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

|(x,t)|2 = g[(x,t),(x,t)]
x2тАУt2 (14a, if x2 тАУt 2 > 0)
=
|(x,t)|2 = тАУg[(x,t),(x,t)]
t2тАУx2 (14b, if x2 тАУt 2 < 0)
=
|(x,t)|2 (14c, if x2 тАУt 2 = 0)
= 0


The new formula for g given in Eqn (13) is known as the Minkowskian, or
тАЬflat spacetimeтАЭ, metric, as opposed to the Euclidean metric of Eqn (1). This metric
meets the same conditions of symmetry and linearity as the Euclidean metric, spelt out in
Eqns (2).
From Figure 6, it's clear that some vectors in spacetime, such as OR, can have
negative values for x2тАУt2, so the new equivalent of Pythagoras's theorem, Eqns (14),
need to take account of this possibility. Although it might seem a shame to have to divide
vectors in spacetime into different classes and treat them each somewhat differently, the
three possibilities involve very real physical distinctions, so it's a good idea not to try to
gloss over the differences.
If x2тАУt2 > 0, the vector (x,t) is called spacelike. A spacelike vector slopes away
from the time axis at a greater angle (and hence a greater velocity) than a light ray. No
object's world line can point in a spacelike direction. In Figure 6, not even a pulse of
light at event O can travel fast enough to reach event Q. For events with spacelike
separation, |(x,t)| is called the proper distance between them; an observer who judges
them to have happened simultaneously measures t = 0, so |(x,t)| = x.
If x2тАУt2 < 0, the vector (x,t) is called timelike. A timelike vector slopes away
 Egan: "Foundations 1"/p.13


from the time axis at a smaller angle than a light ray. Ordinary objects' world lines point
in timelike directions. In Figure 6, there's nothing to stop a spacecraft cruising from
event O to event R тАФ and an observer on the spacecraft would consider that the two
events were separated only in time, not space. For events with timelike separation, |(x,t)|
is called the proper time between them; an observer who judges them to have happened
at the same place measures x = 0, so |(x,t)| = t.
If x2тАУt2 = 0, the vector (x,t) is called lightlike, or null тАФ because the vector's
length is zero, however large its individual x and t components are. Only photons, the
particles of light (and other massless particles) have world lines pointing in null
directions.
The light cone is a real, physical structure, not an artifact of the coordinates you
happen to choose. And since it marks the division between timelike vectors and spacelike
ones, every observer will assign a given vector to the same class. Motionless versus
stationary is a matter of opinion. Timelike versus spacelike is not.
To convert between spacetime coordinates for different observers, we'll need to
be able to project one spacetime vector onto another. It would be nice to be able to use
the equations we've established for vectors in space, like Eqn (7), and simply substitute
the new metric. But rather than doing that blindly, we need to take a closer look at what
the idea of projection really means, in spacetime.