Integration of relativistic equations of motion
CPO Ltd (through F H Read) has developed its own technique for integrating relativistic equations of motion. Here is a description.
Starting with
m = m0/(1-(v/c)2)1/2
we have
m/ = mvv//(c2 – v2)
and
vv/ = Σi vivi/
where x/ signifies dx/dt.
Then
fi = pi/ = m/vi + mvi/ = Σjaijvj/
where
aij = m(δij + vivj/(c2 – v2) )
The solution to this can be shown to be
vi/ = Σjbijfj
where
bij = (c2δij – vivj)/mc2
It is this equation for vi/ that is used in the part of the code that deals with trajectory integration.
The many tests of these equations include test2d05, test3d04 and test3d05.
For 2D non-meridional motion the integration is more difficult, see report 478.
Another useful equation, for the total relativistic velocity v, is
(v/u)2 = (1 + r/2)/(1 + r)2
where
r = K/m0c2
and where K is the kinetic energy, and
u = (2K/m0)1/2
is the nominal non-relativistic velocity.