@A.T.
There is no additional time dilation due to not following a geodesic, at the instant when they pass each other...
... Fair enough... I was a tad uncertain if the "geodesic = path of longest proper time" did anything special here.
@Shaw
A mechanical engineering professor did the Schwarzschild calculation for me which confirmed the fact that gravitational time dilation on the surface equals the time dilation at the escape velocity.
Not exactly verifiable references, but I take it that the calculation I just did is what you were trying to talk about?
Hard to believe it's just a fluke.
... depends what you mean by "a fluke".
I think the idea is that it is not a particularly meaningful concurrence - though it looks suggestive, the result is only for a special case(s?)
Try repeating the calculation for different metrics.
BTW: GR was used to make the prediction ... it was needed to compute the gravitational time dilation.
Anyway: now we know what you are talking about ... what was the question?
Consider a thought experiment where an object accelerates itself by ejecting photons in one direction. The speed of the mass increases as the mass diminishes, until it's almost completely depleted.
As with any rocket ... though there is no need for almost all the rocket to be reaction mass. It sounds like you are thinking of some sort of total conversion - was there a special reason for that.
This will occur at a very high velocity common to all mass.
... well if all the mass were turned into energy, then there would be nothing left to travel. If there were any mass remaining, then it must be traveling less than the speed of light.
We have had rockets which used almost all their mass for reaction-mass but still managed to stay well below light-speed, and different rockets managed to end up with different speeds - not a single common speed.
Why would it be different for total-conversion rockets?
You also forgot to be careful about specifying the observer.
Time dilation should accompany mass depletion in the same proportion. How much mass does it take to accelerate an object to the escape velocity using this scenario? The mass loss will be in proportion to the time dilation.
That could be one way of thinking about it - but consider: the amount of reaction mass used (the mass-loss) to achieve a particular delta-vee (say from rest to a speed that happens to be the escape velocity from some particular distance from the center of mass of some non-rotating sphere we happen to be interested in at the time...) will vary with the final mass that has to be moved.
You use the relativistic rocket equation.
We see inertial mass increase in a cyclotron,...
... are you referring to the "cyclotron effective mass"?
... but the accelerated particles are not isolated. The energy for their acceleration is externally applied. An object falling in a gravitational field can be considered an isolated particle if it's small enough.
Yet the energy for the acceleration is provided by an external field too - or, if you like, the intrinsic curvature of space-time? At what education level do you need these answers?
That's why I ask if anyone has checked to see if there are mass changes associated with gravitational time changes.
I still cannot tell what that means. Mass changes in what? by "time changes" do you mean the time dilation wrt a flat space-time observer?
It is not clear how the gravitational "time change" asked about here means anything in terms of the rocket before.
Please try to be more precise in your wording.
Do you mean to compare the mass used up to make a particular delta-vee in some reference frame compares with the time dilation for the final velocity in the same reference frame? (You just happen to be choosing a delta-vee as the difference between some "escape velocity" and rest) Then you need to look at the relativistic rocket equation.
A time change just by itself seems a little odd to me. It looks like there's a missing piece of the puzzle.
Unfortunately, Nature does not seem to care what you or I find odd. But maybe that's for the best.
I don't know what you mean by a "time change". However - time dilation is not usually "by itself", there is an associated length contraction ... more generally, we talk about the Lorentz transformations and the geometry of space-time rather than just time by itself. Sometimes, though, we just care about comparing the clocks.
Before anyone can hep you, though, you will have to clarify your question, and let us know your education level.
I have a feeling the delta-vee thing is close.
