Hafele and Keating, local gravity field as preferred frame?

[Agerhell]

Please bring me out of my state of confusion if I need to be... The question is how to calculate the rate of an atomic clock (a pendulum clock may work otherwise) on board a vehicle traveling along the surface of the Earth at constant altitude, like a bus, a train or an aeroplane. This was first tested by the Hafele-Keating experiment:

http://en.wikipedia.org/wiki/Hafele–Keating_experiment

As I interpret that experiment one has to use the centre of the Earth as a reference point when determing the rate of clocks on, or in the immediate surrounding of, the Earth.

For instance, at the Equator the Earth is spinning around its axis at about 40000/24 = 1667 kilometres per hour. So If you have a train at the equator traveling westwards the clocks on board that train would tick increasingly faster compared to the clock in a trainstation along the track until the train reaches a velocity of 1667 km/h compared to the surface of the earth. The clocks onboard the train will still tick faster than the clock in the trainstation even if the train travels west at a speed of 3300 km/h compared to the surface of the Earth.

This is because the train station will still have a slightly higher velocity compared to the centre of the Earth.

Is any other interpretation of the Hafele-Keating experiment possible? Sure one would always have to use the centre of the Earth as a reference point when determining the rates of clocks on or near the Earth? (Deliberately ignoring gravitational time dilation)

 

[PeterDonis]

Is any other interpretation of the Hafele-Keating experiment possible? Sure one would always have to use the centre of the Earth as a reference point when determining the rates of clocks on or near the Earth? (Deliberately ignoring gravitational time dilation)

The Hafele-Keating experiment analysis can't ignore gravitational time dilation; the Wikipedia page you linked to includes it (the "general relativity" part), and the actual results don't match the predicted results unless you include gravitational time dilation as well as the "kinematic" effects of SR.

As I interpret that experiment one has to use the centre of the Earth as a reference point when determing the rate of clocks on, or in the immediate surrounding of, the Earth.

The Wiki page may be a little misleading here. You can use any frame you want to analyze the experiment, but a frame at rest with respect to the center of the Earth, and not rotating with the Earth, is easier because the frame is not rotating.

 

[PeterDonis]

For instance, at the Equator the Earth is spinning around its axis at about 40000/24 = 1667 kilometres per hour. So If you have a train at the equator traveling westwards the clocks on board that train would tick increasingly faster compared to the clock in a trainstation along the track until the train reaches a velocity of 1667 km/h compared to the surface of the earth. The clocks onboard the train will still tick faster than the clock in the trainstation even if the train travels west at a speed of 3300 km/h compared to the surface of the Earth.

This is one way in which the Wiki page may be misleading. Even if you remove the effects of changing altitude, and assume everything stays at ground level, the H-K experiment still involves a "twin paradox" situation; the westbound and eastbound clocks have to come back together with the ground-based clock for their readings to be compared. The scenario you are describing in the quote above does not include that. In the scenario you describe above, observers on the train would see clocks in the train station ticking slow, while observers in the station would see clocks on the train ticking slow.

However, if the westbound train went all the way around the world and came back to stop at the same station, the train's clocks would be seen to have measured more total elapsed time relative to the station's clocks (conversely, clocks on an eastbound train that went all the way around the world would be seen to have measured less total elapsed time than the station's clocks). This type of effect is what the "kinematic" numbers in the Wiki page on the H-K experiment refer to.

 

[Dale]

Sure one would always have to use the centre of the Earth as a reference point when determining the rates of clocks on or near the Earth?

You can use whatever coordinate system you like, but if you want to use a simple metric, like the Schwarzschild metric, with spherical symmetry, then you need to use a coordinate system which matches the spherical symmetry. It is just a matter of computational convenience. You can use other coordinates, but then you cannot use the nice symmetry to make the computations easier.

 

[Agerhell]

This is one way in which the Wiki page may be misleading. Even if you remove the effects of changing altitude, and assume everything stays at ground level, the H-K experiment still involves a "twin paradox" situation; the westbound and eastbound clocks have to come back together with the ground-based clock for their readings to be compared. The scenario you are describing in the quote above does not include that. In the scenario you describe above, observers on the train would see clocks in the train station ticking slow, while observers in the station would see clocks on the train ticking slow.

However, if the westbound train went all the way around the world and came back to stop at the same station, the train's clocks would be seen to have measured more total elapsed time relative to the station's clocks (conversely, clocks on an eastbound train that went all the way around the world would be seen to have measured less total elapsed time than the station's clocks). This type of effect is what the "kinematic" numbers in the Wiki page on the H-K experiment refer to.

So I am saying that if you want to calculate the rate of ticking of an atomic clock on a train at any instant in time, what matters is what speed the train has relative to the centre of the earth. Do you agree or do you not agree?

 

[PeterDonis]

So I am saying that if you want to calculate the rate of ticking of an atomic clock on a train at any instant in time, what matters is what speed the train has relative to the centre of the earth. Do you agree or do you not agree?

Rate of ticking relative to what? As observed by whom? There is no such thing as *the* rate of ticking of a clock "at any instant in time"; the answer always depends on the relative motion of the clock and whoever is doing the observing. For an observer at rest relative to the center of the Earth, yes, what matters is the speed of the clock relative to the center of the Earth; but for an observer in the train station, rotating with the Earth, what matters is the speed of the train relative to the station. All of this applies if you're trying to calculate the observed tick rate of a clock "at any instant".

There is a different type of question that does have an invariant answer, an answer that doesn't depend on whose frame you use to calculate it: if I take two clocks that start out together, travel on different worldlines, and then come back together, which one will show the greater elapsed time? The answer to that type of question does not depend on who is doing the observing. So, for example, to figure out which clock will show the greater elapsed time in my example of clocks on trains going around the world, yes, you can use the speed relative to the center of the Earth to get an answer. That's why the clock on the westbound train that goes around the world shows a greater elapsed time than the station clock; the westbound train is not moving as fast relative to the center of the Earth as the station clock is. But the fact that the two clocks come back together again, so their readings can be directly compared, is crucial.

This does mean, of course, that, for example, the observer at the train station can calculate that the westbound train's clock is ticking slow relative to the station clock "at any instant", and yet still find that, when the westbound train has gone around the world and returns to the station, its clock will show more elapsed time than the station clock. This is counterintuitive, but it is not a contradiction. If that seems fishy to you, you might want to check out the following page on the twin paradox:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_intro.html

The situation we've been discussing here is not precisely the same as the twin paradox, but it has enough key features in common with the scenario we're discussing here that it should help to understand it.

 

[Agerhell]

So, for example, to figure out which clock will show the greater elapsed time in my example of clocks on trains going around the world, yes, you can use the speed relative to the center of the Earth to get an answer. That's why the clock on the westbound train that goes around the world shows a greater elapsed time than the station clock; the westbound train is not moving as fast relative to the center of the Earth as the station clock is.

Thank you.

But the fact that the two clocks come back together again, so their readings can be directly compared, is crucial.

This does mean, of course, that, for example, the observer at the train station can calculate that the westbound train's clock is ticking slow relative to the station clock "at any instant", and yet still find that, when the westbound train has gone around the world and returns to the station, its clock will show more elapsed time than the station clock. This is counterintuitive, but it is not a contradiction.

Well, assuming that the trainstation observer knows how fast the Earth is spinnig around its axis and the latitude of the trainstation, then if he assumes that the rate of ticking of his own atomic clock and the rate of ticking of the two clocks onboard the trains depend on their velocities in relation to the centre of the Earth at any given instant of time he would not be particularily surprised to see that the three clocks shows that different amounts of time have elapsed when the trains get back to the station.

 

[PeterDonis]

Thank you.

Well, assuming that the trainstation observer knows how fast the Earth is spinnig around its axis and the latitude of the trainstation, then if he assumes that the rate of ticking of his own atomic clock and the rate of ticking of the two clocks onboard the trains depend on their velocities in relation to the centre of the Earth at any given instant of time he would not be particularily surprised to see that the three clocks shows that different amounts of time have elapsed when the trains get back to the station.

As long as by "rate of ticking" you mean "rate of ticking relative to an observer at rest with respect to the center of the Earth," yes, this is correct. But this "rate of ticking" will still *not* be the same as the rate of ticking that the train station observer would calculate for the westbound train as it chugs away from the station, if the train station observer actually observed the ticks of the westbound train's clock, and corrected for the light-speed travel time between the westbound train and the station. The latter rate of ticking will be *slower* for the westbound train, as seen by the station observer, than for the station clock, which is at rest with respect to the station observer.

 

[pervect]

It's very convenient to use the center of the Earth as a reference point, but you don't _have_ to.

Misner explains some of the philosophy in http://arxiv.org/abs/gr-qc/9508043"

A method for making sure that the relativity effects are specified correctly (according to Einstein’s General Relativity) can be described rather briefly. It agrees with Ashby’s approach but omits all discussion of how, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles.
One first banishes the idea of an “observer”. This idea aided Einstein in building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape into two categories. One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments and the data tables they provide.

...

What is the conceptual model? It is built from Einstein’s General Relativity
which asserts that spacetime is curved. This means that there is no
precise intuitive significance for time and position. [Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?] But one can draw
a spacetime map and give unambiguous interpretations. [On a Mercator
projection of the Earth, one minute of latitude is one nautical mile everywhere,
but the distance between minute tics varies over the map and must
be taken into account when reading off both NS and EW distances.] There
is no single best way to draw the spacetime map, but unambiguous choices
can be made and communicated, as with the Mercator choice for describing
the Earth.

So, the point is that using coordinates relative to the center of the Earth is a convenient choice for making your space-time map, but you can choose any coordinates you want for your map.

In fact, there are rules for how to change one map to another in the details of GR, though of course this starts to get into the actual details of how it works It's good to go into the details, but how much and what to say depends on the persons background.

BTW, some people do like to use the concept of "observers", Misner's preference for "banishing" them is not universal. While I personally agree with Misner's observations that this tends to be more confusing than enlightening, it's not a universal expositional choice.

That said, I would strongly encourage anyone who IS being confused by the concept of an observer (thinking that there is some preferred frame set by the center of the Earth is an exmple of such confusion) to ponder Misner's viewpoint and attempt to think about how things could work WITHOUT them.