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

=
OP2 / OP2
=
= 1


The notation тАЬ(x,y)тАЭ beside the point P is a reminder that points can be referred to
by their x- and y-coordinates, written as an ordered pair. The arrow drawn from O to P
is a reminder that every point can be thought of as defining a vector from the origin to
the point. The advantage of dealing with vectors, rather than just points in space, is that
the same geometry can then be applied to other vectors, like velocity and acceleration.
To make it easier to carry things over from Euclidean geometry to spacetime
geometry, it will help to restate some of these familiar ideas in slightly different language.
In both Euclidean and spacetime geometry, there's a formula for taking two vectors and
calculating a number from them which depends on the length of the vectors and the angle
between them. This formula is known as the metric for the geometry. (You might also
have come across it as the тАЬdot productтАЭ of two vectors.) It's usually written as g:


g[(x,y),(u,w)] = xu + yw (1)


Eqn (1) defines the Euclidean metric. Eqns (2a)-(2c) demonstrate some of its
properties: it's symmetric (swapping the two vectors leaves the value unchanged), and
it's linear (its value is simply multiplied and added as shown, if you apply it to a vector
that's been multiplied by a factor, or had another vector added to it).


g[(u,w),(x,y)] = ux + wy
= xu + yw
= g[(x,y),(u,w)] (2a)
g[a(x,y),(u,w)] = g[(ax,ay),(u,w)]
= axu + ayw
= a (xu + yw)
= a g[(x,y),(u,w)] (2b)
g[(x,y)+(p,q),(u,w)] = g[(x+p,y+q),(u,w)]
= (x+p)u + (y+q)w
= (xu + yw) + (pu + qw)
= g[(x,y),(u,w)] + g[(p,q),(u,w)] (2c)
 Egan: "Foundations 1"/p.6



Eqn (3) is just a restatement of Pythagoras's theorem; the notation |(x,y)| means
the length of the vector (x,y) тАФ also referred to as its magnitude тАФ or if you prefer to
think in terms of the coordinates of a point, |(x,y)| is the distance from the origin (0,0) to
the point (x,y).


|(x,y)|2 = g[(x,y),(x,y)]