Space-time triangles
Gerard Westendorp
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 I happen to be interested in triangulating space-time. How do angles work in for “Lorentzian triangles”? It turns out that important theorems like

Sum of angles = π
 
continue to work. What is really cool is that we can then go on and use Gauss Bonnet to understand curvature! Now that I understand that, I just had to make this page.
 
In the diagram below, we have triangle in space-time, with vertices, (A, B, C). We will call the internal angles at the vertices (a, b, c), and the lengths (LAB, LBC, LCA).
 
Figure1

Figure 1: A triangle in space-time. The 3 dotted crosses represent light cones emitted from the 3 vertices. Note that at vertex B, the triangle crosses a light cone twice.

Lengths.
In classical physics, lengths are invariant under translation, rotations, and changes in velocity. But in relativistic physics, the “length” that stays invariant under changes in velocity is given by 
equation 1
(taking the speed of light c=1 for convenience)
Or, if we think of time as an imaginary-valued space coordinate:
equation 2
 Note: We should not confuse space-time lengths with coordinates. Coordinates change with Lorentz transformations, but lengths and time intervals (sometimes calles eigen-time or proper time) of a given world line are Lorentz invariant. They are physically either a time difference, measured by a clockClock, or a distance, measured by a rulerRuler.
Time intervals have negative squared length, or imaginary length, space distances are real valued.
 
 Rotations and Lorentz transformations.
The space-time analogue of rotations are Lorentz transformations.
A rotation in ordinary space can be written as:
equation 3
 
To get Lorentz transformations, simply replace the the y coordinate by it, and the angle a by iζ:
 
equation 4
 The parameter ζ is called rapidity.  It is essentially a relativistically corrected velocity, as shown in the graph below:
Figure2
 Figure 2: The rapidity(ζ)is a relativistically corrected velocity(v), tanh(ζ)=v

You can add rapidities like angles, but you cannot just add velocities as in classical physics and get the answer you might expect. For example, an object traveling at speed U relative to an abserver traveling at speed V relative to us, is not traveling at speed (U+V) relative to us. The rapidities however, obey ζU+V = ζU + ζV.
Various ways of rewriting Lorentz transformations
Using hyperbolic sines and cosines of real-valued angles:
equation 4b   (eq2)  
Using t instead of it:
equation 5
Using velocity instead of rapidity:

equation 6

From (eq2)  we can see that Lorentz transformations do not only preserve space-time length, they also achieve that space coordinates remain real numbers, and the time coordinates remain imaginary numbers. In other words, they can not change space into time; “break through a light cone”. If we accept fully mixed complex coordinates for N dimensions, we would get a 2N-dimensional space, since complex numbers are 2-dimensional. Instead, time is always imaginary, space is always real.
This is a bit of a complication: if we draw a triangle in Lorentzian space, there may be vertices that connect lines that contain light cones, such as vertex B in figure 1. There is no ordinary Lorentz transformation that  transforms between these lines.
This however, can be done by operating an Euclidean rotation of angle π/2:
equation 8
We 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.
So:
Note on "crossing light cones": At vertex (B) the internal angle (also called interior angle) crosses 2 light cones. What is happening? Actually, you cannot physically cross a light cone, it is an artifact of looking at internal angles (also called interior angles). The external angle (also called exterior or turning angle) corresponds more intuitively to the physical thing happening at the vertex: A slight change in velocity of you worldline corresponds to a small external angle at the vertex, crossing no light cones. But the internal angle is defined between a worldline going to the past and a worldline going to the future! So the internal angle crosses 2 light cones. If you cross only one light cone, you go from a worldline that goes slower than light to one that goes faster than light or vice versa. This is not a physical process, so in practice we see light cone crossings of the internal angle occur in pairs.
At the other vertices (A) and (C) no light cones are crossed. But here we are looking a worldlines of different persons.


Now everything starts to make sense.
equation 9
The twin paradox
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 the 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 edge.
But beware: because of the minus sign in the formula for space-time length, the 2 oblique edges actually represent less time 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
Figure 3: Twin paradox. The left is in the reference frame of the "stationary" observer, the one that does not turn round.
The right is in the reference frame of other observer
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 time 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 eigen-hyperbola)
 
The eigen-hyperbola
The analog of the circumcircle is the "eigen-hyperbola". The formula for it in N-dimensions is the same for any space-time. I have a web page on that here.

Figure 4
 Figure 4: A triangle in 2 space dimensions has an "eigen-circle", the circumcircle. Similarly a triangle in (1+1) space-time has an eigen-hyperbola.
Animation eigen-hyperbola
Figure 5: Animation: The triangle remains on the eigen-hyperbola as it is Lorentz transformed with the hyperbola-center as origin.