An event is a point in space-time. Its coordinates are (t,x,y,z) or (ct,x,y,z). These points are called world points and the path they take is called a world line. A uniform rectilinear motion is a straight world line.
We consider a frame K with axes (t,X,Y,Z) and a frame K' with axes (t',X',Y',Z'), but we choose X to be matching X', while Y is parallel to Y' and Z parallel to Z'. The systems move relative to each other, with constant velocity. Their relative motion is along x, since X and X' are one on top of the other.
In the K system, we observe a signal departing from an event (ct1,x1,y1,z1) to an event (ct2,x2,y2,z2). The *interval* between these events for the K observer is
(ct₂-ct₁)² - (x₂-x₁)² - (y₂-y₁)² - (z₂-z₁)² = 0.
In the K' system, an observer would observe the signal between the same events and would measure an interval
(ct₂'-ct₁')² - (x₂'-x₁')² - (y₂'-y₁')² - (z₂'-z₁')² = 0.
They observe the same interval! Is this a thing that happens only for events separated by signals? Let's wait for the answer. The interval s₁₂ between any two events is
s₁₂ = (ct₂-ct₁)² - (x₂-x₁)² - (y₂-y₁)² - (z₂-z₁)² .
If the events are infinitely close to each other, we write the interval as
ds² = c²dt² - dx² - dy² - dz² .
This is not a usual (Euclidean) way to form an interval, because of signs. We can imagine the distance ds to be living in a four dimensional space-time, but then we must be aware of not adding all terms. This "geometry" is called pseudo-euclidean and was introduced by Minkowski.
We already known that for two signal-separated events we get ds' = ds = 0. If the two events are not separated by a signal, then ds and ds' are infinitesimals of the same order, so they must be proportional to each other, ds² = a·ds'².
What is the order of an infinitesimal? The order of an infinitesimal a wrt to another infinitesimal b is a number n such that the limit of a/bⁿ exists and is not 0 or infinite.
In our case, two infinitesimals are of the same order if the limit of the ratio of both exists and is not 0 or infinite. It is clear that ds'/ds must give a finite number that we call a.
Also, the coefficient a can only depend on the absolute value of the relative velocity between the two inertial systems. Why? Imagine a would depend on coordinates or time. Then, different positions and different moments in time would not be equivalent, in contradiction to the homogeneity of space and time. The value of the interval cannot depend on whether you take the measurement tomorrow or yesterday. Or on whether you are closer or farther from the events. It cannot depend either on the direction of the relative velocity because that would be in contradiction to the isotropy of space.
Let's consider three isor: K, K1 and K2. Let v1 be the relative speed of K1 wrt K and v2 the relative speed of K2 wrt K. Then,
ds² = a(v₁) ds₁² ds² = a(v₂) ds₂² ds₁² = a(v₁₂)ds₂² ,
where v12 is the relative speed of 1 wrt 2. These equations imply
a(v₂) / a(v₁) = a(v₁₂) .
Now notice that v12 = ||v2> - |v1>| so it depends on the relative angle of frames 1 and 2. And this cannot be, due to isotropy of space. This means that a(v) must be a constant, and this constant must be equal to 1. Then,
ds'² = ds² .
This also applies to finite intervals: Δs'² = Δs² and s'² = s².
This is very important. The interval between any two events is the same in all isor.
Now let's look for particular instances of that. For example, given two events (ct1,x1,y1,z1) and (ct2,x2,y2,z2) in sor K, is there any system K' for which these two events occur at the same point of space?
We introduce this notation
t₂ - t₁ = t₁₂ (x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)² = l²₁₂.
Then, the interval between events 1 and 2 is s²₁₂ = c²t²₁₂ - x²₁₂. For the K' system is s'²₁₂ = c²t'²₁₂ - x'²₁₂. But clearly s'=s', so c²t'²₁₂ - l'²₁₂ = c²t²₁₂ - l²₁₂.
If we want both events in K' to be at the same spatial point, we require l'²₁₂ = 0. This implies
c²t²₁₂ - l²₁₂ = c²t'²₁₂ > 0.
This means that such isor K' exists if s²₁₂ > 0. This requires the interval between the two events to be a real number. Real intervals are called *timelike* intervals.
What is the time interval between two events forming in a timelike interval? It is t'²₁₂ = s₁₂/c.
If two events belong to the same body (two points in the same world line), the interval between them is always timelike, since the body cannot go from event to the other with s₁₂ < 0.
Now, let's ask about finding an isor K' such that two events 1 and 2 are seen by K' as simultaneous. Again we have
s'²₁₂ = c²t'²₁₂ - x'²₁₂ s²₁₂ = c²t²₁₂ - x²₁₂.
but now we impose t'²₁₂ = 0, so c²t²₁₂ - x²₁₂ = s²₁₂ = -x'²₁₂ < 0. This requires the interval to be an imaginary number. Imaginary intervals are called *spacelike*.
What is the spatial distance between two events forming a spacelike interval? It is l'²₁₂ = is₁₂.
Since the distinction between timelike and spacelike depends only on the interval, which is invariant, such distinction is also invariant.
We can visualize a sor (ignoring y and z) in two dimensions x and ct. The origin is event O.
ct (···)
| signal
· | ·world
· absolute · lines
· future ·
· | ·
· | ·
· | ·
absolutely · | · absolutely
-----------------O----------------------- x
disconnected · | · disconnected
· | ·
· | ·
· absolute ·
· past ·
· | ·
· | ·
· |
This is what we call a light cone (if we add a y axis, the concept of cone becomes clear by rotating the figure around the ct axis).
Any world line of a particle passing through event O must be contained within the cone, coming from the absolute past and evolving to the absolute future. The two regions outside the cones (left and right) are causally disconnected from the present seen by O, since not even signals can connect events there to O. For a uniform rectilinear motion, we have straight lines in which the inverse of the slope gives us the speed of the moving object. The maximum possible slope is 1, corresponding to signals travelling at 𝑐.
All events within the cones are (wrt O) timelike, and all events outside the cones are (wrt O) spacelike.
All timelike (wrt O) events occurring after (in time) event O have a positive time interval t > 0. We have seen that we cannot find find any isor that allows us to see both events simultaneously (with t = 0). And even more impossible is to find an isor for which we can reverse the sign of the time interval, so that t < 0. This means that the concept of past and future (wrt O) is also absolute. Causality is absolute, like the speed of light. So much for those who say that relativity means that "all is relative"!
Two events can be causally related only if their interval is timelike. Otherwise they are absolutely disconnected. In other words, no interaction can depart from one event and reach the other.
The concepts of earlier and later are also absolute, even if how much earlier or how much later are not absolute quantities.