Clocks Tick Differently Because of Pythagoras?
I perform a thought experiment to study variation in clock ticks in moving bodies at constant velocity in same reference frame, in 3D Euclidean space, with no reference to light, the speed of light, or the invariance of the laws of physics.
Imagine a lighthouse of height T meters, where two friends A, B are stationed at the bottom and top respectively. C is downstairs in a car. C will start travelling at constant velocity 1m/s in a straight line from the lighthouse at time t=0 and distance d=0.
A and B have synchronized their clocks. At every clock tick, they will squeeze a sound horn (of different frequency) that can be heard by C. Imagine for the sake of the experiment, that sound travels at 1m/s.
At the base of the tower at time t=0s, A and B will sound the horn in a synchronized fashion. It should be clear that C will hear A instantly and will hear B after T seconds.
C drives to a distance of d = 1m, A will again sound the horn in a synchronized fashion. C will hear A after 1 second and will hear B after sqrt(1²+T²) seconds (the hypotenuse from B to C).
At every 1 meter distance, C will stop and listen for the horns. While C will hear A in exactly d seconds, where d is distance travelled, C will hear B as the hypotenuse of the right angle triangle converging at that point i.e sqrt(d²+T²) which will not change linearly.
Plotting the graph of the rate of change of hypotenuse gives a correlation to the time taken for the clock ticks to reach C from B compared to A.
As we can see from the graph, the frequency of the clock ticks is non-linear and varies as the hypotenuse does.
So if C were in a phone call with A and B, C will be telling A that he/she could hear A constantly every d seconds, while C will be telling B that he/she could hear B in a changing fashion; quickly ascending but slower than A, and then after a long time, slowly matching closely with the timing of A.
If C was in continuous motion, and A, B were sounding the horns every second, C will hear the horns from A and B at different and varying times. It will appear as though B’s clock runs differently from A though they are synchronized at all times.
This change in clock ticks from A, B is apparently caused purely by how the hypotenuse varies with the length travelled and (apparently) unrelated to physics of spacetime. It may be that mathematics and physics are more interrelated than we thought.
— just a thought experiment.
Continuing…
Let’s say that the car C is doing the honking every second while approaching the lighthouse from a distance of d meters at constant velocity. The car horn would ‘tick’ every second for A, while ticking faster every second for B.
C would seem to be accelerating towards the lighthouse for B, while for A it will be travelling at constant velocity!
So it seems that the notions of difference in simultaneity, constant velocity and acceleration are relative can be deduced purely from the Pythagoras theorem (for a body in motion).
