# Where does -(ct)^2 come from?

nhmllr
I understand that the distance between 2 points in 2 dimensions
sqrt(x^2 + y^2)
and in 3 dimensions it's
sqrt(x^2 + y^2 + z^2)
but in 4 dimensions, with the 4th being the change in time, why is it
sqrt(x^2 + y^2 + z^2 - (ct)^2) ?

Ah, yes! This is perhaps the most important negative sign in all relativity! If it were just normal (ct)^2, then it would be identical to 4-dimensional Euclidean space. I guess it boils down to the fact that time is NOT just another spatial dimension -- there is something unique about it.

If you recall from elementary discussions of the line element, ds^2, what we're really trying to identify is some invariant quantity -- that is, something all inertial observers will agree on. The simple fact is that if you do not use a negative sign, it is not an invariant quantity.

More mathematically speaking, the negative sign comes from the fact that relativity is limited to special sub-class of all riemannian manifolds known as Lorentzian manifolds (those that have - + ++ signature). The language of general relativity (and consequently, special relativity), is formulated in terms of these Lorentzian manifolds which make a distinction between null, timelike, and spacelike events.

soothsayer
Good question, I wouldn't be able to come up with a derivation for that expression, but it has to do with the fact that you want to get an expression for meter out of the temporal part of the spacetime interval equation. It also accounts for Lorentz transformation too, imagine that x, y, and z are actually vxt, vyt and vzt, now time is ct. If you plug in c for v, you'll get S2= (ct)2-(ct)2 = 0 And I think this is due to length contraction and/or time dilation at relativistic speeds.

physeven
i don't recall the derivation in a detailed manner, but if you consider time as a fourth dimension when trying to derive the lorentz factor for time dilation using a light clock you will get -c^2t^2. I assume it works similarly for the lorentz factor for length contraction, the lorentz factor being gamma.