The free, open, and dependable nature of GPS has led to the development of hundreds of applications affecting every aspect of modern life. My results agree with those shown in Figure 2.Like the Internet, GPS is an essential element of the global information infrastructure. Analysis Percentage Frequency Change Due to General Relativityįigure 3 shows my derivation of a formula for the percentage frequency shift due to the effects of general relativity. The clock period correction required for special relativity is in the opposite direction as the correction for general relativity. T SatSpec is the clock period for a GPS clock mounted on a "fast" satellite and measured by an observer on Earth.T EarthSpecis the clock period measured for a GPS clock on Earth.The clock period on the orbiting satellite ( T SatSpec), is longer than the clock period ( T EarthSpec) for the same clock at rest on Earth (Equation 3). Special Relativity Effect on GPS Clock Frequencyīecause a clock on the satellite is moving relative to the same clock on the earth, the clock on the satellite appears to run slower than the same clock on earth. We can compute the clock period shift due to the effects of general relativity by using Equation 2 and working with the differences in the clock periods at the radii of the Earth's surface and the satellite's orbit. However, the the influence of gravity will cause the apparent clock period to change when it moves from the Earth into orbit. While the GPS clock is in orbit, we will need to know its period as measured from the Earth. G is the universal gravitational constant.T Gravity is the clock's period at distance r from the center of the gravitational field.T NoGravity is the clock's period infinitely far from any gravitational field. ![]() We can use Equation 2 to determine the impact of gravitational time dilation on the clock frequency at different distances from the center of a gravitational field. General Relativity Effect on GPS Clock Frequencyīecause a clock on earth is subject to a stronger gravitational field than the satellite, the clock on the satellite will appear outrun the same clock on the earth. d r is the distance error due to the combined relativistic effects.įigure 2: Infographic from a Presentation I Saw on GPS.Δt r is the daily GPS timing correction required for the combined effects of general and special relativity.When I heard this timing correction, I quickly multiplied it by the speed of light to get the equivalent distance error (Equation 1). In this graphic, he mentions a daily timing correction of 38.6 µsec for relativistic effects. Infographic on GPS Satellitesįigure 2 shows an infographic from Hugo Fruehauf, one of the key GPS developers. Figure 2 contains an equation for computing the frequency shift due to general relativity that I have not worked with before and I will derive it as part of my work here. ![]() My approach here will be to work through the equations and material shown in Figure 2 of this post, which presents a slightly different viewpoint on the same material. If you are looking for more background on GPS errors, the Wikipedia as a very good article on the magnitude of the relativistic errors in GPS on this page. So I thought I would work through the math behind these effects and present the results here. ![]() I read a paper quite a while ago on the topic and I recalled that there is ~14 kilometer per day error due to general relativity effects, and ~2 km per day of error in the opposite direction for special relativity, for a total error of ~12 km per day. If general relativity were not taken into account in GPS satellite navigation systems, errors in global positions would accumulate at a rate of about ten kilometers each day. This post will be about the following statement on relativity and the GPS system (Figure 1). However, there were two factoids that I thought were interesting and I would write posts about. I personally would not call the statements in the article shocking, but that is just my opinion. I occasionally read articles on Stumbleupon and I came across an interesting article called "8 shocking things we learned from Stephen Hawking's book" – the book they are referring to is called The Grand Design. These satellitesĮverything from land surveying to car driving
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