Clock Synchronization and Navigation in the Vicinity of the Earth - by Thomas B. Bahder 2008 - Article review

This document contains article review "Clock Synchronization and Navigation in the Vicinity of the Earth" by by Thomas B. Bahder 2008
To order to read the article select:https://arxiv.org/pdf/gr-qc/0405001.pdf

Contents

Reflection

I. Introduction - page 3

Accurate clock synchronization is the backbone of these systems.
Okay
An improvement in the clock synchronization translates directly into an improvement in position accuracy.
Okay
However, most of these schemes have neglected real features of the clock synchronization problem that are essential for real-life applications: the clocks to be synchronized are in relative motion, and at varying gravitational potentials.BR> The fact that clocks are affected by their motion and by the gravitational potential are basic concepts that have their origin in Einstein’s special and general relativity theory.
The true reason is physics, specific if the clocks use lightsignals.
If synchronization of clocks is to be achieved using quantum information concepts, then certain features, such as relative motion of clocks and effects of gravitational potential, must be incorporated in the quantum information approach to clock synchronization.
See D. Quantum Synchronization 22
This article addresses the theoretical problem of clock synchronizing and syntonization (making two clocks run at the same rate), or correlating time on clocks on satellite platforms that are orbiting Earth, or that are near-Earth.
However, Lorentz transformations between two systems of coordinates in relative motion show that space and time are really interwoven.
This is a tricky sentence. The physical issue is that (relative) moving clocks don't stay synchronised. The consequence is that only clocks with stay fixed in space should be used.
We use the fact that space-time is described by a four dimensional metric, but for the most part we do not explicitly use the field equations of general relativity. In this, sense our discussion is not restricted to general relativity.
Okay. But tricky.

II. Hardware Time, Proper Time, and Coordinate Time - page 4

A clock is a physical device consisting of an oscillator running at some angular frequency ?, and a counter that counts the cycles.
This sentence expresses very well that a clock is a physical device and a counter.
The question arises if this frequency is stable or not; as a consequence of external influences.
In this article I distinguish between three types of time: hardware time t*, proper time t, and coordinate time t.
Okay
Hardware time is associated with a real physical device that keeps time, which I call a `hardware clock.
Okay
Specifically, hardware time t* = NT is the time kept by a hardware clock, and is given in terms of the number of cycles N counted by the device.
But this time, measured by a clock, can be influenced by all types of physical processes.
Two hardware clocks will differ in the elapsed time that they indicate between two events, because no two devices are exactly the same.
The assumption is, that two hardware clocks, under the same conditions, will indicate the same time(clock counts).
Proper time is an idealized time interval occurring in the theory of relativity.
The question is how is this time measured or calculated. This is an important issue.
We imagine that there exists an ideal clock (oscillator plus counter) that is unaffected by temperature or vibration.
But such a clock does not exist
However, based on Einstein’s theory of relativity, the ideal clock is affected by gravitational fields, acceleration and velocities.
That means the ideal clock is not as ideal as it seems.
Anyway it is very important to know based on which specific experiments his conclusion are based.
See also: A. Synchronization versus Syntonization 5 where such an experiment is discussed.
According to Einstein’s general theory of relativity, gravitational fields, acceleration and velocities affect all physical processes, and hence these effects are associated with the geometry of space and time.
And this is the most difficult question: if clocks are affected by certain physical processes, why don't you use only clocks which are not affected. This implies only one coordination system, and clocks fixed to that frame.
A basic tenet of the theory is that between any two events that are infinitesimally separated in space-time by dxi, i = 0, 1, 2, 3, there exists an invariant quantity ds called the space-time interval
ds2 = −gij dxi dxj
(2)
where gij is the metric of the 4-dimensional space-time.
The overall question is, how is gij measured resp. calculated. This is a very important practical question
The definition of an ideal clock is one that keeps proper time intervals.
And what is the definition of proper time ?

Page 5

A. Synchronization versus Syntonization page 5

The relation between proper time ∆τ and coordinate time is such that they may "run at different rates".
Okay.
This is clearly the case when the metric of the space-time is not a constant over the integration path.
That is the theory.
What is very important that at least one experiment should be described which explains: proper time, coordinate time and the metric.
The importance of experiments should not be underestimated.
Consider now two ideal clocks at the same location and assume that these two clocks are synchronized to read the same starting time at some epoch, or starting event.
Okay.
Next move the clocks apart (hence they travel on different world lines) and then bring them together once again to a common location.
That means both clocks are subject to different acceleration.
Bringing them together implies a common location.
Why such a complex opperation? Why not keep one clock at a fixed location and move different clocks? When you do that you can compare the results of different clocks, which is extremely important.

III. Choice of a Physical Theory page 5

The second part of Einstein’s general relativity theory consists of the field equations
G ij = −κT ij
(6)

A. Electromagnetic Waves 6

B. The Geometrical Optics Approximation 8

C. Signal Detection and Use of Antenna Phase Center 8

IV. World Function of Space-Time 9

A. Navigation in Curved Space-time 10

In order to extract information from a quantum mechanical system, a measurement has to be performed

V. Physical Measurements 11

In order to extract information from a quantum mechanical system, a measurement has to be performed

A. Observations and Measurements 11

B. Tetrad Formalism 12

1. Construction of the Tetrad: Fermi-Walker Transport 14

2. Fermi Coordinates 16

3. Metric in Fermi Coordinates 16

VI. Reference Frames and Coordinate Systems 16

A. Gravitational Warping of Coordinates 17

VII. Clock Synchronization 18

As discussed in the introduction, accurate clock synchronization is the backbone of applications such as highaccuracy navigation, communication, geolocation, and space-based interferometer systems.

A. Eddington Slow Clock Transport 19

B. Einstein Synchronization 19

C. GPS Clock Synchronization 21

D. Quantum Synchronization 22

VIII. The Syntonization Problem 22

A. Choice of Metric in Vicinity of the Earth 23

B. Integration of Geodesic Equations 25

C. Proper Time Minus Coordinate Time for Various Orbits 26

D. Proper Time Minus Coordinate Time: Numerical Results for Various Orbits 28

E. Observed Doppler and Gravitational Frequency Shift 31

F. Doppler plus Gravitational Frequency Shift: Numerical Results for Various Orbits 33

IX. Geolocation in Curved Space-Time 37

A. Time Difference of Arrival (TDOA) Geolocation 38

B. Doppler Effect in a Gravitational Field 39

C. Observed Doppler Shifts 42

D. Frequency Difference of Arrival (FDOA) Geolocatio 43

X. Summary

Reflection 1 - Clock synchronisation.

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Created: 26 July 2019

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