This document contains comments about the book: "Introducing Einstein's Relativity" by Ray d'Inverno. Reprinted 1998
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### 2.5 The principle of special relativity

This means that, if one inertial observer carries out some dynamical experiment and discovers a physical law, then any other inertial observer performing the same experiments must discover the same law.
This is tricky.
Put another way these laws must be invariant under a Galilean transformation.
That maybe is true, but does that say anything about the experiment itself? about the physical process involved? I doubt that.
In Newtonian theory, we cannot determine the absolute position in space of an event, but only its position relative to some other event.
This is an unlucky description of Newtonian theory. Newton's main discussion is about forces. When the sum of all the forces is zero than the speed (and direction) of an object stays the same *. When this is not the case the speed (and direction) of an object will change. (*) or be at rest.
Thus both position and velocity are relative concepts.
This sentence is empty. No content.
Einstein realized that the principle as stated above is empty because there is no such thing as a purely dynamical experiment.
The issue are the physical process and these processes are dynamic i.e. changing in time.
Even on every elementary level any dynamical experiment we think of performing involves observation, i.e. looking and looking is part of optics not dynamics.
That is why you should try to describe of processes all purely from the physical point of view and remove the step of looking i.e. human observations.
Postulte I. Principle of special relativity
All inertial observers are equivalent
But this does not say anything about the physical processes involved.

### 2.6. The constancy of the velocity of light

However the approach of the k-calculus is to dispense with the rigid ruler and use radar methods for measuring distances.
The use of a rigid ruler is not very practical, specific if you want to measure distances in space. However rigid rulers have one big advantage: You can measure distances simultaneous and that is when you use light signals not the case.
What is rigidity anyway? If a moving frame appears non-rigid in another frame, which, if either is the rigid one?
Tricky sentence
IMO a frame along a straight track defines a rigid frame. A train on that track also.
Thus an observer measures the distance of an object by sending out a light signal which is reflected off the object and received back by the observer.
This sentence should mention the implication if the inner working of the clock used, also involves lightsignals.

### 2.7. The k-factor

Let us assume we have two observers A at rest and B moving away from A with uniform constant speed.
It is tricky to use the word observers. In real it is one object A at rest and object B moving.
However you can also claim that B is at rest and A is moving.
Then in a space-time diagram the worldline of A will be represented by a vertical straight line and the worldline of B by a straightline at an angle to A's as shown in Fig 2.6
It is easy to make such a sketch, but this is very difficult to draw an accurate one based on observations.
 ``` Time ^ |A /B +3 | . | | / +2 | . | | / +1 | . | | / 0-------------- ``` The problem is in the units of the time axis. The Time axis shows the time in days. +1 is at day 1, +2 is at day 2 etc The line A is simple because object A is supposed to be at rest. Line B is much more complex. To draw this line you can send out each day a flash and monitor the reflection. When you send at t0 and you receive at t1 than the total duration of the signal is t1-t0. The maximum distance is at (t0+t1)/2 and the distance is (t1-t0)/2 * c. This calculation requires that the speed of light is c in both directions and that the line A is in "absolute" rest.

### 2.8. Relative speed of two inertial observers

Hence, if v is the velocity of B relative to A we find
v = x/t = (k^2-1)/(k^2+1)

### 2.10 Relativity of Simulatneity

Imagine a train travelling along a straight track with velocity v relative to an onbserver A on the bank of the track.
How is this velocity v measured? v as a function of c?
in the train, B is an observer situated at the center of one of the carriages.
Okay
We assume that there are two electrical devices on the track which are the length of the carriage apart and equidistant from A.
It is also assumed that this is the situation when the train is at rest.
When the carriage containing B goes over these devices, they fire and activate two light sources situated at each end of the carriage (Fig 2.13)
Okay
From the configuration, it is clear that A will judge that the two events, when the light sources first switch on, occur simultaneous.
However B is travelling towards the light emanating from light source 2 and away from the light emanating from light source 1.
Since the speed of light is a constant, B will observe the light from source 2 before observing the light from source 1, and so will conclude that one light source is turned on before the other i.e. these are non-simulataneous events.
This cannot physical be the case considering a universal point of view.
The problem is that also both A and B can be wrong.
• The only thing that is true that definitely both observers cannot claim that they both observe the two lights simultaneous but.
• At the best one observer can claim that he or she observes the light simultaneous.
• Most probably both are wrong in interpreting about what actual happened. The problem is that what each observes (sees) is not what has actual happened.
See Reflection 1 - relativity of simultaneity What makes this whole experiment more complicated is the issue if, in the moving train, physical length contraction can be involved. See also: Clock and Centrifuge

### 2.11. Relative speed of two inertial observers

The moral is that in SR time is a more difficult concept to work with than the absolute time of Newton.
Newton does not use the concept of absolute time nor absolute space. He somehow uses the concepts of time and space and uses somehow only one reference frame. Within that frame everywhere the speed of light is c. I'am specific writing somehow because he only considers the solar system as his reference frame.

### 2.12. The Lorentz transformations

Let event P have coordinates (t,x) relative to A and (t',x') relative to B (Fig 2.17)
How do you know that?

### 9.2 Mach's principle

Let's us ask another fundamental question. If Newton's laws only hold in inertial frames, then how do we detect inertial frames.
The question is more: If Newton's laws are only valid in certain frames and not in others how do we detect in which type of frame we are? (In the sense that Newton's law is valid.
Newton realized that this is a fundamental question and attempted to answer it by devising an ingenious thought experiment - the famous bucket experiment.
This whole problem is related to the question: who is at rest? the moving train or the platform i.e. the Earth.?
The most obvious answer is: the Earth.
The next question is: who is at rest? the Earth or the Sun?
He first of all postulated the existence of absolute space: 'Absolute space, in its own nature, without relation to anything external, remains always similar and immovable'
To define a concept 'absolute space' makes only sense if there is also something as 'space'. Space is considered by definition as empty
An inertial observer then becomes an observer at rest or in uniform motion relative to absolute space.
The difference is impossible to establish.

### 15.2 Classical Kepler motion

 This paragraph shows 4 specific equations: 15.7, 15.10, 15,11 and 15,12. All these 4 equations are simulated in the VB program "VB Mercury numerical" To observe the results select this link: VB Mercury numerical

### 15.3 Advance of the perihelion of Mercury

We now look at the one-body problem in general relativity. We asssume that the central massive body produces a spherically symmetric gravitational field.
The appropriate solution in general relativity is then the Schwarzschild solution. Moreover a test particle moves on a time
It is easy to understand that the reality of the solar system is much more complex.
Moreover, a test particle moves on a timelike geodesic, and so we begin by studying some of the geodesics of the Schwarzschild solution.
To understand the first sentence the definition of a timelike geodesic should be thoroughly understood, other wise it 'explains' nothing.
Also here the reality of the planet Mercury versus the Sun is more complex than a testparticle versus the Sun.
In Newtonian mechanics both objects are considered as spherical. This means that they are pointlike with all the mass at one point, That is already a hugh simplification.

### 15.6 Time delay of light

The idea is to use radar methods to measure the time travel of a light signal in a gravitational field.
That is a very good idea
Because space-time is curved in the presence of a gravitational field, this travel time is greater than it would be in flat space, and the difference can be tested experimentally.
In a more general way, this experiment tests the influence of a gravitational field.
In both Fig. 15.13 one, and Fig. 15.13.1 four of these paths of the light ray are drawn at 4 different locations of the Planet.
We could use our solution (15.42) but since we are only going to work in the first order in m/r it is sufficient to take the straight-line approximation.
The situation showing solution (15.42) is depicted in Fig 15.6. In Fig 15.6 the line is bended. In Fig 15.13 the line is straight.
A more detailed explanation is required

 ``` p1 P2 P3 x3 P4 S E Fig. 15.13 ```
Fig. 15.13 A light ray travelling from a planet to the Earth in the Sun's gravitational field.
 ``` p1x1 P2 x2 x3 P3 x4 P4 S E Fig. 15.13.1 ```
Fig.15.13.1 Shows 4 light rays travelling from a planet to the Earth in the Sun's gravitational field.
Fig 15.13 in the book shows 1 lightray. The ligtray follows the path E,x3,p3,x3 and back to E. The path is not bended.
The modified version Fig 15.13.1 shows 4 lightrays at t1,t2,t3 and t4. 'Each' path is bended.
• A light ray at t1 following the path E,x1,p1,x1 and back to E
• A light ray at t2 following the path E,x2,p2,x2 and back to E
• A light ray at t3 following the path E,x3,p3,x3 and back to E
• A light ray at t4 following the path E,x4,p4,x4 and back to E

The experimental verification of the delay consists of sending pulsed radar signals from the Earth to Venus and Mercury and timing the echoes as the positions of the Earth and the planet change relative to the Sun.
I doubt if this is the correct description of the experiment.
What I expect is that a pulsed radar signal is send from Earth to a planet. This signal is reflected and measured back on Earth. That means both the moments t1 when the pulse is emitted and the moment t2 when the pulse is received are measured.
For Venus, the measured delay is about 200 micro-sec, which gives an agreement with the theoretical prediction of better than 5%.
As said before, this sentence does not give enough detail about the experiment.
It should be mentioned again that the most important is the physical details of the experiment i.e. measurements peformed. The mathematics or the theory behind the experiment comes second. See also Reflection 4 - Shapiro time delay versus speed of light

### Reflection 1 - Relativity of simultaneity.

The first question to answer is: is it possible that two events happening at different locations are simultaneous events. IMO the answer is Yes. At the same time most events that have happened, are happening or will happen are not simultaneous events.
For example: different events happening at the same position, in sequence, are definitely not simultaneous events. In fact each event can be caused by a previous event.
The real problem is that in general it is difficult by one observer to decide if different events are simultaneous or not. The fact that he observes two events simultaneous, does not mean that they are simultaneous.

To decide if events are simultaneous you need a coordinate system, or better a grid, with at equally spaced distances a clock. All the clocks should be synchronised.

### Reflection 2 - Speed of light versus speed of gravity

Studying this book about gravity, specific how the movements of objects influence each other, it is important to make a clear difference between the speed of light and the speed of gravity.

In the next paragraph the concept one world view is introduced. The concept one world view means that at every moment of the physical existance of the universe all the objects in the universe have a specific state and all the events happening at that moment are simultaneous events. That means neither of these events, for example collisions between objects, can influence each other. To be more specific two specific collisions, involving 4 objects, which happened simultaneous can never influence each other. Event A can only influence Event B if event A happened before Event B.

• The speed of light is important, assuming we accept a one world view about the state of the universe, because in order to calculate the actual position of when an event happened (for example a collision between two objects, or in general any observation of an object) the speed of light has to be taken into account.
• The speed of gravity is important because, when there is a collision between two objects in space, to calulate when this collision influences us i.e. an observer on earth, the speed of gravity has to be taken into account.
Also this requires a one world view approach based on the ideas when the events actual happend and when they are observed.
It is important to remark that the speed of light and the speed of gravity are physical two complete different concepts. The speed of light is a much more complex because the path that a light rays follows is influenced by the masses in its immediate environment.
The path that gravity follows is not influenced by any mass and propagates in a sphere around the original source in straight lines. THe strength of this field can flutuate for example when two objets are involved.

### Reflection 3 - Shapiro time delay

The Shapiro effect involves two important phenomena:
• A change in reflection angle of the light path
• A Change in arriving time of the reflected signal
To measure this a pulsed radar signal is send from Earth to a planet. This signal is reflected back towards Earth and measured on Earth. That means both the moments t1 when the pulse is emitted and the moment t2 when the pulse is received are measured.
In principle this measurement should involve one complete revolution of the planet.
Suppose that the trajectory of the planet is a circle.
When the planet is in front of the Sun the path of the light ray is not disturbed by the (gravity field) of the Sun. When the planet is at the back of the Sun the path of the light ray is mostly not disturbed by the the Sun, except when the planet comes the furtest away from the Earth. This is region when Planet, Sun and Earth are almost on one line.
This is the situation sketched in Fig. 15.13.1
In Fig 15.13 (in the book) The path from E to P3 and back to E is a straight line and not bended. This is okay when the position of Planet, Sun and Earth are not in one line, but not when they are almost in one line. This is also very important for a mathematical description of this phenomena.

The experiment should be performed in such a way that over a long period 'every hour' both t1 and t2 should be measured and t2 should be compared with a situation when the gravitational field has no influence.
Unfortunate that information is not supplied.
In a second part t2 should be compared with a theoretical value. For example GR or Newton's Law. Also that information is not supplied.

How ever there is a much more bassic problem. The Spahiro effect is bassicaly a physical effect and involves the behaviour of the path of a light ray very close to an object which also emits light. That means a lot of physical interaction can be involved which can influence this path. The question is in what respect both laws describe this possible interaction. One thing is for sure for both laws and that is that gravity influences the behaviour of the light path of the light ray. One thing that is not know and that is: to what extent light rays emitted by the Sun influence the artificial light rays emitted from earth and reflected by the planet. This is a physical issue.

### Reflection 4 - Shapiro time delay versus speed of light

The Shapiro effect involves two steps:
• First to demonstrate the time delay i.e. to demonstrate that the reflection time is not symmetric. That means is not what is theoretical expected when the light ray passes the Sun at close distance. This happens when the planet is behind the Sun and not in front of Sun, observed from the point of view of the Earth.
• Secondly to quantify this behaviour. This requires the speed of light.
The problem IMO is that Fig 15.13 at page 205 in the book is wrong. The reflected path of the light ray from the Planet towards Earth is drawn as a straight line, and that is wrong? The path should be bended.
In Fig 15.13, the line going from P3 towards Earth is also straight. In Fig 15.13.1, the line going from P3 towards Earth is bended. The same for all the 3 points P2, P3 and P4. The line from P1 towards point E is straight. Figure 15.13.1 is highly exaggerated.
How much bending is involved is different for each light ray. The line p2,x2,E is bended the least. The line p4,x4,E is bended the most. The cause of this bending is physical. There could be two reasons:
• The first reason that the light ray consists of photons and each photon has energy and has mass. This mass is attracted towards the mass of the Sun and as a consequence the path is bended.
• The second reason is that the Sun also emits photons. In fact the photon 'field' emitted by the Sun is much stronger than de photon 'field' embedded in the light ray and original emitted from Earth.
• What is also important is that this bending is happening twice. First when the light ray travels towards the planet and secondly, after reflection when the light ray travels back towards Earth.
What this means, capsulating, is that the bending is very complex physical process.
How closer the Planet, the Sun and the earth are on one line, how more bending is involved.

My estimate is that the speed of the light ray will be effected in both directions, but they will cancel out, meaning that the overall influence will be less.
To assume that the speed of light is physical everywhere the same is wrong. To make things simple as part of mathematical considerations is okay. The most important part lies in the details of the actual measurements involved.

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