Satellite clock drift relative to clocks on the surface of Earth as a function of distance from the center of Earth. 

Context in source publication

Context 1

... transverse Doppler effect is a consequence of special relativity, and it states that as the satellite’s clock is moving fast V relative to a person on Earth V it runs slow. The gravitational effect is a prediction of the general theory of relativity, and it states that a clock in a strong gravitational field is slower than one in free space. A consequence of this prediction is that the clocks on GPS satellites run faster than those on Earth. Fig. 1 is a schematic of the GPS satellite constellation. Sur- prisingly for GPS satellites V orbiting approximately 20 000 km above the surface of Earth V the gravitational effect is about six times stronger than the special relativistic effect. If these effects were ignored, the range error between the satellite and the receiver would be approximately 11 km after one day [2]. In 1977, when the original GPS satellite was launched, a number of engineers working on the GPS design were not convinced that relativistic effects would be observable [3]. Thus, a frequency synthesizer was included in the design to check for a change in the rate of the Caesium clock. It turned out the satellite clock ran about 38 s per day faster than clocks on Earth; relativistic effects are therefore real and observable [3]. It is a fortunate accident of history that the ability to keep clocks synchronized to within nanoseconds was not available when Einstein proposed the theory of special relativity in 1905. Like a good engineer, Einstein proposed an experiment to test his new theory: Take two synchronized clocks and place one on the North Pole and the other on the equator. According to his new theory, the clock on the equator would run slower than the one at the pole. Had this experiment been done, the result would have been that both clocks ran at the same speed [4]. Einstein was not wrong; he just had not discovered general relativity yet. It turns out that the North Pole is about 20 km closer to the center of Earth than the equator and, therefore, is in a stronger gravitational field. According to the general theory of relativity, the stronger the gravitational field the slower the clocks run. Amazingly, the gravitational slow- down at the poles matches the time dilation at the equator, meaning that clocks at the equator and the pole tick at the same rate. Furthermore, all clocks on the surface of Earth tick at the same rate, a fact that is used by international timing agencies [5]. The theory of relativity is not just used for modern advanced time keep- ing systems. Other applications of relativity include nuclear energy ð E 1⁄4 mc 2 Þ , the Doppler effect for electromagnetic radiation, ring laser gyroscopes [6], and signal association [7]. So why is it that relativity is not widely taught in electrical and electronic engineering degrees? I believe the reasons are twofold: historical and cultural. Until the invention of atomic clocks, lasers, and global satellite navigation systems, the only practical application of relativity was nuclear energy V a specialized and controver- sial area of work. To make matters worse, relativity theory has always been taught by theoretical physicists for theoretical physicists, which means that there is a much larger concentration on the mathematics and the apparent paradoxes than there is on the applications. Furthermore, the general theory of relativity requires a good knowledge of tensor calculus, a subject that is not com- monly taught at the undergraduate level, even to physicists. It is my mission to teach relativity to engineers in a way that is comprehen- sible and applicable. I am doing this for two reasons: the first is that advanced engineering is the application of advanced physics and the second is that relativity theory is one of the fundamental scientific theories; thus, we need to be as familiar with it as we are with Maxwell’s equations. The fundamental concepts of relativity theory (both special and general) can be summed up in three concise statements: 1) the laws of physics do not depend on the velocity of those observing them; 2) the laws of physics do not depend on the coordinates they are written in; 3) there is no such thing as a gravitational force. Matter moves in a straight line in a curved space time. The first statement is the basis of special relativity. This statement, along with the requirement that speed of light in a vacuum is a fundamental physical constant, leads to the Lorentz transformations. The Lorentz transformations tell us how to compare the space–time coordinates of events between two moving observers. An application of the Lorentz transformations is the rela- tionship between the interference fringes and the rotation rate in ring laser gyroscopes, used by engineers in inertial guidance systems [8]. The second statement leads to the conclusion that the equations of physics must be tensor equations. Tensor equations, by construction, are independent of the coordinate system they are written in. A conve- nient way to introduce tensor calculus is to consider map projections. Fig. 2 shows the great circle (solid line) and dead reckoning (dashed line) paths from London to Los Angeles. The right-hand side figure is a Mercator projection of the surface of Earth. From this projection one would assume that the dead reckoning path is shorter than the great circle path, whereas we can see from the globe on the left-hand side that it is not. Using tensor calculus we can calculate the optimal path between two points on a curved surface, such as the surface of Earth. In more recent times, geometric algebra (GA) has become a popular method and has been suggested as a possible alternative pathway to intro- ducing relativity to engineers [9]. It has been noted by some authors that the relativistic phenomenon of pre- cession without torque is more easily derived with geometric algebra than tensor calculus [10], [11]. The last statement, pointing out that space time is curved, is incredibly profound and for me is one the greatest conceptual leaps in human thought. It is an idea that must marinate in one’s brain for a while before it can be fully appreciated, however, it has a practical application that can be taught, even to engineers. The general theory of relativity pre- dicts that the rate at which a satellite clock runs is a function of its distance from Earth. The magnitude of this effect is shown in Fig. 3. Depending on the application, modern satellite designs sometimes need to include this effect [12]. Electrical engineers in the final years of their degree can take a tailored course of relativity. That course would not be the same as the one taught at physics departments, but can be designed to emphasize the engineering applications and mini- mize the philosophy. Advanced engineering is the application of advanced physics. It is almost 110 years since Einstein published his seminal paper on the special theory of relativity. It is time to ‘‘get with the program.’’ h The author would like to thank the head of the School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, S.A., Australia, Cheng-Chew Lim, for supporting the ‘‘Relativity for Engineers’’ course at The University of Adelaide. He would also like to thank D. Abbott for both suggesting this article and providing valuable input into ...