dT2 = dt2 - dx2 - dy2 - dz2dS2 = -dt2 + dx2 + dy2 + dz2where c = 1, which is commonly done for convenience.
Both of those sign conventions are in common use. Wald and Misner/Thorne/Wheeler use the second convention, which MTR call the "Landau-Lifshitz Spacelike Convention (LLSC)." The two red pages at the beginning of MTR list the sign conventions used by 37 books and papers.
I think you are right about the c^2 bit (and the typo) but the negative sign still eludes me.
The sign of the time term is different from the signs of spatial terms because the geometry of spacetime is non-Euclidean. If all of the signs were positive, then the equation would define a Euclidean metric and time would behave just like a fourth spatial dimension, which it is not.
No particular problem with what Farsight write above except it is a bit verbose. I think one thing that's not been made as clear as it could is that this 'interval' is zero for any two points on the same light ray and for points in general can be negative. That's ok though - it isn't the same as a distance - it's more a convenient construction for two points that will have the same ds2 for everyone. When a lot of things are relative in relativity finding the things that aren't relative is very handy.
Adding to what
edd said: The equation we're talking about is often called a
line element, and is often used as a succinct description of the spacetime pseudo-metric (where the "pseudo-" refers to the fact that it can be negative, which is impossible for a true metric). In the context of relativity, we often get sloppy and drop the "pseudo-", so you'll see people writing and talking about the metric tensor.
As
edd said, ds
2=0 means the separation between the two spacetime events being compared is light-like: a single ray of light can connect the spacetime coordinates of both events.
With the LLSC convention (which is now more widely known as the MTR convention), a negative value for ds
2 means the separation is time-like: It is possible for a massive (sub-light-speed) particle to participate in both events.
If ds
2 is
non-negative non-positive, so the separation is either light-like or time-like, then there can be a causal connection between the two events: They are close enough in space, and far away enough in time, for a photon or some massive particle to travel from one to the other quickly enough so that one event can influence the other.
A positive value for ds
2 means the separation is space-like: The events are so far apart in space, and so close together in time, that any communication between the two events would have to travel faster than the speed of light. Since (we have good reason to believe) nothing can go faster than light, there cannot be any causal connection between two events whose separation is space-like.
The invariance of ds
2 implies the objectivity of causality: all observers will agree on which causal relationships are allowed by the geometry of spacetime. Relativity allows each observer to choose his/her/its coordinate system almost arbitrarily, so the objectivity of potential causal relationships is an important and non-trivial result.
Wikipedia's
current article on spacetime says pretty much what I wrote above, with more symbols.
Getting back to the subject of this thread: In Einstein's 1916 paper on
Die Grundlage der allgemeinen Relativitätstheorie (The Foundation of the General Theory of Relativity), equation (1) shows the line element
Farsight injected into our discussion of crackpot physics. Einstein's equation (3) shows the fully general form of that line element. That's where
Farsight got lost. There are 72 more numbered equations in that paper.
Farsight has spent years denying several of Einstein's main results in that paper, including the equivalence principle and the admissibility of rather general coordinate transformations. You can read some of that denial by following
Farsight's links in
post #820 above.
[size=-1](The JREF Forum software is no longer rendering LaTeX properly, so some of those posts will contain LaTeX code instead of the mathematical equations you would see if the software were working. I don't know when or whether that software problem will be fixed.) [/size]