In a previous blog, we showed that the log-spiral in configuration space becomes a closed circular curve in velocity space, known as a hodograph. Here we show that the distance is measured by the longitudinal Doppler effect.

The equation of motion we derived was

da/dt+\Omega^2 v=0,

where the frequency is the square-root of the conservation of angular momenta in velocity and configuration space. Now multiply the above equation through by a to get

a^2=2E-\Omega^2 v^2,

after integration where E represents the total energy. Writing a=dv/dt, this becomes

dt^2=dv^2/(2E-v^2),

which is none other than the Poincare metric for his disc model. Taking the square-root and

integrating gives

t=\int dv/(2E-v^2)^1/2=log ([1+u]/[1-u]),

where u=v/(2E)^1/2 is the normalized velocity.

So we have here that Doppler is a measure of exponential time

e^t = [1+v]/[1-v].

The geodesics are semi-circles inserting the disc orthogonally. If \theta is the angle of parallelism then

e^(-v)=\tan[\theta/2].

So the flat rotational curves consist of a non-Euclidean model of a hyperbolic metric of constant curvature in velocity space.

We shall continue the discussion shortly. But, bear in mind that the transfer to velocity space brings in non-Euclidean geometry. What people have been trying to measure in configuration violates thing like the inverse square law because the "real" distance is given by the logarithm of the longitudinal Doppler shift!

The same would be true if we introduce fictional time into Newton's second law. But, there it is the inverse velocity which is a measure of distance. (see J Milnor, "On the geometry of the Kepler problem"). The fictional time ds=dt/r, which has the effect of rendering the elliptical orbits of the Kepler ellipse as circles in velocity space, just as William Hamilton envisioned in 1846.

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