Drawing and Interpreting Space-Time Diagrams

A Space-Time Diagram (STD) is a time versus position graph. For convenience, we plot ct rather than t on the vertical axis so that both axes are in the same units. Distances will often be referred to as lightdays (c-days), lightmonths (c-months), and lightyears (c-years or c-y). Conversely ct can be given in metres, kilometers, lightyears, and so on. STDs are easier to interpret if the intervals on both axes are the same size such as 1000 m per division or 1 c-month per division. A typical STD is shown below. This frame is labelled S. Figure 1 is how an observer in S, let's call him A, sees the universe.

Figure 1 Spacetime as seen by every observer.

The motion of an object is shown as a curve on this graph. We are most interested in objects traveling at constant velocity (since Special Relativity only deals with motion at constant velocity) which means that the curve will normally be a straight line. The curve of the motion is given a special name, it is called the worldline. In Figure 2 below, we show the worldline for a number of particles.  Table 1 tells you how fast each object is moving. Since this is a ct versus x graph, the velocity b = run/rise rather than the usual rise/run used with x- ct graphs.

Figure 2 Worldlines of various objects.

Table 1 Speeds of objects in Figure 2

Object	Velocity	Comment
B	= (2.5/4) = 0.625c
C	= (4/4) = c	Must be a photon.
D	= (-1.25/5) = -0.25c
E	= (4/2) = 2c	Faster than light! Not possible.
F	= (-3/6) = -0.5c
G	= (-1/6) = -0.17c	Travelling backwards in time!

It is very important to understand that everyone, in his own frame, has exactly the same STD shown in Figure 1. Space and time look perfectly normal to a person travelling at a constant velocity; otherwise that person would know he was moving which contradicts the central premise of Special Relativity (SR). That means that all those velocities given in Table 1 are actually relative velocities. In fact a relative velocity is the only kind you can have in SR. Despite the fact that each person sees spacetime like Figure 1 and can only measure velocities relative to himself, two people travelling at different velocities will disagree over how to measure distance and time and consequently velocity. For instance the observer A whose frame of reference is shown in Figure 1 and 2 would say that the measurements of a person in whose worldline is labelled B in Figure 2, call her observer B, would be skewed and distorted as shown in Figure 3. The axes of this skewed spacetime are usually labelled ct¢ and x¢, and the frame is called S¢, to distinguish them from the “center of the universe” observer's frame. Note that the ct¢ axis is actually the worldline for the observer in S', the person on spaceship B. Observer B thinks that she is the “center of the universe” and that everyone moves relative to her. So B is always at the origin of his STD watching time pass. Similarly the ct axis in Figure 1 is actually the worldline of observer A. Notice that the intervals on the ct' and x' axis are not the same width as those in Figure 1. The intervals in Figure 3 are g(b} times the intervals in Figure 1, where b is the speed that A sees for B. The skewing of axes and the distorting of the interval is courtesy of the Lorentz Transfromation.

Figure 3 The spacetime of B as measured by A.

Keep in mind that Figure 3 is what observer A says about observer B's measurements for distance and time; what B actually sees is identical to Figure 1. In fact from B's point of view at the “center of the universe”, A is moving away at -b and she will say that A's measurements are all skewed. When B plots those measurements she gets Figure 4. Note that the ct'' axis now tilts left as B sees A moving to the left.

Figure 4 Spacetime of A as measured by B.

STDs become most useful when we combine more than one frame of reference in the same graph as shown in Figure 5. Figure 5 is ugly and hard to read so we get rid of the skewed gridlines. You must always remember that they are there and must be used when determining the location of a point in the S' frame. Furthermore the scale is usually omitted with only dots for the intervals. The cleaned up STD is given in Figure 6.

Figure 5 Overlapping spacetimes

Always make sure that you have the same scale for each frame, for example 1 c-day per interval. It will make interpreting the diagram much easier.

Figure 6 also shows some points or events as they are called in SR. In Table 2, the location of each point is given for each frame. Note that two events that are simultaneous in one frame (say events E₁ and E₂ in S) are not simultaneous in another. Similarly two events that occur at the same place in one frame (say events E₁ and E₂ in S) don't do so in the other.

Figure 6 Events are not simultaneous in different frames.

Now we can summarize the steps to drawing a proper STD. Consider the case of two observers A and B where you know that the relative velocity of the two is b

1. Take one of A or B to be the stationary, “center-of –the-universe” frame and draw the rectangular ct and x axes.
2. Draw the ct' axis. This is the line whose inverse slope or run/rise equals b. If b is positive, observer B is moving to the right, the line will be in quadrant I. If it is negative, B moving to the left, it will be in quadrant II.
3. The intervals on the ct' axis should be g(b times the intervals on the ct axis.
4. Draw the x' axis. This is the line whose inverse slope or run/rise equals 1/b. If b is positve, observer B is moving to the right, the line will be in quadrant I. If it is negative, B moving to the left, it will be in quadrant II.
5. Both the ct' and x' axis should be symmetric about the worldine of a photon since all observers must measure the same value for c.

Many problems involve objects or spaceships of finite length such as the questions of where the front of one ship is when the tail of the other passes by. In this case it is important that you draw the worldline of both the nose and tail of each object or ship. Since all parts of the spaceship are travelling at the same speed, the worldline of the nose and tail are parallel. Figure 7 shows the case of two spacecraft moving at b = 0.6 with respect to one another (g = 1.25). Both ships are 200 m long in their own frame. Initially the noses are both at the origin. The c-time and position axis are in intervals of 200 metres. The ct axis is the worldline of the nose of the stationary ship. The worldline of the tail is the dashed line parallel to the ct-axis. Note that this spaceship measures all speeds relative to it, so it is not moving. The ct' axis is the worldline of the nose of the moving ship. The worldline of the tail of the moving line is the dashed line parallel to the ct'-axis. From the graph we can read of that the tail of the moving ship will pass the nose at the point labeled X. Reading the scale tells us that this is ct = 260 m and ct' = 340 m.

Figure 7 When the length of an object is important.

Examples

A spaceship travelling at b = 0.6 leaves earth for a planet 4 c-y away. One year later, in the spaceship's frame, it sends a radio signal to that planet and to earth.

Two spaceships are approaching one another. As seen by an observer on Earth the speeds are b_A = 0.4 and b_B = 0.6. Spaceship A radios B to tell him to get out of the way. B immediately radios back to tell A to buzz off. Sadly they crash at the point labelled X.

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A fugitive, A, breaks out of prison and flees earth on a spaceship travelling at 0.4c. Two days later a spacecop, B, leaves earth traveling at 0.8c. The fugitive is caught at the point labelled X.

Using MS Excel to Draw an STD

You can use MS Excel to draw spacetime diagrams for you. It is a matter of using the Lorentz Transform on the coordinates of a moving frame to get the corresponding values in a stationary frame.

The first step is to transform the coordinates of the worldline of the moving particle. Column A and B of Table 3, shows the coordinates (x', ct') of the worldline as seen by the observer in that frame. Columns C and D show how to transform those coordinates into the stationary using the Lorentz Transformation formulas. Columns E and F are the numerical values of those points in the stationary frame. In your spreadsheet, you can change the intervals between points as you wish. For instance, the numbers in column A of Table 3 could have been 0, 10, 20, 30, ….

Table 3 Typical spreadsheet for calculating worldline

The x'-axis is found in a similar manner as shown in Table 4. Note that this is identical to Table 3 except that columns A and B are switched.

Table 4 Typical spreadsheet for calculating x¢-axis