tabletop geodesics, general relativity, embedding diagrams

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's comment:
> ...
> The physics behind this is that the masking tape is essentially modeling a
> photon.  There are no closed orbits for photons, except at the horizon of a
> black hole, and in that case all orbits in the vicinity have the same radius.

I'm sorry, but this is not correct.  Photons do *not* make closed orbits
about black holes at the horizon.  Rather, they execute circular orbits
about a Schwarzschild black hole at r = (3/2)*r_s instead.  Here r is the
radial spacelike coordinate in 'Standard' coordinates, and r_s is the
Schwarzschild radius of the horizon.  If a photon has its periapsis
occur at r = r_p > (3/2)*r_s then the photon escapes to infinity after
its path curves around near the hole.  Let X == r_s/r_p (ratio of
Schwarzschild radius to periapsis, actually, we want X to be the ratio of
r_s to the r-value the photon has when it is locally moving
perpendicularly to the radial direction to the hole), and let [THETA] be
the total scattering angle between the asymptotic direction of an
incident photon coming in from infinity and its asymptotic direction as
it later escapes to infinity.  The closed form formula for [THETA] as a
function of X is very complicated and involves elliptic integrals of
complicated arguments, but in the simplifying limit of X << 1 (i.e. the
weak field limit where the photon doesn't venture very close to the
horizon such as when starlight grazes outside the solar disk) the photon
scattering angle (in radians) approaches just [THETA] = 2*X.  But when X
is very near (but still less than) 2/3 the scattering angle can be
arbitrarily large and the photon can orbit the hole several times
(spiraling in and then later spiraling back out) before escaping back
out to infinity on an asymptote.  In this case the total scattering
(winding) angle [THETA] diverges logarithmically as X approaches 2/3 from
below.

If X = 2/3 photon's distance is at r = (3/2)*r_s when it moves
perpendicular to the radial direction to the hole, then the photon
orbits the hole in a circular orbit at that fixed radius.

OTOH, when X > 2/3 then the photon curves into the hole (and may spiral
in if X is only very slightly greater than 2/3). In this case X
represents the ratio r_s/r_a of the Schwarzschild radius to the
maximal apoapsis distance (rather than r_s/r_p for the periapsis in the
case of an unbound orbit).  In this latter case we have a photon curving
(or even spiraling) outward from near the horizon to a maximal distance
at r_a where the tangent to its trajectory is locally perpendicular to
the direction to the hole.  This maximal r_a is less than (3/2)*r_s
since X > 2/3.  The rest of the photon's trajectory is an inward curve
(i.e. mirror image of the outward curve or spiral) back to the horizon.

David Bowman
David_Bowman@georgetowncollege.edu