[Phys-L] Re: Help on a problem from Goldstein

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Leigh's discussion below:

> John Mallinckrodt writes:
> > Leigh,
> >
> > Apparently the assignment is to apply special, but not general
> > relativity.  I suppose I'd try to solve the equation
> >
> >   -G M r_vec/r^2 = d(gamma v_vec)/dt
> >
> > separating out the radial and azimuthal components and performing
> > something like the usual small amplitude radial oscillation
> > analysis to find the apsidal angle.
> >
> > But maybe that isn't saying anything you hadn't already thought of
> > and the stumbling block lies further along that path.
>
> Thanks. That sounds like a sound approach. I hadn't appreciated the
> applicability of that approach, but I'm ultimately looking for a
> more intuitive description based on special relativistic effects
> like mass increase with velocity*. I will try it with an inverse
> cubic term on the left. (John is acquainted with the problem I'm
> working on.)
>
> and finally Bernard Cleyet chimed in:
>
> > Oh good that's the ed. I can't find. Cost < $10.
> >
> > My first thought was somewhere I'd heard one could expand and use
> > the first and second? terms.  Thereby, adding a cubic term to the
> > force. (quad potential, naturellement).  Some theorem, I think,
> > that only the linear and quad central force result in no
> > precession.  That's why some early on suggested gravity was not
> > exactly inverse square.
>
> Yes. I hit on that technique myself. It is applicable to my problem,
> though an unphysical term, one that seems to depend on the radius of
> the central mass (!), also shows up. I can ignore it, of course, but
> it is larger than the inverse r-cube term I'm keeping, so I feel I
> must be on the wrong track. While the problem I'm working on is
> really unphysical, it should have a reasonable solution lacking such
> terms.
>
> Many thanks for the responses, fellows. It is nice to have
> colleagues on my desktop who will indulge my questions somewhat less
> jusgmantally than the ones I see face to face at school.
>
> Leigh

Leigh, I don't quite know what you're up to with your particular
version of this orbital precession problem and its seemingly manually
inserted counter-terms in the potential function, but I thought I
would point out that it is possible to map the exact *general*
relativistic version of the problem's orbit (in standard
Schwarzschild coordinates) onto a corresponding *Newtonian* problem
with a particularly chosen attractive 1/r^3 potential (1/r^4 force)
counter-term in addition to the actual 1/r^2 potential present.  In
particular, suppose we add to the usual Newtonian potential a 1/r^3
term so the total potential is taken as

V(r) = -m*M*G*(1 + (h/(c*r))^2)/r

*and* then take the Newtonian energy E for this Newtonian problem
and replace it by E --> E'*(1 + E'/(2*m*c^2)) where E' is the
orbiting particle's energy for the general relativistic version of
the problem where the zero level for E' is for a particle at rest at
r = [infinity] (so E' doesn't include any rest energy).  Here the
energy E' is the energy taken with respect to the standard
Schwarzschild coordinate system whose time parameter only really
correctly measures the proper time for observers at rest at
r = [infinity] and whose radial coordinate r also does not really
mean the actual proper radial distance but rather is
r = [circumference of a fixed-r circle]/(2*[pi]) concentrically
inscribed on the curved Schwarzschild geometry for each r-value
labeling the various nested spherical surfaces of constant radius.
The parameter h above is, as before, the orbiting test particle's
orbital angular momentum per unit mass.

If we solve for the orbit of the *Newtonian* problem using the above
V(r) function and make the energy re-definition replacement above
*then* the Newtonian orbit found is also the exact orbit of the
general relativistic version of the problem of a small test particle
in its corresponding orbit about the Schwarzschild geometry using
standard Schwarzschild coordinates.

This is different than what happens for the straight special
relativistic version of the problem in which all the special
relativistic effects can be exactly incorporated in to the
corresponding Newtonian orbit problem without adding any
counter-terms to the potential at all, but rather by simply
renormalizing the values of the parameters h, M, E, and the
angular variable [phi] (of polar coordinates, i.e. the azimuthal
angle) as we turn on a finite value for 1/c^2.

David Bowman
_______________________________________________
Phys-L mailing list
Phys-L@electron.physics.buffalo.edu
https://www.physics.buffalo.edu/mailman/listinfo/phys-l