poeteye said:
If gravity and acceleration are "cousins," How much faster does time proceed on Earth than floating motionless (or rather without acceleration) outside the Earth's gravitational field?
Accelerating clocks tick slower, right? So too for clocks on Earth's surface, then?
Well, relative to a given reference frame, clocks tick slower based only on their velocity, not acceleration. But the twin paradox demonstrates the frame-independent fact that if two clocks compare readings, move apart, and then later reunite and compare readings again, then if one clock moved inertially between meetings while the other accelerated at some point, the one that accelerated will have elapsed less time. Even here though it's really the "shape" of the path through spacetime that matters, the magnitude or length of acceleration doesn't determine the difference in ages...one can compare it to the fact that in ordinary 2D geometry, a straight-line path between two points will always have a shorter length then a path between the same points that has a bend in it (for more on this geometric analogy, see [post=2972720]this post[/post]).
Anyway, to answer your question, in general relativity it's always a little tricky to talk about the rates of ticking of different clocks since this depends on the choice of coordinate system and there are an infinite number of equally valid coordinate systems you can use. But if we're talking about the gravitational field generated by a spherically symmetric nonrotating mass, this gives the
Schwarzschild metric, and the most common coordinate system to use would be
Schwarzschild coordinates. So relative to these coordinates, if you have one observer at rest at radius R and the other infinitely far away from the mass, the clock of the observer at radius R is ticking slower than the clock of the distant observer by a factor of \sqrt{1 - \frac{R_0}{R}} (see
here), where R
0 is the
Schwarzschild radius of the spherically symmetric mass, equal to 2GM/c
2 (G is the
gravitational constant, M is the object's mass, c is the speed of light).
poeteye said:
Also, with regard to the twin paradox, if one twin is on a circular orbit that brings him back to the other twin, but is not accelerating, do they experience time differently?
If one twin is not moving relative to a nonrotating sphere (say, standing on the surface of a nonrotating planet, or standing on a platform attached to it) while the other departs the nonmoving twin, makes an orbit, and then returns, the twin that orbits will have aged less. The idea that two observers in relative motion can each say the other is aging more slowly only works when you can analyze the situation from one of two
inertial reference frames. If you use general relativity to model the gravity of the sphere, then spacetime around it is curved and no coordinate system covering a large region of curved spacetime can qualify as "inertial" (though you can have 'locally inertial' frames on infinitesimally small regions of curved spacetime, and objects in freefall like the orbiting twin are moving in a 'locally inertial' way, see the bottom part of
this article for details). Even if we ignore gravity and just imagine the sphere is massless and the other twin is using rockets to travel in a circle around it rather than orbiting naturally, then although there will be no spacetime curvature here, the one that moves in a circle is moving non-inertially (accelerating) so this is just a variant of the twin paradox where the twin that accelerates between two meetings is guaranteed to have aged less than the one that moved inertially.
