Re: I Owe Einstein an Apology. Sorry Albert!

From: Henri Wilson (H_at_..(Henri)
Date: 12/16/04 

Date: Thu, 16 Dec 2004 20:09:44 GMT

On Wed, 15 Dec 2004 14:16:10 +0100, "Paul B. Andersen"
<paul.b.andersen@deletethishia.no> wrote:

>The Ghost In The Machine wrote:
>> In sci.physics.relativity, Paul B. Andersen

>
>Indeed.
>Which make it rather complicated to calculate
>the gravitational potential of the orbit exactly.
>And then there is the Sun and the Moon..
>
>But the main reason why the satellite clocks have to
>be corrected is however the finite precision of the clocks.
>The clock drift per day will normally be bigger than the error
>of the GR correction. Thus the limiting factor is the precision
>of the clocks, not the precision of the GR correction.
>It is correct within the precision of the clocks.

HaHaHa!
The truth has finally emerged.

You really don't have any evidence that the clock rate shift in free fall IS
the same as the GR prediction.

>
>>>>>>>>SR predicts moving clocks can't keep good time?
>>>>>>>>I have two garden hoses an egg timer and a bag
>>>>>>>>of marbles that says they must be broken.
>>>>>>>>http://www.boulder.nist.gov/timefreq/time/commonviewgps.htm
>>>>>>>
>>>>>>>The SR correction is t' = (t-vx/c^2) * gamma .
>>>>>>
>>>>>>>The vx/c^2 is merely a reflection that the two clocks are
>>>>>>>communicating through speed-of-light (e.g., radio), but
>>>>>>>the gamma is the killer; gamma = 1/sqrt(1-v^2/c^2).
>>>
>>>The vx/c^2 is a reflection of the simultaneity of relativity.
>>>It has nothing to do with "communication through speed of light".
>>
>>
>> Hmm...well, if two clocks are spatially separated in an idealized
>> Galilean space, and moving with a velocity v, then the first
>> clock will see the second clock at position x, but the first
>> clock has an observation delay of x/c and the second clock will
>> have moved vx/c.
>>
>> All these are of course crude first-order approximations.
>
>Moving relative to what?
>You mean something like:
>Given two clocks moving at v in an inertial frame.
>The clocks are separated by the distance x measured
>in this frame.
>If a light beam is sent from one clock to
>the other, then the trinsit times measured in this frame
>will be x/(c+v) = ca. c/x - vx/c^2 ?
>
>OK.
>But the equation t' = gamma(t - vx/c^2)
>is a coordinate transformation which has nothing
>with "communication through the speed of light"
>to do.
>
>>>>>>>It's a very slow killer, of course; the typical speed
>>>>>>>of a spacecraft is on the order of 9 km/s = 3 * 10^-5 c, which
>>>>>>>results in a gamma of about 4.5 * 10^-10. The GPS delta is
>>>>>>>almost exactly this: 4.46 * 10^-10. However, this is at
>>>>>>>best a very very rough estimate, just to give one the feel.
>>>>>>>It's also the wrong sign. :-)
>>>
>>>Which should tell you something. :-)
>>
>>
>> Yeah, it tells me I've not studied Hafele & Keating and/or
>> Old Man's equations enough yet. :-)
>
>I was rather thinking it should tell you that
>your "speed term" 4.5*10^-10 is very wrong.
>The orbital speed of the satellites isn't 9 km/s,
>it is 3.87 km/s.
>
>>>It is actually quite simple to make a first order
>>>calculation of the rate of the GPS satellites.
>>>
>>>The relative difference in the rate of a clock in circular orbit
>>>compared to a clock on the surface of the Earth is according to GR
>>>to a first order approximation:
>>>(Approximation of the Schwarzchild solution)
>>>
>>>(f2 - f1)/f1 =
>>> (G*M/(c^2*r1) - G*M/(c^2*r2)) - (0.5*v2^2/c^2 - 0.5*v1^2/c^2)
>>>
>>>Where G = gravitational constant, M = mass of the Earth,
>>>r1 = radius of the Earth, r2 = radius of the orbiting clock's orbit,
>>>v1 = speed of the Earth clock in ECI frame,
>>>v2 = speed of the orbiting clock in ECI frame
>>>
>>>Since we have G*M/r1^2 = g, acceleration at Earth's surface, we have:
>>>(G*M/(c^2*r1) - G*M/(c^2*r2)) = (g/c^2)*r1*(1-r1/r2)
>>>
>>>Altitude of GPS satellites = 20200 km
>>>Orbital period = half sidereal day
>>>Radius of the Earth r1 = 6.37*10^6 m
>>>Radius of GPS orbit r2 = 26.57*10^6 m
>>>g = 9.81 m/s^2
>>>
>>>Inserting these numbers, we find that the rate difference
>>>due to gravitation is: 5.28*10^-10 (+45.6 us/day)
>>>
>>>So to the speed part:
>>>v1 = 40000km/(23h 56m) = 4*10^7/86160 = 464 m/s
>>>v2 = 2*pi*r2/(11h 28m)= 3.87*10^3 m/s (-7.1 us/day)
>>>
>>>0.5*v2^2/c^2 = 0.83*10^-10
>>>0.5*v1^2/c^2 = 1.2*10^-12
>>>Thus the rate difference due to the speed will be: -0.82*10^-10
>>>
>>>The combined rate difference: (5.28-0.82)*10^-10 = 4.46*10^-10
>>>Note that the orbiting clock runs _fast_.
>>
>>
>> Right. Nice piece of work. :-) Saves me a wee bit o' trouble.
>
>But note that this calculation is rather crude.
>I will estimate the precision to be something like +/- 0.01*10^-10.
>To get the answer with 5 significant digits, a much more
>elaborate calculation will be necessary.
>Specifically, the oblatness of the Earth make it difficult
>to find the gravitational potential correctly.
>See:
>http://relativity.livingreviews.org/Articles/lrr-2003-1/index.html
>
>>
>>>During one day, the difference in proper times will amount to:
>>>4.46*10^-10*86400 s = 38.5*10^-6 s = 38.5 us
>>>
>>>According to:
>>>http://vishnu.nirvana.phys.psu.edu/mog/mog9/node9.html
>>>the factor used in the GPS satellites is 4.4647*10^-10.
>
>That pointer is outdated.
>Should be:
>http://www.phys.lsu.edu/mog/mog9/node9.html
>
>>>>>>>An accurate clock left running for a year in a GPS orbit
>>>>>>>will gain about a hundredth of a second.
>>>
>>>You mean a "normal clock" not slowed down
>>>by the factor 4.4647*10^-10?
>>>Yes. It would gain ca. 14 ms a year.
>>
>>
>> Correct. Of course, there are some issues regarding the actual
>> clock manufacture; "fountain clocks" don't work in space, AIUI.
>>
>>
>>>[..]
>>>
>>>>>>>(SR predicts loss
>>>>>>>of time, but then SR requires straight-line freespace
>>>>>>>travel. :-) ). Therefore, GPS clocks are "broken" by
>>>>>>>design, coding in this adjustment factor -- and even
>>>>>>>then, they have to be steered from the ground using
>>>>>>>synchronization signals from the TAI.
>>>
>>>1. "GPS time" is a coordinated time where the coordinate
>>> system in question is stationary in the ECI-frame.
>>> The coordinate time is per definition such that
>>> clocks on the geoid will stay in sync (run at the same
>>> rate) with this coordinate time.
>>> "GPS time" is a theoretical time, derived from
>>> all the clocks in the system, that is all the satellite
>>> clocks and all the ground station clocks.
>>>
>>>2. This "GPS time" is steered so that it (but for a number
>>> of whole seconds) is equal to UTC. The spec says it should
>>> be within 1 us, but in actual practice, it differs but
>>> few ns. This difference is known by the system, and each
>>> satellite will transmit the difference GPS-time - UTC
>>> so that a receiver can calculate the correct UTC.
>>>ftp://tycho.usno.navy.mil/pub/gps/utcgps30.dat
>>
>>
>> Another poster mentioned a delta of 50 ns.
>
>Yes, that's taken from here:
>ftp://tycho.usno.navy.mil/pub/gps/gpstt.txt
>" GPS time is automatically steered to UTC(USNO)
> on a daily basis to keep system time within one
> microsecond of UTC(USNO), but during the last
> several years has been within 50 nanoseconds."
>
>But you can see from the actual data:
>ftp://tycho.usno.navy.mil/pub/gps/utcgps200.dat
>that the current difference is 6 ns, and that
>it has been less than 20 ns during the last 180 days.
>
>> That translates into
>> an accuracy of about 7.5 meters (if one assumes +25 / -25).
>
>No, no!
>This is one of the misconceptions I tried to clear up!
>The difference GPS time - UTC does not affect the precision,
>that's why it can be allowed to be as big as 1 us.
>
>The receivers use GPS-time only in the calculation of
>the position, all times are in GPS-time.
>What determines the precision is how much the time
>of the individual satellites deviates from GPS-time.
>See point 4. below.
>
>The difference GPS time - UTC is only of interest when
>the receiver shall calculate the UTC-time. But since
>the satellite knows and transmits the difference,
>it does not really matter what it is, the receiver
>can calculate it correctly anyway.
>(It is still the error in the four used satellites
> that limits the precision.)
>
>BTW, in addition to the few ns difference we are talking
>about here, there are a difference of 13 whole seconds.
>That's because leap seconds are not inserted in GPS-time
>like it is in UTC. (And 13 leap seconds are inserted
>in UTC since GPS-time was defined.)
>
>>>3. The GPS satellite clock are built to run slow
>>> (compared to a clock using the SI definition of a second)
>>> by the factor 4.4647*10^-10 prior to launch.
>>> In orbit, they will thus run synchronously to GPS-time.
>>
>>
>> Like I said, "broken" -- but only because they have to run slow
>> up in space to compensate for the SR+GR effects. There's not
>> a lot one can do about it, except, erm, "break" them. :-)
>>
>>
>>>4. The precision of the clocks is in the order of 10^-13.
>>> That means that they during one day may drift off sync
>>> by several ns - or even tens of ns. The most that can
>>> be tolerated is a few tens of ns.
>>> (Less than 100ns at the very most)
>
>In fact it is the difference in the errors of the four satellites
>that matters for the precision of the position.
>If all four clocks are off sync with GPS-time by the same amount,
>then it will not matter because the satellites still would be
>in sync with each other, and the receiver would not know that
>that the time reported by the satellites are not correct.
>It would still calculate the distances to the satellites correctly.
>But if the satellites are off sync by different amounts, then
>they are not in sync with each other, and the distance to the
>satellites will be calculated wrongly.
>
>>>
>>>5. To correct for the latter, ground stations are monitoring
>>> the satellite clocks, and are uploading correctional data
>>> typically once a day. The "clock offset" is simply how
>>> much the clock is off sync. For some satellites, this may
>>> be up to milliseconds. The satelite clocks are not corrected,
>>> but the "clock offset" is transmitted together with the
>>> "clock time", and the receiver calculates the corrected time.
>>>
>>>
>>>>>>Indeed:
>>>>>><< GPS TIME STEERING
>>>>>>=================
>>>>>>GPS time is automatically steered to UTC(USNO) on a daily basis to keep
>>>>>>system time within one microsecond of UTC(USNO), but during the last
>>>>>>several years has been within 50 nanoseconds. The rate of steer being
>>>>>>applied is +/-1.0E-19 seconds per second squared.
>>>>>>ftp://tycho.usno.navy.mil/pub/gps/gpstt.txt
>>>
>>>Right.
>>>But note that as stated above. this is steering of
>>>the GPS-time, not of the individual satellites.
>>>You have two different coordinated time systems,
>>>each "living its own life" which should be made
>>>to run equal. The +/- 1.0E-19 seconds per second squared
>>>is the maximun "speed" with which the GPS time is changed.
>>
>>
>> My brain's beginning to hurt admittedly, but I know that TAI
>> has some interesting issues (Boulder runs fast); presumably
>> GPS does, too.
>
>You can see from:
>ftp://tycho.usno.navy.mil/pub/gps/utcgps200.dat
>that GPS time sometimes run slow, sometimes run fast
>compared to UTC (or TAI).
>(Look at the slope).
>
>You will find a lot of data about the GPS at:
>http://tycho.usno.navy.mil/gps_datafiles.html
>
>Paul

HW.

www.users.bigpond.com/hewn/index.htm

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