth noting that this reverses the usual logic about light: instead of saying that light travels along geodesics, we define geodesics as the path followed by light (limit messenger). Once we have geodesic coordinates, we can use them to characterize the position of observers in rigid reference systems (those in which the distance between observers does not vary over time). 

Inertial reference systems appear as a subset of rigid reference systems: they are those for which there exist geodesic coordinates such that the Pythagorean theorem is valid. In this situation, the geodesic coordinates coincide with the Cartesian coordinates. 

Finally, we have discussed what the coordinate changes allowed between different inertial reference systems are. The requirement that the Pythagorean theorem and causality are preserved limits these transformations to the inhomogeneous orthochronous Lorentz group (Poincaré) multiplied by dilations, a result obtained by the physicists Alexandrov and Zeeman last century. 

In short, the work offers a conceptually clear perspective: relativity does not depend on light, but on causality and the geometry of space, reinforcing its fundamental and universal nature.