On 5/16/23 2:24 AM, Richard Livingston wrote:
Recently I've been thinking about planetary aberration. This is the
effect where the finite speed of light causes an orbiting body to see
the opposite body on its past light cone, and thus displaced from its
"now" position directly opposite the center of gravity. The problem
is that if the body it attracted directly towards this displaced
position, there is a torque on the system that would cause it to
speed up.
This clearly does not happen. [...] My question for this group is why
not?
In Newtonian mechanics, the gravitational interaction between sun and
earth is instantaneous action at a distance, so the force on the earth
always points directly to where the sun is now, at each point around the
orbit.
In GR this is different, as there is no instantaneous action at a distance:
In the full theory, treating the earth as a massless "particle" with
negligible effect on the geometry [#], the earth follows a timelike
geodesic through spacetime, and that geodesic is completely independent
of where the earth is located. No force is involved, the earth just goes
as "straight as possible" through the geometry, which is an orbit around
the sun [@].
[#] This is an excellent approximation, as the earth
mass is just ~ 0.000003 of the sun's mass.
[@] Considered in approximately Minkowski coordinates
in which the sun is at rest, this path is an elongated
helix along the time axis, with radius ~ 8 light minutes
and period 1 light year -- just a few parts per million
different from the straight line of the sun.
In the PPN approximation to GR, one uses a Minkowski metric on spacetime
(i.e. flat, with clear simultaneity in the sun's rest frame). One can
identify a term in the equation as a force -- this force points not to
the current location of the earth, but rather to the 2nd-order
extrapolation of where the earth will be when the force (propagated at
c) reaches the earth. Remember than an ellipse is a 2nd-order curve, so
that extrapolation is extremely good, and the interaction differs by an extremely small amount from the instantaneous interaction of Newtonian mechanics -- the gravitational effects of other planets are enormously
larger.
[This is reminiscent of classical electrodynamics, in
which the Lienard-Wiechert potentials perform a
similar extrapolation to first order.]
Tom Roberts
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