Quantum Gravity 285.4: Derivation of the Mass Cone m^2 = r^2 + t^2 From Circle Composition (Star) Product
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Thu, 14 Aug 2008 13:45:40 -0700 (PDT)
From Osher Doctorow
Let's transform the Jacobson Radical circle composition product:
1) m = t o L (m = mass, t = time, L = length or principal radius or
scale factor, since r will be used for something else below)
to slightly different coordinates by the transformation:
2) m(t, L) --> m(r, theta), t = rcos(theta), L = rsin(theta)
Then (1) becomes:
3) m(r, theta) = t + L - tL = rcos(theta) + rsin(theta) - r^2
sin(theta)cos(theta)
But we have:
4) sin(2 theta) = 2sin(theta)cos(theta)
and therefore:
5) sin(theta)cos(theta) = sin(2 theta)/2
so we can write (3) as:
6) m(r, theta) = rcos(theta) + rsin(theta) - (r^2/2) sin(2 theta)
Choosing 2theta = pi, so that theta = pi/2, we know that cos(pi/2) = 0
and sin(pi/2) = 1, while sin(2 theta) = sin(0) = 0. So (6) becomes:
7) m(r, theta) = r
But in polar coordinates, r^2 = t^2 + L^2, r = sqrt(t^2 + L^2) from
(2), so we have:
8) m(r, theta) = sqrt(t^2 + L^2)
We designate m(r, theta) by z:
9) z = m(r, theta) (since m(r, theta) can be regarded as the "z"
coordinate)
and from (8) by squaring both sides we get:
10) z^2 = x^2 + y^2
which a cone in three dimensional Cartesian coordinates.
Osher Doctorow
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