Re: I Owe Einstein an Apology. Sorry Albert!

From: Paul B. Andersen (paul.b.andersen_at_deletethishia.no)
Date: 12/13/04 

Date: Mon, 13 Dec 2004 15:28:17 +0100

Excuse me for butting in, but I find so much
confusion about the GPS in this thread that I
feel the need for clearing up a few misconceptions.

jahn wrote:
> "The Ghost In The Machine" wrote:
>>jahn wrote:
>>>"The Ghost In The Machine" wrote:
>>>> jahn wrote:
>>>>>"The Ghost In The Machine" wrote:
>>>>>>If the clocks are moving then SR predicts they will not remain
>>>>>>synchronized.

Clocks moving in a frame of reference will not
generally stay synchronous in that frame,
but they may do so in special cases.
For example, clocks moving in a circle with
the same speed will stay synchronous to each
other in the "stationary frame". They will
run slow compared to stationary clocks,
but they will stay in sync.
This is the case in the GPS.
All the GPS satellite clocks stay in sync
to each other in the ECI frame because
they all move in circular orbits with the
same speed and at the same gravitational potential.

>>>>>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".

>>>>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. :-)

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_.

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.

>>>>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.

[..]
>>>>(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

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.

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)

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.

>>>It must take a REAL good mathematician to get 7 microseconds
>>>per day out of that. ;-)

.. and a real bad physicist to try. :-)

>>I get 38.53 microseconds per day out of the 4.46*10^-10 corrective factor.
>>It's a simple enough multiplication:
>>
>>4.46*10^-10 sec/sec * 86400 sec/day

Right.

> The altitude effect isn't cumulative.

Utter nonsense.

> That is, you can move
> from Honolulu to Tibet and make one adjustment to the
> length of a pendulum. You don't have to do it every day.

But the error is still cumulative, isn't it? :-)

> The 7 ns per day is for transverse doppler effect (less
> frequently called the SR correction )

3 orders of magnitude wrong.
(But we will call this a typo.)

> It was included in
> the preset because it does save the receiver some calculating
> (there may have been some political reason to have all
> the "relativistic" effects in one synthezer too [shrug] )
> Anyway the 7ns error would accumulate if it represented
> a clock rather than a propagtaion effect.

Of course a rate error of 7 us/day would accumulate 7 us a day.

> I think you can tell at a glance that +/-1.0E-19 steering
> and +/- 50 ns / daily won't on the worst day add up to
> the 7ns necessary to atrribute to an SR clock anomaly.

This confusion is too gigantic to try to remedy. :-)
You really have no clue whatsoever, have you jahn Sue?

> It's all public information.
> http://tycho.usno.navy.mil/gps_datafiles.html
> http://tycho.usno.navy.mil/systime.html

But you have no idea of what they mean?

Paul