The Global Positioning System (GPS) consists of a network of 24 satellites in ~12-hour orbits, about 20,000 km above Earth’s surface. Each satellite carries an atomic clock on board. The clocks must all be synchronized with each other and also to the U.S. Naval Observatory Master Clock. A GPS receiver with an accurate terrestrial clock can then measure the signal propagation time and hence the distance from a few satellites and determine its position.
The satellite clocks are apparently built with a 38 microseconds per day lag relative to the master clock, so that when they are in orbit, they run at the same rate as the master clock. What puzzles me is that the clocks are apparently still re-synchronized every day, otherwise errors build up. Is there a discrepancy in relativity theory, or are the clocks just not accurate enough?
SL
Hi SL, interesting new topic!
I think the reason for the regular resynchronization is inaccurate orbits rather than inaccurate atomic clocks. Another possible source of difference is the fact that Earth is not a perfect sphere and the orbits have all sorts of orientations relative to the equator.
Something that puzzled me is why 12 hour orbits and not orbits at about 1.5 times the Earth’s radius, period 2 hours 38 min? There orbiting clocks and ground clocks would run at roughly the same rate automatically, because velocity time dilation and gravitational redshift would cancel each other.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
12 hour orbit vice 2.6 hour orbit because you need to be able to easily measure the time the signal took to reach the receiver. A lower orbit would have a much shorter travel time from transmitter to receiver and be more difficult to measure with the necessary precision and accuracy. Remember this was all done with late eighties technology in mind.
Burt, I’m sure that even the selected orbit you note would not be perfect enough that the cancellation would be exact.
It seems to me that re-synching is going to be a practical necessity no matter what the orbits.
I don’t know much about the GPS system, but the 12-hr. periods must have some practical advantage (Fewer satellites needed, maybe? Closer synch to Earth’s rotation makes system design better?).
SL, I love your questions!
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Burt said: ” why 12 hour orbits and not orbits at about 1.5 times the Earth’s radius, period 2 hours 38 min? There orbiting clocks and ground clocks would run at roughly the same rate automatically, because velocity time dilation and gravitational redshift would cancel each other.”
1. How would the 2h 38m orbits make clocks in orbit and the ground clocks run at the same rate?
2. What difference would it make if there are still corrections due to “inaccurate orbits” required?
3. Why bother with the relativistic corrections if you are going to re-sync every day anyway?
Sorry to sound terse, but these things are issues that puzzle me. That’s why I started this thread. (Plus your special “2h 38m orbit” – another puzzle!)
SL
but I could hazard a couple reasons:
The first being agreement with the first comment about irregular orbits.
The second being that assumption of correctness (via coded algorithms) can never compete with empirical data. So, a trusted source of real data is far better than a computer’s model of the truth.
Third – computer hardware in space is expensive and I would assume they would want most redundant and radiation hardend components. These probably are not super capable in functionality and I would also assume that any self correcting positional algorithm would need a lot of computing power for three dimensional space – factoring in a number of outside variables.
Finally – it just has to work. (part and parcel with #3 I guess) The system needs to always be up with as little maintence as possible – the military is depending on it heavily – so it should ALWAYS be correct. (part of #2) Additionally, I would also have to say that ‘control’ factors into it as well. I couldn’t imagine a scenario where the military would allow self managed satelites to be such a low-level and trusted infrastructure component.
Eric
Eric, I tend to agree with you, but they have enough fancy hardware up there – I think it is likely just the orbits that cause a drift in the clocks.
SL
Hi SL.
I’m short of time and will only respond briefly to your no. 1 question for now: “1. How would the 2h 38m orbits make clocks in orbit and the ground clocks run at the same rate?”
At the GPS altitude, clocks run 45 microseconds per day faster than on Earth due to the lower gravity. The velocity time dilation is some 7 microseconds per day (clocks slower than on Earth). This gives the 38 microseconds faster per day that you have stated originally.
As you lower the orbit, gravity gets stronger and the 45 microseconds (faster) per day reduces. At the same time, the orbital velocity goes up, causing more velocity time dilation (the 7 microseconds per day goes up), until at around 1.5 Earth radii, the two are equal.
Sorry to be a bit cryptic for now, but if you want, I’ll get back with more detail.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Craig wrote: “A lower orbit would have a much shorter travel time from transmitter to receiver and be more difficult to measure with the necessary precision and accuracy.”
The travel time for a 20,000km high orbit is about 66ms and for a 3,150km orbit it is about 11 ms. I don’t think this would make any accuracy difference, because GPS receivers measure time to sub-microsecond accuracy.
There must be another reason for the 12 hour orbits, perhaps that each satellite would pass over the same spot twice a day?
SL
A major reason to prefer 12 hour orbits ofer 2.6 is that satellites in a 2.6 hour orbit are much lower, and have a much smaller visible footprint on the ground. You would need to have a lot more satellites in orbit to guarantee multiple satellite visiblity to any particular point on earth. Consider the Iridium satellite telephone system. They are in 1.7 hour orbits, and they need 66 satellites.
Another point about why they correct the clocks for time dilation, and make continuous adjustments instead of periodic time steps, is because GPS is not only a position service; It is also a frequency standard. It is used to transfer time and frequency between clocks on the ground, and is used as a reference to compare them. Therefore, it is vital that the frequency be as stable as possible. The fundamental accuracy of the Cesium and Rubidium clocks used is the limiting factor involved. They steer these clocks in increments of femtoseconds to keep their frequencies locked.
Hi SL, getting back to your other two questions.
You wrote: “2. What difference would it make if there are still corrections due to “inaccurate orbits” required?”
Yep, you’re right. Perhaps a little saving in the production time (cost) of the system, but that’s notmally not an issue with space-borne equipment. They are mostly costly “one-ofs” anyway.
Your: “3. Why bother with the relativistic corrections if you are going to re-sync every day anyway?” is a different matter.
To maintain military precision during the course of day, the clocks must be as accurate as possible. You cannot tolerate tens of microseconds errors. So they must be relativistically corrected upfront.
This is just about the only useful aspect of Einstein’s relativity that the general population experience directly.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Burt wrote: “To maintain military precision during the course of day, the clocks must be as accurate as possible. You cannot tolerate tens of microseconds errors.”
Ah, I see, thanks. Even if synchronized every 12 hours, the uncorrected clocks could have been 19 microsecs out every 12 hours. That translates to, hmmm… 19,000 ft errors (given that light travels about a foot per nanosec). I guess the software could have corrected that as well, but I agree it was better to correct the gross errors upfront.
Coming back to your 2h38m orbits vs. the “harmonically synchronized” 11H58m orbits: I think the fact that the same satellite returns to roughly the same spot above Earth every 23h56min is the main reason for this choice of altitude.
SL
If the time dilations meet in the middle and cancel out.. Is that technically an orbit at which time is no longer relative? I mean if the Earth was a perfect orb, would that exact orbit literally be relativity free?
My real question is, suppose you created a similar orbit in space where there was no planet or gravity well or immeadiate source of gravity. Would we merely see the difference of the red shift? Or would there be a bigger change without gravity pulling down on all of the materials in the ship and clock?
MainFragger
Hi MainFragger, interesting question!
No, I would not say the 1.5R orbit is “relativity free”, but yea, if you sit on Earth and monitor a natural oscillator on the satellite, you will not detect any relativistic time dilation, because the gravitational redshift cancels the velocity time dilation.
You asked: “My real question is, suppose you created a similar orbit in space where there was no planet or gravity well or immediate source of gravity. Would we merely see the difference of the red shift?”
I’m not sure I understand your question, but if you would go and sit stationary inside such a (powered) orbit in free space, there will be no gravitational redshift to cancel out the velocity time dilation and you would detect the velocity time dilation only. The two situation are not equivalent.
In both cases there may be Doppler shift due to relative motion, but that’s not a strictly relativistic effect.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
I guess what I am asking is more about the clocks than anything. What I am trying to figure out is that each force gravity and movement have a relativistic effect on the clocks, but are some of those forces cumalitive? In otherwords.. The red shift and the movement each create relativity in the clocks, but is the effect of both at the same time actually more than the sum of each of the forces?
My even more long term goal is to figure out if time is actually relative, or if the materials we are using to measure time are just subject to laws of gravity and movement, and will make it look like there is relativity, when in fact, there was just forces interacting with the materials of the clock..
Burt, something in connection with MainFragger’s post that puzzles me:
I agree with your ” if you sit on Earth and monitor a natural oscillator on the satellite, you will not detect any relativistic time dilation, because the gravitational redshift cancels the velocity time dilation.”
Now consider an observer in the satellite monitoring an identical terrestrial oscillator. The orbiting observer is entitled to view herself as stationary and that it is Earth spinning with a 2h38m period below her. The terrestrial oscillator is deeper into Earth’s gravitational well and hence redshifted (=lower frequency). At the same time, it is moving in the observer’s inertial frame and is time dilated (=lower frequency). Here both effects work in the same direction, hence the relativistic effects do not cancel!
Is this another way of proving that the scenario cannot be “relativity free”?
SL
Hi SL. You wrote:
“Here both effects work in the same direction, hence the relativistic effects do not cancel! Is this another way of proving that the scenario cannot be “relativity free”?”
Nope, I don’t think so. The ‘reversibility’ that you described is a purely special relativity issue and cannot be applied when there is gravity around. The orbiting clock suffers velocity time dilation relative to the ground, never the other way round.
The orbiting observer would have seen the ground clock redshifted (running slow), but her own velocity time dilation cancels that at 1.5 Earth radii. This velocity time dilation is observed on the ISS, which is far lower than 1.5R, where the gravitational redshift does not cancel it.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
OK, I follow.
Burt mentioned Doppler shift before. “In both cases there may be Doppler shift due to relative motion, but that’s not a strictly relativistic effect.”
The GPS satellites are generally moving fast relative to GPS receivers on the ground. Are corrections for the Doppler shift built into the receiver software? Or doesn’t it play a role?
SL
Hi SL. You asked about Doppler shift and GPS.
AFAIK, it does not play a role, because the GPS transmitters send a time stamped signal and the receivers simply subtract that time from their reception time to get distance. Hence Doppler shifts should not matter.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Hi MainFragger.
You wrote: “What I am trying to figure out is that each force gravity and movement have a relativistic effect on the clocks, but are some of those forces cumulative?”
The gravitational field and the velocity each influence the rate of clocks and clock rates obviously have a cumulative effect on the time that the clocks record or display.
“My even more long term goal is to figure out if time is actually relative, or if the materials we are using to measure time are just subject to laws of gravity and movement, and will make it look like there is relativity, …”
Time is very relative. Every inertial frame and every frame sitting inside a gravitational field has its own perception of time.
Forces interacting with atomic clocks do not change their time keeping. This has been shown by subjecting atomic clocks to tens of g’s in centrifuges. Gravity does, but then gravity is not a force…
Hope this helps.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Hi MainFragger. You wrote: “If those pressure forces don’t affect the clocks, why does the height of the clock become an issue at all?”
As you speculate on later, the crux is that gravity is not the same thing as a force. Gravity is caused by curved spacetime and curved spacetime also affect clocks. The ‘deeper’ a clock is in a ‘gravitational well’, the slower it ticks, compared to a clock that is not so deep in the ‘well’.
“What effect does the dark energy and dark matter have on all of this. ”
Not much – dark matter ‘deepens’ the gravitational ‘wells’, because it also clumps together like ordinary matter. Dark energy on the other hand is thought to be evenly distributed throughout the universe and hence does not influence time from place to place.
“Maybe its not time that is relative, but space. ”
Both are relative! What we perceive as space depends on our frame of reference. Observers in other frames of reference perceive both time and space differently than us.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
I don’t get it. If those pressure forces don’t affect the clocks, why does the height of the clock become an issue at all?
I wonder if its because gravity is different. After all, if a shell is strong enough, you can put great pressure on it, and anyone inside the shell wouldn’t feel a thing.
On the other hand, gravity seems to exist through structures of every kind that we know of. Otherwise, you’d become weightless just by stepping into a house. So it would be in that same shell. No matter how strong we make it, we are still pulled to the floor by gravity.
What effect does the dark energy and dark matter have on all of this. If the universe is expanding at the urging of dark energy, and dark matter is always tyring to fill the empty space in our stretchable universe, isn’t that a factor in how relative time would seem to? Maybe its not time that is relative, but space. Maybe with space constantly stretching (albeing in extremely small, but always accelerating increments) it just takes longer to reach from point A to B than we’d expect, because space literally stretches imperceptably during the trip.
MainFragger wrote: “What I started to wonder about is how the relativity of time is affected by near or overlapping gravity wells.”
The depth of a gravitational well scales approximately with GM/R, where G is Newton’s gravitational constant and R the scalar distance from the center of a mass M. When more than one mass are involved, you can add up the GM/R’s for each and know the depth of the well at any point. The gravitational time dilation is determined by the depth of the gravity well, while gravitational acceleration is determined by the slope of the well at that point.
I do not think the rotation of planets is determined by the gravity wells, but more by their formation history and collisions that happened.
You also wrote: “Also, I am curious as to whether the size of the 1.5 orbit canceling out the speed and distance dilations is consistent with larger planets and/or moons.. Or does it just happen to be 1.5 with Earth.”
For any spherically symmetrical and roughly uniform density object, the 1.5 radii is where the velocity and gravitational time dilations cancel, relative to the surface.
“I also wonder if you took a much larger planet than earth, and ten drilled into it just enough to get to the surface of what would be the earth and put a clock there.. And then put a shuttle in orbit around that planet at the same height as previous relativity tests on earth…”
No, this will create a new scenario that is not equivalent to Earth. The gravity and time dilation inside a planet follows different rules than on the outside, hence the 1.5 radius cancellation rule only works relative to the surface of a planet.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
I am sorry its taken some time to write back about this. I have been busy, and its also taken me time to do some brushing up and thinking about time dilations, and what you said about gravity wells being curved time/space.
What I started to wonder about is how the relativity of time is affected by near or overlapping gravity wells.
For example, the Milkyway is part of a gravity well, our solar system is part of a smaller gravity well on that gravity well, the each planet has a gravity well, and each moon has a gravity well. And I purposely list it that way for heirarchal purposes, because I picture each of the smaller gravity wells as being stacked on the other larger ones. So technically, a moon is in 4 gravity wells at the same time.
To make matters worse, the smaller gravity wells move around the larger ones. And not all (if any) of the orbits are perfect circles. Which means that one well may have a stronger or weaker connection to another well according to position.
Maybe thats also part of why some planets rotate in reverse. Another opposing gravity field… Orrr.. The mass of the planet causes back spin instead of forward spin. If the mass of the planet barely surpasses the coefficient of gravitation friction..maybe you get a skittering effect that actually culminates in the planet spinning the wrong way..
Also, I am curious as to whether the size of the 1.5 orbit cancelling out the speed and distance dilations is consistant with larger planets and/or moons.. Or does it just happen to be 1.5 with Earth.
I also wonder if you took a much larger planet than earth, and ten drilled into it just enough to get to the surface of what would be the earth and put a clock there.. And then put a shuttle in orbit around that planet at the same height as previous relativity tests on earthn… Would the earth calculations of doppler shift still match up? Would the 1.5 orbit still work?
So the 1.5 dilation is a constant.. Interesting. This makes me wonder if the 1.5 dilation is the edge of where the curvature in space becomes extreme, or if it is at the event horizon of the curve. I am partially wondering about that becuase I had heard that the penny wells that you see at the mall appear perfectly circular, but are actually purposely slightly mishapen. Because if it was perfectly circular, there is a chance that some coins might hit a sweet spot where they can spin for an extremely long time (I mean hours) before they actually slide down the well. I’m wondering if in planetary terms, that 1.5 dilation is that sweet spot..
Is there a vast or minor difference in gravity of say.. The Moon, when it is on the side of the earth farthest out from the Sun at the Earths farthest portion of its orbit from the sun, and when the earth is closest to the sun in its orbit with the moon on the side of the earth closest to the sun?
MainFragger
MainFrageers wrote: “This makes me wonder if the 1.5 dilation is the edge of where the curvature in space becomes extreme, or if it is at the event horizon of the curve.”
No, no, 1.5 times the radius of the planet is simply where the two time dilations cancel out from the perspective of a static observer on the surface of a planet.
It has nothing to do with extreme curvature, event horizons and/or that “sweet spot”. You can read some discussion and the equations in this Blog: http://cr4.globalspec.com/blogentry/3754
You asked: “Is there a vast or minor difference in gravity of say.. The Moon, when it is on the side of the earth farthest out from the Sun at the Earths farthest portion of its orbit from the sun, and when the earth is closest to the sun in its orbit with the moon on the side of the earth closest to the sun”
Due to the large distance from the Sun, compared to the Moon’s orbital diameter, there is very little difference in the Sun’s gravitational pull in such cases. Interestingly enough, when the Moon is on the far side of the Earth relative to the Sun, the Moon ‘falls’ fastest towards the Sun. This is because it then ‘feels’ the combined gravity (spacetime curvature) of the Sun and the Earth. It is the opposite when on the side nearest to the Sun, where the Earth’s gravity subtracts from the Sun’s.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Now consider an observer in the satellite monitoring an identical terrestrial oscillator. The orbiting observer is entitled to view herself as stationary [. . .] Here both effects work in the same direction, hence the relativistic effects do not cancel!
If the effects cancel in one frame they have to cancel in the other. The key here is that the assumption that the orbiting viewer is in an inertial frame is incorrect — it’s a rotating frame, and with an orbital period of 2h38m, and is traveling faster than an observer on the earth, relative to an inertial observer. The orbiting observer will see the earthbound clock as running faster; one must use an inertial observer to reconcile this. The orbital observer is moving faster than the earthbound observer, so the earthbound clock runs faster due to the kinematic term. This will cancel the gravitational term