What are spacetime intervals?
In: Physics
In our normal existence, we typically think of physical distance and duration as separate concepts, for example, we might say that Alex left their home, and 10min later, they arrived at the park 8 km away.
The issue is that due to relativity, different observers can disagree on how far apart two things are how much time passes between two events.
Ok, to see the effects of relativity better, we’ll need larger numbers. So let’s say we on earth observe Alex leave earth and arrives at the sun (about 150M km away) 10min later. To Alex themself, they haven’t moved at all, and the trip took about 6 minutes. Some other interplanetary traveller might observe that the trip took 8min, but earth and the sun are only about 100M km apart.
It turns out, though, if everyone puts their distance and time measure into a certain formula, they all get the same number; This is the “spacetime interval”. For Alex’s trip, this number is about 6min (exactly the same as Alex’s perception; this is not a coincidence).
—here ends the eli5 explanation—
The formula (or rather, one formula, depending on choice of unit) is:
sqrt(t^2 – d^2 / c^2 ) where t is the observed time, d is the observed distance, and c is the speed of light.
One thing we can use the result for is to determine whether something happens before something else. If the result is a real number, then yes. Take the Alex example above; even though observers disagree on the distance traveled and time taken, they all agree that Alex left earth before they arrived at the sun. We call these events “time-like”
If the result is an imaginary number, then different observers could disagree on which event happened before the other. For example, suppose Alex left earth, and from Earth’s perspective, 2 minutes later, there’s a solar flare. The spacetime interval is about 7.75i min. What this tells us is that some space traveler could observe both events happening at the same time, or even the solar flare happening before Alex’s departure. We call these events “space-like”.
An important consequence of two events being space-like, is that they cannot be causally related, since information cannot travel faster than light, and they’re separated by more distance than light can travel in the time between the events.
E: formula formatting
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