Special relativity diagrams

Diagrams of special relativity

These diagrams are a continuation of the classical relativity diagrams.

Unlike classical relativity diagrams where space and time are different entities, special relativity diagrams are a visualization of space-time, since the temporal component is one of its dimensions.

As with classical relativity diagrams, the ultimate goal is to understand the implications that special relativity has on our conception of reality. We'll see it in the different "worlds" of Diagram 10.

Relativistic speeds

They are speeds close to the speed of light (c). All the speeds that we're going to see in the special relativity diagrams are relativistic.

Diagram 5

Along with observers A (grey color) and B (blue color) in Diagram 1, we have now a photon (red color) that moves in the same direction as observer B.
We start from the initial situation where observers A, B and the photon are located at the origin (0). A and B synchronize their clocks.
We have that after a certain time T has elapsed according to observer A, observer B is at a distance vT and the photon at a distance cT.
What does observer B measure? We're going to analyze it using the Minkowsky diagram.

Diagram 6: ct instead of t on the time axis

On the time axis, instead of t we're going to measure ct, which is the temporal component of the interval, the invariant of space-time.
At time T, a photon travels the distance cT. Since the two axes (ct, x) measure this same distance, the light path is at 45º.
Although the ct axis has distance dimensions, the corresponding units of time are usually indicated. If we measure seconds, the x-axis will then measure light-seconds; if we measure years, the x-axis will measure light-years, and so on. The units in the example diagram are seconds and light-seconds.

Diagram 7: Galileo's transformation is not valid for relativistic speeds

Let's see how the classical (Galileo) transformation doesn't keep the speed of light constant for observer B.

Since the two axes (ct, x) measure the same dimension (distance), the speed $(v= x/ct)$ is a dimensionless that corresponds to the percentage of c. In the example diagram, observer B has a speed $v = (3/5) c$.
Note that according to this diagram, in event E2, observer B would measure a light-speed of: $v' = (cT-vT)/cT < 1$, therefore the classical (Galileo) transformation is not correct for relativistic speeds since it doesn't keep the speed of light constant (c).
We can see that the condition for observer B to measure constant light-speed (c) is that the projections of the photon's event (E1) on the axes (ct', x') be equal so that $v' = x'/ct' = 1$. We have to rotate the x'-axis so that the light path is the bisector of (ct', x'). We'll see it in the next diagram.

Diagram 8: Minkowsky diagram

The Minkowski diagram, also known as the space-time diagram, is the geometric representation of the properties of space and time that emerge from the postulates of special relativity. It allows to easily visualize the phenomena that are deduced from the equations: relativity of simultaneity, time dilation, etc.

As we've seen in the previous diagram, we have to rotate the x'-axis so that the light path is the bisector of the axes (ct', x'). This way both observers A and B will measure constant light-speed (c).
We see that both observers can only measure constant light-speed (c) if they measure different times for the E1 event. The constant speed of light (c) implies that there is no an absolute time, each observer measures his own time in his own frame of reference.
The path that an object follows through space-time is called its world-line.

Lorentz transformation

The special relativity equations that relate the coordinates of the two reference frames A and B are known as the Lorentz transformation. Unlike Galileo's transformation, these equations do take into account the constant speed of light.

Historically, they predate the concept of space-time. It was Hermann Minkowsky who gave the geometric interpretation, who saw that these equations are a consequence of and are derived from the geometry of space-time.

$$t'= \gamma(t -vx/c^2)$$ $$x'= \gamma(x -vt)$$ $$\gamma = \sqrt{1-v^2/c^2}$$

The slopes of the axes (ct', x')

Applying the Lorentz transformation we can mathematically obtain the slopes of the axes (ct', x') of the Minkowsky diagram:

Slope of ct'-axis
The ct'-axis corresponds to events where x' = 0

$x' = \gamma(x-vt) = 0$
$\implies cx - cvt = 0\implies \boxed{ct = (c/v) x}$

Slope of x-axis
The x'-axis corresponds to events where t' = 0

$t'= \gamma(t - vx/c^2) = 0$
$\implies ct -vx/c = 0 \implies \boxed{ct = (v/c) x}$

Diagram 9: the relativity of simultaneity

Lines parallel to x-axis are the set of simultaneous events for observer A at a certain time T.
Lines parallel to x'-axis are the set of simultaneous events for observer B at a certain time T'.
Lines parallel to ct-axis are at the same distance from A over time.
Lines parallel to ct'-axis are at the same distance from B over time.
Notice how simultaneous events E1 and E2 for observer B are not simultaneous for observer A. This is known as relativity of simultaneity: events that are simultaneous (occurring at the same time) in a frame of reference may not be so in a different framework. Simultaneity is therefore relative, it depends on the observer. It is impossible to say absolutely that two events happen at the same time.

Diagram 10: the space-time continuum

In Diagram 4 we've seen the concepts of "world" and "now" applied to classical relativity. Now we'll see how they apply to special relativity.
When observer B is next to the house (event E1), the "now" of observer A is world 1, while the "now" of observer B is world 2.
We see that in world 1 the photon is located before the tree (event E2), but in world 2 it's alredy surpassed the tree (event E3). We have therefore that when B is next to the house (E1), the photon exists simultaneously both before (E2) and after (E3) of the tree.
Assuming that we are observer A, our intuition tells us that when B is next to the house (E1) there only exists world 1. But we see that B lives in a different world (world 2) formed by events that are in our future and others located in our past. We are witnessing the most surprising consequence of special relativity: the existence of the space-time continuum, where all past, present and future events exist simultaneously.
Let's analyze it a little more in detail:
- We're studying what inertial observers A and B in relative motion measure when observer B is next to the house (event E1).
- Remember that we start from the initial situation where A, B and the photon are in the origin event (0, 0) with the clocks of A and B synchronized.
- Observe how when B is next to the house (E1), the photon in E2 implies constant speed of light measured by A.
- Observe how when B is next to the house (E1), the photon in E3 implies constant speed of light measured by B. The distances (0-E1) and (E1-E3) are equal.
- In short, for both A and B to measure the same light-speed (c) at event E1 (observer B next to the house), the photon must exist simultaneously in both E2 and E3 events. But not only the photon, we've also the two events of the tree and the two events of observer A: all events exist in the four-dimensional space-time continuum.