"Greg Egan - Foundations 1 - Special Relativity" - читать интересную книгу автора (Egan Greg)

Then s is just the slope of the x2 axis as a line in the (x1,y1) reference frame, and
the vector (1,s) points along the x2 axis, while the vector (тАУs,1) points along the y2 axis.
These are not unit vectors, but we can apply Eqn (7) and divide out their lengths:


x2 = g[(x1,y1),(1,s)] / |(1,s)|
(x1 + sy1) / тИЪ(1 + s2)
= (11a)
y2 = g[(x1,y1),(тАУs,1)] / |(тАУs,1)|
(y1 тАУ sx1) / тИЪ(1 + s2)
= (11b)


Spacetime Geometry
 Egan: "Foundations 1"/p.10



The fact that spacetime has four dimensions, as opposed to the three of space alone, is
important, but it's far from being the distinguishing feature of spacetime geometry. Our
simple problem in interstellar travel involves only one dimension of space and one of time
тАФ a two-dimensional тАЬsliceтАЭ through four-dimensional spacetime тАФ but the geometry
that applies to that slice is not the same as the geometry of the familiar two-dimensional
Euclidean plane.




In Euclidean geometry, given two fixed points, the (x,y) coordinates you give
those points will depend on the reference frame you choose. The coordinates you give P
and Q in Figure 5 depend on where you stand, and the direction you're facing. But the
distance between the points, PQ, which can be calculated with Pythagoras's theorem,
must always be the same. A quantity like this, which every observer agrees on, is called
an invariant. That the distance between points is an invariant in Euclidean geometry
seems almost too obvious to mention, but it's worth checking that both Eqns (10) and
(11) yield the result:


x22 + y22 x12 + y12
=


Points in spacetime are usually called events, to distinguish them from points in
space. Events can be specified by giving their space and time coordinates, (x,t),
according to a particular observer. For example, in Figure 1, if the event of the
spacecraft's launch is (0, 0), the event of its arrival would be (8.7, 14.5).
The question is, is there a тАЬspacetime distanceтАЭ between these two events, which
 Egan: "Foundations 1"/p.11