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.
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