Relativity

Spacelike, Lightlike, and Timelike Intervals

Another absolute quantity in special relativity is the spacetime interval between two events, which is defined as follows:

spacetime interval = d² – c²t²

where d is the distance between the events (according to a given reference frame), t is the time between the events (in that frame), and c is the speed of light. Remarkably, all inertial reference frames agree about this quantity, even though they disagree about distance d and time t.

The spacetime interval indicates how two events are related to each other in space and time. It can be positive (spacelike), negative (timelike), or zero (lightlike):

Spacelike intervals: If the spacetime interval is positive, this means you’d have to travel faster than the speed of light to get from one event to the other. As we’ll see, reference frames may disagree about the order in which these events occurred. (In other words, one observer may think event A happened first, while another observer thinks event B happened first, and yet another may think the two events were simultaneous. And according to Einstein, there’s no fact of the matter who’s right!) But all reference frames will agree that the events occurred at different locations, hence the events are said to have spacelike separation.
Timelike intervals: If the spacetime interval is negative, this means it’s possible to get from one event to the other traveling slower than the speed of light. Reference frames may disagree about whether these events happened at the same location. But all reference frames will agree that the events occurred at different times, and they’ll also agree about the order in which these events occurred, hence the events are said to have timelike separation.
Lightlike intervals: If the spacetime interval is zero, this means a beam of light could travel directly from one event to the other. All reference frames will agree that the two events are lightlike separated, and they’ll also agree about the order in which these events occurred. Events are said to have lightlike separation when they are related in this way.

In the special theory of relativity, the distances and times between events are relative matters, depending on the observer’s frame of reference. However, the spacetime interval between two events is the same for all inertially moving (non-accelerated) observers: it is an absolute quantity. In other words, whether two events have timelike, lightlike, or spacelike separation is an absolute matter, independent of the observer’s reference frame.

What is the spacetime interval between the following two events?

Event A: a solar flare erupts on the sun
Event B: an astronomer witnesses the flare from an observatory on Earth

Since light travels directly from event A to event B, these two events must have lightlike separation, so the spacetime interval is zero. To confirm this, let’s do the math. The sun is approximately 150 million kilometers from Earth (as judged from our reference frame), and it takes about 500 seconds (8.3 minutes) for light to travel that distance, so the astronomer will see the flare 500 seconds after it occurred. The spacetime interval between events A and B is:

d² – c²t² = (150,000,000 km)² – (300,000 km/s)² (500 s)²
= 0

For another example, suppose the astronomer sneezed five minutes after the flare occurred. (Sometimes looking at the sun makes you sneeze.) Let’s call this event C:

Event C: the astronomer sneezes 5 minutes (300 seconds) after the solar flare occurred (200 seconds before she sees the flare)

It takes more than 8 minutes for light to travel between the sun and Earth, so you’d have to travel faster than light to get from the flare to the sneeze. Therefore, events A and C have spacelike separation, and the spacetime interval must be positive. Let’s check. The spacetime interval between event A (the solar flare) and event C (the sneeze) is:

d² – c²t² = (150,000,000 km)² – (300,000 km/s)² (300 s)²
= 1.44 × 10¹⁶ km

What about event C (the sneeze) and event B (the astronomer seeing the flare), which occurred 200 seconds apart? Since they occurred in the same location, these two events must be timelike separated. (Indeed, they would still be timelike separated if they had occurred at different locations, since the astronomer—who was present at both events—moves slower than light.) Let’s do the math to verify that the spacetime interval is negative:

d² – c²t² = (0)² – (300,000 km/s)² (200 s)²
= -3.6 × 10¹⁵ km

Now, imagine an alien spaceship cruising past the solar system at a significant fraction of the speed of light. If the aliens happen to witness events A, B, and C, they will disagree with the Earthling astronomer about the times and distances between those events. They may even disagree about the order in which events A and C occurred, since those two events have spacelike separation: from an alien’ s perspective, perhaps the sneeze occurred before the solar flare! Nevertheless, aliens and Earthlings will agree about the spacetime intervals. Not only will they agree about which events are spacelike, timelike, and lightlike separated; they will agree about the precise values of those intervals. That is, they will agree that the interval between A and B is exactly zero, and that the interval between A and C is 1.44 × 10¹⁶ km, and that the interval between B and C is -3.6 × 10¹⁵ km.