What are spacetime intervals?
In: Physics
As you may have heard, in Einstein’s special theory of relativity, space and time are relative. The faster you are moving relative to something, the more your time is dilated and length is contracted relative to what they experience.
We describe events by where they took place (within the 3 spatial dimensions) and when they happened (within the 1 temporal dimension). In relativity, both of these things are relative, but remember that Einstein told us we shouldn’t treat time and space as separate things and we should instead keep them together as spacetime. Well in space we calculate the distance between things using the Pythagorean theorem: d^(2) = x^(2) + y^(2) + z^(2). Well if we want to include time, we can multiply it by a factor of c (the speed of light) to get something that has the same units as distance, and then we can include that:
s^(2) = (ct)^(2) – x^(2) – y^(2) – z^(2).
We can also do s^(2) = -(ct)^(2) + x^(2) + y^(2) + z^(2), both of these will give you a result that is not relative. These two different ways of doing the negatives are called the “metrics”. The reason there’s a negative in there is that when you change the reference frame you are increasing (or decreasing) both of those things, and as we are trying to make a construct that is invariant we want one to increase and the other to decrease, and with these signatures they cancel each other out.
Now that’s all very well to say that it’s a thing that is invariant, but what is its use? Well one thing it does is tell us whether the things could be causally linked or not, that is to say whether or not light could have travelled from A to B in time to cause the event at B.
If the events are separated in such a way that light would get to B as B happens, then the spacetime interval is 0, if it would take longer than that then the spacetime interval is greater than 0, and if it takes less time than that then the interval is less than 0.
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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