From: carlip@landau.ucdavis.edu (Steve Carlip)
Newsgroups: sci.physics
Subject: Re: Spin
Date: 8 Oct 92 07:00:29 GMT
Organization: Physics, UC Davis

In article <Bvon89.I70@well.sf.ca.us> metares@well.sf.ca.us (Tom Van Flandern) writes:
>[much deleted...]
>     If gravitation propagated at the speed of light or anything comparable,
>it too would be subject to aberration.  If an attractive force eminates from
>the Sun with some speed g, then we can represent the force by a velocity
>vector from the Earth toward the true Sun with magnitude g.  Call this vector
><EG>.  Then on a moving Earth with velocity <EV>, the apparent direction of
>the force will be <EV> + <EG>.  Observations flatly contradict this, so we
>can be sure that gravity does not propagate at the speed of light or anything
>comparable.
>[...]
>     Although I'm trying out the following wording for the first time, I
>believe it to be accurate.  Consider two large masses in close orbit; for
>example, a binary pulsar.  Each body responds gravitationally to the true,
>instantaneous position of the other (not its position retarded by lighttime,
>nor its extrapolated retarded position; but its exact, true position, as
>required by observations).[...]  

This is a bit tricky --- a couple of the "obvious" terms in this description
are highly ambiguous in general relativity.  To me, the continuing discussion
about the "speed of gravity" is largely a cautionary tale about the dangers
of trying to apply flat space concepts to curved spacetime.

For instance: "a vector from the Earth toward the true Sun," presumably a
spatial vector at a fixed time, makes sense if space is flat and you know
what a "fixed time" means.  But in general relativity, space isn't flat,
and there are an infinite number of "time-slicings" of spacetime, each with
a different definition of a "fixed time."  Until you specify a coordinate
system, you don't know what <EV> and <EG> mean.  Similarly, "the true, 
instantaneous position" is something that is simply not defined in general
relativity.  Even in special relativity, the relativity of simultaneity
makes "instantaneous position" ambiguous; in general relativity, it is
much more so, since you don't even have a set of inertial frames.

For another instance: "an attractive force" (of gravity) presumably means
something that produces acceleration, where acceleration is the second
derivative of position with respect to time.  But this is coordinate-
dependent, too --- you have to tell me the coordinates you're using to
describe position and time.  Again, this is easy in a flat spacetime,
where you can refer back to an inertial frame (say, the frame in which
the sun is at rest).  But there is simply no unambiguous generalization
to curved spacetime.  In fact, this "Newtonian" acceleration isn't even
covariant; it can be zero in some coordinates and nonzero in others.
There is a covariant definition of acceleration (the second covariant
derivative of position with respect to proper time), but in a gravitational
field, this is _zero_.

What Tom is saying is true, if you add the qualifier "in some coordinate
system" all over the place.  I'll even go for "in some coordinate system
in which the equations of motion are comparatively simple for a certain
class of problems."  

But if you really want to surprise me, you would have to find some way of 
restating these results in a coordinate-independent manner, in terms of 
genuine, physically measurable quantities.  Try this --- you'll see it's
a lot harder than you might think.  (For instance, a candidate for "a vector
from the Earth toward the true Sun" would be "a tangent vector to a spacelike
geodesic between the Earth and the Sun."  But which spacelike geodesic?
For that, you need to specify a time coordinate.  Maybe "specify a spacelike
hypersurface S orthogonal to a timelike Killing vector, and connect the
intersection of the Earth's world line with S to the intersection of the
Sun's world line with S by a spacelike geodesic."  This might work, but is
it the same as the direction Tom is talking about?  Now try "acceleration"
...)

Steve Carlip
carlip@dirac.ucdavis.edu
  
Xref: sparky sci.physics:16218 sci.math:12942 soc.men:17497 soc.women:17501