According to general relativity (and its still-viable metric-theory
competitors, e.g. the Brans Dicke theory) the 'speed of gravity' is the
same as the speed of light which is the same as for the motion of any
massless particle/excitation. As far as I know this particular prediction
of GR has not been verified experimentally yet via a direct measurement.
Even the existence of propagating gravitational waves has not been
*directly* observed, although there is substantial indirect evidence for
them, especially in the observed orbital decays of bound strongly
gravitating systems, (e.g. binary pulsars) for which the orbital decays
follow the precise predictions of GR for the energy loss from the system
via gravitational radiation. Even though the 'speed of gravity' has not
been directly measured, virtually no physicist thinks it is not
c = 299792458 m/s.
>The hypothetical question I have posed to physicists is the following: if
>the sun all of a sudden disappeared, it would take ~8 minutes for us to be
>engulfed in darkness. Would it also take 8 minutes for the earth to fly off
>in space (and would we have 8 minutes to see the sun become progressively
>smaller, or would it happen sooner?
The Sun could not just disappear all of a sudden without the performance of
a major miracle, since such a disappearance would strongly violate the laws
of nature. Once the laws of nature are violated we can't say "what would
happen" under such circumstances. In fact, there is a theorem in GR,
called Birkoff's theorem, that states that a spherically symmetric
gravitational field in empty space is, necessarily, static (i.e. it cannot
change in time). The gravitational field in the vicinity of the Sun is
to a *very* good approximation spherically symmetric. It's external
gravitational field is, therefore, constant in time. So if the Sun (and
its mass) instantly ceased to exist we would have a physical contradiction
where the remnant gravitational field would indicate a mass that didn't,
in fact, exist. *But*, be that as it may, *if* the miraculous aspect of
this disappearance was confined to just the vicinity of the present/former
location of the Sun in spacetime, and *if* some way could be found around
Birkoff's theorem (maybe the Sun could be caused to disappear over a short
non-zero time in a significantly non-spherically symmetric way), then the
rest of the solar system would orbit as it has been for a few minutes
until the gravitational shock wave from the Sun's disappearance reached
each planet. For the Earth this would be the same 8 minutes that the light
from the Sun takes to get to the Earth. After this time interval the Earth
would, presumably, go off on an orbital tangent (after its motion stablized
after the passage of the shock wave). The type of gravitational radiation
used for this, necessarily, anisotropic shock wave would have to have a
very peculiar (quadrupole) polarization relative to the anisotropy in the
shape of the the shock wave. I'm not quite sure how (or if) it could be
made to work.
>For that matter, is there any reason, physical or otherwise, why nothing can
>go faster than the speed of light?
There is a reason, but it is not true that *nothing* can go faster than c.
The constraint is that no causal or informative influence can travel
locally faster than c. If some phenomenon does not involve the propagation
of causal influences then that phenomenon may be allowed to go faster than
c. The physical reason for this constraint is simply that there appears to
be a speed limit for causation in nature, and this speed limit happens to
be c. Experimentally, there is no instantaneous interaction at a distance.
All interactions over a distance d between two physical entities are
delayed by the time t it takes for the influence of changes in one of the
entities to propagate to the other entity, where t obeys the weak
inequality t >= d/c. A couple of phenomena that may happen faster than
c are: 1. the collapse, upon measurement, of a quantum wave function, and
2. the rate of recession of very distant parts of the universe (due to the
Hubble expansion of the universe) from us that happen to be well beyond the
causal horizon which exists as a remnant of the Big Bang, which happened
in a finite amount of time in the past. There are also other phenomena
which may happen faster than c and still obey all the laws of nature.
> I understand that mass approaches
>infinity as the speed of light is approached, but does this apply to gravity
>as well? (whatever gravity is, anyway)
This is a poor way of looking at the situation. It is better to understand
the mass of an object as a property of that object and *not* as a joint
property of the object and the state of relative motion between the object
and a potential observer. (The modern definition of mass requires that it
is an invariant scalar under Lorentz transformations.) It is true,
however, that the faster an object moves relative to an observer that the
object's inertial property (i.e. resistance to acceleration, F/a)
increases with speed. We just do not call that property 'mass'. After
all, an object's resistance to acceleration is separately dependent on the
directions of the net applied force *and* the direction of the velocity.
In general, at high speeds, the direction of the net applied force is not
even parallel to the direction of the object's acceleration.
We consider gravity as the manifestation of the curvature of the spacetime
manifold. Understanding this statement requires a familiarity with the
mathematics of differential geometry.