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I am confused by the explanations of time dilation and gravity vs. acceleration. The consensus (and apparently experiments) is that time dilation in a gravitational field (not free-falling) is "as if" one were accelerating. Examples given for sending light signal each way in a gravitational gradient, one sees blue shift the other red shift just as if accelerating in space due to velocity transformation. But is there a paradox? If actually accelerating, the velocity will increase and so will the time dilation relative to origin. If in a gravity field, the effect on clock time will be a constant difference relative to free-fall (e.g., orbit). Satelite clocks are apparently corrected for both velocity and gravity density which should be equivalent to acceleration. If a satelite were moving above escape velocity but held to the earth on a tether, would its time dilation only be a difference from the velocity, or would the force of the tether imply higher gravity density? Any thoughts?

Qmechanic
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John Kineman
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You have to be very careful with statements like “The consensus (and apparently experiments) is that time dilation in a gravitational field (not free-falling) is ‘as if’ one were accelerating”. There is a grain of truth to it, but as stated it is a little problematic.

Time dilation in a gravitational field is locally the same as time dilation in a uniformly accelerating reference frame in flat spacetime. This distinction is important because it avoids your subsequent problem: “If actually accelerating, the velocity will increase”. In the accelerating reference frame the accelerating observer remains at rest.

In both an accelerating reference frame and in a gravitational field the time dilation depends on the potential. Locally that is $gh$ where $g$ is the gravitational acceleration in the gravitational field or the acceleration of the accelerating reference frame, and $h$ is the height difference in the gravitational/acceleration field. I am not sure if you mean $g$ or $gh$ or something else by “gravity density”, but I would recommend avoiding that term since it is not standard terminology

Dale
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  • So, you are saying that the potential in one case is gh, whereas the potential in the other case is v? Is that the analogous potential in the case of acceleration? – John Kineman Oct 21 '18 at 13:17
  • No, the potential is $gh$ in both cases. In the accelerating reference frame $g$ is the “artificial gravity” in the reference frame. – Dale Oct 21 '18 at 13:20
  • Thank's Dale, but what does it correlate with in the case of accelerating motion? Closest thing would be force, I suppose, holding gravitational and acceleration force equal. Then force is: F = m2 x a = M x m2 x G X h / r^2 where M is a real or presumed gravitational mass, r is a distance across the gradient of the potential field, which is the equivalent distance in a gravitational field to produce a. So we are constructing a pseudo gravitational field. – John Kineman Oct 22 '18 at 17:02
  • Then my question is really about time dilation in each case. In the gravitational case I understand time is less in a gravitational field and in Lorentz relation with velocity. But experiments show no time dilation with near instantaneous acceleration. Is there no time dilation with near instantaneous gravity? And in gravitational time dilation is it cumulative as it would be in the case of acceleration (due to increasing velocity)? – John Kineman Oct 22 '18 at 17:06
  • It seems like a contradiction where either time dilation in a constant gravitational field would increase with time, or there would be an additional effect in the case of acceleration, added to the effect of velocity - but experiment doesn't confirm either one. – John Kineman Oct 22 '18 at 17:08