Re: Is LIGO just observing viscosity of the vacuum?

Richard D. Saam <rdsaam@xxxxxxx> wrote:
[[about "gravity propagates at the speed of light"]]
How does this all fit within the context of:

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

Gravity travels at the speed of light BUT:
"the "force" in GR is not quite central--it does not point directly
towards the source of the gravitational field--""it depends on velocity
as well as position. The net result is that the effect of propagation
delay is almost exactly canceled, and general relativity very nearly
reproduces the newtonian 'instantaneous' result." 'in the constant
velocity condition'.

In this constant A,B velocity condition,
gravity interaction between bodies A & B
is instantaneous across the universe?

It's complicated. :) But basically, the answer is "no".


More precisely, John Baez's (excellent!) web page which you cited
is referring specifically to the case where we have two masses in
orbit about each other, and where that system is "nearly isolated"
(in the sense that we can neglect outside influences on it), and
where the two bodies are orbiting relatively slowly compared to
the speed of light. This could, for example, describe the orbits
of planets in our solar system around the Sun, or the orbital motion
of a binary black hole or binary neutron star any time before the
bodies get "too close" to each other (where "too close" depends
on just how much slower than the speed of light we want).

The underlying conclusions I'm going to state are valid for any
kind of orbits, but for simplicity I'm only going to consider the
circular-orbit case hereinafter.


If you consider Newtonian mechanics, then the orbit is stable
(it doesn't decay), essentially because the Newtonian gravitational
force is always central, i.e. parallel to the (instantaneous) vector
between the two bodies, and thus perpendicular to the instantaneous
velocity vector. Thus the dot product of the force and the velocity
vector (which gives the work done by the force) is zero.

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.

It's this remarkable series of cancellations that the web page
mentioned above was discussing.


Finally, I emphasize that all of these calculations were done assuming
a pair of bodies in a bound orbit around each other, in an otherwise-
-empty asymptotically-flat spacetime (i.e., one where, far from the
bodies, the spacetime is almost Minkowski). It would definitely *not*
be correct to extrapolate these results to a pair of bodies which are
any nontrivial distance apart in a more complicated (realistic)
cosmology.


Another way to look at the original poster's question is to consider
the following gedanken experiment:

Consider two small masses joined by a compressed spring.
Because the masses are "small", we can approximate spacetime as
very nearly Minkowski. This lets us set up a special-relativity--style
(almost) inertial coordinate system system of observers, and equip
each observer with a gravitational-wave detector.

Now at time t=0 we release the spring, so the two masses start
oscillating farther-closer-farther-closer-... from each other.
This emits gravitational radiation. We can then ask, at what time
does an observer a distance r away from the two masses, first detect
that gravitational radiation?

The answer is, r/c. That is, gravitational radiation travels at
the speed of light.

Making this statement more precise is technically a bit tricky,
but it can be done, and the basic results don't change.


ciao,

--
-- "Jonathan Thornburg [remove -animal to reply]" <jthorn@xxxxxxxxxxxxxxxxxxxxxxx>
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"C++ is to programming as sex is to reproduction. Better ways might
technically exist but they're not nearly as much fun." -- Nikolai Irgens

.

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