Alice checks the round-trip time for photons to and from two distant points. If these times are the same, then the points are both the same distance from Alice.
Drawn once more against a different backdrop, with time ticking away up the screen and space running out to the left and the right. Alice does not move right or left, so the line tracing out her path runs straight up the page (that's her worldline). The speed of light is constant, and so the photon paths always trace out a 45-degree line on this backdrop: both out and back.
Move the two waypoints further away from Alice: the trip time (out and back) increases.
But the waypoints cannot be anywhere at all in this space: the clocks must tick at the same rate: so photons leaving Alice at one time must return at the same time as each other. And a now for Alice must lie within what Alice can be affected by (that was a now) or affect (that will be a now).
Everything along the line joining the waypoints is a now
for Alice: a now line
.
Identical out and back trip times for a pair of photons only happens for some placements of the waypoints: what counts as now
is constrained by the ticking of the clocks, as it should be.
Now Alice notices a moving Bob. His nows, as seen by Alice, are also constrained by his ticking clocks. His velocity affects where the mirrors must be for the light to bounce back at the right time — light still travels at the same velocity.
Alice‘s nows are usually not the same as Bob‘s nows. Their clocks tick at different rates, but they disagree about the present: Alice’s before and after are not the same as Bob‘s before and after.
The relative velocity is what's important: there is nothing special about Alice or Bob. You can choose to see the world from either viewpoint.
But you do have to choose a viewpoint: one local witness. You cannot step outside of space and time to get an independent
or global
point of view. There are only local witnesses, so their light cones define the structure of space and time.
You can extend this to see what Alice notices about both the nows for both Bob and Charlie.
You can allow each of Alice, Bob and Charlie to set several nows.
So far, you’ve explored how what might count as now depends on velocity in just one spatial dimension. But of course, space is three dimensional. You can go one better by visualising two dimensions of space.
Alice Bob and Charlie will mostly disagree about when things happen. Chaos is avoided because the light cones show the causal structure of the universe, and you can figure out what’s going on by working out what’s going on by figuring out what the Alice, Bob or Charlie moving along with the process records, using the connections between the now lines
. That’s not to say it’s easy, but it can be done – watch carefully (notice and record) to gather data to be able to work out what the local witness notices and records.