[GR] Which clock is ahead?

Question

Precise clock A is fired from a cannon vertically upward.
Another twin clock B remains at rest near the cannon.
After clock A lands, which clock is ahead?

Clocks are insensitive to acclerations.
Air resistance is absent.

People, did you ever hear of twin paradox? The only question here which clock is in inertial frame of reference, and which is not (the tricky part).

Answer 1

There are two competing phenomenon here

Time dilation due to velocity: the flying clock will experience a slowing of time wrt to the stationary clock due to its velocity and accelerated path

This time dilation is proportional to sqrt(1-v^2/c^2)
where
v is velocity
c is speed of light

Time dilation due to gravity: the clock near the cannon is closer to earth's center and thus experiences also some slowing of time wrt to the other clock due to the higher gravity field.

This time dilation is proportional to sqrt(1-2GM/(rc^2)) where
G is gravitational constant
M is earth's mass
r is distance to earth center
c is speed of light
Since both clocks are in the gravitational field, we will need to divide the two terms: dilation at ground elevation divided by dilation of flying clock.

Of course the exact time dilation experienced by the clocks is across different speed and different heights throughout the whole trajectory. But if we can show that at the maximum velocity and maximum height, one effect is considerably larger than the other, we will be able to answer quickly (and most of all easily).


Let's assume v = 1000m/s
so maximum height is 50000m

Dilation due to maximum velocity is
Dv = sqrt(1-1000^2/3E8^2) = sqrt(1-1.1E-11)
Dv = 1-5.5E-12

Dilation due to gravity
Dilation of ground clock
Dg = sqrt(1-2*6.7E-11*5.9E24/6.40E6*3E8^2)
Dg = sqrt(1-1.37257E-9)

dilation of flying clock at maximum altitude
Df = sqrt(1-2*6.7E-11*5.9E24/6.45E6*3E8^2)
Df = sqrt(1-1.36193E-9)

Dilation of ground wrt to flying is
Dg/Df = sqrt(1-1.37257E-9)/sqrt(1-1.36193E-9)
Dg/Df = 1 - 5.3E-12

Uhoh!!! The two terms are very similar. And they are at other speed (e.g. at 100m/s, Dv=1-5.5E-14, Dg/Df=1-5.3E-14).

Now someone needs to get down and dirty and integrate these terms along the course of the flight. I am not going to do it but looking at the curves of velocity vs time (an inverted triangle) and height vs time (a parabola), it seems the gravity factor will win for the reason given by scythian: hang time.

Answer 2

If clock A is shot up, then 1) it would be slowed relative to B because of Lorentz time dilation, and yet 2) it would speed up relative to B because of gravitational time dilation. This is not unlike that atomic clock experiment they did back in the 1970s where they carried them on planes around the world. What's the answer? I don't know, I think I would need the clock's trajectory as a function of time B on the ground. I would guess that clock A would be speeded up relative to clock B, because clock A will have a certain "hang time" with little kinematic effect while at a higher gravitational elevation.

Answer 3

If the clock is fired with a velocity v close to c, the velocity of light in vacuum, there will be a time dilation given by the Lorentz-Fitzgerald equations. But for normal gravity, the velocity and acceleration are negligible compared to c and hence can be ignored. In such cases, both the time pieces will show the same time.

Answer 4

time is slower in the air because of less gravity so, clock b would be ahead.

Answer 5

Clock B is ahead because Clock A is smashed to pieces!!!

I dunno!