Relativity and the GPS
30 Jun 2005 8:11:16 UTC
Topic 189451
(moderation:
Nowadays the best evidence for the superiority of GR over Newton comes from the GPS.
GPS works with very accurate clocks, and you know where you are because your GPS receiver works out the difference between the times encoded into signals from four satellites. From the time differences it figures out where you are in three dimensions.
To do this it has to know where the satellite is, so it needs to know the staellite's orbit, and it also needs to know the time the signal left each satellite.
Now, obviously you put a clock on the satellite to do this. Trouble is, Einstein says moving clocks run slow.
If the satellite is moving, you speed up its clock to compensate for special relativity, and the time dilation effect slows it down again, and it all comes right.
Wrong. Turns out to give a result that is even more wrong that simply believing Newton.
Try again. Special Relativity doesn't work in this case, try General Relativity.
GR also says moving clocks run slow, but it disagrees with SR about which clock is the moving clock. From the viewpoint of GR, the stationary clock is the one in free fall, and the moving clock is the one that feels the pull of gravity.
The clock in the ground receiver is running slow (according to GR) as is every clock on the Earth's surface. Set the satellite clock to run slightly slow and things will be fine.
Turns out that GR gives the right answers. Either Newton or SR would produce errors of kilometers well within the first day.
Doesn't "prove" that GR is right - but it does mean that if GR is wrong the correct theory must be closer to GR than it is to Newton or to SR.
~~gravywavy
Relativity and the GPS
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If the arena for GR is 'curved spacetime', and the arena for SR is 'flat, Minkowski spacetime', what arena does the entire GPS operate in?
Both SR and GR are 'frameworks' for Classical Physics, which is to say that the momentum of an object is always a product of the mass times the velocity – quite Newtonian...
RE: If the arena for GR is
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If GR is right, there is no such force as gravity. Curved space does not just make the orbit precess, it makes the orbit happen at all. Without curved spacetime we'd fly off in a straight line right away.
For convenience, we tend to do the sums according to Newton and Kepler, and then add in a small correction ("perturbation") to account for the difference between Newton and GR. A GR purist would say that we are using Newton's equations as a convenient approximation that holds when spacetime is only slightly curved, and that we are still working in curved spacetime.
A mathematician might say that if s/he is doing the sums in a flat mathematical space then they are working in a flat space.
It is a matter of taste which you prefer.
~~gravywavy
RE: the momentum of an
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This is a subject of philosophical debate amongst relativists.
The relativistc expression for momentum is
p = gamma m0 v
where p is momentum, m0 is the invariant mass (or rest mass or proper mass), and v is the observed velocity
Some say that there is an "observed mass" or "relativistic mass" m where
m = gamma m0
and then of course p = m v
Others (including me) are not comfortable with the idea of an observed mass, and say that the idea leads to confusion in other areas of applied relativity. As you say it is "very Newtonian", which makes the first steps of learning SR easier. The downside is that it leads the learner into making other Newtonian assumptions that are *not* valid. The relativity course I teach is very strict about this, my colleagues and I never talk about relativistic mass or observed mass. We only mention the invariant mass and usually simply call it the mass.
The algebra is the same in either case, it is a question of how we intepret the algebra in our heads. Again a matter of taste how we talk about the maths we do.
A huge part of any theory is not the maths, but how we think about the maths. A way of thinking about the maths that helps us 'gravitate' towards the right assumptions is one that will lead to the better future development of the subject. Everyone agrees on that, we disagree amongst ourselves about which way of thinking comes out top in that test.
Thank God us scientists don't burn each other at the stake when we hear heresies.
~~gravywavy
GravyWavy -- Interesting
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GravyWavy --
Interesting discussion, and I took the liberty of reviewing the course outline you you referred to ... too bad I can't apply for it here in the USA. Maybe by 2007 that will change.
If I've lived this long - I gotta be that old!
RE: Others (including me)
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What happens when mass is replaced in the equations by its equivalent energy (using m = E/c^2, and setting c=1 to use “geometrized units”) in those 'other areas of applied relativity'?
What are some Newtonian assumptions that turn out to be not valid (I am a learner)? And is it a (greater) underlying principle that invalidates the assumptions, or is it experimental observations?
I thought that if one wanted to do calculations, then coordinate bases should be used, and if one is interested in physical interpretations (how we 'think about the maths') then orthonormal bases should be used. Re: the implementation of GPS (or LIGO), we have to use both, right?
I would certainly take your “Space, Time and Cosmology” course if it should be made available to persons with few pounds, little time, and who live in the USA...
RE: What happens when mass
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(m c)^2 = ( E / c )^2 - p^2
where m is rest mass, E energy, p momentum
In c=1 units,
m^2 = E ^2 - p^2
These are invarioant equations: the thing on the left is the same for all observers, whereas people moving differently will observe differing values for energy and momentum. But any pair of those values measured by the same observer can be used to derive the invariant mass.
There are several other invariants. The most useful of all is that if the distance between two events is x and the time delay between them is t, then the value s is invariant where
s^2 = (t/c)^2 - x^2
even tho different observers have differing observations about the time interval and the spacing, all will agree on the value of s.
Interstingly, all the invariant equations are in the form of something^2 minus something^2, and in every case the first something is the square of a scalar, and the second something is always the square of the modulus of a vector. Physicists find such patterns interesting. Students like them too bacuase it makes remembering the equations easier.
Newton assumed:
1. if things happen at the same time for me, then you agree that they happended at the same time
2. if I measure the length of an object, then someone moving past will measure it to be the same length
3. that there exist inertial frames, that is frames of reference in which his first law holds
4. that 3 is exactly true, and that all inertial frames can in principle be extended out to infinity
5. that the laws of mechanics are the same in all inertial frames
And borrowing from Euclid, he assumed
6. parallel lines never meet, all non parallel straight lines in a single plane will meet if extended far enoug
7. a point has no size and a straight line has no thickness
Special Relativity drops 1 and 2, and keeps the rest. It adds to assumption 5, now saying that all laws of physics (including things like electromagnetism) are the same in all inertial frames.
I find it interesting that SR deepens one assumption at the same time as challenging others.
General Relativity drops 1, 2, 4 and 6 and shares the SR extension to 5.
GR also changes the meaning of 3 from an exact claim that reaches out to infinity, and makes it an approximate claim that is only valid locally. Imagine a farmer assuming his field is flat to work out its area, when in fact he knows he lives on a non-flat Earth. If the field was big enough, the size would be wrong cos the roundness of the Earth adds a bulge in the middle. In the same way, an inertial frame is only valid for a limited volume of space and only for a limited duration. Then you need to swap to a new inertial frame.
Dropping 6 also creates some stange results: in different cases you can get lines that are not parallel that still don't meet; or lines that are parallel that don't!
String theory drops 7 as well, and so do other modern theories that try to combine GR with quantum effect. Those theories claim there is a smallest possible thickness of a physical line or point. Almost everything they tell you at school is doubtful....
OK folks: that is as far as I want to go teaching relativity here. Thanks for the questions, I've enjoyed answering them but can't spend the rest of my life doing this unpaid!
Should I write an eBook about this stuff? If I did, how much would you pay to download it (in UK pounds, US dollars, or EU Euros?)
If you live in the European Economic Area you could sign up for our course next time it comes around (sign up mid 2006, course next starts Feb 2007). I warn you, if you live outside the UK it is not cheap!
Sadly, our govt won't fund financial support for people living outside the EU, and your govt won't fund undergraduate study at a Uni outside the US. Even if you were rich and wanted to study it, our funding people won't allow us to offer it in your country, not even at full economic cost. The Higher Education Funding Council for England and Wales paid for the courses to be created and they can't be offered to those outside the Eoripoean Economic Area (which is the EU plus a few more countries). Doesn't politics stink!
I was whingeing about this to my boss not six hours ago, and he agrees totally but there's nothing he can do either.
~~gravywavy
RE: 5. that the laws of
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To be sure, but what affect (if any) do things like inflation, accelerating expansion, dark energy, and dark matter have regarding the “depth” of the assumption?
A recent book purchase of mine was “What is Quantum Mechanics?” (ISBN 0-9643504-1-6, soft cover, 566 pages) and it cost ~ (US)$30. – I'd pay more than that for an eBook of yours. (But considering the first rate job you did with my questions, perhaps you might also consider offering your services as a tutor/consultant?)
RE: A recent book purchase
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That is encouraging... I will think seriously about doing that.
and that... maybe around UK£30/hour in my home area (Manchester England), *lots* more away from home.
~~gravywavy
RE: If GR is right, there
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I was able to reach the same conclusion, but not by using GR. It was a line of reasoning that started out something like, “Energy and spacetime don't mix. There are apparent interactions between the two, especially with energy in motion. Energy displaces spacetime...” (This was contemplating the nature of fundamental particles in flat space, a few years back, after reading most of Feynman's books.)
But I sort of decided that just because one can paint a strange picture of an interaction that results in preservation of the laws of motion (including GR), and yields the same math for 'gravitational attraction', doesn't mean that's how things really are. (Not to mention that it seemed rather like heresy to say, “Mass isn't attracted to mass – it's repulsed by space.”)
Now, however, I wonder what other explanations could there be?
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