I was reading Classical Mechanics : The theoretical minimum by Leonard Susskind, and he says
Assume that two clocks at different places can be synchronised.
I don't understand why one should do that. Can't one clock at the origin be enough? Whenever I try to start on special relativity, this crops up. Can someone explain this to me or at least point me towards any resources which explain such issues in detail? Especially special relativity related.
All of special relativity is based on the assumption that any observer can set up a coordinate system and then label spacetime events with their coordinates in that system. Then we can use the Lorentz transformations to transform between the coordinate systems of different observers.
The positions of events are easy because I have an infinite number of rulers and simply by laying them one after the other I can create a grid that fills all of spacetime. Then when some event happens my colleague who happened to be standing where the event happened can just look at my rulers and note down the position.
But the time is trickier. Time measurements are easy for events at my position because I just look at my clock and note the time. But for any distant event I have to ask my colleague next to the event to note the time on their clock. I could wait for the light from the event to reach me, and subtract off the travel time to get the original time of the event, but this is now an indirect measurement of the time. This workable in SR, but in GR light travel times are impossible to calculate unless I know the exact trajectory the light took, and indeed the light could reach me by multiple paths as happens in gravitational lensing.
So the only safe option is to put a clock at each point of my grid of rulers then synchronise them all. That way the event coordinates can be recorded by a colleague standing at that point. But this only works if all the clocks can be synchronised, and this is harder that it appears at first sight. If I move my clock to yours so we can synchronise them my clock will be time dilated by the motion and this spoils the timing. That's why we resort to protocols like Einstein synchronisation.
Now this is all conceptual rather than realistic and we clearly don't actually measure events this way. However it is a concept that is at the heart of special relativity, and that's why books on SR tend to labour the point.
Two clocks are required to show that the time and place where an event occurs is relative to the two observers (one moving and one stationary), even if the two clocks were synchronized in the first place. This is usually expressed using the Lorentz transformation, \begin{align} t' &= \gamma\left(t-vx/c^2\right) \\ x' &= \gamma(x-vt) \end{align} where $(x,\,t)$ are the coordinates in the stationary frame, $(x',\,t')$ the coordinates in the moving frame (moving with velocity $v$) and $\gamma^2=1/(1-v^2/c^2)$.
For an arbitrary time $t>0$, we find that $(x,\,t)$ and $(x',\,t')$ have different values, indicating that each of the two observers sees the same event happening at a different time and place as the other observer. Hence, two clocks would be required to analyze the scenarios, rather than a single one.
When we want to compare clocks, there must be a method to compare them. Since time is not absolute in relativity theory, different observers will disagree on what events they consider simultaneous. So comparing clocks is no longer unambiguous.
To be able to compare clocks, therefore, we must have a method that yields the same outcome for all observers, regardless of these observers’ own motions. This is clock synchronization.
