Published at : 10 Nov 2021

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If you’ve seen a distance-time graph before, you’ll pick up spacetime diagrams very quickly… and understand their true power in visualizing special relativity!

A spacetime diagram is any diagram that shows the behavior of an object through space and time. However, the term most commonly applies to Minkowski Diagrams, which have position plotted on the horizontal axis, and (speed of light x time) plotted on the vertical axis.

We start by seeing that a stationary object (in our reference frame) can be represented with a vertical line on our spacetime diagram. This line is known as the "world line" of our object, and shows how it moves through space and time. We also see that objects moving at a constant speed will have straight lines (not necessarily vertical) as their world lines, and objects accelerating will have curved world lines.

Next, we consider a pair of events to show on our diagram. Events are defined by 4 coordinates in relativity - one time coordinate, and three spatial coordinates. To keep things simple we just consider 1 time and 1 spatial coordinate, and assume the other 2 spatial coordinates are the same for all objects being considered.

An event is just shown as a point on our spacetime diagram. And when we plot two events (such as a switch being flicked, and a light bulb switching on), we can see whether or not the two events are "causally connected". In other words, can the first event communicate with the second event before it happens, and therefore is it possible for the first event to cause the second one?

On our spacetime diagrams, the vertical axis shows ct (speed of light x time) rather than just time. For this reason, a photon (or any other object travelling at light speed) will show up as a straight line at 45 degrees to either of the axes (in our stationary reference frame diagram anyway). This is the fastest any object is allowed to travel in the theory of relativity.

This means that we can draw 45 degree lines out from the first event, and see if the second event can be causally linked (or causally connected) to it. If the second event lies to the top left of the 45 degree line, this means some signal could be sent from event 1 before event 2 occurs (and therefore could be the cause of event 2). However if the second event lies to the bottom right of the 45 degree line, then a signal would have to travel faster than the speed of light (not allowed) for the two events to be causally linked. In other words, these events are independent.

With a more rigorous approach, we see how events can be described as being in the "future", "past", and "elsewhere" of a particular event. We also realise that if we extended our diagram to more spatial dimensions, the light lines would actually make a cone - this is known as the light cone of the event we are considering.

Next, we see how the spacetime diagrams for two observers moving relative to each other (at a constant speed) can be plotted on the same graph. For an observer moving relative to us, the axes representing their frame of reference will be "squished", towards the 45 degree photon line. The faster the second observer moves relative to us, the more "squished" their axes.

We also see how we can find a reference frame where two events occur at the same time for the moving observer, even if they appear to occur at different times in our original reference frame. Similarly, we can find a reference frame where two events happen at the same spatial coordinate, even though they happen at different places for the original observer. Both observers are "correct" in both cases, as there is no one preferred inertial reference frame in special relativity.

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Timestamps:

0:00 - What are Spacetime (Minkowski) Diagrams?

1:46 - World Lines, and Why Spacetime Diagrams are Useful

3:25 - Events in Special Relativity - (t,x,y,z) Coordinates in 4 Dimensions

4:42 - The Causal Connection Between Events (and the Speed of Light on Spacetime Diagrams)

9:27 - The Light Cone of an Event (Future, Past, and Elsewhere)

10:17 - Spacetime Diagrams for Observers Moving Relative to Each Other

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