Time Dilation: Is it Counterintuitive?

[Happy Jack]

Between meetings, more time elapses for the oscillating clock A than elapses for the central clock D.

What about gravity, as the stationary clock D has no gravity, where the osccilating clock ranges from 1g to 0g to 1g? In orbit, the clocks speed up (relative to the surface) as gravity drops, and slow down as the speed increases.

Has this been tested, the effect of decread gravity on a clock in a deep hole/mine?

 

[yuiop]

...

clock A is thrown straight up from the surface and returns to the surface;
clock B is dropped from rest through a tunnel that goes through the centre of the planet;
Clock C remains on the surface;
clock D remains at the centre of the planet;
clock E orbits the body right at the surface.

Assume A is thrown up as E passes C so that A returns to C at the same time as E returns.

These are the results of some calculations I have done for the proper times of the clocks where R is the Radius of the planet and H is the Height that clock A reaches at its apogee when thrown upwards. F is the coordinate time according to a clock at infinity. I have not calculated the proper time for clock B oscillating inside the tunnel as that will require a real expert to figure out ;)


\begin{tabular}{l l l l l l l}<br /> R : 2.13m &amp; 2.80m &amp; 3.21m &amp; 4.93m &amp; 45.23m &amp; 448.85m &amp; 44,847.79m\\<br /> H : 3m &amp; 5m &amp; 6m &amp;10m &amp; 100m &amp; 1,000m &amp; 100,000m\\<br /> \hline<br /> A : 7.52 &amp; 19.32 &amp; 25.95 &amp; 57.79 &amp; 1882.18 &amp; 59658.94 &amp; 59673959.04<br /> \\<br /> C : 4.81 &amp; 15.71 &amp; 22.11 &amp; 53.07 &amp; 1868.45 &amp; 59615.90 &amp; 59673528.99<br /> \\<br /> D : -2.52 &amp; 8.87 &amp; 12.82 &amp; 45.19 &amp; 1847.07 &amp; 59549.27 &amp; 59672863.68<br /> \\<br /> E : (12.48) &amp; (7.90) &amp; 9.13 &amp; 43.08 &amp; 1846.72 &amp; 59549.16 &amp; 59672863.67<br /> \\<br /> F : 19.52 &amp; 29.41 &amp; 36.06 &amp; 68.83 &amp; 1911.19 &amp; 59749.16 &amp; 59674859.61\\<br /> \end{tabular}

Note that tossed clock A consistently records the longest proper time period than any of the other clocks, (except for the coordinate clock at infinity of course.) Orbiting clock E consistently records the least proper time. (The figures in brackets for clock E should not be compared as they are virtual periods because there are no real circular orbits at radii less than the photon orbit radius at 3m.). Centre clock D always records less proper time than stationary clock C at the surface as would be expected but it is somewhat surprising to me that also generally records more proper time than orbiting clock E. The proper time for clock D is calculated using the interior Schwarzschild solution. Generally the interior solution requires knowledge of the density distribution of the mass of the planet but in the case of a stationary clock exactly at the centre, the enclosed mass is zero and the density distribution becomes irrelevant.

The next table compares the coordinate spatial distances for the paths of the various clocks. Clocks C, D and F have not been included as they are always stationary.

\begin{tabular}{l l l l l l l}<br /> R : 2.13m &amp; 2.80m &amp; 3.21m &amp; 4.93m &amp; 45.23m &amp; 448.85m &amp; 44,847.79m\\<br /> H : 3m &amp; 5m &amp; 6m &amp;10m &amp; 100m &amp; 1,000m &amp; 100,000m\\<br /> \hline<br /> A : 1.74 &amp; 4.40 &amp; 5.59 &amp; 10.14 &amp; 109.54 &amp; 1102.30 &amp; 110304.42\\<br /> B : 8.52 &amp;11.19 &amp; 12.82 &amp; 19.73 &amp; 180.92 &amp; 1795.40 &amp; 179391.15\\<br /> E : 13.38 &amp; 17.58 &amp; 20.14 &amp; 30.99 &amp; 284.18 &amp; 2820.21 &amp; 281786.96\\<br /> \end{tabular}

Now it can be seen that clock A not only consistently has the greatest proper time but it also has the shortest spatial path distance and orbiting clock E that always has the least proper time also always has the longest path distance.

The geodesics of clocks A and E are the only ones that are directly comparable as they are the only geodesics that pass directly through the same point C. The clock at C is not on a geodesic as it is not inertial. The path of clock B is a geodesic that passes through point C but it remains to be determined what its proper time period is and whether there is a density distribution that allows it to return to point C at an interval that coincides with clocks A and E. An experiment could be set up to send clock A up to an altitude such that it returns to point C at the same time as clock B and clock E could be sent on an approximately elliptical orbit that has its short axis of the same length of the tunnel, but the maths would be very complicated and the fact that orbit of clock E would be precessing would add further complication.

Note: The negative proper time for clock D when R=2.13 is not a typo. The proper time of a clock at the centre of a gravitational mass is negative always negative for radii less than 2.25m or less than \frac{9}{8} R_{s}

 

[Stoonroon]

A reference that may clarify and tie loose ends in this thread is a 1961 paper by N. W. Taylor, 'Note on the Harmonic Oscillator in General Relativity,' in the Journal of the Australian Mathematical Society, vol. 2.

Google: journals.cambridge.org/abstract_S1446788700026677

Taylor's calculation refers to the difference in elapsed time between a clock at the center (clock D, in the above discussion) and a radially oscillating clock (clock B, in the above discussion). [These designations correspond to Dr. Jones' "revised" notation used in the latter half of this thread.]

The calculation is based on an approximation of the Schwarzschild interior solution for a uniformly dense sphere. Except for a difference of order v^4/c^4, the result (which corroborates an earlier analysis by O. Bergmann) is that clocks B and D show the same elapsed times when clock B is at the center and at the limits of its path.

It is not clear to me how this difference would be affected by a calculation based on the exact interior solution. In any case, it shows that Epstein's intuitively deduced result is at least very nearly correct. It would be interesting to hear back from Dr. Jones as to the magnitude of the time difference that his numerical integration yielded for this problem.

On the basis of general relativity, the agreement between clocks D and E – the latter being in a grazing circular orbit -- can be easily seen for the following reason. The frequency of a clock at the center (D) is supposed to be slower than a clock at infinity by the square root of the temporal coefficient (1 – 3GM/rc^2) from the Schwarzschild interior solution. And the frequency of the orbiting clock is slower by the square root of (1 – 2GM/rc^2 – GM/rc^2). These coefficients are obviously equal. In the latter coefficient the middle term represents the effect of gravity, while the right hand term represents the effect of the orbiting velocity.

Interesting as this all may be theoretically, gaps in the empirical evidence must be pointed out. We have ample evidence regarding clocks E (orbiting) and C (on the surface). But we have no evidence for clocks A, B, and D. Garth has mentioned Gravity-Probe A (the Vessot-Levine experiment) which surely involved a tossed clock. But the elapsed time on this clock was never measured; only its apparent frequency was deduced from a distance. So we do not have direct empirical evidence bearing on the time given by clock A (tossed straight up).

Furthermore, nobody knows for certain that a clock at the center ticks at the predicted rate. The times that would be given by clocks B and D are quite unknown. We cannot even claim to know for certain whether a clock dropped into the hole oscillates according to the textbook prediction. As scientists, we cannot presume to know that the book answer agrees with Nature's answer until we check.

It would be possible to find out with a laboratory experiment (using, e.g., a modified Cavendish balance).

http://www.ptep-online.com/index_files/2011/PP-24-03.PDF

If we would at least confirm that the oscillation prediction is correct, then this could be adduced as indirect evidence of the clock rate prediction. As it presently stands, the interior solution predictions regarding clock rate and simple harmonic motion have no empirical support; they are only mathematical extrapolations.