Atomic clocks: day – night speed variations

[Alvydas]

Hello.
How big is day – night speed variations for stationary atomic clock placed somewhere on equator?
And what experiment can measure such variations?

 

[K^2]

Depends on frame of reference. Relative to observer on Earth, there are no variations. Relative to the motion of the Sun, just use the Lorentz time dilation formula.

To measure these, you need to place a very precise clock far outside Solar System. There might be some pulsars out there that could work.

 

[Alvydas]

Depends on frame of reference. Relative to observer on Earth, there are no variations. Relative to the motion of the Sun, just use the Lorentz time dilation formula.

To measure these, you need to place a very precise clock far outside Solar System. There might be some pulsars out there that could work.

Yes, pulsars. I forget about them.
But the problem could be that after rotation we see an other part of the Sky.
I mean would be difficult to see the same pulsar.

My question is targeting more to gravitational clock dilation due Sun's influence.
At day time we are closer to the Sun.
Therefore as I understand clock could go slower a bit (due bigger Sun's gravitational potential).
Is this measurable?

But we may also try to place the clock on a satellite at Geostationary orbit.
http://en.wikipedia.org/wiki/Geostationary_orbit

Now we would have much bigger variations of Sun's gravitational potential.
But stationary placed clock on the Earth can be much better quality.

So which setup would be more realistic to detect clock's variations due variations of Sun's gravity.
Lets say we compare clock's rate with pulsar.

 

[russ_watters]

It would be negligible.

 

[willem2]

Yes, pulsars. I forget about them.
But the problem could be that after rotation we see an other part of the Sky.
I mean would be difficult to see the same pulsar.

My question is targeting more to gravitational clock dilation due Sun's influence.
At day time we are closer to the Sun.
Therefore as I understand clock could go slower a bit (due bigger Sun's gravitational potential).

The Earth is 5 million km closer to the sun in januari, then in july. On an average day the Earth gets about 28000 km closer or further away, so this is larger effect on most days.

 

[mfb]

Just to give an order of magnitude: t ~ $\sqrt{1-\frac{r_0}{r}} \approx 1-\frac{r_0}{2r}$ with the Schwarzschild radius r₀ (3km for our sun) and the distance r. Putting 1 astronomical unit in the formula, you get 2*10^(-8). But this is relative to the interstellar area - the difference between "1 AU" and "1 AU + 100000km" (a bit above the geostationary orbit) is just 1.3*10^(-11). Enough to be measurable with precise atomic clocks, but I would expect that a lot of other effects (non-circular orbits due to moon/sun/earth, variable velocity relative to other objects) are more important. For day/night on earth, the difference is one order of magnitude smaller. In addition to the things willem2 mentioned, the Earth is moving around the center of mass of the earth/moon system - this is smaller than the day/night difference, but it is comparable.

 

[Alvydas]

Just to give an order of magnitude: t ~ $\sqrt{1-\frac{r_0}{r}} \approx 1-\frac{r_0}{2r}$ with the Schwarzschild radius r₀ (3km for our sun) and the distance r. Putting 1 astronomical unit in the formula, you get 2*10^(-8). But this is relative to the interstellar area - the difference between "1 AU" and "1 AU + 100000km" (a bit above the geostationary orbit) is just 1.3*10^(-11). Enough to be measurable with precise atomic clocks, but I would expect that a lot of other effects (non-circular orbits due to moon/sun/earth, variable velocity relative to other objects) are more important. For day/night on earth, the difference is one order of magnitude smaller. In addition to the things willem2 mentioned, the Earth is moving around the center of mass of the earth/moon system - this is smaller than the day/night difference, but it is comparable.

For the case of not so strong gravity your formula can be transformed to
t = t0*(1+Φ/c^2)
And this is well tested for one body case.
However I do not know any experiment which would test it for two bodies
when clocks is between these two bodies.
Something like (Earth) (clock)...(Sun) - day version
compare to opposite (clock)(Earth)...(Sun) – night version (which can be seen like 1 body case).

Yes the case mentioned by willem2 must be accounted too.

In terms of differences this is worth to test,
because there is another viewpoint which predict bigger dilation for night than day.
Because here is used vectors instead of Φ and the sum of these vectors becomes smaller for day version (because these vectors have opposite directions)

Of course you may say it is not worth to test,
but in this case would be very interesting to know how you determinate which experiment is worth to test, and which is not ?