Understanding special relativity

Let's look at the basic concepts needed to understand special relativity. The first one is the concept of frame of reference. The second one is the key to understanding all relativity: the invariant. Starting from these concepts, we'll view:

How classical relativity conceives space and time. Which are its invariants.
How special relativity modifies these invariants and the consequence that derives from them: space-time.

Frame of reference

In order to describe the trajectory of an object in motion we need to know its position at a given moment. This requires another object that will act as a point of reference (origin) and a clock to measure time. Once an origin has been chosen, we can specify the position of the object through a coordinate system. We thus have our frame of reference.

An inertial frame of reference is a frame that has no acceleration and therefore moves with uniform rectilinear motion. Two inertial frames are in a state of constant, rectilinear motion with respect to one another.

When we speak in relativity of an observer we're thinking about the frame of reference associated to this observer. The observer is said to be at rest in this frame.

In this diagram, we have an observer with a reference frame of two spatial dimensions: (x, y).
He measures that in his frame of reference, the coordinates of point P are (Px, Py).

Invariant

It is a quantity (value, measurement) on which all observers (frames of reference) agree.

1. Classical relativity

It is the view of the world, space and time as we perceive it with our intuition.

The first hypothesis on relativity was enunciated by Galileo. He illustrated it with this example: given two observers, one inside a boat that moves at a constant speed and another at rest on the shore, the observer inside the boat can't feel that he's moving unless he looks out of the window. Galileo came to the conclusion that the person inside the ship can't do any mechanical experiment to realize that he's moving. He enunciated his hypothesis of relativity in the following way:

Galileo's principle of relativity

The laws of mechanics are equal in all inertial frames of reference.

In classical relativity, this idea has a very important consequence: speed is not absolute. It can only be measured in relation to (it's relative to) a certain object (frame of reference). If we change the frame of reference, the speed may be different.

This is what our common sense tells us. If someone inside a train traveling at 100 km/h launches forward a flashlight at 50 km/h, an observer on the platform will notice that the flashlight moves at a speed of 100 + 50 = 150 km/h.

We'll see the temporal and spatial relationship between these two frames in Diagram 3: Galilean transformation

Space and time are NOT related to each other

As we've alredy said, classical relativity is a view of the world as seen with our intuition. Therefore, it considers space and time as two independent entities. It also assumes the existence of an universal time, which is the same for all observers.

The coordinates of a reference frame in space determine positions.

The invariants of classical relativity

Remember that an invariant is a quantity (value, measurement) on which all observers agree.

Time is itself an invariant, since there's the same universal time for all observers.
The invariant of space is distance. It is easily calculated with the Pythagorean theorem from the components measured on the axes of the reference frame. In a three-dimensional frame (x, y, z) the distance between two points is:

$$\boxed{d^2 = Δx^2 + Δy^2 + Δz^2}$$

Distance invariance

In this diagram we have two observers A (grey color) and B (blue color) with two-dimensional frames of reference.
Let's notice that frame B has translation and rotation with respect to frame A.
Observer A measures in his frame of reference that the distance is: $d^2 = Δx^2 + Δy^2$
Observer B measures in his frame of reference that the distance is: $d^2 = (Δx')^2 + (Δy')^2$
We can see that both observers measure the same distance (invariant), although in their respective frames of reference the components $Δx, Δy, Δx', Δy'$ are different.

2. Special relativity

The postulates of special relativity, stated by Einstein in 1905, extend Galileo's principle of relativity from mechanics to all the laws of physics. But they introduce a fundamental change: speed is not relative, there's a maximum limit that cannot be exceeded.

Postulates of special relativity

Special Principle of Relativity. The laws of physics are the same in all inertial frames of reference. In other words, there is not a privileged inertial frame that can be considered as absolute.
Invariance of the speed of light. The speed of light in vacuum (c) is a universal constant.

These postulates modify the classical conception of space and time. Speed is not relative to the observer as classical relativity asserts, but there's a universal constant more fundamental than space and time: the speed of light. Speed as conceived by classical relativity is a correct approximation as long as the speeds considered aren't close to the speed of light (c).

In our previous example of the train, if we launch a photon of light instead of the flashlight, the observer on the platform will always measure the same speed of light, whatever the speed of the train.

Space and time are related to each other

Notice that the constant speed of light (c) relates distance and time: $c = Δd / Δt$. Therefore distance and time are not invariant as in classical relativity, they must be adjusted so that c remains constant, and also they are related to each other. Because space consists of 3 dimensions, and time is a 1-dimensional, in relativity space and time are therefore interrelated in a four-dimensional geometric structure that we call space-time.

Let's remember that the coordinates of a reference frame in space determine positions. The coordinates of a reference frame in space-time determine events.

The invariants of space-time

Relativity is a geometric theory and therefore has its foundation in the equation that describes the geometric space studied by the theory. The equation that gives us the distance between two points in a given space is called the metric of that space. That's why the metric is of fundamental importance in relativity.

Let's recall that the metric of the flat 3-dimensional space of classical relativity (Euclidean space) is:

$$\boxed{d^2 = Δx^2 + Δy^2 + Δz^2}$$

The metric of the flat space of special relativity (also called Minkowski space-time) is:

$$\boxed{Δs^2 = -c^2 Δt^2 + Δx^2 + Δy^2 + Δz^2}$$

In geometry, a line segment is a part of a line bounded by two end-points. In space-time, the line segment between two points (events) is called the interval $(Δs^2)$. In its infinitesimal form, it is called the line element.

We have that the interval, being the distance between two points (events) in space-time, is an invariant in the same way that the distance between two points is an invariant in space. This is remarkable: just as in space there exists a constant in the distance between two points, in space-time there exists a constant between the "distance" of two events. This invariant is a direct consequence of the postulates of special relativity and can be directly derived from them. It reveals fundamental physical properties related to the nature of space and time.

Notice that the temporal component of the interval appears with dimensions of length $(cΔt)$ and with an opposite sign to the spatial components. Different observers may not agree on the magnitude of this component between two events in space-time. This is the phenomenon known as time dilation.

The Minkowski diagram will allow us to visualize space-time and all the phenomena related to special relativity without the need for mathematical equations. We'll see it in the special relativity diagrams.