## Engineer's Book

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## Navigation 10 – GPS Clock Variation ? General Relativity 1

Written By: Jean-Paul Cipria - Nov• 03•12

How the electromagnetic clock sended by GPS is modified by general relativity ?
How terrestrial stations send corrections to GPS to annihilate relative phenomenon ?

« Nuage Position GPS » – Clock variation are very week and can’t be seen in this picture. Therefore the « cloud » position acuracy (average 9 m in this measurement) is by narrow band and satellite position incertitudes : Ionosphere effect : 5 m – Satellite Position Accuracy : 2 m. Relativist Frequency : 1 mm/seconde or 6 cm/min or 3,6 m per hour !

# GPS Clock ?

## Remote clock from earth

GPS clock systems are embedded on satellites at 36 000 km from the earth. Then when you receive electromagnetic waves from this device the precise clock you need to calculate the distance from emetter to receptor varies. Why ?

## Inertial fied

We are all include in a inertial field, the gravitational field. Then from the earth surface we interact with the earth at the radius distance from the center. Therefore GPS satellite interacts from R+36000 km from the earth center. Field gravitation is different for the clock device inside the GPS satellite. Understand ? Capito ?

Then we will explain you how to calculate the clock variation sended by the terrestrial station to satellite to compensate this relativist clock deviation.

Go ?

## Theory of Relativity ? Why ? And when ?

### Why ?

It is important to have some notions with theory of relativity. Why ? First to understand some posts in this web site there, second to evaluate how you can make mistakes in position accuracy.

### When ?

Then we do not need a tensor of the sixth order to make certain calculations even in PhD (doctorate) !!! We can do it with a college level ! Then my french, and only french, academic « superiors » are not agree with me. This is not as difficult as inertial moment that « terminale scientifique » student does in mechanics when he just uses formulas without understanding anything ?

# Gravity Energy Potential

## General Relativity

The general relativity expresses that the gravity potential energy acts on all the particles according to positions in the potential field.

• $E_B(r_2)=E_A(r_1)+\Delta{E}_{r_1}^{r_2}$ .          $\Delta$ is a Difference and not a Laplacian !

## Initial Axioms on quantic physics

Plank-Einstein relation :

• $E=h.\nu$      (or De Broglie with $k$ wave vector).

Terrestrial Potential Energy

• $E_A=G.m_T.m/r$

## Physics Constants

Universal gravitation constant :

• $G=6.67428.10^{-11}~N.m^{2}.kg^{-2}$

Earth mass :

• $M=5.9736.10^{24}~kg$

Light velocity :

• $c=299792458~m.s^{-1}$

• $R=6371.10^3~m$

# Potential difference for a photon ?

## Potential Difference Energy

We calculate the potential energies difference when a photon climb from position A to B.

• $\Delta E=G.m_T.m_B/(R+H) - G.m_T.m_A/R$
.
• $\Delta E=G.m_T.[m_B/(R+H) - m_A/R]$
.

Replace left term by quantic De Broglie equation :

• $h.\nu_B-h\nu_A=G.m_T.[m_B/(R+H) - m_A/R]$
.

We know that « frequency » photon energy is egal to mass energy :

• $E_A=h\nu_A=m_A.c^2$    It is De Broglie and Einstein equivalence.
.

Trick of the trade  (Astuce) : Divide left by $h.\nu_A$ and second term by $m_A.c^2$

• $\frac{(\nu_B-\nu_A)}{\nu_A}=\frac{G.m_T}{m_A.c^2}.[m_B/(R+H) - m_A/R]$
.
• $\frac{(\nu_B-\nu_A)}{\nu_A}=\frac{G.m_T}{c^2}.[\frac{m_B}{m_A(R+H)} - \frac{1}{R}]$
.
• $\nu_B-\nu_A=\nu_A.\frac{G.m_T}{c^2}.[\frac{m_B}{m_A(R+H)} - \frac{1}{R}]$
.
• $\nu_B=\nu_A.\left[1+\frac{G.m_T}{c^2}.[\frac{m_B}{m_A(R+H)} - \frac{1}{R}]\right]$
.

In first approximation compared to value 1 :

• $m_A\approx{m_B}$ Then $m_B/m_A\approx{1}$
.
• $\nu_B=\nu_A.\left[1+\frac{G.m_T}{c^2}.[\frac{1}{(R+H)} - \frac{1}{R}]\right]$
.
• $\nu_B=\nu_A.\left[1+\frac{G.m_T}{c^2}.[\frac{1}{(R+H)} - \frac{1}{R}]\right]$
.
• $\frac{\nu_B-\nu_A}{\nu_A}=\frac{G.m_T}{c^2}.[\frac{1}{(R+H)} - \frac{1}{R}]$
.

We retreive this kind of formula expressed at the first order (see $m_A \approx m_B$ !) in the relativist tensor.

## Atomic Time Reference

The reference time is qualified with the Cesium energy frequency. Then if we consider Cesium at altitude R then what is the frequency at altitude R+H ?

If $\nu_b$ is less than $\nu_A$ then $T_B$ is more long than $T_A$.

• $\frac{T_B-T_A}{T_A}=\frac{G.m_T}{c^2}.[\frac{1}{(R+H)} - \frac{1}{R}]$
.

## GPS Time Reference ?

In the 36 000 Km altitude GPS Cesium clock have a relative difference time of 3,912E-012 compared to the earth clock. We must correct this difference if we receive clock from this altitude. But we don’t do like this. We correct frequency directly from terrestrial station to the satellite. Then clock satellite are modified and we receive correct clock frequency on our own receptor. Why ?

Your own GPS receptor have very bad clock generator and are unable to do some acurate operations. Then it is easier to correct it directly on satellite by terrestrial atomic reference clock.

• $T_{Earth}=\frac{T_{Cesium_{GPS}}}{1+3.912.10^{-12}}$
.
• $\Delta{r}=1.2~mm$ The variation on distance on 1 second is very weak.

## Examples

 Constants M : Earth Mass 5,973600E+024 Kg G : Gravity 6,674280E-011 N·m2·kg−2 C : Light Velocity 2,997925E+008 m/s R : Earth radius 6,371000E+006 m Photons Relative Times Différences H : Altitude H (Tb-Ta)/Ta 1 m 1,0 1,093E-016 10 m 10 1,093E-015 Small hill 100 1,093E-014 Airplane 1000 1,093E-013 Jet 10000 1,091E-012 Satellite 36000 3,912E-012

# Relativity for position impact on particles

## Speed impact on particles

Because when you send high intensity laser to a plasma then electrons were sudenlly accelerated $a=22~G$ then after a short while particles speeds come near to light velocity and we can not negligeate relativistics effects. The restricted relativity expresses the influence of the relative speed of two marks on the energies of particles. You can discover this calculation on the following link :

## Cesium Atomic Clock

With  Cesium atomic clock there is a different problem, Cesium atoms have not a so high speed, first because they do not be accelerated so much by their own temperature (90 °C), second cause is they have a heivy mass, higher than electrons ones. But atomic clock have an relative accuracy around $10^{-16}$ and then the cesium atom position varies about 60 cm in the earth gravitation. Then we can estimate the small gravity impact on the particle physics behavior with general relativity. You can discover this small calculation on the following link :

# References

.

Jean-Paul Cipria
Research Engineer
03/11/2012 : Creation.
26/06/2016 : Presentation Modifications.

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