Does gravity slow time's passage?

[poeteye]

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?

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?

 

[poeteye]

I meant to indicate:
Let's say the orbiting twin is traveling near light speed, but is not accelerating.

 

[JesseM]

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 R₀ is the Schwarzschild radius of the spherically symmetric mass, equal to 2GM/c² (G is the gravitational constant, M is the object's mass, c is the speed of light).

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.

 

[Nabeshin]

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?

I think this is buried in JesseM's reply, but must not the twin on circular orbit be accelerating simply because it is a circular orbit?

 

[JesseM]

I think this is buried in JesseM's reply, but must not the twin on circular orbit be accelerating simply because it is a circular orbit?

In special relativity circular motion would be acceleration, but in curved spacetime a freefall orbit (freefall=motion not under the influence of any non-gravitational force) would be "locally inertial" motion according to the equivalence principle. That's why astronauts feel weightless for example--not because gravity is much less at the distance they orbit, but just because they're in constant freefall.

 

[Nabeshin]

In special relativity circular motion would be acceleration, but in curved spacetime a freefall orbit (freefall=motion not under the influence of any non-gravitational force) would be "locally inertial" motion according to the equivalence principle. That's why astronauts feel weightless for example--not because gravity is much less at the distance they orbit, but just because they're in constant freefall.

Obviously. But is this what was meant by the OP?

Also this would imply that the "stationary" twin would be the one accelerating, since they meet back up after one orbit. Either way, someone has nonzero acceleration, so not sure what was meant by the OP's not accelerating remark... From the OP's clarification of near light speed, it seems to me he is interested in the special relativistic scenario... I won't say any more without clarification of which situation he's actually interested in.

 

[JesseM]

Also this would imply that the "stationary" twin would be the one accelerating, since they meet back up after one orbit. Either way, someone has nonzero acceleration, so not sure what was meant by the OP's not accelerating remark... From the OP's clarification of near light speed, it seems to me he is interested in the special relativistic scenario... I won't say any more without clarification of which situation he's actually interested in.

Well, the OP talked about one twin being in a circular orbit "but not accelerating"--it's possible the poster just wasn't aware that change in direction in an inertial frame is "acceleration" even if the speed is constant. Clarification is needed, yes.

 

[yuiop]

Obviously. But is this what was meant by the OP?

Also this would imply that the "stationary" twin would be the one accelerating, since they meet back up after one orbit. Either way, someone has nonzero acceleration, so not sure what was meant by the OP's not accelerating remark...

This is an interesting point. If one twin is hovering at a given height but motionless relative to the CMB and the other is in a natural orbit at the same height, then the orbiting twin experiences the least proper time when they meet again. The difference in elapsed proper times of the twins in this case is purely a function of their relative velocities. This is slightly odd, because it is a rare example of the observer that experiences proper acceleration that experiences the most elapsed time. The opposite is nearly always true.

It is clear that the OP meant proper acceleration, because he said it is the orbiting twin that is not accelerating.

 

[poeteye]

Well, the OP talked about one twin being in a circular orbit "but not accelerating"--it's possible the poster just wasn't aware that change in direction in an inertial frame is "acceleration" even if the speed is constant. Clarification is needed, yes.

I assumed that an orbit was actually a straight path through curved four-dimensional space.
However, this aside, can you tell me if I got the following correct?

SPECIAL RELATIVITY
-- James Ph. Kotsybar

No longer absolute, we now know time
is altered by travel’s velocity
and will even change, the higher we climb --
a relational curiosity
of gravity and acceleration
called the equivalency principle.
Though some would view this as aberration,
since it implies Newton’s famous apple,
in its free-fall, would maintain the notion
that Newton’s noggin, resisting Earth’s pull,
was the object exhibiting motion.
Thus, common sense is just ostensible.
Falling fruit sustains a fixed point in space,
while we sense it’s time-sense starting to race.

 

[yuiop]

Well, the OP talked about one twin being in a circular orbit "but not accelerating"--it's possible the poster just wasn't aware that change in direction in an inertial frame is "acceleration" even if the speed is constant. Clarification is needed, yes.

As you yourself pointed out earlier, the orbiting twin is not in an inertial frame. He is following a geodesic which is equivalent to a straight line in curved space and so experiences no proper acceleration, (but can be said to have coordinate acceleration).

 

[poeteye]

Can you explain coordinate acceleration with regard to its difference to proper acceleration's influence on the (perceived) passage of time?