Posted on 04/06/2016 7:36:51 PM PDT by MtnClimber
Very good. But only true in the Newtonian approximation. I always think that we should say E = gamma m c², which gives E = mc² + 1/2 m v², in the second order approximation of gamma ( which is 1 to first order in v )
It's all fun.
bfl
That's correct. Einstein's Relativity had to be applied. And not just because of the high rate of speed of the satellites, but also the time-dilation effects of Earth's gravitational field. Time apparently ticks differently at the surface as compared to higher altitudes.
According to the theory, a clock in motion ticks out time more slowly than a clock at rest. Likewise a clock experiencing greater gravitational effects. So, in the case of a fast moving satellite, these two effects compete against each other. ie, the rapid motion slows the satellite clock while the greater surface gravity slows the land-based clock. In the case of GPS, gravity actually wins out over speed, so the net result is that the ground-based clock is a wee bit slower than the clock in the satellite. And the rate of time of the signal to and from the satellite is how distance, and thus position, is determined. Without applying Relativity, position would not nearly be so precise.
True, that solving for m in both Newton's F=ma (force = mass times acceleration) and Einstein's E=mc^2 (energy = mass times the speed of light squared) (f/a = E/c^2) can be set equal to each other. However, I don't believe neither F=ma or E=mc^2 is used in GPS calculations. The Einstein component applied is Relativity due to the high rate of speed of the satellite and also the stronger gravitational effects at the surface vs high altitude (see above)..
Sorry, at 4:30AM here on the east coast, I lost track of what the thread was actually about (gravity, not GPS).
I think so.
People often ask me "What good is Relativity?" It is a commonplace to think of Relativity as an abstract and highly arcane mathematical theory that has no consequences for everyday life. This is in fact far from the truth.
Consider for a moment that when you are riding in a commercial airliner, the pilot and crew are navigating to your destination with the aid of the Global Positioning System (GPS). Further, many luxury cars now come with built-in navigation systems that include GPS receivers with digital maps, and you can purchase hand-held GPS navigation units that will give you your position on the Earth (latitude, longitude, and altitude) to an accuracy of 5 to 10 meters that weigh only a few ounces and cost around $100.
GPS was developed by the United States Department of Defense to provide a satellite-based navigation system for the U.S. military. It was later put under joint DoD and Department of Transportation control to provide for both military and civilian navigation uses.
The nominal GPS configuration consists of a network of 24 satellites in high orbits around the Earth, but up to 30 or so satellites may be on station at any given time. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour (the orbital period is roughly 12 hours - contrary to popular belief, GPS satellites are not in geosynchronous or geostationary orbits). The satellite orbits are distributed so that at least 4 satellites are always visible from any point on the Earth at any given instant (with up to 12 visible at one time). Each satellite carries with it an atomic clock that "ticks" with an accuracy of 1 nanosecond (1 billionth of a second).
A GPS receiver in an airplane determines its current position and course by comparing the time signals it receives from a number of the GPS satellites (usually 6 to 12) and trilaterating on the known positions of each satellite[1]. The precision achieved is remarkable: even a simple hand-held GPS receiver can determine your absolute position on the surface of the Earth to within 5 to 10 meters in only a few seconds. A GPS receiver in a car can give accurate readings of position, speed, and course in real-time!
More sophisticated techniques, like Differential GPS (DGPS) and Real-Time Kinematic (RTK) methods, can deliver centimeter-level positions with a few minutes of measurement. Such methods allow GPS and related satellite navigation system data to be used in precision surveying, autodrive systems, and other applications requiring greater position accuracy than achieved with standard GPS receivers.
To achieve this level of precision, the clock ticks from the GPS satellites must be known to an accuracy of 20-30 nanoseconds. However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy.
Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture).
Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.
Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground.
A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.
The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38). This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time. This kind of accumulated error is akin to measuring my location while standing on my front porch in Columbus, Ohio one day, and then making the same measurement a week later and having my GPS receiver tell me that my porch and I are currently somewhere up in the air many kilometers away.
The engineers who designed the GPS system included these relativistic effects when they designed and deployed the system. For example, to counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations their clocks would appear to tick at the correct rate as compared to the reference atomic clocks at the GPS ground stations. Further, each GPS receiver has built into it a microcomputer that, in addition to performing the calculation of position using 3D trilateration, will also compute any additional special relativistic timing calculations required [2].
Relativity is not just some abstract mathematical theory: understanding it is absolutely essential for our global navigation system to work properly!
source:
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
Wow. Nice post. I have to admit that I have not looked closely into the time error issue. As a result, I am quite surprised by the magnitude of the corrections needed.
Thanks for posting.
Relativity is an abstract PHILOSOPHICAL theory! I was quite impressed by the "thought experiment" that Einstein describes in his "popular exposition" book, RELATIVITY. He is arguing for, or illustrating, the Principle of Equivalence, and describes the POV of someone in a rotating frame of reference, which they take to be stationary, of course. They would notice time dilation effects which, according to the Principle of Equivalence, would have to be attributed to the "gravity", as experienced in this frame but which we would ascribe to Special Relativity from the POV of an inertial frame.
Einstein points out that General Relativity has to accommodate this POV and agree with Special Relativity in that case.
Also, I suppose that the required GPS adjustment might have been noticed experimentally, and could have been the seed for a General Theory of Relativity, if we didn't have one already. Of course, this made it a lot easier!
It's been supported by facts in multiple areas. We wouldn't have many of today's technologies if it weren't for the "theory" being valid.
Sure. But I’m just saying it shows you the power of Einstein’s essentially philosophical point of view. It led him straight to it. You know, he went searching for the mathematics to support his idea, and had to learn it from Riemann. But he already knew where he was going. He just needed the math to get there.
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