Generalized Kepler's Second Law and Lorentz factor

I was wanting a bridge between Newtonian gravity and GR.
In the following s.p.r. thread

http://groups.google.com/group/sci.physics.relativity/browse_thread/thread/c0f787a8c1fa68de#

I've pointed out that the momentum of a moving body can be
expressed as p = mc sinh(v/c). Thus, any relativistic momentum
p_r = mv/sqrt(1 - v^2/c^2) is only an approximation of the former,
but now addition of velocities remains strictly as an euclidean sum
of vectors.

Let's see how Kepler's Second Law can be expressed under
this enhanced Galilean relativity, assumed gravitational motions
take place in a 3-d flat manifold plus time. Kepler's Second Law
is expressed in a concise form as

h = r^2 w
where
h is angular momentum per unit mass,
r is radial distance of the orbiting body, and
w is angular velocity

(this angular momentum per unit mass, h, is preserved
along any orbit).

Now, under this enhanced Galilean relativity, since the
momentum is p = mc sinh(v/c), where v is tangential velocity,
we get angular momentum

L = mcr sinh(v/c),
and angular momentum per unit mass
h = L/m = rc sinh(v/c),

thus, Kepler's Second Law can be written

rc sinh(v/c) = r^2 w,
c sinh(v/c) = r w,

which shows a deeper relation between v and w.

w = (c/r) sinh(v/c),
v = c arcsinh(rw/c).

For strictly pure Newtonian gravity, we can consider
the limit c --> oo, therefore sinh(v/c) --> v/c, so the above
equation reduces to

w = v/r.

But, if we consider a finite c, then Kepler's Second Law,
which is a rigorous consequence of angular momentum
conservation, must hold for any angular momentum

L = mcr sinh(v/c),

and this implies that elliptical orbits must exhibit periastron
precession. The amount of periastron precession can then
be deduced at any instant as

dw = w - w_n,
where w_n is Newtonian angular velocity,
defined as the limit of w when c --> oo,
so it is w_n = v_n/r,
where v_n is Newtonian tangential velocity

Thus, the precession dw is

dw = (c/r) sinh(v/c) - v_n/r,
dw = (c/r)(sinh(v/c) - v_n/c)

We must recall that v is not the Newtonian tangential
velocity v_n, because in general, it would be v > v_n
for any finite speed of light c. So, for any finite radial distance
r along an orbit, the values of v and c 'conspire' in such
a way that precession dw yields always a non-zero value.
The question is, if c is tunned to be the invariant standard
value for the speed of light c_0 = 299792.458 km/s, what is the
relation between v and v_n, if any? By the observation of the
extra Mercury's perihelion precession, which is about 43.1
arcseconds per century, we can attain an ansatz as

dw = (4 pi c_0 / r)(sinh(v_n/c_0) - v_n/c_0)

valid for any orbit, where v_n = sqrt(GM/r). This ansatz tells us
that,

(4 pi c_0 / r)(sinh(v_n/c_0) - v_n/c_0) = c (sinh(v/c) - v_n/
c),
(4 pi c_0 / c )(sinh(v_n/c_0) - v_n/c_0) = sinh(v/c) - v_n/c


An approximation to first order is

(4 pi c_0 / c )(v_n/c_0)^3 = (v/c)^3,
v/c = (v_n /c_0) (4 pi)^(1/3).

Let's return again to the equation c sinh(v/c) = rw. Divide it
by v, it yields

(c/v) sinh(v/c) = rw/v,
call it gamma' factor,
gamma' = (c/v) sinh(v/c).

Thus, Kepler's Second Law can be written as

h gamma' = r^2 w,
where h = v r, so
v gamma' = r w,
c sinh(v/c) = r w


Then, if we apply the standard Lorentz factor to Kepler's
Second Law, it would yield

v gamma = r w,
v/sqrt(1 - v^2/c^2) = r w.

But, this latter would be a wrong answer, since a Lorentz factor
can only be applied to inertial bodies in Minkowski manifolds,
not to bodies orbiting in gravitational fields. I have showed above
that radial distance, r, tangential velocity, v, and angular
velocity,
w, of an orbiting body, are related as

c sinh(v/c) = r w

in any gravitational field. GR was Einstein's attempt to accomplish
a full description of gravitation, but failed, because there is a
more
simple and beautiful way to accomplish the gravitational issue.
Generalized Kepler's Second Law, is a good example.


.

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