Sometimes people have underlying assumptions of how coordinates in a coordinate system should relate to measurements.
My question is whether the following requirements would be so restrictive as to apply solely to inertial coordinate systems, or if there happens to be some other class of coordinate systems that could meet these requirements as well.
Assume we have some collection of identical standardized rulers and clocks. Now consider a coordinate system $(x^0,x^1,x^2,x^3)$ with the following properties:
- "one coordinate matches clock readings"
for a clock sitting at constant $x^1,x^2,x^3$ coordinates, $\Delta x^0$ between events on the clock is the time measured by the clock - "three coordinates match rulers and grade-school geometry"
for a ruler with one end stationary at $x^1_a,x^2_a,x^3_a$ and the other end at $x^1_b,x^2_b,x^3_b$, the distance as measured by the ruler is $(x^1_b - x^1_a)^2 + (x^2_b - x^2_a)^2 + (x^3_b - x^3_a)^2$ - "clock synchronization can be checked by carefully moving a clock over to another"
if a clock at $(x^0_a,x^1_a,x^2_a,x^3_a)$ is set to $x^0_a$, then for any point $(x^0_b,x^1_b,x^2_b,x^3_b)$ which it can be moved to with speed much less than c, $x^0_b$ will be equal to the clock's reading in the slow transportation limit.
Follow-up question:
In either a Newtonian world, or in special relativity, is this actually overly restrictive and one of the three can actually be deduced from the other two?
If it is actually under-restrictive, what additional restrictions would limit it to inertial coordinate systems?
