Re: Is LIGO just observing viscosity of the vacuum?
- From: Igor Khavkine <igor.kh@xxxxxxxxx>
- Date: Sat, 2 May 2009 20:18:21 +0200 (CEST)
On May 2, 5:57 am, "Jonathan Thornburg [remove -animal to reply]"
<jth...@xxxxxxxxxxxxxxxxxxxxxxx> wrote:
Suppose you naively try to insert a finite-speed-of-light correction
into Newtonian gravity, by saying that the gravitational force points
in the "retarded" direction, i.e., in the direction where the other
body was a time r/c ago (where r is the distance between the two bodies,
i.e., the orbital diameter). Then it's easy to see that you've
effectively introduced a drag proportional to v/c into the Newtonian
orbital dynamics. This would make orbits decay very rapidly.
(This is not observed: the Earth has been in roughly its present
orbit around the Sun for 4+ billion years. So we conclude that
the retarded-Newtonian-gravity theory doesn't accurately describe
orbital motion.)
In general relativity, it turns out that this v/c drag cancels out.
I think (but am not quite sure) that the reasons for this are somewhat
similar to those for a similar effect in (flat-spacetime) Maxwell
electromagnetism, where the electric field from a charge which is
moving at a uniform velocity, and has *always* been moving at that
same (vector) velocity, points towards/away from the *current* position
of the charge, not the retarded position of the charge. [The "has
always been moving at that same (vector) velocity" part is essential
here!]
In general relativity, it turns out that any drag proportional to
(v/c)^2 also cancels out. [Alas, I don't have a nice intuitive
explanation of why this is so -- it comes out of a long & tricky
calculation.]
Remarkably, the (v/c)^3 and (v/c)^4 terms also cancel out.
[Once again, I don't have a nice intuitive explanation for why.]
In fact, in general relativity, the lowest-order "drag" effect
is damping due to emission of gravitational radiation, which
appears at (v/c)^5.
Landau & Lifshits have some remarks about these cancellations in
Sections 65 and 106 of volume II of their Course of Theoretical
Physics. These sections derive the effective Lagrangian for slow
moving electromagnetic and gravitational point charges, respectively.
In either case, it is possible to ignore the field degrees of freedom
and express the effective Lagrangians in terms of just the particle
positions and velocities only to a finite order in v/c.
Those orders are (v/c)^2, for electromagnetism, and (v/c)^4, for
gravity (as you've already pointed out). The reason they give is that
radiation effects appear at the (v/c)^3 and (v/c)^5 orders and beyond
those orders the field degrees of freedom can no longer be kept out of
the effective Lagrangian. The respective powers are explained by the
fact that electromagnetic radiation couples to charge dipoles, while
gravitational radiation couples to charge quadrupoles. Following
similar reasoning, if there were a field that could only couple to
charge octupoles (or higher moments), then point charges interacting
with that field could have an effective Lagrangian up to (v/c)^6
order.
Igor
.
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