allow this transformation, because although it can change space into
time and vice versa, it does create mixed complex
coordinates. (i.e. time remains imaginary, space real)
This works also in space-time with more dimensions, because an angle
between 2 lines defines a plane, either with 2 space-like coordinates,
2 time-like coordinates, or 1 of each. In the latter case, we
should take note of the light cone originating from the vertex.
- Angles in (1+1) dimensional
space-time are imaginary valued
times the number of light cones crossed.
Now everything starts to make sense.
The twin paradox
- The internal angles of a triangle sum to π,
because you always have to cross 2 light cones, and the
imaginary parts (rapidities) sum to zero.
- In a triangulation of flat (not curved)
angles at a vertex sum to 2π.
- In curves space-time, there is a
non-zero angular deficit
at vertices. There may be a net rapidity, a time-like curvature, or and
angle, a space like curvature.
- Angles (rapidity differences) are
Lorentz invariant. The
are related to the Lorentz invariant space-time lengths by the law of
There is a way of interpreting the twin paradox as a triangle, as shown
in figure 3. Imagine one observer traveling along the vertical side of
yellow triangle. The twin brother travels along the 2 oblique edges.
All edges are time-like in this case, their space-time length represent
a time difference on the watch of an observer who travels along the
But beware: because of the minus sign in the formula for space-time
length, the 2 oblique edges actually represent less
on the observes watch.
The twin who
"turns around" is always younger when they reunite.
According to this Wikipedia
article on triangle inequality
the twin paradox can be viewed as
an example of the triangle inequality, which reverses if all paths are
time-like and in the same light cone.
Figure 3: The yellow triangle corresponds to the twin paradox. Because
of the minus sign in the formula for the space-time length, length
ratio's are different to what we see visually, which is the ordinary
space of our picture! An
observer traveling via 2 time-like edges always takes less
than an observer traveling to the same event along 1 time-like edge.
The observed age difference is Lorentz invariant. (The blue line is an
analog of the circumcircle is the
"eigen-hyperbola". The formula for it in N
is the same for
any space-time. I have a
web page on that here
4: A triangle in 2 space
dimensions has an "eigen-circle",
circumcircle. Similarly a triangle in (1+1) space-time has an
Figure 5: Animation: The triangle
remains on the
eigen-hyperbola as it is
transformed with the
hyperbola-center as origin.