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Tamminen wrote: ↑October 8th, 2018, 4:49 pm So can we say that the relative time dilation between Alice and Bob is one and the same function f(a,h,t) when moving in flat or curved spacetime, where a = acceleration relative to an inertial frame, h = the distance betweeen the clocks and t = travel time? Is this what the equivalence principle means in this case?Those are all locally measurable things, so yes, but a=acceleration or strength of the gravitational field. Alice and Bob cannot tell the difference between those. In the gravity field, they're not really accelerating.
Halc wrote: ↑October 8th, 2018, 6:12 pmBut if we look at this from the point of view of the inertial frame: in the case of the flat spacetime it is moving at a constant speed and in the case of the curved spacetime it is free fall. In the latter case a = acceleration against gravitational potential, so that for instance when the rocket is standing on the ground in a gravitational field of 1 g, it really accelerates at a = 1 g. So it does not matter if the spacetime is flat or curved or how curved it is, the relative time dilation is the same if a is the same. Right?Tamminen wrote: ↑October 8th, 2018, 4:49 pm So can we say that the relative time dilation between Alice and Bob is one and the same function f(a,h,t) when moving in flat or curved spacetime, where a = acceleration relative to an inertial frame, h = the distance betweeen the clocks and t = travel time? Is this what the equivalence principle means in this case?Those are all locally measurable things, so yes, but a=acceleration or strength of the gravitational field. Alice and Bob cannot tell the difference between those. In the gravity field, they're not really accelerating.
David Cooper wrote:For reference, this is the post we're talking about here:Steve3007 wrote:Which particular part(s) or that post of mine is/are, in your view, "not good enough"?Where does it address the issue?
Halc wrote:Gravitational force also does not cause dilation. I have a clock here on Earth, and one on my lab on Uranus where I weigh about 8/9th of Earth. But the Uranus clock is dilated more, despite the weaker force of gravity. It feels like less acceleration, yet the dilation effect is more. Clearly the acceleration, or a gravitational field that feels like acceleration, plays no direct role in the dilation. It is caused by the negative gravitational potential, which is far greater on Uranus (about 3x Earth), and nonexistent in a centrifuge.Out of interest, I added Uranus to the application that I wrote and which I discussed in my previous post. The result is shown below. As you can see, the gravitational time dilation calculated between a point just above the Uranus cloud tops and a point much further away is the same when calculated using General Relativity and when calculated using numerical integration of a long series of small vertically stacked boxes within which the gravitational field can be taken, to a high enough level of accuracy, to be uniform.
Tamminen wrote: ↑October 9th, 2018, 4:18 am But if we look at this from the point of view of the inertial frame: in the case of the flat spacetime it is moving at a constant speed and in the case of the curved spacetime it is free fall.Our examples correspond to neither of these cases. From the POV of flat spacetime, we're in a rocket accelerating, so no constant speed. The observer feels a 1g force. From the POV of the curved spacetime (on Earth), the observer is not in free fall, but has a force of 1g applied on him from the ground underneath him, which is indistinguishable from the thrust of the rocket.
So it does not matter if the spacetime is flat or curved or how curved it is, the relative time dilation is the same if a is the same. Right?Yes. It takes multiple measurements to observe the the relative dilation: two clocks that we can compare to each other.
Steve3007 wrote: ↑October 9th, 2018, 5:57 am Output01_Earth.jpgSo I looked at this (and the Jupiter one). Not sure what all these numbers are meant to demonstrate. It just shows dilation due to varying points of negative gravitational potential. I don't contest any of it. I didn't pick Jupiter in my example since the gravity there is so much larger than on Earth. Saturn is the closest, but I chose Uranus because you actually weigh less there.
Steve3007 wrote: ↑October 9th, 2018, 6:00 am The second method uses the results of Special Relativity to calculate the time dilation between two clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field. It uses the equation that was derived mathematically ( as equation 8 ) in this article:I see. The point of the output was to have the two numbers be the same (between upper and lower clocks).
https://thecuriousastronomer.wordpress. ... elativity/
...and iterates over a very large number of small boxes. If such a small box were in the presence of a radial gravitational field, the field within the small volume of each individual box could be regarded as uniform and therefore the equivalence principle applies. So iterating over a large number of boxes, each with a slightly different uniform acceleration/g field yields a value for the time dilation in a radial gravitational field where the field strength is a function of radius.
Steve3007 wrote: ↑October 9th, 2018, 6:53 am The SR numerical integration method is affected by this slower reduction in 'g' with increasing 'r' because it integrates over a long series of small volumes from a small 'r' to a large 'r'.The SR method should have no reduction in g at all since the formula is for a uniform gravitational field. There is no r in that formula, just like there is no h in the GR formula above it. You sure you used it in the bottom calculation?
Halc wrote:I see. The point of the output was to have the two numbers be the same (between upper and lower clocks).No, the two numbers were calculated using the two methods, as described. The point was to see if this resulted in them being the same. It did.
Halc wrote:The SR method should have no reduction in g at all since the formula is for a uniform gravitational field. There is no r in that formula, just like there is no h in the GR formula above it. You sure you used it in the bottom calculation?
Steve3007 wrote:The second method uses the results of Special Relativity to calculate the time dilation between two clocks at either end of an accelerating room/box/elevator/rocket which is not in a gravitational field. It uses the equation that was derived mathematically ( as equation 8 ) in this article:I can post the code to make it clearer.
https://thecuriousastronomer.wordpress. ... elativity/
...and iterates over a very large number of small boxes. If such a small box were in the presence of a radial gravitational field, the field within the small volume of each individual box could be regarded as uniform and therefore the equivalence principle applies. So iterating over a large number of boxes, each with a slightly different uniform acceleration/g field yields a value for the time dilation in a radial gravitational field where the field strength is a function of radius.
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