e2m2a said:
Ok this concept of proper time is obviously subtle. They say its the time of a moving clock along its worldline. But the velocity of this worldline, the ratio of its space/xt coordinates should differ with respect to different inertial observers. So wouldn't each inertial observers deduce a different proper time for the clock?
The problem with learning (special) relativity is that you have to relearn the concepts behind how to measure spatial distances and how to define time at different places.
On the other hand special-relativistic spacetime, the socalled Minkowski space has a lot in common with Galilei-Newton spacetime. First of all by assumption Newton's 1st law is valid in SR as well, i.e., there exists a class of inertial reference frames, where bodies which are not interacting with anything else, move in straight lines with constant velocity. Also by assumption for any observer at rest wrt. an inertial frame the geometry of space is Euclidean and there exist standard clocks, which are in practice realized with very high precision by atomic clocks (like the "cesium standard" defining the second as the unit of time via the frequency corresponding to a hyperfine transition of the Cs-133 atom).
The difference between Galilei-Newton spacetime and the spacetime model of special relativity then comes with Einstein's postulate about the "invariance of the speed of light in vacuum", i.e., the independence of the speed with which electromagnetic waves propagate in the vacuum from the velocity of the light source relative to an inertial observer.
Now this can be used to construct spacetime measures by a convention, how to synchronize standard clocks, all of which are assumed to be at rest wrt. an inertial frame of reference at different places. With one standard clock, say located at the origin of the inertial frame in question, you can only measure the time at this one place, i.e., the duration between two events taking place at the origin of the reference frame. This was Einstein's great insight into the problems of electrodynamics with regard to the non-invariance under Galilei transformations. To synchronize clocks at different places you have to use the postulate of the invariance of the speed of light, and to define a "global" time in an inertial reference frame you need to choose some convention. The convention is that you consider all clocks to be at rest within the inertial frame under consideration and then they are tuned such as to be consistent with the invariance of the speed of light. This can be done by using one reference clock at the origin and sending a light signal to another clock at a given distance from the origin, letting it reflect back to the origin and measuring the time it takes for the signal to propagate to the distant clock and back and assume that the time it takes to move from the origin to the distant clock is half of the time measured for the light signal to go forth and back. This is in accordance with the assumed symmetries of the Euclidean space (isotropy and homogeneity) used to define the distance between the clocks. In this way you can prepare the clocks in advance, putting them to times ##r/c##, where ##r## is the distance of the clock from the origin, and ##c## the speed of light in a vacuum, send a light signal from the origin to all the distant clocks, and start them when this light signal arrives at their corresponding places. Then by this convention (!) these clocks, all at rest within the inertial frame where this procedure is done, are all synchronized.
From this operation it is now clear that for any other inertial frame, where one has done the same synchronization procedure, that the clocks in different inertial frames are not synchronized relative to each other, but when transforming between different inertial frames, being in uniform motion relative to each other, you have to transform both the time, defined for each inertial frame by the synchronization procedure, and the spatial coordinates of "events" from one frame to the other. That results in the socalled Lorentz transformation, which substitutes the Galilei transformation of Newtonian mechanics, and with this transformation you can derive all the kinematic effects resulting from the symmetry assumptions of the Minkowski spacetime model, including the invariance of the speed of light in vacuum:
-the "relativity of simultaneity": To distant events which are at the same time as observed in one inertial frame are usually not simultaneous as observed in another inertial frame of reference moving with respect to the other.
-"time dilation": A standard clock moving wrt. an inertial frame with velocity ##v## "ticks slower" by a "Lorentz factor" ##\gamma=1/\sqrt{1-v^2/c^2}## compared to the time defined in this inertial frame (by the clock-synchronization convention explained above).
-"length contraction": When measuring the length of a rod, which is at rest wrt. an inertial frame ##\Sigma##, ##L_0##, the length of the as measured in another inertial frame ##\Sigma'## moving with velocity ##v## relative to ##\Sigma##, is shorter by an inverse Lorentz factor, ##L'=L_0/\gamma=L_0 \sqrt{1-v^2/c^2}##.
This results from the fact that the observer in ##\Sigma'## measures the length of the rod by marking the coordinates of its two end points simultaneously within his frame. These "measurement events" are not simultaneous when observed from the inertial frame ##\Sigma##, and this is the origin of the length-contraction effect.
For the mathematical details of SR, see
https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
although there the Lorentz transformation is derived in a somewhat different more formal way than using Einstein's clock-synchronization procedure. It is well worth to read the first paragraphs about this "kinematical part" of SR in the original paper:
https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bodies
