Mitchell,

Thanks making my point that spacetime coordinates are

clumsy and

unintuitive. I believe this is how things evolved.

Minkowski: Einstein's former professor realized brilliance in Special Relativity and the need for a graphical representation. To give Minkowski spacetime a Cartesian (or "pseudo-Euclidean") representation, he applied i = √-1, typically with speed limit

c, to the temporal axis as

ict. This meant temporal and spatial components could be inserted in the Pythagorean formula for an interval (

∆d):

∆d² = ∆x² + ic∆t)² = ∆x² – (c∆t)²Implicit: As people struggled conceptualizing √-1, a trend developed to drop the term and instead emphasize underlying

hyperbolic geometry, where the

minus sign is implicit to the relation of space, time and interval. So

i tended to disappear from the coordinates along with

c, which was conveniently expressed as 1 in natural units. (Similarly, deltas are often dropped.)

Stonewall: What you'll currently find in

Wikipedia is a

denial that the interval

∆d has meaning of itself. Instead, it suggests the preferred thinking is to consider only ∆d² to avoid the conceptual problem that you mentioned above,

negative intervals.

"

Rather than deal with square roots of negative numbers, physicists customarily regard [∆d²] as a distinct symbol in itself, rather than the square of something."

That makes about as much sense as suggesting that we only deal with meters² or seconds² if the span happens to be a hypotenuse. LOL

Declaration: I find this more sensible approach in modern literature. The idea is to adopt the appropriate convention for the task at hand. This recognizes Minkowski's application of √-1 to the time coordinate as

arbitrary. He could just as well have applied it to the three spatial coordinates. And we're always free to make the change as convenient. Spacetime interval spans depend on the

difference of the squares of its components but not their

order. For example, adopting a "spacelike convention" (where ∆x > ∆t):

∆d² = ∆x² – ∆t² is used, while adopting a "timelike convention" (where ∆t > ∆x) uses:

∆d² = ∆t² – ∆x². Thus, negative intervals are avoided. Remember, unless we are all superimposed, spacelike separation is real (and not negative). We can't travel at spacelike speeds but the separations exist.

Phyxed: By selecting Euclidean, interval-time coordinates in the first place, there is no

hyperbolic geometry, √-1 or

minus sign to deal with:

∆x² = ∆d² + ∆t². Interval-time coordinates are a

Euclidean lens with which to resolve many outstanding mysteries in physics. As I've said, without non-Euclidean distortion, we see the lightlike interval plainly as interval

contact. And of course, there is

no negative interval, any more than there is negative length.

As spatial hypotenuse rotates through vertical, it simply changes the side on which the interval is depicted (same as triangles in ordinary plane geometry).But wait, …there's more! We have a reason for a universal speed limit!

c is the

absolute speed limit because

contact is the

absolute proximity limit. You can't get closer than contact. The speed limit is

invariant because contact is

invariant.

That's just for starters. There's lots & lots to look at with a

Euclidean lens, especially if you're the only kid on your block who seems to have one. The

Phyxed channel is committed to explaining a new "mysterious" phenomenon every two weeks.