The GPS satellites are constantly moving in their various orbits around Earth.

For a given observer on the surface of the Earth, there must be at
least one satellite that is racing away in one direction, and at least
one that is rapidly approaching from another. And there is probably at
least one satellite (on the far side of the Earth) that is doing
neither (it has stopped racing away, and is about to return).

(1) Does the satellite that is racing away appear to operate slower to
the observer? If so, is it simply because the distance is increasing
(taking longer for communication to occur), or is it also because the
satellite is moving and therefore operating slower?

(2) Does the satellite that is approaching appear to operate faster to
the observer? If so, is it simply because the distance is decreasing
(taking less time for communication to occur)? Since the satellite is
also moving, does it tend to operate slower and thus offset some of
the effect of decreasing distance?

(3) Does the satellite that is on the opposite side of the Earth
appear to operate normally (because it is neither receding nor
approaching)? Or does it still appear to operate faster because it
is moving?

Note that I am a layman, so please do your best to provide simple answers.

the speed of light     = 299 792 458 m / second
speed of orbiting sat  = 4,722  m / second

light goes at almost 300 million meters per second, the satelite here
is barely moving at about 1/100,000 the speed of light, way to slow to
have any effects. You would need some of the most sofisticated caesium
clocks to measure it.

And furthermore, the GPS receives data from at least three  - maybe as
many as seven -  satellites to calculate its position with software
that comes up with one  position.  Someone may know if it discards
data that doesn't agree closely with that from the other satellites or
just averages all data.

"You would need some of the most sofisticated caesium
clocks to measure it."

   In fact, cesium clocks are available to measure the effects. Each
GPS satellite carries an atomic clock, which is what allows it to make
such precise determinations of surface positions. And of course, the
question asked is not how large the effects will be, but rather what
their nature is.

   As it turns out, a satellite approaching a ground based observer
will display a blue shift in its frequency, proportional to the ratio
of its speed to c. If going away, it'll have a similar redshift. This
is the familiar Doppler shift. But the "Einsteinian" effects will be
general relativistic, because the two systems (satellite/observer) are
in non-flat spacetime; there is gravity at work. The higher the
strength of gravity in the neighborhhood, the slower time will be
relative to that for a region of lower strength. In other words,
clocks on the ground run slower than they do on satellites at high
altitudes. (Or, for that matter, than on airplanes, or the top floors
of buildings.)

   An observer on an orbit which just barely grazes the event horizon
of a black hole may see whole galaxies come & go in the time taken to
change underwear. An observer far away from the black hole may see
that the other one is taking just about forever to change underwear.

   Special relativity is at work in curved spacetime, but only
locally. That's because SR is only absolutely, strictly true for flat
spacetime, where motion is inertial. In a curved region of spacetime,
such as surrounding a massive object (the gravity field), flatness is
only approachable as the size of a region approaches infinitesimally
small. (If the curvature is very small for all practical purposes,
then it's approximately flat, and an object can travel approximately
inertially. Thus, astronauts are practically in free-fall, even though
they & their accoutrements are of finite size.)

   So anyway, the software onboard the GPS satellites must factor in
the GR transformations. Even though they are small in the neighborhood
of Earth, over time the error can accumulate to the point that
coordinate measurements will be way off.

Hi qed100, 

You are correct, I don't dispute any of what you've said. However he asked 

"appear to operate slower to the observer? "
"appear to operate faster to the observer?"
"appear to operate normally"

However true your statements might be, nothing of what you've said
applies to his question. The qualifying word here is 'appear' clearly
he is defining a human perspective. If that was not his intent, then
the question could be reprhased.

Good post though,

Eli

"The qualifying word here is 'appear' clearly
he is defining a human perspective. If that was not his intent, then
the question could be reprhased."

   Certainly. But I've spoken with the OP in another thread, and in
his case I think I understand that he wants to know what Einsteinian
theory has to say. I could be wrong.

-Mark Martin

Yes, I would like to to know what Einsteinian theory has to say.

The clocks on-board each satellite tick faster than identical clocks
on the ground by about 38 microseconds per day. 38 microseconds is
38,000 nanoseconds.

From Astronomy.mps.ohio-state.edu

"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 (among other things)
performs the necessary relativistic calculations when determining the
user's location."

To markvmd-ga:

Thank you for your comment.

If I understand you correctly, the mere fact that the satellites are
moving (relative to an observer on the surface of the Earth) means
that their atomic clocks operate at a different speed compared to the
observer?s clock.

What shocks me about your comment is that they will operate FASTER
than the observer?s clock.

I thought that things that accelerate and move (relative to an
observer) always operate at a SLOWER speed than the observer.  I mean,
isn?t that the crux of the Twin Paradox?  (The man who travels through
space at high speed returns to Earth to find that he hasn?t aged very
much, but that his Earth-bound twin is now an old man.) The traveler
ages SLOWLY.

So, how come the GPS satellites operate FASTER?

"So, how come the GPS satellites operate FASTER?"

   See my earlier post. The satellites aren't in flat spacetime; it's
non-linear due to the presence of mass, i.e., there is gravity. This
means they are subject to the principle of general relativity
globally. The satellites are in a weaker gravitational potential than
stuff down on the ground, so there is a time differential between
them. Time on the ground, where gravity is stronger, is slower than up
high. And as it turns out, in the case of GPS satellites orbiting with
Earth, the GR time transformation is greater than the SR
transformation due only to their local inertial motion.

From Astronomy.mps.ohio-state.edu

To achieve their level of precision, the clock ticks from 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. 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.

BUT... 

The satellites are in high orbits, 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.
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

Utterly fascinating!  Please tell me if I understand correctly:

SPECIAL RELATIVITY: Because the GPS satellites are moving (relative to
an observer on the ground), their clocks will operate about 7,000
nanoseconds per day more slowly than they would if they were not
moving.

GENERAL RELATIVITY: Because a clock on the ground experiences more
intense gravity than clocks in high orbit, it will operate about
45,000 nanoseconds per day more slowly than it would without such
intense gravity.

Although clocks in orbit operate more slowly than they would if they
weren?t moving, they nevertheless experience much less gravity than a
clock on the ground, and the net result is:  they operate faster. The
difference is 45,000 ? 7,000 = 38,000 nanoseconds per day. That is,
clocks in orbit operate 38,000 nanoseconds per day faster than a clock
on the ground.

To eliminate this 38,000 nanosecond discrepancy, the clocks in the GPS
satellites were deliberately slowed down before they were launched.

Yes, that's pretty much the size of it.

As Qed100 said, that's the size of it. Quod erat demonstrandum. The
adjusted clock speed and the computers (don't forget the computers)
compensate for the differences caused by relativity.

That fellah who figgered this out was a real genius. No wonder they
called him Einstein!

For folks learning science in Kansas and Pennsylvania, they compensate
for the Flying Spaghetti Monster's noodling. Or maybe they are just
for show. It's a mystery. Try THAT on a spelling test!



For more on the Flying Saghetti Monster and how His Noodly Appendage
created all of us, visit http://www.venganza.org

Isn?t the Flying Spaghetti Monster a tongue-in-cheek alternative to
Intelligent Design (which, in turn, is a proposed alternative to the
teaching of evolution)?  Was Einstein deeply involved with any of
this? (Did he evolve into the Flying Spaghetti Monster?)

My thanks to qed100-ga and markvmd-ga for your very helpful comments.
Five stars to you both!

Yes, the FSM is a viable alternative to Intelligent Design and just as
accurate. Einstein had nothing to do with it, and couldn't possibly
evolve into the FSM because evolution doesn't exist in Kansas and
Pennsylvania (maybe just 1600 Pennsylvania Ave). I just wanted to poke
a little fun at people who seem have their collective heads in the
sand... er, who may have an alternate explanation for this so-called
relativity thing.

Thanks for teh kudos. I'm a little surprised an official Answerer
didn't take this question on, though the commentators seem to have
jumped on it rather early.

--Mark

You can no more believe in evolution than you can believe in gravity.