Planetary Aberration

[Richard Livingston]

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. A quick calculation for the earth-sun
system shows that the angular momentum would double in about 800 years,
which clearly has not been happening.

My question for this group is why not? I'm not equipped to calculate GR
results for this problem, but I have calculated for electromagnetic
attraction, such as for positronium (an electron and positron in mutual
orbit). Even including magnetic effects does not displace the direction
of attraction towards the center of gravity. Likewise the Lorentz
transform and relativistic doppler shift and "search light" effect also
do not solve the issue, and if anything make it worse.

I've been considering the possibility that each object is attracted to
images of the other object on both the past and future light cones, but
this has the issue that oscillating motion of the future image should
radiate signals to the object in the past, like a radio transmitter,
which also clearly does not happen.

Does anyone here know of any papers treating this? I've been searching
on scholar.google.com, arxiv.com and the APS journal web site, but so
far I haven't found anything that addresses this issue.

Rich L.

[Tom Roberts]

Post by Richard Livingston
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

[Steven Carlip]

Post by Richard Livingston
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. A quick calculation for the earth-sun
system shows that the angular momentum would double in about 800 years,
which clearly has not been happening.
My question for this group is why not?

I've done the calculation, for both electromagnetism and GR,
in detail, in https://arxiv.org/abs/gr-qc/9909087.

Steve Carlip