Science B18

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Global Positioning Satellites and Relativity

By Arlon Staywell RICHMOND — The notion that data from Global Positioning Satellites can more certainly establish that the theory of relativity is true and accurate has become rather widespread lately. Perhaps that is not surprising since the system can establish a position on Earth with amazing accuracy. And many witnesses can see the results on their own GPS receivers.
How GPS Works
To determine the unknown position of the GPS receiver, known positions of several satellites are used and the distance to each. Each satellite broadcasts a radio signal announcing the time on board it with some means to identify its position at that time. The receiver compares that time with its own time. Any difference is the time it took the radio signal to reach the receiver. Radio signals travel at the speed of light. The distance to a particular satellite is found by multiplying the speed of light by the apparent time difference in the clocks.
So far that is no proof of relativity, no one has suggested it is. We all know that is still purely newtonian physics, which has the assumption that the clocks on the satellites and the receiver experience time the same. The effects of the dilation of time predicted by theory of relativity have not been taken into account at this stage. Let's continue with the newtonian physics until we determine the unknown position of the receiver by that means as we might need it in later computations.
With information from only one satellite we only know its position, call it point A, and the distance to it, call it dA. We might be anywhere on an imaginary, "mathematical" sphere centered at Point A with a radius of dA. With information from a second satellite at point B and at distance dB we might be anywhere common to the spheres (center A, radius dA) and (center B, radius dB). Those common points describe a circle in most circumstances. A third satellite gives the receiver the sphere (center C, radius dC). In most circumstances that will narrow the possibilities to two points common to the circle and the third sphere. Information from a fourth satellite can determine the location. Call this position point E. Point E is the "newtonian" position of the receiver, as no adjustments have yet been made for relativity.
Adjusting for Relativity
But according to the theory of relativity, objects in motion with regard to each other do not experience time the same. A critical assumption above was that they did. A reason given for the necessity of adjustment is

The combination of ... 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 (Pogge)

.
While the "atomic clocks" presumably on board the satellites might indeed be accurate enough to be adjusted for that, it is not really useful if the clock in the receiver is not also an atomic clock. If its clock, like most quartz crystal clocks, drifts far more than 38 microseconds a day, correcting for relativity that way does not appear possible.

Further, each GPS receiver has built into it a microcomputer that (among other things) performs the necessary relativistic calculations when determining the user's location. (Ibid)

.
Analyzing how that might be done is rather complicated. So let's take it one step at a time. Before any adjustments can be made for the relative motions of the individual satellites to the receiver, you must have at least an approximate location of the receiver. The simple distance to the satellite is not enough. That requires the calculation of the newtonian location, point E, as was described previously.

Recall that the computation yielded the following

4 points in space; A, B, C and D.
4 distances dA, dB, dC and dD.

dA and dB determine a circle such that each point on the circle is dA from A and dB from B.
(The intersection of 2 spheres whose radii are dA and dB respectively)

dC determines 2 points on that circle, which are dC from C, call these E and F.
(The intersection of the circle and a sphere whose radius is dC)

dD determines 1 of these points, which is dD from D, call it E.
At point E the distance d(E-A) = dA, d(E-B) = dB, d(E-C) = dC, and d(E-D) = dD.

Maintaining Convergence
Point E is the newtonian "convergence" point of the four lines from the satellites, the point we need before can begin adjusting for relativity.

Consider the following statement (Its "usefulness" will be explained later):
#1 It is impossible, with one irrelevant exception, to change any 1 of the four distances, for example dA, without losing convergence.

Why?
Because the other three distances still converge at E and if dA does not equal d(E-A) it cannot converge there.
What about that exception?
The exception is if you change dA to dF. Remember that three distances determined 2 points, E and F, so the three unchanged lines converge there also. And if d(E-A) = d(E-F) then dA converges there as well.
Why is the exception "irrelevant"?
Because it does not allow you to make a change based on the adjustment for relativity unless that also happens to be dF, which is unlikely.
Please consider this "proof" of statement #1 above.

So, since you cannot adjust the distances for relativity one at a time, then we'll just have to change more than one at a time. That is the "usefulness" of the statement just proved.

Now consider another statement:
#2 Changing all but one of the distances is the same as changing only one distance.

Why?
Assume that if any shape consisting of line segments changes size by a factor of k, the length of all the lines also changes by a factor of k, and the reverse. This is very basic geometry.
Then by changing all but one, that is three, lines by a factor of k, they have the same shape and still converge. But the shape made with the lone segment is not proportional, so the effect is that three have not changed and the lone segment has, which is not allowed by statement #1.
This eliminates the possibility of changing all but one by any proportion k. (Never mind where k=1 because that is NOT changing anything.)

Then we'll just have to change all but one in a disproportional way, such as using the different adjustments based on the different relative velocities of the satellites to the receiver.

But if you change three lines in a *dis*proportional way, they won't have the same shape and won't converge with themselves, never mind the fourth.
This eliminates all possibilities of adjusting all but one of the distances.
Please consider this "proof" of statement #2 above.

What about changing two at a time?
The proportionality problem exists there as well.
Writing a Computer Program
Note that a computer program that starts making adjustments to the newtonian distances loses track of the convergence point and any further calculation will introduce errors comparable to any it removes.
Setting the Receiver's Clock
In the newtonian mechanics problem the clocks on the satellites can be assumed correct, since they are highly accurate and no account of relativity is made yet. Then there is only one unknown (not five as above) to solve, the time at the receiver, since it has a less accurate clock. A possible means to correct it is as follows. If the clock on the receiver is fast (The arrival of the signals will seem slow.) the position will seem farther away from all the satellites and lower than the real position, possibly way beaneath the surface of the Earth. If it is slow (The arrival of the signals will seem fast.) it will appear higher, possibly in the stratosphere. Even slower and the signal will appear to arrive before it left which is not possible. Barring those conditions the latitude and longitude will be fairly accurate so long as the satellites are somewhat evenly distributed, which is recommended anyway. All you need is the altitude. Knowing the latitude and longitude gives the distance from an Earth based time signal such as WWV in Fort Collins, Colorado, for example. That distance is not effected much by the altitude. The receiver simply needs to adjust its clock to that one accounting for the distance to it. Further accuracy might be obtained using the now more accurate latitude and longitude.
Since there is in theory only one altitude at which true convergence occurs, it might be possible to run through all the altitudes until that is found without consulting a ground based time signal. But because of the limitations of the accuracy of the calculations a rather broad range of altitudes might appear to satisfy convergence. And apparent convergence might fade in and out again several times over the steady adjustments in receiver clock time.

Before the clock in the GPS receiver is set and you don't have true convergence how do you find the approximate point of convergence?
The first two distances yield a circle just as with convergence, but the third distance might not touch the circle in two points. There will however likely be two points on the circle whose distance from C is closest to dC. They will be easily found in two "troughs" in the data. Neither of those two points might be exactly dD from D, but the point whose distance from D is closer to dD is an approximate convergence point. The computer program is the same as for true convergence which should not attempt to exit a program loop on an equality anyway. That can often lead to infinite looping.
Ways to improve the accuracy before consulting a ground based time standard include 1) whenever two points that should be the same aren't using the midpoint between them and 2) comparing the satellite data in different orders.
Remember that using an approximate convergence point is only practical when solving for one unknown. Solving for five unknowns is not practical.

While the clocks onboard the satellites probably are more expensive, elaborate and accurate than the clocks in every little GPS receiver, they are probably not truly "atomic" clocks. Rather they are likely regularly updated by an atomic clock on Earth along with regular course corrections and program changes.
There are no adjustments for relativity to that yet. And the other errors in the system make it difficult to measure what relativity might actually play in it all.
Still Another Question?
Good Lord! Okay then, another question, why can't people who believe in relativity use the newtonian convergence point to simultaneously adjust for relativity and show that convergence improves? Because the same "relativity errors" that presumably make the true location appear not to perfectly converge can make some nearby false location appear to converge better, then adjusting for relativivty might give a more accurate location, but the apparent convergence got worse, not better. Then why not use receivers with known locations to check the accuracy of the adjustments? Because in making its guess of the approximate convergence point the program could accidentally guess better than worse, since it might just as well do either. Then the adjustments for relativity could give a less accurate location, or perhaps a not measurably different one.
And using a receiver with a known location to calibrate the system corrects for a multitude of errors from a multitude of sources, and as noted before it is difficult to say what part relativity might have played.
There are far more accurate ways to measure what effects relativity might have on various clocks sent for example to the moon and back. Albeit more expensive, there should be data from the various lunar missions already undertaken.
Conclusion
That data and reasoning of Professor Pogge do not appear able to support a case for the Theory of Relativity. Rather than making a case for relativity the "amazing accuracy" of the GPS system indicates that the adjustments necessitated by relativity are either too trivial on this scale or non-existent. If the adjustments required were large then the newtonian "approximate" convergence point would be farther off and the adjustments for relativity based on it would necessarily show an error.

Pogge
Richard W. Pogge is a professor of Astronomy at Ohio State University's Department of Astronomy.
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

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