I Time dilation for the Earth's orbit around the Sun

[DanMP]

The rotating worldline referred to above is one that is going around a rotating object at the right speed and direction to minimize time dilation. Due to rotational frame dragging this is not a stationary worldline, but one that slowly revolves around the central object. This scenario was identified in the Wikipedia article on frame dragging.

It is not a clock orbiting the sun at 1 AU. So my comment was correct, that worldline is longer,

So, if a clock is seen from the Sun's perspective as revolving around the Earth (at the right speed and direction), its worldline is longer than the one of a stationary clock (in the same frame), but if the clock is revolving around the Sun, its worldline is shorter than the one of a stationary clock. For me it's quite confusing ... (I'm not saying that it's wrong, just that it is confusing), so I'll try again without worldlines:

From Sun's perspective, I see a clock revolving (at the right speed and direction to match the frame dragging) around the stationary/hovering Earth and another clock stationary (in the same - Sun's - frame), and contrary to everything I know, the stationary one is more time dilated than the moving one, while the gravitational time dilation is the same, both clocks being at the same distance from de Earth/Sun. That makes me think that a clock being in that rotational dragged frame is "more stationary" than the one that appears stationary from Sun's perspective. Now, if the Earth is orbiting the Sun, that frame is co-moving with it (while dragged around the Earth) and the question is if a clock stationary in the dragged frame would be influenced by the orbital speed of the Earth around the Sun, or not, being again "more stationary" that it appears to be.

GR has already been tested for multi-body systems, using numerical solutions. The solar system and binary pulsar systems are two well known examples.

Ok, but how are these tests/examples related to my scenario? How can we measure/detect, without atomic clocks, the way is time passing on the surface of a distant massive body?

 

[PeterDonis]

if a clock is seen from the Sun's perspective as revolving around the Earth (at the right speed and direction), its worldline is longer than the one of a stationary clock (in the same frame), but if the clock is revolving around the Sun, its worldline is shorter than the one of a stationary clock. For me it's quite confusing ...

That's because you keep switching scenarios, and the rest of us are having trouble keeping up. From what I can see, you have brought up, at one time or another, at least seven different clocks:

(1) A clock at rest relative to the Sun's center of mass (i.e., not rotating about the Sun at all), at the same distance from the Sun as the Earth's orbit, but not anywhere near the Earth (so it's unaffected by the Earth's gravity well).

(2) A clock moving around the rotating Sun at the "minimum time dilation" angular velocity due to the frame dragging of the rotating Sun, at the same distance from the Sun as the Earth's orbit, but not anywhere near the Earth.

(3) A clock in a free-fall orbit around the Sun at the same distance as the Earth, but not anywhere near the Earth.

(4) A clock orbiting the Sun at the same distance as the Earth, near enough to the Earth to be affected by the Earth's gravity well (for definiteness, say about 100 miles above the Earth's surface, roughly the altitude of low Earth orbit), but not orbiting the Earth at all (i.e., its orbit is the same as that of the Earth's center of mass around the Sun).

(5) A clock moving around the rotating Earth at the "minimum time dilation" angular velocity due to the frame dragging of the rotating Earth, at the same distance from the Earth as clock #4.

(6) A clock at rest relative to the rotating Earth (i.e., it stays above the same point on the Earth as the Earth rotates--think of it as being at the top of a 100 mile tall tower that rotates with the Earth) as it orbits the Sun, at the same distance from Earth as clock #4.

(7) A clock in a free-fall orbit around the Earth at the same distance from Earth as clock #4.

Some of the above you might not have brought up explicitly, but you've used ambiguous language that could imply several of the above possibilities.

To the best of my understanding, the ordering of the time dilation of these clocks, from "minimum" (longest elapsed time) to "maximum" (shortest elapsed time) is as follows:

#2 - #1 - #3 - #5 - #4 - #6 - #7

Note that the ordering of #7, relative to #6, depends on the altitude; as we increase the altitude above the Earth, clock #7's elapsed time will become greater than that of #6. (This happens at an altitude of half the Earth's radius, IIRC.)

 

[PeterDonis]

How can we measure/detect, without atomic clocks, the way is time passing on the surface of a distant massive body?

By looking for spectral lines in the light coming from the distant body, and comparing them with similar spectral lines measured in the lab to obtain a frequency shift. If we know the motion of the distant body relative to Earth, we can subtract out the frequency shift due to the Doppler effect; what is left will tell us the gravitational redshift of the body, which in turn tells us how time is passing on its surface.

This method is not viable for all distant bodies, because we can't always detect spectral lines in the light coming from them. But it's the only method we have. It's been done for the Sun and for many other stars; in fact this method was how the mass of the white dwarf Sirius B was determined (because its radius was known from telescopic observations, and knowing the radius and the redshift let's you calculate the mass).

 

[DanMP]

That's because you keep switching scenarios, and the rest of us are having trouble keeping up.

Sorry about that. I'll try again:

Here (what is between { } was added today):

From Sun's perspective, I see a clock {A1} revolving (at the right speed and direction to match the frame dragging) around the stationary/hovering Earth and another clock {B} stationary (in the same - Sun's - frame), and contrary to everything I know {e.g. Hafele-Keating exp.}, the stationary one {B} is more time dilated than the moving one {A1}, while the gravitational time dilation is the same, both clocks being at the same distance from de Earth/Sun. That makes me think that a clock being in that rotational dragged frame is "more stationary" than the one that appears stationary from Sun's perspective. Now, if the Earth is orbiting the Sun {the second scenario}, that frame is co-moving with it (while dragged around the Earth) and the question is if a clock {A2} stationary in the dragged frame would be influenced by the orbital speed of the Earth around the Sun, or not, being again "more stationary" that it appears to be.

there are 2 scenarios. In both of them the Sun is not rotating, in order to leave out its rotational frame dragging. On the other hand, Earth's rotation (and/or mass) is "enhanced", in order to have a significant rotational frame dragging.

In the first scenario, the Earth is not orbiting the Sun, "hovering" at the same distance from the Sun as it was when orbiting (this may be possible using huge thrusters, at least if the Earth was orbiting the Sun very far from it). Here we have two clocks, both flying around the Earth, over the equator (at the same altitude, say 10 km):
- clock A1, flying at exactly the right speed and direction to match the Earth's rotational frame dragging (seen from the Sun)
- clock B, flying at exactly the right speed and direction to appear stationary from the Sun's perspective, at the same distance from the Sun as clock's A1 average distance from the Sun

In the second scenario, the Earth is orbiting the Sun and we can consider three clocks:
- clock A2, flying around the Earth, over the equator, at exactly the right speed and direction to match the Earth's rotational frame dragging (seen from the Sun)
- clock C, in a free-fall orbit around the Sun at the same distance as the Earth, but not anywhere near the Earth
- clock D, at one of the Earth's poles, at the same height/altitude as clock A2

I added clocks C and D because I mentioned them in the OP, but, if I'm not mistaking, clock D should tick with the same rate as A2, so the OP problem/question is to compare clock A2 with clock C, keeping in mind, from the above first scenario, that a clock matching the Earth's rotational frame dragging seems to have less kinematic time dilation than it would appear from Sun's perspective ...

 

[PeterDonis]

clock A1, flying at exactly the right speed and direction to match the Earth's rotational frame dragging (seen from the Sun)

Ok, so this clock, relative to Earth's center of mass, would be moving in the same direction as the Earth's rotation, but much slower. Relative to someone standing on the surface of the rotating Earth, this clock would be moving in the opposite direction as the Earth's rotation.

clock B, flying at exactly the right speed and direction to appear stationary from the Sun's perspective, at the same distance from the Sun as clock's A1 average distance from the Sun

And this clock, relative to Earth's center of mass, would not be moving at all. Relative to someone standing on the surface of the rotating Earth, this clock would be moving in the opposite direction to the Earth's rotation, somewhat faster than clock A1.

clock A2, flying around the Earth, over the equator, at exactly the right speed and direction to match the Earth's rotational frame dragging (seen from the Sun)

So relative to Earth, this clock moves the same as clock A1, correct? (Its motion relative to the Sun is different since now the Earth is orbiting the Sun.)

clock C, in a free-fall orbit around the Sun at the same distance as the Earth, but not anywhere near the Earth

No problem here.

clock D, at one of the Earth's poles, at the same height/altitude as clock A2

So this clock is not orbiting the Earth, correct? In other words, relative to the Sun, its motion is the same as that of the Earth's center of mass.

For comparison, I will add two more clocks:

First, the one I defined as clock #1 in post #52 (at rest relative to the Sun, at the same distance as the Earth's orbit, but nowhere near the Earth).

Second, a clock moving the same relative to Earth as clock B, but for the case where the Earth is orbiting the Sun. (This clock is not the same as clock D because it is hovering over the equator, not one of the poles.) Call this clock E.

Assuming my statements above are all correct, then the ordering of time dilation of these clocks, from "minimum" (longest elapsed time) to "maximum" (shortest elapsed time) would be, I think:

Clock 1 - Clock A1 - Clock B - Clock C - Clock D - Clock A2 - Clock E

Note that this is just off the top of my head using heuristic estimates of the time dilations involved; I have not done the detailed math. Some key points that I used:

I am estimating that the additional time dilation due to the Earth orbiting the Sun, as compared with being at rest relative to the Sun at the distance of the Earth's orbit, is larger than the time dilation due to the Earth's gravity well at the Earth's surface or in low Earth orbit. The numbers are pretty close here but I think this is right. (This is why clocks A1 and B come between clock 1 and the rest of the clocks.)

For an object above one of the poles of a rotating object, the time dilation is smaller (longer elapsed time) than for any state of motion in the equatorial plane. (This is why clock D comes before clocks A2 and E.)

 

[PeterDonis]

I'm not mistaking, clock D should tick with the same rate as A2

You're mistaken. See the last point in my previous post. The Kerr metric is not spherically symmetric, so being above one of the poles is not the same as being above the equator.

 

[PeterDonis]

a clock matching the Earth's rotational frame dragging seems to have less kinematic time dilation than it would appear from Sun's perspective

This is not correct, because, as I have pointed out before, in the Kerr metric, i.e., in the presence of rotational frame dragging, it is no longer possible to have purely "kinematic" time dilation (i.e., time dilation that depends only on speed relative to observers "at rest").

 

[pervect]

This is not correct, because, as I have pointed out before, in the Kerr metric, i.e., in the presence of rotational frame dragging, it is no longer possible to have purely "kinematic" time dilation (i.e., time dilation that depends only on speed relative to observers "at rest").

The Hafele-Keating experiement [for this post, the HK expriement] experiment is, I think, I good illustration of this point. If one slow-transport a clock around a spinning massive object in different directions (in the HK experiment, this spinning object is the Earth], then reunite the clocks so that they meet again and are at rest relative to each other and to the spinning surface of the object, said clocks will not be synchronized. The clock that moves in the direction of the spin will have a different reading than the clock that moves in the direction opposite to the spin.

This is an experimentally confirmed prediction of special relativity. It'd be overstating the case to say this singe experiment is conclusive, but at this point I don't want to get into a full review of the experimental tests of Special Relativity.

The point I want to make is that if one's understanding of special relativity does not match the results of the HK experiment, if one does not understand why the theory predicts that the clocks will have different readings when re-united, one's understanding of special relativity does not match the current understanding of the professional community.

Presenting the current understanding of the professional community is our mission goal here at PF.

An equivalent experiment to the HK experiment could be created with the Earth, and two spaceships that move in powered trajectories clockwise and anti-clockwise along the Earth's orbit. This modified version detects the frame-dragging effects of the spin of the sun, just as the HK experiment detects the effects of the spin of the Earth.

The approach favored by the OP, where one focuses on "Time dilation", is particularly vulnerable to such confusions, this is a fundamental limitation of the approach which typically neglects the relativity of simultaneity. I suspect that may be an issue for the OP - it's a very common issue. However I have not been following this thread in detail.

 

[PeterDonis]

The Hafele-Keating experiement [for this post, the HK expriement] experiment is, I think, I good illustration of this point.

Not of the point under discussion in this thread, no. The HK experiment was nowhere near sensitive enough to spot the tiny effects on time dilation of the frame dragging due to the Earth's rotation. It was only testing the time dilation effects of the Earth's static gravitational field and of motion relative to an Earth-centered non-rotating frame. The latter type of motion includes the motion of a clock at rest on the surface of the rotating Earth; but the time dilation effect due to this motion is not caused by the frame dragging due to the Earth's rotation. It's caused by motion relative to a non-rotating Earth-centered frame.

To put it another way, to analyze the HK experiment to the current experimental accuracy, it is sufficient to use the Schwarzschild metric (which is static, so a clean split between "gravitational" and "kinematic" time dilation can be made) with some static correction factors due to the Earth's quadrupole moment (which don't change the fact that the metric being used is static, so they don't affect the clean split just mentioned). We would need a number of orders of magnitude better accuracy to spot the time dilation effects of frame dragging, and therefore to require a non-static (but still stationary) metric for the analysis, in which the clean split is no longer possible. I am assuming such a metric in my posts in this thread, but that is only theoretical; experimentally we have no reason to have to do this yet, nor will we for quite some time.

If one slow-transport a clock around a spinning massive object in different directions (in the HK experiment, this spinning object is the Earth], then reunite the clocks so that they meet again and are at rest relative to each other and to the spinning surface of the object, said clocks will not be synchronized.

This is the Sagnac effect, which is a real effect, yes, but is not due to rotational frame dragging. (This is obvious from the fact that the Sagnac effect is present in flat spacetime, where there is obviously no rotational frame dragging.)

The approach favored by the OP, where one focuses on "Time dilation", is particularly vulnerable to such confusions, this is a fundamental limitation of the approach which typically neglects the relativity of simultaneity.

I agree that neglecting relativity of simultaneity can cause problems, but I don't think the OP is making any errors attributable to that in this thread. The "time dilation" the OP is describing is invariant, not frame-dependent (so "time dilation" might be a bad term to use, "differential aging" might be better).