Twin Paradox: Which Twin is Older?

[virgil elings]

Maybe someone can explain the following twin paradox. There are two twins A and B. Twin A is shot straight up into the air with a very short and powerful cannon. Twin B remains on earth. Twin A goes up a distance and then falls back to earth. Which twin is now older?

 

[A.T.]

Which twin is now older?

Captain Cannonball

 

[DaveC426913]

Captain Cannonball

Two forms of dilation are in play, both will cause twin A to age less.

time dilation due to speed (Twin A will age less while making a round trip)
time dilation due to gravitational potential (Twin A will age less while spending time in a lower gravity well)

 

[A.T.]

time dilation due to speed (Twin A will age less while making a round trip)

A is the inertial one, while B experiences proper acceleration all the time. So the round-trip argument says that A ages more. However, you have to be careful with this argument in curved space time, because in some cases (not this one) the inertial world-line accumulates less proper time.

time dilation due to gravitational potential (Twin A will age less while spending time in a lower gravity well)

A is flying up, and spends time at higher gravitational potential, than B. So that argument again says that A ages more.

 

[harrylin]

Maybe someone can explain the following twin paradox. There are two twins A and B. Twin A is shot straight up into the air with a very short and powerful cannon. Twin B remains on earth. Twin A goes up a distance and then falls back to earth. Which twin is now older?

This has been tested with a clock in a rocket in the "gravity probe A" experiment (Vessot 1980). The rocket engine was stopped and the rocket fell back to earth. What makes this test even more interesting than the answer you are seeking, is that they had found a smart solution to monitor the clock rate during the flight.
As reckoned with the reasonably inertial ECI frame of B (thus also correcting for the Earth's rotation), the gravity probe experimenters expected that clock A would tick faster when going up due to increased gravitational potential, but they also accounted for the reduced ticking rate of clock A due to its speed. The end result (just before crashing) was that A was measured to be "older" than B by the expected amount.
PS: I did not try to derive if there is an extreme case in which the end result would be the inverse.

 

[Dale]

time dilation due to gravitational potential (Twin A will age less while spending time in a lower gravity well)

Twin A is shot up, so twin A is higher in the gravitational potential and ages more. Clocks run slow (in Schwarzschild coordinates) on a big planet or near the EH.

 

[DaveC426913]

Man, I bolluxed that up badly...

 

[stevendaryl]

Man, I bolluxed that up badly...

No, what you said was correct. Assuming that all velocities are much smaller than $c$, and that the Earth's radius is much larger than its Schwarzschild radius, then we can approximate the effects of GR on the proper time of a clock as follows:

$\delta \tau = \delta t (1 - \dfrac{T - V}{mc^2})$

where $T$ is the Newtonian kinetic energy of the clock, $V$ is the Newtonian gravitational potential energy, and $m$ is the mass of the clock, and $t$ is the time as measured by a clock of zero velocity at a location where $V=0$. So $T$ being larger makes $\delta \tau$ is smaller--a velocity-dependent time dilation. $V$ being larger means $\delta \tau$ larger--a location-dependent time dilation. So the total time dilation can be thought of, in this approximation, as a sum of a velocity-dependent part and a "gravitational time dilation". The full GR treatment doesn't really separate out a gravitational part from the velocity part, but it makes sense to do that as a nonrelativistic approximation.

 

[A.T.]

No, what you said was correct.

Including his conclusion that the cannonball-twin will age less?

 

[stevendaryl]

Including his conclusion that the cannonball-twin will age less?

Whoops. No, he was mistaken about that. So he gets partial credit for correctly enumerating the two effects, but not full credit, because he didn't correctly compute those two effects. The cannonball twin travels higher in the gravitational potential, rather than lower.

 

[m4r35n357]

There is a nice discussion of the combined effects of speed and gravity in this article: http://mathpages.com/rr/s6-05/6-05.htm

 

[DiracPool]

There is a nice discussion of the combined effects of speed and gravity in this article: http://mathpages.com/rr/s6-05/6-05.htm

That's a good resource. If the OP wants a less mathy treatment of the subject, here's a fun video that discusses the combined effects in the context of GPS tracking. Trivia question, can you state whether GR or SR has a greater effect on the time differences between satellite and ground units, and by how much each? The answer is at 4:40.

 

[A.T.]

Assuming that all velocities are much smaller than $c$, and that the Earth's radius is much larger than its Schwarzschild radius, then we can approximate the effects of GR on the proper time of a clock as follows:

$\delta \tau = \delta t (1 - \dfrac{T - V}{mc^2})$

where $T$ is the Newtonian kinetic energy of the clock, $V$ is the Newtonian gravitational potential energy, and $m$ is the mass of the clock,

Doesn't the mass of the clock $m$ cancel out here?

 

[harrylin]

Doesn't the mass of the clock $m$ cancel out here?

Yes it does, but then the equation has to be written differently: m is also inside T and V.
It's a very neat equation, you can also write it all as energies - just replace mc² by E.

 

[stevendaryl]

Doesn't the mass of the clock $m$ cancel out here?

Yes, but the nice thing about writing it this way is that it's clear that maximizing proper time is the same (in the nonrelativistic approximation) as minimizing the classical action, which is of course, is the same as obeying Newton's laws of motion. So to this approximation, geodesics are the classical paths.

 

[A.T.]

Yes, but the nice thing about writing it this way is that it's clear that maximizing proper time is the same (in the nonrelativistic approximation) as minimizing the classical action, which is of course, is the same as obeying Newton's laws of motion. So to this approximation, geodesics are the classical paths.

Thanks for the clarification. At first sight it might look like the mass of the clock is relevant, even in this weak field approximation.

 

[IgorM101]

This has been tested with a clock in a rocket in the "gravity probe A" experiment (Vessot 1980). The rocket engine was stopped and the rocket fell back to earth. What makes this test even more interesting than the answer you are seeking, is that they had found a smart solution to monitor the clock rate during the flight.
As reckoned with the reasonably inertial ECI frame of B (thus also correcting for the Earth's rotation), the gravity probe experimenters expected that clock A would tick faster when going up due to increased gravitational potential, but they also accounted for the reduced ticking rate of clock A due to its speed. The end result (just before crashing) was that A was measured to be "older" than B by the expected amount.
PS: I did not try to derive if there is an extreme case in which the end result would be the inverse.

I think the extreme case would be if the canonball dude remained kinda stationary for a relatively amount of time in space the twin on Earth would age much faster and hence they might be the same age or the Earth twin might be older due to gravity time relationship

 

[stevendaryl]

I think the extreme case would be if the canonball dude remained kinda stationary for a relatively amount of time in space the twin on Earth would age much faster and hence they might be the same age or the Earth twin might be older due to gravity time relationship

No. When the "cannonball" twin is at the top of his trajectory, his speed is lowest and his altitude is the highest. Both effects would make him age faster than the Earth-bound twin (as measured in Earth-centered coordinates).

 

[harrylin]

No. When the "cannonball" twin is at the top of his trajectory, his speed is lowest and his altitude is the highest. Both effects would make him age faster than the Earth-bound twin (as measured in Earth-centered coordinates).

Indeed; and the issue with this set-up is that a very fast rocket that is shot straight up will never fall back to earth. And of course, if we change the question then it's easy to get different results.