Re: Does motion affect the operation of clocks or not ?

The figures you are quoting would apply if one would compare the
satellite clock directly with the receiver clock. However, this is not
how GPS works in practice. The receiver clock is in fact very much
irrelevant, as the position is obtained by observing the difference
between the time signals obtained from a number of different
satellites.

Consider for simplicity a one dimensional problem where the receiver
is located somewhere on the line connecting the two transmitters. In
this case the signal from transmitter 1 reaches the receiver at time

(1) t1 = t0+ x1/c

and the signal from transmitter 2 reaches the receiver at time

(2) t2 = t0+ x2/c ,

where t0 is the time the signal is being sent out (assuming both
transmitter clocks are synchronized), x1 is the distance of the
receiver from transmitter 1, x2 the distance of the receiver from
transmitte 2, and c the speed of light.
Now if one subtracts Eqs.(1) and (2) one gets

(3) x1-x2 = c*(t1-t2) .

One knows therefore the position of the receiver just by comparing the
time signals from the two transmitters (if t1=t2, the receiver would
be exactly in the middle between the two transmitters). The receiver
clock is thus completely irrelevant for determining the position.

If one assumes now that the transmitter clocks are running fast or
slow by a relative factor (1+r), one has instead:

(4) x1-x2 = c*[(1+r)*t1 -(1+r)*t2] = c*(1+r)*(t1-t2)

which means that the position will simply be wrong by a relative
factor r, but there is obviously no accumulation as the transmitter
clocks run at the same rate relatively to each other.

Now the usually quoted relativistic overall correction of 38
microseconds/day corresponds to r=4.4*10^-10. As the satellites are at
a distance of around 20000 km (=2*10^9 cm), the positional error due
to relativity should actually only be 4.4*10^-10 *2*10^9 cm = 0.8 cm.
This is much less than the presently claimed accuracy of the GPS of a
few meters, so relativistic effects should actually not be relevant at
all (which would be anyway largely due to General Relativity, i.e. not
be associated with the motion).

Thomas

.

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