Time Dilation on Earth: Why Hemisphere Matters

[russ_watters]

A clock at the Earth's center would also be at rest in the above frame, but it would run at a different rate than the surface clocks.

Why?

 

[A.T.]

Why?

On the actual Earth the center clock runs slower than the surface clocks because of gravitational time dilation. In flat space time the center clock runs faster than clocks revolving around it. The "reason" depends on the frame: Kinetic time dilation in the non-rotating frame. "Gravitational" time dilation from the centrifugal potential in the co-rotating frame.

So the argument "The clocks have some common rest frame, so they run at the same rate" doesn't work here. It works only for common rest frames that are inertial.

 

[mfb]

Thanks again for the replies,so my (uneducated) understanding now is that:-
from an observers point of view on earth, as we are all rotating at the same rate around the centre there would be no difference in time /tick rates. (the orbit around the sun can be ignored, as from the Earth bound POV the sun may as well be orbiting us??sorry if this is complete rubbish!)
However from a point of view outside of the earth, the orbits and velocities observed are complex and there would be constant accelerating and decelerating, with the associated time dilation effects. These would obviously average out over a 24hr period. Am i on the right track?

Right.

You cannot say "the sun orbits earth" as accelerations are absolute, but that acceleration is small.

There would be another effect: gravitational time dilation due to the different distance to sun. That should lead to an effect that you could observe on earth, at least in theory.

 

[PeterDonis]

On the actual Earth the center clock runs slower than the surface clocks because of gravitational time dilation.

This is actually more complicated than it appears. If we pick an Earth-centered, non-rotating frame, then the center clock is at rest at the "bottom" of the gravitational potential well, and the surface clock is moving at a higher altitude in the potential well. So there are two competing effects: the surface clock runs faster because it's higher, but it runs slower because it's moving. The speed of the surface clock, in the non-rotating frame, is slow enough that the altitude effect wins out; at least, that's what I remember from a thread a while back where we did approximate computations of various cases. IIRC, at least to a first approximation, the rate of clock in low Earth orbit came out the same as the rate of a clock at the Earth's center; and a clock at rest on the surface of the (rotating) Earth is moving slower than the orbiting clock, so will have a faster clock rate.

(Btw, the definition of "clock rate" I am using here, even though it looks frame-dependent, actually has an invariant meaning, because, at least in the approximation we are using, the spacetime has a time translation symmetry, which let's us define an invariant notion of "clock rate" among other things. The Earth-centered, non-rotating frame is basically the one that matches up with the time translation symmetry.)

 

[A.T.]

This is actually more complicated than it appears.

Yes, but even if we avoid GR complications, by replacing the Earth with a mass-less rotating beam, the clocks attached to it at different positions will run at different rates. So one cannot conclude that clocks run at the same rate, just because they have some common rest frame.

 

[A.T.]

You cannot say "the sun orbits earth" as accelerations are absolute, but that acceleration is small.

Proper acceleration is absolute, which is zero for both, sun and earth. So, why can't you say "the sun orbits earth"?

 

[harrylin]

[..] *Just to be sure I'm clear on what this looks like, this frame is drawn with me at its center, rotating in place, right? So at any instant the other observer has a particular velocity in this frame. The catch is that he also has an acceleration that we are ignoring for the instantaneous picture, right?

The clocks are both on the equator so that they may be assumed to be at the same gravitational potential. As a result we can stick to SR only for those clocks; that simplifies the discussion.
The acceleration is not ignored for the instantaneous picture; time dilation is a function of velocity, not acceleration. Some types of clocks are affected by acceleration (and of course also by the apparent acceleration from gravitation), but of course we should assume identically built clocks. Then that influence from acceleration is the same for both clocks so that it drops out of the comparison. For simplicity that effect is usually ignored in examples with different acceleration ("clock assumption").

 

[harrylin]

If we are hovering over the Earth, then all of the equator clocks tick at the same rate, but not toward the poles, eh? These would have less and less dilation approaching the poles.

[..] clocks at the equator and the poles run at slightly different rates relative to one another so they cannot both agree with the standard time, and in practice neither one does.[..].

In practice all clocks tick at slightly different rates and even fluctuate but the SR prediction for clocks at sea level is wrong.
To very good approximation the Earth's shape is adjusted to its rotation in such a way that for those clocks the SR effect from speed is exactly compensated by the GR effect from gravitational potential: the clocks at the equator move "faster" (making them tick slower) but are also "higher" (making them tick faster). There is an interesting relation between time dilation and potential and kinetic energies.

 

[1977ub]

Proper acceleration is absolute, which is zero for both, sun and earth. So, why can't you say "the sun orbits earth"?

For one thing, if you choose the Earth rather than the sun for your frame, that is a frame much farther from being an IF, so there will be more deviations from exact SR calculations you would perform in an IF.

 

[A.T.]

if you choose the Earth rather than the sun for your frame, that is a frame much farther from being an IF

Why? The centers of both are inertial.

so there will be more deviations from exact SR calculations you would perform in an IF.

There are no SR calculations for gravitation.

 

[1977ub]

Why? The centers of both are inertial.There are no SR calculations for gravitation.

Both Earth and sun centers are *more* inertial than points at their respective surfaces. However each is moving wrt true inertial frame, since both motions curve wrt solar system barycenter, and beyond that galactic core, supergalactic, etc.

 

[mfb]

Proper acceleration is absolute, which is zero for both, sun and earth. So, why can't you say "the sun orbits earth"?

We consider points on Earth with a separation. Those are not inertial, and they will feel different effects from the sun.

 

[PeterDonis]

each is moving wrt true inertial frame

There is no such thing as "true inertial frame". There are only local inertial frames, and the only real difference between them is how large of a patch of spacetime they cover to some desired level of approximation.

 

[1977ub]

There is no such thing as "true inertial frame". There are only local inertial frames, and the only real difference between them is how large of a patch of spacetime they cover to some desired level of approximation.

Ok. I guess my only point then is that there are better approximations to IF within the solar system.

Hmm. If we are not speaking in absolutes, then every frame such as center of Earth or even a point on Earth's surface is a "local inertial frame" (albeit not a very "good" one)?

 

[PeterDonis]

every frame such as center of Earth or even a point on Earth's surface is a "local inertial frame"

Points by themselves aren't frames, but yes, you can construct a local inertial frame using any point in spacetime you like as the origin. However, that local inertial frame might not be quite what you are thinking of.

For example, if I pick a point on the Earth's surface at some instant of time and call that event the origin of a local inertial frame, then an object at rest in that frame will be in free fall--i.e., it will be "hovering" just at the Earth's surface at the chosen instant of time, but then it will start falling downward (and accelerating), and it will have arrived at the surface at the chosen instant of time by free-falling upward (and decelerating to a stop just at the chosen instant). An observer at rest relative to the Earth's surface will be accelerated in this frame; in fact he will be a Rindler observer from the standpoint of this frame.

 

[A.T.]

We consider points on Earth with a separation. Those are not inertial, and they will feel different effects from the sun.

Points on the Sun's surface are also non inertial. So I still don't see why you cannot say "the sun orbits earth".

 

[A.T.]

Ok. I guess my only point then is that there are better approximations to IF within the solar system.

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits. You seem to use Newtonian notions of inertial frames.

 

[1977ub]

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits. You seem to use Newtonian notions of inertial frames.

I think that "approximate" is exactly what you *can* do in the solar system... you just have to keep in mind that is what you are doing. It's true I wasn't taking gravitation into account.

https://en.wikipedia.org/wiki/International_Celestial_Reference_Frame

The International Celestial Reference Frame (ICRF) is a quasi-inertial reference frame centered at the barycenter of the Solar System, defined by the measured positions of 212 extragalactic sources (mainly quasars). Although general relativity implies that there are no true inertial frames around gravitating bodies, the ICRF is important because it definitely does not exhibit any measurable angular motion since the extragalactic sources used to define the ICRF are so far away. The ICRF is now the standard reference frame used to define the positions of the planets (including the Earth) and other astronomical objects. It has been adopted by International Astronomical Union since 1 January 1998. ICRF had a noise floor of approximately 250 microarcseconds (µas) and an axis stability of approximately 20 µas; this was an order-of-magnitude improvement over the previous Fifth Fundamental Catalog (FK5)

 

[PeterDonis]

In GR you cannot approximate an inertial frame for the solar system, unless you want to ignore gravity altogether, but then you cannot explain orbits.

Actually, you can, by using a weak-field approximation in which gravity is viewed as a tensor field on a flat spacetime background. This is basically what the ICRF that 1977ub mentioned does. It is called "quasi-inertial" because it is an inertial frame with respect to the flat spacetime background, not with respect to the full curved spacetime (as you note, the solar system is too large for any local inertial frame with respect to the full curved spacetime to cover it). Gravity is weak enough everywhere in the solar system that this works fine.

A similar frame can be constructed centered on the Earth, called an Earth-Centered Inertial (ECI) frame:

http://en.wikipedia.org/wiki/Earth-centered_inertial

The difference here is that, since the Earth only takes a year to orbit the Sun, while the solar system takes about 250 million years to orbit the center of the galaxy, an ECI frame has non-negligible acceleration (i.e., is not really inertial) on many time scales of interest (the Wikipedia page talks about this). But it's still basically the same thing as above: gravity is weak enough everywhere in Earth's vicinity that it can be treated as a tensor field on a flat spacetime background, and you can construct "quasi-inertial" frames that are inertial relative to the flat spacetime background.

 

[1977ub]

Points on the Sun's surface are also non inertial. So I still don't see why you cannot say "the sun orbits earth".

Heliocentric and geocentric are models, not truths. So you can use either, depending upon your practical purpose. Most people balk at using the geocentric model to describe the motions of solar system bodies.

https://en.wikipedia.org/wiki/Deferent_and_epicycle

 

[mfb]

Points on the Sun's surface are also non inertial.

Yes but the influence of Earth on them is 6 orders of magnitude smaller than the other direction.

 

[harrylin]

Ok. I guess my only point then is that there are better approximations to IF within the solar system.

Hmm. If we are not speaking in absolutes, then every frame such as center of Earth or even a point on Earth's surface is a "local inertial frame" (albeit not a very "good" one)?

Clarifying a little more: PeterDonis isn't talking about Galilean frames but about the "local inertial frames" of GR. Those are free-fall frames that mimic inertial (Galilean) frames locally - for example the inside of a freely falling elevator is such a frame. The "Earth Centered Inertial frame" is (even when correcting for its orbit) not such a "local inertial frame" because objects are accelerated towards the center. It is only far away from heavy mass that "inertial frames" and "local inertial frames" match with each other. And many people who are versed in GR mean such "local inertial frames" when they speak of "inertial frames".

 

[PeterDonis]

The "Earth Centered Inertial frame" is (even when correcting for its orbit) not such a "local inertial frame" because objects are accelerated towards the center.

Correct; the terminology is somewhat confusing but it's the standard terminology so we just have to deal.

 

[stevendaryl]

When I say the speed is zero, I mean nothing more or less than that the distance between them is not changing with time.

What you're talking about is sometimes called the "closing speed" (the rate at which the distance between objects changes). That's not the same thing as "relative velocity", which is the quantity that is important for relativity.

On a piece of graph paper, set up the x-axis horizontally and y-axis vertically. Draw a circle centered on x=0, y=0 of radius 1 inch. Now, imagine two ants on opposite sides of the diameter: one at x=+1, y=0, the other at x=-1, y=0. They start moving around the circle counterclockwise at a rate of 1 inch per second.

Then the distance between the ants never changes. But their relative velocity is not zero. The ant on the left, at x=-1 has initial velocity -1 in the y-direction. The ant on the right, at x=+1 has initial velocity +1 in the y-direction. Their velocities are not the same. The relative velocity has magnitude 2 inches/second.

 

[stevendaryl]

Why? The centers of both are inertial.

Well, there is a quantitative difference between the sun-centered frame and an Earth-centered frame, which is that a sun-centered frame has approximately a constant metric, while an Earth-centered frame has a time-dependent metric (that is, the components of the metric are time-dependent). Both metrics are time-dependent, but in the case of the sun-centered case, the time-dependence is a small correction.