36.7.2 The Spacetime Interval

Forget relativity for a minute and think about ordinary geometry. FIGURE 36.27 shows two ordinary coordinate systems. They are identical except for the fact that one has been rotated relative to the other. A student using the xy-system would measure coordinates (x1, y1) for point 1 and (x2, y2) for point 2. A second student, using the x′y′-system, would measure (x′1, y′1) and (x′2, y′2).

FIGURE 36.27 Distance d is the same in both coordinate systems.

The figure consists of two parts that illustrate the measurement of two coordinates in two coordinate systems where one is rotated relative to the other.
FIGURE 36.27 Full Alternative Text

The students soon find that none of their measurements agree. That is, x1≠x′1 and so on. Even the intervals are different: Δx≠Δx′ and Δy≠Δy′. Each is a perfectly valid coordinate system, giving no reason to prefer one over the other, but each yields different measurements.

Is there anything on which the two students can agree? Yes, there is. The distance d between points 1 and 2 is independent of the coordinates. We can state this mathematically as

d2=(Δx)2+(Δy)2=(Δx′)2+(Δy′)2 (36.16)

The quantity (Δx)2+(Δy)2 is called an invariant in geometry because it has the same value in any Cartesian coordinate system.

Returning to relativity, is there an invariant in the spacetime coordinates, some quantity that has the same value in all inertial reference frames? There is, and to find it let’s return to the light clock of Figure 36.20. FIGURE 36.28 shows the light clock as seen in reference frames S′ and S″. The speed of light is the same in both frames, even though both are moving with respect to each other and with respect to the clock.

FIGURE 36.28 The light clock seen by experimenters in reference frames S′ and S′.

The figure illustrates the light clock seen by experimenters in reference frames S prime and S double-dash.
FIGURE 36.28 Full Alternative Text

Notice that the clock’s height h is common to both reference frames. Thus

h2=(12c Δt′)2−(12Δx′)2=(12c Δt″)2−(12Δx″)2 (36.17)

The factor 12 cancels, allowing us to write

c2(Δt′)2−(Δx′)2=c2(Δt″)2−(Δx″)2 (36.18)

Let us define the spacetime interval s between two events to be

s2=c2(Δt)2−(Δx)2 (36.19)

What we’ve shown in Equation 36.18 is that the spacetime interval s has the same value in all inertial reference frames. That is, the spacetime interval between two events is an invariant. It is a value that all experimenters, in all reference frames, can agree upon.

EXAMPLE 36.6 Using the spacetime interval

A firecracker explodes at the origin of an inertial reference frame. Then, 2.0 μs later, a second firecracker explodes 300 m away. Astronauts in a passing rocket measure the distance between the explosions to be 200 m. According to the astronauts, how much time elapses between the two explosions?

MODEL The spacetime coordinates of two events are measured in two different inertial reference frames. Call the reference frame of the ground S and the reference frame of the rocket S′. The spacetime interval between these two events is the same in both reference frames.

SOLVE The spacetime interval (or, rather, its square) in frame S is

s2=c2(Δt)2−(Δx)2=(600 m)2−(300 m)2=270,000 m2

where we used c=300 m/μs to determine that c Δt=600 m. The spacetime interval has the same value in frame S′. Thus

s2=270,000 m2=c2(Δt′)2−(Δx′)2=c2(Δt′)2−(200 m)2

This is easily solved to give Δt′=1.85 μs.

REVIEW The two events are closer together in both space and time in the rocket’s reference frame than in the reference frame of the ground.

Einstein’s legacy, according to popular culture, was the discovery that “everything is relative.” But it’s not so. Time intervals and space intervals may be relative, as were the intervals Δx and Δy in the purely geometric analogy with which we opened this section, but some things are not relative. In particular, the spacetime interval s between two events is not relative. It is a well-defined number, agreed on by experimenters in each and every inertial reference frame.