Re: Does Newtonian gravity bend light?

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Roger Haar's comment:
> ...
>        In Misner Thorne and Wheeler (pages 177-186)
> deals with fails in simpler models for gravity.
> There is no deflection of light if gravity is a
> scalar field.

This gets at the comments in my case 2a, and is what I meant by a
"Nordstrom-esque SR-based theory".  Such a theory *is* a scalar theory
in that the gravitational potential is a 4-scalar, and its source is the
scalar (rest) mass of the massive particles present.  It is fomulated on
the flat Minkowski spacetime of SR.  Such a theory has light completely
obliviously unaffected by any gravitational field present, and EM fields
do not serve as sources for the dynamical gravitational field.  Since
such a relativistic theory doesn't have light being deflected, it stands
to reason that its Newtonian limit also doesn't have light deflected by
a gravitating body (assuming of course that the limit is taken in such a
way that light waves themselves somehow remain in tact in that limit).

>        As I understand the difference deflection is
> about a factor of two.  The difference comes about
> because in the GR case, 4-space is NOT Euclidean.

The way I interpret the situation is that in GR there are *two* causes
for the gravitational bending of a light ray.  One of them is due to
the spatially dependent gravitational time dilation factor which has
the ordinary Newtonian potential result from it in the non-relativistic
limit.  It causes the effects we see for small-mass test particles moving
through the gravitational field of a gravitating body. Such a non-
relativistic limit has the test particles themselves moving slowly wrt c.
Their deflections are caused by Hamilton's principle making the
trajectory chosen the one that minimizes the action by maximizing the
elapsed proper time.  Since there is a spatially dependent time dilation
factor due to the gravitational field, this causes the path to refract
(sort of like like how Fermat's Principle makes the light rays of a mirage
refract when going through an inhomogeneous air medium with a spatially
dependent index of refraction).  The result of all this is in the
non-relativistic limit is ordinary Newtonian gravity with ordinary
Newtonian particle dynamics.  In this limit the effects of the GR-caused
distortions in the geometry of space itself is negligible because the
nonrelativistic test particles are moving so slowly that their
world lines are nearly parallel to the time axis and they travel through
much more time than they do through space, so that only the GR-caused
distortions of time (i.e. dilations) have a cumulative effect that is
strong enough to show up in the particle's trajectory in the non-
relativistic limit.

If we take the resulting limiting theory described above and pretend that
light is made of nonrelativistic photons that happen to travel at speed
v=c, it gives 1/2 of the correct deflection when such a photon passes a
gravitating body.  The reason for the error is the fact that for light
there is a *second* cause of deflection of the beam that an ordinary
nonrelativistic particle does not experience.  This second cause of
deflection is the geometric distortion of space itself--which for
slow (compared to c) particles has neglibible effect in the nearly
flat spacetime limit of Newtonian gravity.  Recall that light rays do
*not* move slowly wrt c, but move *at* speed c on null geodesics.  This
means they are moving at a 45 deg angle wrt the time axis in spacetime.
As they go they sample just as much 'time' as they do 'space'.  For
a vacuum solution of Einstein's GR equations it can be shown that the
amount of distortion of time is the 'same strength' as the distortion
of 'space' (typically the timelike coefficient of the metric tensor is
the reciprocal of the radial spacelike metric coefficient in the
spherically symmetric Schwarzschild geometry).  This means that a light
ray will be affected just as much by the distortions of space as it is
by the distortions of time, and these distortions have equal-strength
deviations from the flat Minkowski spacetime case.  Thus the amount of
response of the light ray to those distortions is *twice* as much as
would be the case if the distortion in space was ignored when minimizing
the action for the trajectory.  This makes the light ray deflect by
twice the prediction of the naive Newtonian limit.

David Bowman
David_Bowman@georgetowncollege.edu