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Re: Does Newtonian gravity bend light?



Regarding Larryy's question:

One of the textbooks on my desk says regarding predictions of GR "Gravity
should bend light rays, an effect not predicted by Newtonian mechanics
because light has no mass." Yet
<http://www.theory.caltech.edu/people/patricia/lclens.html> claims that
Newtonian gravity bends light, just less than Einsteinian gravity does.
What gives?

Both are right and both are wrong. The problem is that the question, i.e.
"Does Newtonian gravity bend light?" is ill posed. The answer depends on
just *how* the Newtonian limit is taken. Note that Newtonian physics
deals with the limit of relevant speeds being much slower than c, but
a light beam travels *at* speed c, *not* at an infinitesimal fraction of
that speed. Thus the result will depend on how electromagnetism is
embedded in an otherwise Newtonian version of classical physics.

Case 1. If we consistently take the limit 1/c^2 --> 0 limit of *all* of
relativistic physics *including* Maxwell's equations as we take the
Newtonian limit, we find that not only does Newtonain gravity not bend
light, but that there is no such thing *as light* in the first place,
because in this case all of electromagnetism collapses into
electrostatics where the electric field provides instantaneous
interaction at a distance between charges, and the E field and its
potential can be treated as auxiliary fields that do not possess any
intrinsic dynamical degrees of freedom of their own, and are just
mathematical constructions from the instantaneous Coulombic
configuration of the charge sources.

Case 2. We take the Newtonian limit of particle mechanics and Lorentz
transformations to get Newtonain mechanics that transforms between frames
invariantly using Galilean transformations, but we leave Maxwell's
equations alone. In this case we have to deal with the absolute rest
frame of the Luminiferous ether which has Maxwell's equations remain as
they are. But for all intertial frames in uniform motion w.r.t. that
frame, the form of Maxwell's equations is *different*, and light's
behavior is frame dependent. The amount of deflection of light one
get's *may* depend on the details of the state of motion of the
gravitating body w.r.t. the ether-at-rest frame. The reason I said "*may*
depend" is that there are at least 2 different subcases to consider.
case 2a. Here we assume that a particle feels a gravitational force
proportional to its rest mass and not to its total
relativistic mass/energy before we take the Newtonian limit.
In this case light is *not* bent by gravity at all. The
limit in this case is a Newtonian limit of a Nordstrom-esque
SR-based gravity theory rather than a limit of GR.
case 2b. Here we assume that a particle feels a gravitational force
proportional to its (total energy)/c^2 as the 1/c^2 limit is
taken. In this case I believe the amount of deflection of
light depends on how the gravitating mass is moving w.r.t.
the ether frame.

Case 3. One treats light photons as merely fast-moving Newtonian test
particles that happen to be moving past the massive gravitating body at
speed c. One just calculates the asymptotic deflection for a hyperbolic
conic section (orbit for unbound positive energy solution) of a test
particle moving (Newtonianly) past the gravitating body with a given
impact parameter and plug in v = c into the equation. Here we assume the
photon test particles locally experience an instantaneous acceleration
given by the local value of the gravitational field g of the gravitating
body as they go by. The result is that the deflection is 1/2 of the
angle predicted by GR.

I'm sure there are other limiting cases with other answers that one could
dream up as well.

David Bowman
David_Bowman@georgetowncollege.edu