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Assuming that there are no other planets or other gravitational sources around the observer in empty space, would the observer's very fast circular motion create GR gravitational or else called acceleration time dilation?

I am not interested on the SR or else called kinetic constant velocity time dilation, only for the acceleration time dilation? Even if the angular velocity is kept constant, since this is a circular motion the velocity vector direction is changing all the time and therefore must be considered as an accelerated motion. Right? So therefore there must be also a GR gravitational time dilation component together with the SR kinetic time dilation due to the constant angular velocity of this assumed circular motion.

The only reason I could think of for no gravitational time dilation to exist is because the observer's stable path circular motion and angular velocity, the centripetal cancels out with the centrifugal and therefore there is no gravitational time dilation component.

The reason why I am asking this question is because I was puzzled lately by the fact that by the recent g-2 Fermilab muons experiment lecture, in the experiment's muon storage ring (i.e. cyclotron) the only time dilation present they have announced was about (64-2.2)=61.8μs and was purely a SR kinetic time dilation. They never said anything about any GR acceleration time dilation effect due the circular motion, present on the experiment?

Dale
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Markoul11
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  • Short answer, yes. There will be – silverrahul Apr 25 '21 at 19:55
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    if there are no gravitational sources, why would you need general relativity? – Kosm Apr 25 '21 at 20:02
  • It's not clear what your question is. The iconic experiment that corroborated gravitational time dilation is the Pound-Rebka experiment. The top of the tower is at a higher gravitational potential than the base of the tower. According to GR: if you would construct a wheel large enough to pull 1 G of acceleration (at the perimeter), and the diameter of the wheel is large enough to accomodate the same tower height as in the Pound-Rebka experiment, then you can replicate the Pound-Repka experiment. In that storage ring however, the distance of the muons to the center of rotation is constant. – Cleonis Apr 25 '21 at 20:04
  • It is my understanding that gravitational or else called acceleration time dilation is caused because accelerated masses and mass inertia and does not need necessarily a planet. Even so in the presence of Earth we have g=9.81m/s^2 . Gravity is acceleration. – Markoul11 Apr 25 '21 at 20:14
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    @silverrahul: No, acceleration does not produce time dilation. MTW says: "...experiments do suggest that the times measured by atomic clocks depend only on velocity, not on acceleration" Gravitation, 2017 edition, section 38.4, p. 1055. They refer to Farley et al. 1966. See also Dale's answer below. – Eric Smith Apr 25 '21 at 20:53

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So therefore there must be also a GR gravitational time dilation component together with the SR kinetic time dilation due to the constant angular velocity of this assumed circular motion ... the only time dilation present they have announced was about (64-2.2)=61.8μs and was purely a SR kinetic time dilation. They never said anything about any GR acceleration time dilation effect due the circular motion, present on the experiment?

There is no accelerational time dilation. This known as the clock hypothesis and has been experimentally verified up to about $10^{18}$ g (see: Bailey et al., “Measurements of relativistic time dilation for positive and negative muons in a circular orbit,” Nature 268, July 28, 1977, pg 301).

Gravitational time dilation does exist, but it is not determined by acceleration. Gravitational time dilation is determined by gravitational potential. In an inertial frame there is no gravitational potential, so regardless of the acceleration there is only the kinematic time dilation. The analysis you mention is correct.

Now, per the equivalence principle, it is possible to treat a uniformly accelerating reference frame like a uniform gravitational field. This does not help with uniform circular motion because the acceleration is not uniform, and in particular the Coriolis force complicates things. You can use the mathematical framework of tensors etc. to calculate time dilation in a rotating reference frame. You will get a term that has the form of a gravitational time dilation due to the centrifugal force, but the remainder of the terms do not follow the usual kinematic time dilation formula. So overall it is not particularly helpful to think of a rotating reference frame in terms of gravitational time dilation.

Dale
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  • " This does not help with uniform circular motion because the acceleration is not uniform, " I dont get this , if it is uniform circular motion, does it not mean uniform constant acceleration ? – silverrahul Apr 25 '21 at 21:01
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    @silverrahul No, in exactly the same way that the velocity isn't constant. – ProfRob Apr 25 '21 at 21:55
  • @silverrahul the difference between vectors and their length. – anna v Apr 26 '21 at 05:39
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    So, the magnitude is same, but direction is not ? correct ? – silverrahul Apr 26 '21 at 06:23
  • Yes............ – Dale Apr 26 '21 at 11:20
  • "Gravitational time dilation is determined by gravitational potential." But do you have any thoughts (even if they're speculative) on what it's physically caused by? – Gumby The Green Mar 10 '22 at 09:53
  • @GumbyTheGreen that sounds like a good question. You may want to check for previous similar question, and if you don’t find one then ask it directly – Dale Mar 10 '22 at 11:56
  • Your definition of the clock hypothesis (which I've seen elsewhere) works in the inertial frame, but in the accelerating frame, the acceleration itself has to cause some time dilation in order for the twins to agree on their ages at the reunion. And aren't both frames equally valid? The definition given in this paper—that the hypothesis "equates the proper time... with the length of that curve as determined by the metric"—works in both [...] – Gumby The Green Apr 15 '22 at 08:28
  • [...] frames. In the accelerating frame, you of course have to use a metric like the Rindler metric during the acceleration (per the paper, "the restriction to Minkowski spacetime and inertial motion has been dropped"). You then find that the inertial curve is longer and thus that the inertial clock speeds up during that time to a degree that's equivalent to gravitational time dilation. – Gumby The Green Apr 15 '22 at 08:28
  • @GumbyTheGreen again, please ask this as a question. In a non-inertial frame time dilation depends on velocity and position, not acceleration. The explanation why doesn’t fit in comments – Dale Apr 15 '22 at 11:04
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    Ok, I've posted one here. – Gumby The Green Apr 20 '22 at 10:24
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Different worldlines have different lengths, and the elapsed time on a worldline is the length of the worldline. Names like "special-relativistic time dilation" and "gravitational time dilation" are given to ratios between lengths of worldlines in specific geometric arrangements. They're special cases, not different phenomena.

A circling particle has a helical worldline. In Euclidean 3-space, if you've got a straight line and a helix heading in the same direction, the helix is longer. If the slope of the helix relative to the straight line is $m$, then it's longer by a factor of $\sqrt{1+m^2}$. In spacetime, by convention the slope is called $v/c$, and because of the flipped sign in the metric, the helix is longer by a factor of $\sqrt{1-(v/c)^2}$ (which is less than $1$, so it's actually shorter).

General relativity adds nothing to special relativity except that spacetime curvature is related to energy-momentum density. In this problem, we're taking curvature to be negligible, so the geometry is the same as it was in the special-relativistic case, and the ratio of lengths is the same.

See this answer for more about the geometry of gravitational time dilation and this answer for special-relativistic time dilation.

benrg
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