What is the reason of different clock's measurements in different frames? [closed]

Question

I tried to understand what is the reason of different clock's measurements in different frames.

For this I looked at easy example with distance S traveled by man with moving clock (velocity V1=2m/s) and distance traveled by man with another moving clock(velocity V2=1m/s). If clock is moving with velocity V1 then measurement T1=S/V1 and for V2 we have measurement T2=S/V2. First clock is more slowly than second clock.

It was bad example but something like example with time dilation effect where relation between velocity V and time T is more complicated (example with light clock, Pythagorean theorem).

Am I right that reason of different clock's measurements is velocity or am I missing something?

Answer 1

In physics there are two different concepts of time. “Proper time” is the physical time, denoted $\tau$, read by a good clock. Proper time is invariant, meaning that all reference frames agree on what a given clock reads. “Coordinate time” is the mathematical time, denoted $t$, used in some coordinate system. Different frames use different coordinates, so they disagree on coordinate time.

If the coordinate system is a standard inertial frame then the relationship between the two is $$c^2 d\tau^2=c^2 dt^2-dx^2-dy^2-dz^2$$$$\frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$$$\frac{d\tau}{dt}=\sqrt{1-\frac{v^2}{c^2}}=\frac{1}{\gamma}$$