system
Given the following post, respond as if you're commenting on Reddit:
human
If the distance between 2 points in spacetime is sqrt(x^2+y^2+z^2-ct^2), does that mean that the further away 2 events are in time, the closer they are in spacetime?

The minus sign in front of ct^2 is really throwing me off.
gpt
What you've written down is called a spacetime interval, 

> s^2 = x^2 + y^2 + z^2 - (ct)^2

The s^2 is not a spacial distance, but a spacetime distance, in that it incorporates the distance between two events including time. So for instance, if a grandfather clock rings every hour, then the spacetime interval separating any two rings will be,

> s^2 = - (c\*1 hour)^2 = roughly -10^18 km^2 or -1 lighthour^2 .

So here we are with a spacetime distance that has zero spatial separation from the reference frame of the clock.

So to answer your question, no, that's not how the formula is used. Now to answer *why* the time component has a different sign that the spatial component is because while time is a dimension like the z-direction, it behaves differently than say a fourth spacial dimension. Essentially, when we're asking what Pythagoras's theorem looks like in 4D spacetime, this is the form the equation takes.

Let's go over the different interval types:  

* s^2 < 0 (like our clock) is a timelike interval. The two points are causally linked.

* s^2 > 0  spacelike interval. The two points are not causally linked, not enough time has past to connect them.

* s^2 = 0 lightlike interval. The two points are connected causally, but only by information traveling at lightspeed. This represents the farthest two events can be spatially to affect each other in a given timespan.