In an inertial frame K, duration can be measured at each single location by using a uniformly ticking clock that is fixed right there.

**Note:** uniformity is ensured by producing all ticks by the very same method, so that we cannot think of any reason why one tick should take longer than the other.

The simultaneity of events that happen directly to entities __at rest__ relative to K is measured as follows:

**(Definition)** Two events are simultaneous in K if and only if a symmetrically placed observer in K sees them, by the naked eye (through vacuum), happen simultaneously.

This definition is compatible with the classical conception of time.

**Notes:** (a) by saying "observer in K", it is meant that the observer is also at rest relative to K, (b) we can imagine that each entity sends a light signal to the observer when its local event happens, (c) here and in the following, locations, events, entities, observers, signals and clocks are, unless explicitly stated otherwise, always meant to be point-like, (d) accordingly, by "light signal" it is understood just the tip of a ray of light.

Instead of light, other signals could also be used, e.g. pistol bullets or even carrier-pigeons, as long as the symmetry of their propagation can be assumed. Since we are talking about mechanics, it seems reasonable to require only the symmetry of the net forces acting upon the chosen pair of (identical) "messengers". Einstein's original approach eliminates this requirement elegantly: it uses light signals and assumes that __nothing__ whatsoever can make a difference to their propagation in vacuum. The same is definitely not true for e.g. a bullet whose motion is affected by a multitude of gravitational forces at the very least.

**(Definition)** Two clocks (at rest) in K are synchronized if the same positions of their hands are simultaneous events in K.

**Notes: **(a) in general, by saying "E in K" it is meant that E is at rest relative to K, (b) we'll always assume that all clocks are of the same construction.

Using mostly symmetry-based reasoning, we infer all the below:

**(Theorem)** If one position of the hands of two clocks in K are simultaneous events, then all positions are.

**(Proof) **There is no such difference in the situations of the two clocks that would explain why a symmetrically placed observer would see them showing different times, ever.

**(Lemma)** If a light signal is sent earlier from one location to another in K, it also arrives earlier.

**(Proof)** If synchronized clocks are used at the two locations, the difference between the time values at sending and arrival, as shown by the corresponding co-located clocks, is always the same. This can be seen if we add another pair of synchronized clocks that, when sending the second light signal, both show the time as it was when the first one was sent. The original and the added clocks run then in parallel, and there is no reason why the latter would not show the same time difference for the second light signal as it was for the first one.

**(Theorem)** All symmetrically placed observers in K judge the simultaneity of two events the same way.

**(Proof)** Let o_{1} and o_{2} be two symmetrically placed observers, and let o_{2} send light signals s_{1} and s_{2} to o_{1} upon seeing the two events e_{1} and e_{2}, respectively. Due to symmetry, o_{1} will observe the same time difference between seeing s_{1} and e_{1} as that between seeing s_{2} and e_{2}. Thus e_{1} and e_{2} are simultaneous to o_{1} if and only if s_{1} and s_{2} are sent by o_{2} at the same moment.

**(Theorem)** Simultaneity in K is transitive.

**(Proof)** Let event e_{1} be simultaneous with e_{2}, and e_{2} with e_{3}. If any two of the events are co-located, the statement is trivial. Otherwise, create an event e_{4} which is simultaneous with e_{2} but not located on any of the e_{1}e_{2}, e_{1}e_{3}, e_{2}e_{3} lines. Then, an observer in the circumcenter of the triangle e_{1}e_{2}e_{4} will see that e_{1} and e_{4} are simultaneous. Similar is true for the triangle e_{4}e_{2}e_{3}, i.e. e_{4} and e_{3} are simultaneous too. Finally, an observer in the circumcenter of the triangle e_{1}e_{4}e_{3} will see that e_{1} and e_{3} are simultaneous.

The last two theorems basically say that the definition of simultaneity is consistent.

For simplicity, it is assumed in the following that in every inertial frame, all clocks are already ticking in sync… just by coincidence.

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