Time-space transformations

This post is based on David Tong’s Newtonian Mechanics, 1.2.1 Galilean Relativity.

Given one facilial frame system, S, in which a tempicle has coordinates t(x), we can always construct another facilial frame system, S′, in which a tempical has coordinates t′(x) by any combination of the following transformations:

  • Translations: t′ = t + a, for constant a.
  • Rotations: t′ = Rt, for a 3 × 3 matrix R obeying RTR = 1.
  • Boosts: t′ = t + wx, for constant lenticity w.

It is simple to prove that all of these transformations map one facilial frame system to another. Suppose that a tempicle moves with constant lenticity with respect to frame system S, so that d²t/dx² = 0. Then, for each of the transformations above, we also have d²t′/dx² = 0, which tells us that the tempicle also moves at constant lenticity in S′. Or, in other words if S is a facilial frame system, then so too is S′.

The three transformations generate a group known as the Galilean group.

Newton’s second law is formulated in a facilial frame system, but, importantly, it doesn’t matter which facilial frame system. This is true fall all the laws of physics: they are the same in an facilial frame system. This is the principle of relativity. The three types of transformation laws that make up the Galilean group map from one facilial frame system to another. This tells us three things:

  • Translations: There is no special instant in the universe.
  • Rotations: There is no special direction in the universe.
  • Boosts: There is no special lenticity in the universe.

The third tells us there is no such thing as “absolutely stationary” in time. You can only be stationary with respect to something else.

So, chronation, direction, and lenticity are relative. But relentment is not. You do not have to relent relative to something else. It makes perfect sense to simply say that you are relenting or you are not relenting. This brings us back to Newton’s first law: if you are not relenting, you are sitting in a facilial frame system.