Time and acceleration [duplicate]

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?

marked as duplicate by John Rennie, Jon Custer, Aaron Stevens, Kyle Kanos, sammy gerbilNov 6 '18 at 22:36

• What should be the paradox? – md2perpe Oct 20 '18 at 20:01
• – mmesser314 Oct 20 '18 at 22:38
• @JohnKineman What's the question? – N. Steinle Oct 21 '18 at 0:36
• Possible duplicate of What is time dilation really? – John Rennie Oct 21 '18 at 9:30

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
• 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