Tying it together

 

First, some light background on orbital mechanics 

A really great, and quite humorous video about orbital mechanics can be viewed below:

(PLEASE SEE “GR IN GPS IMAGE RESOURCES” PAGE FOR VIDEO SOURCE)

The video is a very short and condensed. What we learn from it (and may be obvious to the reader): objects (in certain orbits) move faster in relation to Earth’s surface; and orbit higher out of Earth’s gravitational “well”. What this tells us: there should be relativity effects due to the increased speed and the orbital radius. And there are!


 

A GPS Clock

The clocks aboard a GPS satellite, as stated, are very accurate. But just how accurate? How do they even work? I would go to length to explain this to you, but a marvelous video is posted below:

(PLEASE SEE “GR IN GPS IMAGE RESOURCES” PAGE FOR VIDEO SOURCE)


Time Dilation due to Orbital Speed

Each GPS satellite has an extremely accurate atomic clock on board. The clocks on GPS satellites, as expected from relativity, will tick slower due to their increased speed. But how much slower? According to the webpage “What the Global Positioning System Tells Us about Relativity” by Tom Van Flandern, special relativity predicts that GPS clocks will tick slower by 7200 nanoseconds per day compared to a clock on the ground (a handy link: here). This must be taken into account when programming the clock for a GPS satellite.

Time Dilation due to Orbital Radius

We saw earlier that the equation for time dilation due to a gravitational source is: T =  T0 √(1-2GM/Rc2)

According to Flandern, GPS clocks run faster by a rate of 45,900 nanoseconds per day. Often, the clocks onboard GPS satellites are adjusted to their orbital rate prior to launch, so when the desired orbit is reached, the clock is ticking synchronously with Earth. Flandern goes on to explain that data indeed shows that the GPS clocks behave as predicted, but slightly different. Remember, the rates are adjusted for orbit, and sometimes the desired orbit isn’t exactly obtained.


SHIFTS DUE TO ORBIT AND EARTH’S SHAPE

On the previous page, I talked about the oblate shape of the Earth. Ashby cites in his paper that this shape of the planet causes a fractional frequency shift every 6 hours, with an amplitude of 0.695*10-14 meters.

In the last section I discussed Earth’s geoid. Surprise surprise, it comes up in this section as well. We have to consider oblateness and the geoid model of the planet in order to make use of relativity for GPS.

Below is an equation, derived by Ashby, that details the fractional frequency shift for a satellite in circular orbit in around Earth’s geoid.

Image courtesy of Neil Ashby: "General Relativity in the Global Positioning System".
Image courtesy of Neil Ashby: “General Relativity in the Global Positioning System”.

The only term you may not be familiar with in this equation is Φ0. This term is simply the effective gravitational potential on Earth’s rotating geoid. The “r” term is simple the orbital radius (which is constant, in this case). The f in the denominator of the fraction on the left is the frequency for a reference clock fixed on Earth’s geoid.

But, if the orbit of the satellite is Keplerian (an ellipse, parabola, or hyperbola), the equation changes to a different form:

https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/An_image_describing_the_semi-major_and_semi-minor_axis_of_ellipse.svg/350px-An_image_describing_the_semi-major_and_semi-minor_axis_of_ellipse.svg.png Image courtesy of Neil Ashby: "General Relativity in the Global Positioning System".
The first two terms are applied to GPS clocks prior to launch, to make them tick in sync with clocks on Earth’s surface. The rest of the equation depends on “r” and “a”, and are where our relativistic effects get involved. Image courtesy of Neil Ashby: “General Relativity in the Global Positioning System”.

Another term creeps up here, the “a”. This term refers to the semi-major axis of the orbit. The semi-major axis, put simply, is the greatest distance of an orbiting body to the source of gravity along it’s path. See below:

Image courtesy of wikipedia.org
Image courtesy of wikipedia.org

A WEBSITE ON GENERAL RELATIVITY WOULD NOT BE COMPLETE WITHOUT MENTIONING BLACK HOLES. NEXT, WE DO JUST THAT!

Overview