Special Relativity - events and measurements

Question

I'm trying to re-ask the Q based on the feedback I got on this forum. Here is the premise for the Q.

1. Consider an IRF, S. Assume only 2-D i.e (x,t). And the observer, O, is as assumed typically "fixed" at the origin of S.

2. Let an event, E, happen in the 'space' which we are trying to 'measure' using S and O (which I understand is a mathematical abstraction to study the laws of physics)

3. The observer O assigns (and thus records in his notebook) coordinate (x0,t0) to the event E.

My questions is how does he do that? (What is the precise physical procedure, even if idealistic, followed to measure x0 and t0?)

Some challenges I'm dealing with to understanding this (and to give a background why I'm asking this Q)

• Observer O (origin of reference frame S) and E are separated by 'space' - How does information about E reach O? Does this non-locality has greater implications?

• What is the significance of 'clock' present at every 'x'? And t0 is the time shown by local clock present at E? Are all the clocks in S synchronized? Why can't observer O, use his local clock at the origin (say when information about the event E has happened reaches O)?

• Similarly how does O measure x0? How does the reading on his 'meter sticks' reach him? etc.

I have been struggling to properly define the act of 'measurement' - and I'm unable to do so. Most books also do not cover this in adequate detail and for me it is quite important to understand this Q - to understand SRT

Thanks

Answer 1

Imagine empty universe with some rigid grid connected to you and you are not accelerating. If the rigid grid is perfectly symmetrical, that is the material used is everywhere the same and at every place the forces are perfectly balanced so that the grid is indeed rigid, then in STR we assume the evolution of this grid in time will make no observable changes. Therefore you can pick some stick with certain length and use this stick to actually measure wheter distances between grid points are everyhwere the same. No hurry, you have infinite amount of time and the measurements results are independed on the time by assumption.

You don't even need to care about nonlocality of the distance measurement, because if you attach the stick to the grid, you can just check where the one end of the stick is at one time and where the second end is at another time because there is no time evolution by assumption.

Now you can attach the clocks everywhere on the grid. The best clocks are light clocks because light is assumed to have constant speed. You use light to calibrate the clocks, like this:

I send light signal from point $P_1$ at time $t_1$ of my clocks of the grid to the point $P_2$ of the grid at time $t_2$ of those clocks and there it gets reflected back and is recieved at time $t_3$ of my clocks.

Because everything is perfectly symmetrical (no point is different from any other), we assume all clocks tick at the same rate and so the time $t_2$ should be exactly in the middle of the times $t_1$ and $t_3$: $$t_2=t_1+(t_3-t_1)/2$$

This is all that is to it. Now you should check the consistency of this whole procedure, like if clocks 1 and 2 are calibrated and clocks 2 and 3 are calibrated so are the clocks 1 and 3 and so on. You can also check what happenes between two grids moving with respect to each other.

The trick is assuming perfect symmetry of spacetime and evolution of the grid and the rest is easy. Which is of course not possible in reality to do, even if there is no gravitational field. But you use this abstraction to define mathematical model that is something like background on which the whole physics to build. Any discrepancies between this model and physics you attribute to some additional "forces" present in real world and you try to find formulas and sources for these forces and so on. Which is not always possible and the background model must be then rethinked, like it is in the case of gravity