Groups > sci.physics.relativity > #674401

Re: GPS clocks should run Slow Not Fast.

From	"Paul B. Andersen" <paul.b.andersen@paulba.no>
Newsgroups	sci.physics.relativity
Subject	Re: GPS clocks should run Slow Not Fast.
Date	2026-10-13 22:32 +0200
Organization	A noiseless patient Spider
Message-ID	<11am4e6$stm6$1@dont-email.me> (permalink)
References	<18dd9f33b63d4fa4$584$2574533$4006de53@news.newsgroupdirect.com> <11ajabc$3ss4t$1@dont-email.me> <11akq10$2mvnq$1@gwaiyur.mb-net.net>

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Den 13.10.2026 10:27, skrev Thomas 'PointedEars' Lahn:
> Paul B. Andersen wrote:
>>
>> The gravitational blue shift of the SV signal is Δf/f =  5.2996e-10
>> The kinetic red shift (due to motion) is         Δf/f = -0.8349e-10
>> The total(gravitational + kinetic) blue shift is Δf/f =  4.4647e-10
> 
> Note that this is just a crude, at best first-order, approximation.

It is a very precise approximation. The error is less than 1e-15.

> To obtain the exact values according to GR, you cannot simply add
> "gravitational" and kinetic "time dilation" or (here) frequency shifts;
> you have to use the full (Schwarzschild) spacetime metric (at least).

See:
https://paulba.no/pdf/Clock_rate.pdf
Equation (18) can be written:
  Δf/f₀ = δ_utc - GM/(c²r) - GM/(2c²r)

Gravitational blueshift Δf/f₀ = (δ_utc - GM/(c²r)) =  5.299580e-10
Kinetic redshift        Δf/f₀ = - GM/(2c²r)        = -0.834852e-10
Total(gravitational + kinetic) blue shift:         =  4.464728e-10

Now you can wonder why such a simple equation can give so precise result.
> Note also that not all GPS satellites have the same orbital altitude or
> orbital direction (and the orbits are only approximately circular), so
> both the "gravitational blueshift" and the "kinetic redshift" varies
> between satellites and between locations on the ground (they also vary
> at least with latitude due to different rotation speeds).

Don't be ridiculous.
The blue shift doesn't depend on the location on the ground.
The speed and orbital radius of all the satellites are equal
so the blue shift of the signal from all the satellites
is equal.

We calculate the blue shift from the assumption that all
satellites are in their nominal orbit.
Six orbital planes 60 degrees from each other, four satellites
in each plane, planes 55 degrees inclination to equatorial plane.

> 
> The infinitesimal elapsed proper time in central-circular motion in the
> Schwarzschild spacetime is given _exactly_ by
> 
>      ds² = −dτ² = −(1 − 2m∕r) dt² + (1 − 2m∕r)⁻¹ dr² + r² dΩ²,
>        m = G M∕c²,
>      dΩ² = dθ² + sin²(θ) dφ²,
> 
>    ⇔ dτ² = (1 − 2m∕r) dt² − (1 − 2m∕r)⁻¹ dr² − r² dΩ²
>          = dt² [1 − 2m∕r − (1 − 2m∕r)⁻¹ (dr/dt)²
>                   − r² (dθ/dt)² − r² sin²(θ) (dφ/dt)²]
> 
>    dr/dt = 0
>    ⇔ dτ  = dt √[1 − 2m∕r − r² {(dθ/dt)² + sin²(θ[t]) (dφ/dt)²}].
> 
> To calculate the elapsed proper time Δτ = ∫_W dτ along a worldline
> segment W becomes a little difficult without making simplifying
> assumptions such as a constant polar angle (dθ/dt = 0) or an equatorial
> orbit (θ = π∕2).
> 
> In any case, you should substitute the orbital parameters in an ECI
> frame (in the case of a GPS satellite), and the components of the
> rotation velocity of a point on the terrestrial surface in the same
> frame, respectively.
> 
> The exact blueshift (according to GR) follows from that, as I derived in
> <mid:119a9s8$182fs$2@gwaiyur.mb-net.net>.

So is your very exact blue shift different from 4.4647e-10 ? :-D


-- 
Paul

https://paulba.no/

Thread

GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-12 00:01 +0000
  Re: GPS clocks should run Slow Not Fast. Emanuele Babuh <uh@elbbem.ru> - 2026-10-12 07:40 +0000
  Re: GPS clocks should run Slow Not Fast. "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-10-12 20:55 +0200
    Re: GPS clocks should run Slow Not Fast. Maciej Woźniak <mlwozniak@wp.pl> - 2026-10-13 07:30 +0200
      Re: GPS clocks should run Slow Not Fast. Clancy Balanovski <av@sysal.ru> - 2026-10-13 07:07 +0000
    Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-13 10:27 +0200
      Re: GPS clocks should run Slow Not Fast. "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-10-13 22:32 +0200
        Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-14 14:20 +0200
          Re: GPS clocks should run Slow Not Fast. "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-10-14 21:08 +0200
            Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-14 23:08 +0200
              Re: GPS clocks should run Slow Not Fast. "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-10-15 23:10 +0200
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-15 22:05 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-18 16:54 -0700
                Re: GPS clocks should run Slow Not Fast. Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-10-19 03:09 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-19 09:50 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-19 23:20 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-21 12:25 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-22 11:34 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-22 12:28 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-22 14:30 -0700
                Re: GPS clocks should run Slow Not Fast. Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-10-22 19:46 -0700
                Re: GPS clocks should run Slow Not Fast. The Starmaker <starmaker@ix.netcom.com> - 2026-10-22 22:14 -0700
    Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-13 15:20 +0000
      Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-14 14:12 +0200
        Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-16 10:36 +0000
          Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-19 03:28 +0200
      Re: GPS clocks should run Slow Not Fast. "Paul B. Andersen" <paul.b.andersen@paulba.no> - 2026-10-14 21:07 +0200
        Re: GPS clocks should run Slow Not Fast. Wildo Zhmulev <vedlml@hlhiudlo.ru> - 2026-10-15 07:52 +0000
        Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-16 10:39 +0000
  Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-13 09:42 +0200
    Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-13 15:54 +0000
      Re: GPS clocks should run Slow Not Fast. Python <python@cccp.invalid> - 2026-10-13 16:01 +0000
        Re: GPS clocks should run Slow Not Fast. Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-10-13 09:10 -0700
          Re: GPS clocks should run Slow Not Fast. Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-10-14 06:18 -0700
      Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-14 17:30 +0200
        Re: GPS clocks should run Slow Not Fast. Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-10-14 09:32 -0700
  Re: GPS clocks should run Slow Not Fast. Mikko <mikko.levanto@iki.fi> - 2026-10-21 11:04 +0300
    Re: GPS clocks should run Slow Not Fast. Maciej Woźniak <mlwozniak@wp.pl> - 2026-10-21 10:07 +0200
    Re: GPS clocks should run Slow Not Fast. ram@zedat.fu-berlin.de (Stefan Ram) - 2026-10-24 20:18 +0000
      Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-27 11:25 +0000
    Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-27 21:32 +0100
      Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-27 23:03 +0000
        Re: GPS clocks should run Slow Not Fast. Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-29 00:47 +0100
          Re: GPS clocks should run Slow Not Fast. HenryWilson <Hgw@home.com> - 2026-10-29 16:41 +0000

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