Sundial & Relativity: Inertial Frame Differences?

[mishima]

If one super accurate sundial could be constructed on a high mountaintop and one at sea level, would it be possible to detect the effect of time dilation?

Or does the sun's role in both sundials make both inertial frames the same...?

 

[pervect]

I would say that the sundials would not be able to measure time dilation.

The sundials would define the time of a day between noon and noon, an atomic clock would show a different number of seconds in a day depending on altitude, the sundial would define this amount of time as one day regardless of altitude.

There really aren't any inertial frames in this analysis - hopefully I didn't misunderstand the original question.

 

[Vanadium 50]

No. Consider a more extreme case - would it ever be noon at the top of a mountain and midnight at the bottom?

 

[mishima]

No. Consider a more extreme case - would it ever be noon at the top of a mountain and midnight at the bottom?

Oh, maybe I'm totally mistaken then, I thought this actually happened (on a nanosecond scale). I thought any periodic activity slowed as the gravitational potential decreases (gravitational time dilation). Then I was thinking a sundial is a periodic activity; but it was confusing me how the sun played its part in it all.

 

[pervect]

Local clocks of any sort will be affected by time dilation in a uniform manner. But the Earth isn't a local clock.

The solar day varies in length, gradually slowing down due to tidal interactions with the Moon (and other reasons, but this is the dominant one). Modern atomic clocks are good enough to detect this "slowing down". To keep atomic time synchronized with the solar time, leap seconds are added as needed to TAI time (which is the name for the atomic time standard).

Note that this time (TAI time) is a coordinate time. Coordinate times are used to "label" events with time values. In general, because space-time is curved, coordinate times are different from the times that clocks actually keep. The ratio between the coordinate clocks and the actual physical clocks is called "time dilation". The definition of the coordinate system makes the two times equal for some special locations - for TAI time, the special location is the surface of the Earth, specifically the surface of constant gravitational potential, the geoid.

 

[HallsofIvy]

No. Consider a more extreme case - would it ever be noon at the top of a mountain and midnight at the bottom?

Since we are talking "extreme cases", yes- if the mountain is tall enough that it takes light 12 hours to go from the top to the bottom.

 

[DaveC426913]

Oh, maybe I'm totally mistaken then, I thought this actually happened (on a nanosecond scale). I thought any periodic activity slowed as the gravitational potential decreases (gravitational time dilation). Then I was thinking a sundial is a periodic activity; but it was confusing me how the sun played its part in it all.

GR really does come into play when looking at differences in gravitational potential. It's just that a sundial is not an appropriate device for measuring local time. The sundial receives light from outside the gravity well, so the sun is synced with that frame of reference, not the frame of reference at the bottom of the grav well.

 

[jambaugh]

I suggest rephrasing the problem.

a.) Given a planet is rotating with a certain period as seen by a distant observer, (r->infinity) what is the period for an observer a distance r from the center.

b.) As a function of time of day (let's say in radians) what is the apparent position of the sun for the observer a distance r from the center relative to noon. (One must take into account the bending of light as it falls through the gravity well at an angle.)

The answer to a.) is given by the standard GR calculation of time dilation in a gravity well (plus relativistic effects of rotational velocity if you want to figure that... assume the "planet" is a neutron star.

The answer to b.) shouldn't be too terribly difficult conceptually. I believe the answer will show that the apparent motion of the sun is not constant... note also that the arctic circle will be a bit farther from the pole in a deep gravity well since the bending of light of the almost setting sun causes it to appear higher above the horizon.


There is a third question one could ask (a') which is how fast does the observer at height r above the center see the planet rotating with respect to his gyroscopes. That will depend on the mass of the planet and be affected by gravitomagnetic force (frame dragging).

 

[mishima]

GR really does come into play when looking at differences in gravitational potential. It's just that a sundial is not an appropriate device for measuring local time. The sundial receives light from outside the gravity well, so the sun is synced with that frame of reference, not the frame of reference at the bottom of the grav well.

Thank you, sir. That's exactly what I was trying to question. So I need to remember that when books say stuff like "anything can be a light-clock" there is a caveat about local time.

Incidentally, how is this different from biological metabolic processes that depend on the sun for energy? Is it that when the light transfers into energy on the molecular level it "localizes"?

 

[pervect]

The frequency of the incoming light as measured by local clocks does change as the result of gravitational time dilation.

I'd describe what's going on as follows. The Earth is to big to be considered as a single inertial frame, even if we ignore the fact that it's rotating , which definitely makes it non-inertial.

You can label the Earth with time coordinates - in fact,that's the job of a coordinate time. But the resulting coordinate system is not an "inertial frame".

Because it's not an inertial frame, the coordinate time differs from the proper time, depending on one's position. In this case, altitude is the important element of position. We call the difference between coordinate time and proper time (the time measured by clocks) "time dilation".

As far as biological effects, some elements of biology will be driven by light and sunlight, but for the most part these effects could be separated out by putting someone in a sealed room and using non-natural forms of illumination. IT doesn't really have anything to do with time, just light.

Once upon a time, the Earth's rotation was the best timekeeper available, but that has long since ceased to be the case. Atomic clocks can and do measure the slow rate of change of the Earth's day, they are fundamentally more precise than the Earth's rotation.

Also note that the solar day was never precisely constant. http://www.larry.denenberg.com/earliest-sunset.html does a good job of explaining the issues, one being due to the tilt of the Earth's axis, the other being due to the shape of the Earth's orbit.