GR: Inertial Systems & Direction Changes of Motion

[Ratzinger]

When a canon shoots a ball straight up (one dimension), the ball only accelerates as long as the nongravitational force of the canon acts on it. After that, the ball decelerates in the gravity field until the point it turns back and accelerates in the opposite direction as it falls back on earth.

So is it right to say that according to GR, not only the freely falling body, but also the upmoving and 'turning point' frame are inertial, since no non-gravitational force acts on them?

But doesn't the ball notice the direction change of its motion? Is that because there are (at least) three local inertial systems?

 

[masudr]

I'm pretty sure you're confusing special relativity's inertial frames with GR's inertial frames. In GR, in an inertial frame, particles don't go in straight lines, they go in geodesics. In this case, the ball's trajectory is a geodesic.

 

[Garth]

When a canon shoots a ball straight up (one dimension), the ball only accelerates as long as the nongravitational force of the canon acts on it. After that, the ball decelerates in the gravity field until the point it turns back and accelerates in the opposite direction as it falls back on earth.
So is it right to say that according to GR, not only the freely falling body, but also the upmoving and 'turning point' frame are inertial, since no non-gravitational force acts on them?
But doesn't the ball notice the direction change of its motion? Is that because there are (at least) three local inertial systems?

In the ideal case of no air resistance: in GR there is only one locally inertial frame of reference in this case: that of the cannon ball.

The cannon ball's acceleration w.r.t. the supported cannon's non-inertial frame of reference is always negative - 'downwards', whereas, initially, the cannon ball's velocity is 'upwards'.

That 'upwards' velocity is first reduced to zero by the negative acceleration before becoming a 'downwards' velocity and then eventually it would fall straight back down the cannon's mouth. [Well you did say: "a canon shoots a ball straight up (one dimension)"!]

In the cannon ball's frame of reference it is in free fall and feels no acceleration.

Therefore the cannon would not notice the change of direction unless it had an altimeter on board, when the reading would reach a maximum, the cannon ball is in free fall all the time. It is the ground and the cannon that are not in an inertial frame of reference, that is why the cannon weighs so much!

I hope this helps.

Garth

 

[Ratzinger]

I hope this helps.

It did. No it looks easy. Thanks, Garth.