Classical relativity diagrams

Diagrams of classical relativity

Let's look at a successive series of space and time diagrams in order to learn how to interpret them. We're in classical relativity, so although the way of looking at these diagrams will help us understand the diagrams of special relativity, a fundamental difference must be taken into account: in classical relativity, space and time are two separate entities.

The ultimate goal is to understand the implications that classical relativity has on our conception of reality. We'll see it in the different spaces or "worlds" in Diagram 4.

Diagram 1

We have two inertial observers A (grey color) and B (blue color) in relative motion between them.
Let's consider a single spatial dimension (x).
Let's take observer A as a reference source. Observer B therefore moves with uniform rectilinear motion relative to A.

Diagram 2: Events and Trajectories

Let's measure the temporal dimension (t) on the vertical axis.
Let's take as origin of coordinates (t, x) = (0, 0) the moment when A and B share the same x position.
In this coordinate system, points represent events. They determine the x position of an object at a certain T time.
The blue line therefore shows the trajectory of B in the reference frame of A. For example, at moment t = 3, observer B is at position x = 1.8 (event E1).

Diagram 3: Galilean transformation

Let's take the path of B as the axis (t') to measure distances from observer B. (In this example diagram, the distance to the tree).
Lines parallel to t-axis are at the same distance from A over time.
Lines parallel to t'-axis are at the same distance from B over time.
Lines parallel to x-axis are the set of simultaneous events for A and B at a given time T.
Notice that the x-axis of observer B's reference frame matches the x-axis of observer A.

Galilean transformation

We take into account that space = speed x time. Therefore, an object moving at constant speed v for a time Δt travels the distance vΔt. We can easily see the equations that relate the coordinates of the two reference frames A and B, known as the Galilean transformation:

$$x'=x-x_{B}=x-vΔt$$ $$t'=t$$

Diagram 4: worlds formed by simultaneous events

A space or "world" is the set of simultaneous events for a certain observer at a certain time T. It's the equivalent of a film frame.
The "now" of a certain observer is the "world" that he considers as his present.
This diagram shows an example with four worlds. It represents the classical view that our common sense tells us. There is only one "now", which we call the present, and it's the same for both observers. Events that correspond to past and future worlds don't exist in our present. In the special relativity diagrams, we'll see that this view is not correct.