*"The good mathematician generalizes from one example." (Source unknown)*

## Purpose

This essay is to distill the philosophical content of Space and time in special relativity. It can be achieved by focusing solely on the time aspect of special relativity, while at the same time making basically no use of formal proofs.

## Time in classical mechanics

Time is homogeneous and constantly passing. It consists of durations that events can occupy. To identify (measure) durations, clocks are used.

Time is absolute, it permeates the whole universe. All physical objects, even the most remote ones, are connected through time. They all "experience" the very same time.

## Inertial frames

**(Definition)** A reference frame in which the law of inertia holds is called an inertial frame.

The law of inertia states that the velocity of a particle (= point-like rigid object) may only change if there is force acting upon the particle.

**Note:** velocity is speed together with its direction; i.e. it's a vector.

Intuitively, this means that if all forces acting upon a particle were "switched off", it would continue to move at the constant velocity it has reached until then. Thus, there is a physical reason to think that not all reference frames are "equal", but some of them are "special", namely the inertial frames.

It is true in both classical mechanics and special relativity that:

**(Theorem)** Given any one inertial frame, exactly those reference frames are inertial frames which are moving at a constant velocity relative to it.

## Principle of relativity

The classical laws of motion are exactly the same in every inertial frame. That is, with respect to classical mechanics, inertial frames are all equivalent.

The following, plausible generalization is called the principle of relativity, and its validity is generally assumed in physics:

**(Principle)** The laws of nature are exactly the same in every inertial frame. That is, inertial frames are all equivalent in describing __any__ physical phenomena.

## The scope of a theory

**(Definition)** The circumstances under which a theory can be considered correct are called the scope of the theory.

The originally intended scope can shrink as new facts come to light. This happens when it turns out that some (often tacit) assumptions cannot be considered correct in all situations. In special relativity, two such assumptions are Euclidean geometry and continuous quantities. Both have already been challenged by later developments in physics.

There are plenty of other, tacit assumptions which we don't even notice. We can rest assured that all those will be challenged some day. Still, it is more constructive to say that the assumptions are correct within limits, rather than saying they are just incorrect.

## Time in special relativity

### Speed of light

It is a well-established experimental fact that:

**(Law)** The tip of a ray of light propagates in vacuum at a constant c ≈ 300'000'000 m/s speed, always along a straight line, relative to __every__ inertial frame.

**(Corollary)** The speed of light in an inertial frame is independent of the state of motion of the entity that emits it.

This is contradictory, since according to classical mechanics the speed of a given entity, i.e. that of the tip of the ray of light in this case, should depend on the reference frame in which it is observed. To eliminate the contradiction, it was necessary in special relativity to revisit and challenge our intuitive concepts about space and time.

In order to ensure that we are not misled by intuition, Einstein suggested that all definitions in physics must be based on measurements which are, in principle, feasible.

### Simultaneity 1

Let K denote an inertial frame.

**(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.

**Note:** by saying "observer in K", it is meant that the observer is also at rest relative to K.

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.

### Coordinate time

It can be shown that it's possible to synchronize all clocks of interest in an inertial frame, so that they are all ticking in sync.

**(Definition)** In an inertial frame K, the time read off from its synchronized clocks is called the coordinate time of K.

The coordinate time of an event can be read off from the clock in K that is momentarily co-located with the entity to which the event happens.

As long as there is only one inertial frame considered, time seems no different from that of classical mechanics.

### Experience vs. definition

As for the simultaneity of two or more events, naked-eye observers placed at arbitrary (non-equidistant) locations would typically come to different conclusions, due to the finiteness of the speed of light. This suggests that __inertial frame wide simultaneity is not a direct experience but a mere definition__, based on an agreed way of measurement. Similar is true for coordinate time, since it also builds on the concept of simultaneity.

**Note:** already in the case of duration, two co-located observers need to agree on using an independent device (i.e. a clock) in order to avoid ambiguity.

It does not harm to imagine that the above concepts describe a directly intangible "objective reality" in an inertial frame, but as a matter of fact they are essentially just tools that help us in calculating answers to questions about our (more) direct experiences. The most important question to answer is this: between two events that happen directly to an observer, how much time elapses according to the observer's own clock?

### Causality

The following suffices for our purposes:

**(Assumption)** Particles can be destroyed: if a particle gets destroyed, its trajectory is discontinued in every inertial frame. That is, if t_{K} denotes the time of destruction relative to inertial frame K, then the trajectory will not exist for any t > t_{K} in K.

**Note:** backed by the fact that the speed of light is finite, similar can be assumed about signals too.

**(Theorem)** The order of two events e_{1} and e_{2 }that happen directly to a given particle is the same relative to every inertial frame.

**(Proof)** Let K and K' be two inertial frames, and let e_{1} and e_{2} happen at t_{1} and t_{2} in K, respectively, with t_{1} ≤ t_{2}. Let t_{1}' and t_{2}' denote the corresponding time values in K'. Moreover, let the particle get destroyed in K at t_{2}, which corresponds to t_{2}' in K'. Then, since the particle does exist in K' at t_{1}', t_{1}' ≤ t_{2}' must hold. Finally, t_{1}' = t_{2}' if and only if t_{1} = t_{2}, so t_{1}' < t_{2}' if and only if t_{1} < t_{2}.

The argument can be applied to signals too:

**(Theorem)** The order in which a signal meets two given entities is the same relative to every inertial frame.

**Note:** the tip of the signal plays the role of the particle.

### Upper limit of speed

Let K and K' denote two inertial frames, with K' moving along the x-axis of K in the positive direction, at a constant speed v.

**Note:** vectors are written in boldface, e.g. velocity **v**, while scalars in plain text, e.g. speed v = |**v**|.

Let AB be a segment of non-zero length in K, parallel to the x-axis, with x_{A} < x_{B}. Let A' and B' denote the corresponding points in K', marked out at some given time t in K.

In K', let's send a light signal from A' toward B'. It will arrive at B' after some time Δt' > 0. For the corresponding duration in K, Δt > 0 must hold due to causality. (Relative to K', the light signal meets first A' and then B', so it happens in the same order relative to K too.)

Both A' and B' are moving at **v** relative to K, so when the light signal is emitted, B' is ahead of A' in K. Later the light signal meets B', which means it is traveling in K along a straight line parallel to the x-axis, in the positive direction. The fact that it does catch up with B' implies:

**(Theorem)** v < c.

As for the speed limit of signals (i.e. not inertial frames), it can be shown that due to causality no signal can travel faster than light.

### Time dilation

Combining the principle of relativity with the constancy of the speed of light leads to a conclusion about time that is foreign to classical mechanics:

**(Theorem)** If an observer is moving at velocity **v** relative to an inertial frame K, then after Δt time elapsed in K, the observer's own clock will show only a corresponding Δt' = Δt · √1 – v2 / c2 time elapsed.

Thus, in general, the (coordinate) time that elapses between two events depends on the inertial frame in which it is measured. The effect is called time dilation since Δt > Δt' holds whenever v > 0 and Δt' > 0. In other words, the observer moving relative to K always sees as if time was passing faster in K.

**Note:** time dilation is unnoticeable in our everyday life.

The formula works for polygonal trajectories as well, provided that the speed is the same along all edges.

**(Assumption)** Any motion (of a point-like entity) can be approximated with arbitrary accuracy by polygonal motion.

This makes the formula valid for any motion of constant speed; even for circular ones as a matter of fact.

### Simultaneity 2

Let K and K' again denote two inertial frames, with K' moving along the x-axis of K in the positive direction, at a constant speed v > 0.

A consequence of time dilation is that:

**(Theorem)** At any given time t in K, the corresponding coordinate time t' in K' that is read off by an observer in K decreases as the x-coordinate of the observer's location increases. For an increase of Δx, there is a decrease in t' by Δx · (v / c^{2}) / √1 – v2 / c2 .

So two events that are simultaneous in K do not happen at the same (coordinate) time in K', unless they have the very same x-coordinate.

### Interpreting simultaneity

One might think of simultaneity as a bond between certain events. If two events in an inertial frame K happen at the same coordinate time, they are connected by "now".

However, two events that happen at different times in K can also be connected by "now", provided there exists another inertial frame relative to which the very same two events happen simultaneously. Moreover, assuming that the bond is transitive, it can be shown that in fact any two events are connected this way, i.e. all the events of the world (ever) are simultaneous. Yet at our outset, non-simultaneous events did seem to exist.

Instead of spending a lot of time figuring out what this really means, we rather throw out this elusive, action-at-a-distance-flavor bond and interpret inertial frame wide simultaneity as a mere definition without any immediate physical content.

### Absolute time?

Absolute time can be represented by a geometrical straight line T. Points on T correspond to moments, while distances to durations. The idea behind absolute time is that every event can be mapped to exactly one point on this single T, because there exist absolute temporal relationships among events.

However, time dilation revealed that the coordinate time of any one inertial frame is not absolute time, since both simultaneity and duration are inertial frame dependent. To be precise, from the perspective of absolute time what was demonstrated is this: when two inertial frames are moving relative to each other, it cannot be that the coordinate time in __both__ of them measure absolute time. Yet the principle of relativity suggests that the situation of the two inertial frames must be symmetrical. It would not fit in the picture if it was possible that the coordinate time in one frame is absolute, while in the other it's not… and to make matters worse, there would be no known way to tell which frame is which.

This is the point where we must realize that time dilation has left us with no tangible evidence but only one thing supporting the idea of absolute time: our imagination, i.e. the intuition that we've gained from the limited spectrum of our day-to-day experiences. Time dilation has refuted the strongest argument we thought we had for absolute time. Namely, it's not true that two clocks always show the same time elapsed between any two of their encounters. To test this, we don't even need to define simultaneity and coordinate time, all we need is the two clocks.

That's why, since it does not seem to help to go back and try to adjust the simple and robust definitions that led us here, the choice has been made in special relativity to rather give up absolute time.

## Conclusions

What we've seen so far of the time aspect of special relativity allows us to already formulate quite a few philosophical ramifications.

### Do we understand relativity?

The major obstacle in coming to terms with relativity theory is the objection from our own intuition. In addition to that, the formulas of special relativity are apparently more complicated than those of classical mechanics. This all happens because we seek an understanding of new phenomena in terms of such mathematical and physical concepts, and even senses, that developed for a very long time against a fundamentally different backdrop.

It does not help either that special relativity discovers more the "what" than the "why". As a logical process, it derives the strange consequences of a counter-intuitive assumption about light. It would be better for understanding if one could derive the same consequences from a more intuitive assumption instead. (It's not as hopeless as it sounds, take for example the many surprising consequences of the non-surprising law of conservation of angular momentum.)

In the future, when relativistic effects become more and more a common experience, the concepts and formulations of the theory, as well as our intuition, will adapt and hopefully make relativity look simpler and more intuitive to grasp. Nevertheless, as suggested by von Neumann, we never really understand a theory. It can only become intuitive at best, once we've managed to get used to it.

### Reality and illusion

In the light of relativity theory, we can say that our perception of absolute time is an illusion: beyond the scope of classical mechanics, empirical evidence does not support it any longer.

The illusion arises due to restrictions prevailing in our environment: the relatively low speeds and short distances of physical objects in our everyday life, coupled with the limited accuracy of our senses and measurements. As soon as the restrictions are relaxed, we cease to perceive time as an absolute entity: there is no such thing anymore as an observer-independent, globally passing time.

In a sense, everything, or rather, every property we perceive is an illusion and ceases to exist as soon as the horizon of our perception and measurements sufficiently broadens. (Another example: is it due to causality that no signal can travel faster than light, or is it because nobody has ever seen a signal traveling faster than light that causality appears as a law of nature?)

Does then special relativity capture the real nature of time? Well, on the one hand it definitely does, in its own scope. But on the other hand it does not, for the real nature of time is that eventually, it doesn't exist.

## Related reading

A.J. Christian (2003), *Mit keresett Isten a nappalimban?*

Y. Lin (1948), *The Wisdom of Laotse*