Spacetime diagram

The work of a curious fellow
Ignoring two dimensions and still getting the message...

First of all, read the spacetime notes to get the necessary background. It is not convenient to plot more than two dimensions on a flat screen so the spacetime diagram normally plots the x and t dimensions of spacetime, with x horizontal and t vertical. Spacetime diagrams represent a snapshot at a particular instant. Below is a basic spacetime diagram showing two events in spacetime, one marking a commonly used zero on the time axis, the other perhaps less widely known.

Basic spacetime diagram

In addition to isolated events we can plot the events encountered by an object on its journey through spacetime. The resulting path is called the object's worldline. Shown below are the worldlines of three objects. The red line represents an object at rest in the reference frame of this diagram. The blue object is moving left at a constant speed from the positive x direction. The green object is moving right at a constant speed from the negative x direction. Ignore the dashed lines through each object for the time being.

Worldlines on the diagram

Notice in the diagram above the horizontal line at a particular time, which was the age of the universe at time of the snapshot this spacetime diagram represents. The age of the universe might be a good candidate for the time we call "now". The Big Bang marks the beginning of time as we know it and since it happened everywhere at once we should find the same age for the universe everywhere. This is confirmed by the cosmic microwave background radiation. It remains to be demonstrated that this universal age is the time known to us as "now".

A more complex spacetime diagram would represent a satellite in orbit about Earth. The satellite would be moving in three spatial dimensions as well as drifting up the t-axis at the speed of light. Our diagram only shows the x component of the satellite's spatial position. The diagram refers to two clocks one on the satellite and the other in my office. The range of motion of the office is so small compared to that of the satellite that the office clock appears to be moving right up the t-axis. Again ignore the dashed lines and the two star marked events A and B. We will open that particular can of worms later.

Satellite spacetime diagram

Let's use spacetime diagrams to think about the nature of "now", that being the only time in which anything can happen. The past is gone and the future has not arrived so "now" is the only time available to carry out actions. Classically, before we had to think about relativity effects, we thought of "now" as a slice through time, separating past from future, sort of as shown below.

Classic picture of 'now'

With understanding of the relativity principle the picture got more complicated. With that we have to consider a thing called a light cone, or null cone. Look at the spacetime diagram below. It is arranged so that the same units are applied to both the time and space axes, meters for example.

Relativistic picture of 'now'

With time and distance both in meters, what would be the worldline of a photon of light? Well it would be a line up and to the right or left at a 45 degree angle. The above image shows a spacetime diagram including the worldlines of four photons. One is outgoing to the right from the green object sitting at the uptime terminus of its worldline. One is outgoing to the left. One is incoming from the left. The last is incoming from the right. The first two photon worldlines form a 90 degree angle opening upward. The second two form a 90 degree angle opening downward. The four photon worldlines divide spacetime into four regions. In the two dimensional spacetime diagram we are using here, the regions are triangular. If we add a dimension the regions are cones. Including the third spatial dimension the regions are spheres. Because we used light photons to establish these regions we get the name "light cone".

The region labeled "Past" contains all the volume of spacetime from which events had any chance of causing an effect on the green object at its spacetime location. Events outside the "Past" region simply did not have time to communicate with the object, given the fixed speed of light demanded by relativity. The region labeled "Future" contains the volume of spacetime where any action by green object at its spacetime location might have caused an event. Our object is too late to cause any events outside the "Future" area. The two areas that are neither "Past" nor "Future" contain events that cannot have an effect on our object, nor be affected by it. Those events are separated from our object by a spacelike interval.

Because our purpose here is familiarity with spacetime diagrams, I will leave the discussion of the ramifications of this for a later time... except to mention that it appears in relativity that "now" is no longer a surface in spacetime but a separate point for each object. That is the only place where the object's past and future are joined.