#008: Special relativity as a theory of integration
Category: Special relativity - Tags: Classical mechanics, Riemannian geometry, Measure theory
See how much of special relativity can be recovered by positing spatially integrable distributions for all inertial observers.
Mathematical problem. The idea is that while space and time variables have the same role when identifying an event (i.e. they are all coordinates), they play different roles under integration. If we have a probability distribution, for example, the constraint is not\[\int_X \rho(x, t) dx dt = 1\]
but rather\[\int_U \rho(x, t) dx = 1\]
for all surfaces \(U\) at equal time. A boosted observer will see time and space mix, but the constraint will have to apply at equal time of the boosted observer.
Alternatively, the distribution could be assumed to be in position/time/velocity as these would be the variables needed to specify the initial conditions.
Physical significance. The goal is to remove the postulate of the invariance of the speed of light. The ability to express constraints, such as unitary probability, in a coordinate independent way would be a requirement for the principle of relativity itself. The idea is that integrating in time vs space becomes relative to the observer and therefore there must be a unique way to change units of space into time. The speed of light, then, would not be a speed but the constant that translates time intervals into spatial intervals (i.e. it is the ratio between the measure on space and time).