General Relativity Experiment Query

Question

Could we more accurately test General Relativity's time dilation predictions by comparing an atomic clock on a geostationary satellite with a corresponding clock on Earth's surface? Given that both would traverse a more controlled/similar region of relative spacetime curvature, would this provide a more isolated test of gravitational time dilation in GR compared to current experimental setups? Would this setup also reduce confounding special relativity time dilation effects as both geostationary and earth based atomic clocks would pass through the same relative amount of space-time curvature.

Answer 1

would this provide a more isolated test of gravitational time dilation in GR compared to current experimental setups?

My understanding of your question is that you wish to eliminate velocity in order to have a “pure” measure of gravitational time dilation, and to be able to run the experiment for a long time. The hope appears to be that this will give a measurement with higher precision than previous measurements, with no confounders.

Unfortunately, this approach will not eliminate the velocity as a confounding factor. Neither a clock on the surface of the earth nor a clock in a geostationary orbit are at rest in the Earth centered inertial frame.

Nor are they at rest in Schwarzschild coordinates. Nor are they at rest in the usual Kerr coordinates.

You could not consider it a motion-free test of General Relativity without transforming to some arbitrary coordinates where they are at rest. But if you are willing to do that, then any experiment where they are relatively stationary will serve.

Such experiments have already been done. From a precision standpoint, while a geostationary orbit has the advantage of a large difference in gravitational potential, it has the disadvantage that it will be continuously perturbed. Those perturbations will be much larger than those between the roof and basement of a building or between the top and bottom of a mountain.

The time is not an advantage since an experiment in a building or on a mountain could also be done for a long time.

It is unclear which effect would dominate the error budget. A detailed calculation would be necessary to determine if it is advantageous or not.

Answer 2

Could we more accurately test General Relativity's time dilation predictions by comparing an atomic clock on a geostationary satellite with a corresponding clock on Earth's surface?

Yes. This has been done innumerable times, namely any time anyone has ever checked their location with a GPS. GPS satellites are high enough up (and the timing needs to be precise enough) that they have to account for time dilation between their clocks and the clocks on the surface of the Earth.

When you use a GPS, the timing information that your device gets is corrected for general relativity, and it produces correct results (citation: your GPS works to within a few meters, if not better). Those corrections implicitly contain the relativistic calculations that tell the difference in tick rate between your ground clock and the satellite clock. Since the corrections produce correct results, the conclusion is that those calculations are valid, and that general relativity made a successful prediction.

Other satellites, like Gravity Probe A and B, tested the effect to much higher precision and got results in agreement with general relativity.

Answer 3

This was done in two cases. First with the Pound-Rebca experiment using clocks at the base and top of a tower at Harvard university. Einstein’s law says that g=Rc^2/D where g is the rate of gravitational acceleration (9.8m/s^2 on Earth), R is the fractional difference in clock ticking rates (2.43x10^-15 in this case) and D is the distance between them (22.3 metres in this case). Pound-Rebca found 2.43x10^-15 x c^2 / 22.3 = 9.8 m/s^2 which is the acceleration due to gravity on Earth. The accuracy of this was improved with the Gravity Probe A experiment using a satellite at 10,000 km altitude and a clock rate of 30 microseconds per day faster than the Earth bound clock.

Answer 4

About Geostationary orbit:

As we know, the Earth and the Moon are orbiting their common center of mass.

Earth Radius is about 6400 kilometer, and the common center of mass of the Earth-Moon system is aboat 4700 kilometer away from the Earth's center.

The orbit of any Earth satellite is affected by the dynamics of the Earth-Moon system in two ways: Specific for the case of geostationary orbit:

For geostationary orbit heigth above the Earth center is about 42000 kilometer. Earth Moon distance is about 384000 kilometer. Gravitational effect from the Moon is small, but not negligable.

The satellite is attracted to the Earth's center of mass, but the Earth is a moving target. The center of the Earth is orbiting the common center of mass of the Earth-Moon system.

For any satellite: the altitude is not a highly stable quantity. There are always perturbations. Some of the time perturbation tends to make the orbit elliptical, some of the time there is on balance a circularization effect. Averaged over a long time interval the altitude is constant, but there are fluctuations.

For the purpose of testing General Relativity:
When monitoring the amount of proper time that elapses onboard a geostationary satellite I expect that multiple orbit perturbing factors must be accounted for.

To give an idea of the extent of perturbing factors: geostationary satellites have onboard thrusters. The thrusters are there to maintain position relative to the Earth, in response to perturbation of the orbit.

Over its working lifetime the propellent onboard the satellite is used as economically as possible. When the onboard propellent is close to being depleted the last remaining propellent is used to boost the satellite to a higher orbit, so as to clear the space at geostationary altitude.

Summary:
While a satellite in geostationary orbit (24-hour orbit) will be in a somewhat quieter space than the 12-hour orbits of the satellites of the GPS system, geostationary orbit, like any other orbit, features changes of orbital velocity and orbital height.

In all, having an atomic clock onboard a geostationary satellite would not present a significantly better opportunity than the orbits of the GPS satellites.