NPPF: CANTO ONE: Lemniscate and Spacetime

[David Morris]

http://www.mathpages.com/home/kmath170.htm

The Lemniscate and Spacetime

The advance of the quantum phase of a system over a timelike interval
of spacetime equals the Lorentz-invariant magnitude of that interval.
In the xt plane the square of the magnitude of the interval from 
the origin to the event (x,t) is s^2 = t^2 - x^2.  Similarly we can
define the squared magnitude of a spacelike interval from the origin
to the event (x,t) as s^2 = x^2 - t^2.  

[...]

This means that if we select an event whose coordinates are (x,t)
with respect to a given inertial system, the interval from the origin
to this event makes an angle q relative to the positive x axis in
the xt plane, and the squared Lorentz-invariant magnitude of the 
interval is given by (x^2 + t^2) cos(2q).

Consequently the "circular" locus of events such that x^2 + t^2 = r^2
for any fixed r can be represented in polar coordinates (s,q) by the 
equation
                      s^2  =  r^2 cos(2q)

This is the equation of a lemniscate, illustrated in the figure below
for several values of r.


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