Why is (ct)^2 negative in 4-dimensional space-time?

[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) ?

Can somebody please explain?

 

[Nabeshin]

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 v_xt, v_yt and v_zt, 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.

 

[K^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.

Except, it's not that straight forward. If you have two reference frames that are moving relative to each other, time coordinate of one has projection onto spatial coordinates of the other, and in both, you'll have exactly the same negative sign in the metric. So it isn't something unique about a particular direction in space-time. It's something about the structure of space-time that gives you an ability to pick one of infinite number of possible directions to be different than the rest of the directions you are going to pick.

And yeah, this doesn't pack very well in a brain that's used to Eucledian space. I've seen pure mathematicians get downright upset when I suggested that a metric does not have to be positive.