1.5.1 Inadequacy of the galilean transformation law.

An inertial reference system is defined to include an inertial reference frame and a system of synchrous clocks at rest in that frame.

My understanding is, that all these clocks, at each instant (of time) show instantaneous the same clock time.

Further more we can imagine that there are spectators (observers) at every point of the reference frame who can determine the position (x,y,z) and time t of each event coincident with its occurence.

All these observers have each, the same understanding that when their clock shows 1 o'clock all the other clocks in the universe show simulataneous this same clock time.

However, it must be remembered that such an observer is always present at every event; he is not seated at one point in space watching distant events when their light reaches his eye.

It is important to note that each observer, observes the clocks which are at the same distance from his position, simultaneous. That means all these clocks show the same clock time. What is important all these clocks run behind equally (as a function of distance).
In fact this allows an observer to identify simulataneous events which happened in the past from his position.

We consider two inertial reference systems S and S', with S' moving at the constant speed V, relative to S, along the direction of the positive x axis

My understanding is that all clocks (using light signals) with a speed V will run behind clocks at rest in S.

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1.5.2 Time dilatation and Lorentz contraction

The Lorentz transformation law, which is valid for all possible speeds V, is derived on the bassis of the kinematic postulate that a light signal traveling with speed c realtive to S also travels with speed c relative to S'.

Observer S determines the length of A'B' by measuring the positions of A' and B' simultaneously relative to S.
Let A" be the position of A' relative to S when A' crosses the y axis, and let B" be the corresponding position of B' (Fig 1.42).
Since C'B' = C'A' and since the direction of motion is perpendicular to the S' rod, then A"C' must equal B"C', etc

I would write: then C'B" must equal C'A". This means point C' is in the middle.

Consider a light signal emitted at C at such a time that the signal that the signal reaches B' when B' coincides with B".

This is a lightsignal which moves in the S frame from C to B. At the same time the point B' of S' moves to B of S

Test1

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1.5.3 A special set of Lorentz transformation equations

The Lorentz transformation law, which is valid for all possible speeds V, is derived on the bassis of the kinematic postulate that a light signal traveling with speed c realtive to S also travels with speed c relative to S'.

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1.5.4 Derivation of time dilatation and Lorentz contraction formulas from Lorentz transformation equations

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1.5.5 The general form of the Lorentz transformation equations

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1.6 The visual Appearnce of Moving Objects

Until 1959, it was believed that the lorentz contraction was the sole effect that needed to be considered when describing how you actually would see an object moving past you at a high speed.

Okay

According to this belief, since an object moving past (you) with speed V, is shortened by a factor sqrt(1-V^2/c^2) in the direction of motion, a viewer would see the object distorted by this compression.

The question is how the speed V is calculated in your frame, including the speed of light.

This belief was maintained until 1959, when it was pointed out that the visual appearance is not determined by the simultaneous positions of all points on the objects

Okay

Rather, the visual appearance is determined by the light from all points on the object that arrives simultaneously at the eye.

That is correct for all what we see. Our visible world view is an image of all photons that arrive simultaneous at both our eyes. This has nothing to do with

1.6.1 The difference between the instantaneous location of an object and its visual appearance

The American physicist N. James Terrell showed in 1959 that the Lorentz contraction is not the only effect that determines the visual appearance of a rapidly moving object.

Reflection 1 - Lorentz contraction

• In vertical direction.
  

  Consider a straight rod in vertical direction (of length 2*l) at t0 wich connects the three points: (y0,x0), (y1,x0), (y2,x0). This is confirmed by three observers near these three points at t0. This rod moves with a constant speed V in the x direction and arrives at t1 at the three corresponding points: ((y0,x1), (y1,x1), (y2,x1) This is confirmed by three observers near these three points at t1. Question: Is this conform to reality? Answer:Yes. See Page 42 That means for a vertical rod, which moves horizontal there is no length contraction involved.

  y2 | --> | y1 | --> | y0 | --> | x0 x1 Figure 1

• In horizontal direction.
  

  Consider a straight rod at rest in horizontal direction (of length 2*l) at t0 wich connects the three points: (y0,x0), (y0,x1), (y0,x2). This is confirmed by the three observers at these points at t0. This rod moves with a constant speed V in the x direction and arrives at t1 at the three corresponding points: ((y0,x1), (y0,x2), (y0,x3) This is confirmed by the three observers at these points at t1. Question: Is this conform to reality?

  y00 |----R1---->----R2---->----R3----> t0 y01 | ----R1---->----R2---->----R3----> t1 y02 | ---R1--->---R2--->---R3---> t1 y02 | ---R1--->---R2--->---R3---> t2 y02 | ---R1--->---R2--->---R3---> t3 y02 | ---R1--->---R2--->---R3---> t4 x0 x1 x2 x3 x4 Figure 2

  Figure 2 shows 3 rods, identified as R1,R2,R3, in y0, at three different situations. The line y00 identifies the situation for t0. This is the starting position. The line y01 shows the situation at t1 when length contraction is not involved. The 4 lines y02 shows the situation at t1,t2,t3 and t4 when length contraction is involved There are also 4 observers. O1 is at the point x1, O2 is at the point x2. O3 is at the point x3, O4 is at the point x4 All these 4 observers show something specific when length contraction is involded! O1 observes at t1 the back of R1 coincides with the point x1. O2 observes at t2 that the front-of-R1/back-of-R2 coincides with the point x2. O3 observes at t3 that the front-of-R2/back-of-R3 coincides with the point x3. O4 observes at t4 that the front-of-R3/back-of-R4 coincides with the point x4. These 3 rods move with a speed V towards the right.

  The first question is which situation is correct: y01 (no length correction) or y02 (length correction)
  The second question is: When y02 is selected, what is the physical explanation of a length contraction?
  The explanation of the observation that a moving clock runs slower, compared with a clock at rest is that the lightpath of a moving clock is longer than of a clock at rest. This explanation does not involve any form of length contraction.

Reflection 2 - Time dilation.

The problem with time dilation is that there exists not something as time-dilation or time-dilatation i.e. that time runs slower or faster.
What exist that in the universe, all physical objects, including positions, shapes and composition, are constantly changing.
This change we humans can observe and experience and we call that the positions change in time. The issue is that the universe as a whole, changes. Time used in this way is called: universal time or the age of the universe.
A whole different issue is that moving clocks, based on light signals, tick slower as clocks at rest. That says something about clock time. Clock time is a parameter of (the behaviour) of a physical process. Gravity is also a parameter of a physical object.

Reflection 3 - Length contraction & Length expansion a visual illusion?

Consider a train (which consists of loco in front and cariages) of length l, which travels along a platform in a straight path. At the platform is an observer. At t4 the back of the train passes the observer, what will the observer see of the position of the loco of the train?
Assume at t4 the loco issues a light signal. That light signal has to travel the distance l. This signal will arrive at the observer at t6 i.e. later as t4. That means at t4 the observer will not see where the loco is at that instance but at an instance earlier than t4, from a distance closer than l. That means the observer will observe length contraction of the train.

' t6 x b-----------f t5 b--x--------f t4 b-----------xf t3 b--------x--f2 t2 b-----------f1 t1 b-----------f t0 b-----------f O-------------> Figure 3 >

Figure 3 shows a train, passing along a platform in the right direction, with an Observer at O. At t0 the front f of the train is at the position O of the Observer. At t4 the back of train b is at the position O of the Observer. At that same moment also a light signal is emitted from the from the front, identified as the point xf At t5 the emitted light signal is at the point x between the front f and the back b of the train. At t6 the Observer sees at t4 the front of the train emmitted at the point xf. But that is to late. For the observer to see the front of the train at the light has to be inbetween the f1 and f2.

We all know that we still can see a train pulling away at a far distance, the reason is that the speed of light is much greater than the speed of the train.
Suppose an observer stands on a platform and the back of the pulling away train is exactly at his position, now. At that same moment can the observer see the front of the train?
Yes and No. He cannot see the front of the train where the train is now, because it still takes time from that position for an image to reach his eyes. He can see the front of the train, earlier, a shorter distance away. That means the observer observes length contraction.

Consider the same train of length l, which travels in a straight path towards an observer at a plaform. At t4 the loco of the train passes the observer, what will the observer see of the position of the back of the train a distance l away.
Assume at t4 the back issues a light signal. That light signal has to travel the distance l. This signal will arrive at the observer at t6 i.e. later as t4. That means at t4 the observer will not see where the loco is at that instance but at an instance earlier than t4, from a distance futher away than l. That means the observer will observe length expansion of the train.

t8 f-----------b t7 f-----------b t6 f-----x-----b t5 f--------x--b t4 f4----------b4 t3 f--x--------b t2 f-----x-----b t1 f--------x--b t0 f-----------b0 -----------O<----------------------- Figure 4

Figure 4 shows a train approaching a platform from the left. The Observer is at O At to the front of the train is at f and the back at b0. At t0 there is also a light flash at b0 towards the front of the train. This light flash follows the path indicated by x. At t4 the front of the f4 is at the position O of the observer. At that moment the Observers sees the back of the train b0 at t0. This indicates Length Expansion. At t4 there is also a light flash at b4 in the direction of the front of the train. This light flash follows the path indicated by x. At t6 this light flash from the back of the train b4 is observed by O, but too late. For the observer to see the back of the train the light flash has to be emitted earlier. As explained above this has to be at t0.

We all know that we can see an approaching train from a far distance, the reason is that the speed of light (towards us) is much greater than the speed of the train (towards us).
Suppose an observer stands on a platform and the front of the train is exactly at his position, now. At that same moment can the observer see the back of the train?
Yes and No. He cannot see the back of the train where the train is now, because it still takes from that position for an image of the back to reach his eyes. However, he can see the back of the train earlier, a further distance away. That means the observer observes length expansion.

Consider a LHC storage ring of 100 km. Consider there exists a rails alongside which also has a length of 100. Place 100 trains on this rails. Give all these train the same speed V, and increase this speed slowly such that each train still has the same speed. The question is: will there slowly become a certain space between each train? The answer is no.

Reflection 4 - World view.

In order to understand the universe and how the unverse evolves, you must have a certain picture, or a world view of the universe.
Starting point of this picture must be that at any instance in time there exists a universe, which consists of objects and events. All events happening at that instance are happening simultaneous throughout the universe. With event I mean any specific or identifiable change.
What is important that all these simultaneous physical changes are caused by previous physical changes and in turn will cause physical changes in the future.
A typical case are all the stars in our Galaxy, which are influenced by all the stars and inturn will influence all the stars.

It is very important that humans, just by observing with their eyes, can not influence any physical change, specific concepts like length contraction or length expansion .
It is important to consider that there are two different definitions of time:

• The first definition of time is like physical time. This time is connected with the existance of the universe, also called the age of the universe or universal time.
  Physical time is based on the concept that at any moment the universe has the same time. It is based on what we humans experience, with our brain. What we experienced are events that have happened in the past and that there is a now. Based on these experiences we can predict, to a certain extend, what will happen in the future and take certain actions, now.
• The second defintion of time is clock time. However clock time is established by a physical process, which in principle are unreliable.
  Moving clocks, using light signals, run slower than a clock at rest. That means only clocks at rest can be used to understand the laws of physics.
• My understanding is, that in order to understand/explain any physical process in the universe, no physical clock is required. The processes, which resembles the behaviour of clocks, are cyclic processes, like the seasons here on earth, but to understand that you don't need a clock.
  Predicting something requires a clock.
• What is also important if you want to measure the size of an object, specific the length of an object you must measure the position of the points, linked to the object, simultaneous. That problem is 'solved' if you consider the object as a point.
  For a more general approach, if you want to simulate the movement of stars using Newton's law, you must start with a sequence of equally dt spaced, measurements, for all objects involved. The result are a sequence equally dt spaced predictions, each showing the positions of all the (moving) stars at a certain time (of the day).
  If you want to simulate a train which goes from A to B in a straight line you follow this approach:

  First you draw a horizontal line from A to B which shows the track. Next you draw a vertical line through the point A which represents the time. Next you draw a curved line starting at point A, horizontal towards the right. The first section follows the circle from the bottom of a clock going from 6 to 5. The second section is straight under an angle of 30 degrees. The third section follows the circle from the top of a clock going from 11 to 12, after which you reach the destination point B. This is the path of the back of the train. What the curve shows is that in the first section the speed of the train increases, stays constant in the second section and decreases in the third section and stops when point B is reached. To draw the front of the train, you start with a second line from a point 1 cm to the right of point A. This second line is a copy of the first line, 1 cm towards the right. The second line ends, is at a point 1 cm to the right of point B.

  ^ | .-----. | .-----. | .-----. t| .-----. i| .-----. m| .-----. e| .-----. A.-----.-----------track-----------B------> Figure 5

  To validate Figure 5 requires the same rods, clocks and observers as described in figure 1.38.
  What figure 5 shows is that a moving train does not show any physical length contraction or length contraction.
  That is different from what an Observer at A or at B observe.
  An observer at A can observe Length Contraction and an Observer at B can observe Length Expansion, but that is not physical. It is an visual illusion. See also: Reflection 3 - Length contraction & Length expansion a visual illusion?

However there exist a much more bassic issue. The issue is that we want to understand the physical processes that are happening in the universe, how they evolve in time, how each started, its cause and how they will end. For example a star can explode.
Understanding these physical processes starts by making observations by humans. This involves the use of light or light signals. The tricky part is that (passive) observations is not a direct way to understand (almost) any physical process. The most important way to understand physical processes is by performing experiments. Specific by performing additional experiments as a result of previous experiments. By doing that you create a model of your processes about the different parts of the process studied and how these parts interrelate and influence each other. This strategy is for example, the major approach to study medical issues.
One major field of study are astronomical issues, specific the behaviour of stars. To study stars, starts with what is called vissible matter, however that is only part of the problem, the real issue is all matter, all baryonic matter of which a part is invisible. The major physical issue to understand are forces, or how by means of forces, masses influence each other masses at a distance i.e. propagate through space. The most important parameter is the speed of gravity. This fact that influence propagation takes time is something Newton did not take into account.
However, now comes the other part of the coin, the concept light, in general electro-magnetic radiation, specific the speed of light, is much less important as previous thought to understand and explain the (physical) behaviour of stars.

That does not mean that light is not used to understand this behaviour. If you want to predict accurate positions of the stars you need accurate measurements of the actual, present position of the (vissible) stars, and in order to do that you must understand the behaviour of light i.e. photons.

However if you read the text available from the book "Relativity for scientists and engineers" it raises certain issues:

• Why do they discuss different reference frames? I don't know. IMO if you want to predict the positions of stars and if the assumption is that stars influence each other than you need a strategy that starts from a common reference frame which includes the vissible stars and even better which includes all masses estimated.
• How important is mathematics?. In my opinion much less than commonly assumed. I have a great doubt if you need complex numbers, like general relativity does. The world view of Newton is too simple, because he assumed that gravity acts instantaneous. At the same time I have a doubts to what extend you can simulate the movement of the stars around the BH Sagittarius A* using GR in its full glory. The problem is that all these stars influence each other as rather simple can be demonstrated using Newton's Law.
• A more tricky discussion are the postulates of SR, including equivalence principle and the importance of SR.
  Starting point of the discussion should be our objective, which is the trajectory of an apple falling from a tree or more general how to calculate the trajectories of any binary system i.e. two stars, based on vissible observations. One of the most important parameters are the two masses of the objects considered or better the ratio between these two masses. My impression is that this exercise only requires the gravitational mass of the two objects and not the inertial masses. This reduces the problem to the ratio of the gravitational masses.
  My impression is also that this problem can be solved using one reference frame. That means to decide which laws are relevant in this reference frame. This problem problem belongs to mechanics i.e. Newton's mechanics or GR, ( https://en.wikipedia.org/wiki/Mechanics) but I guess that GR can be simplified, but complex enough that it should include more than 2 large masses even Black holes. What I expect that the problem can be solved without using the concept space-time, that means complex numbers, but I'am not sure.
• A whole different issue to what extend Lorentz transformation equations have to be used in this exercise. The problem is: speed is a calculated parameter based on two events: The position of a an object (or point) at one instance and the position (point) of that same object at a second instance. IMO the only way to do that is to define a 3D grid of rods in empty space at rest. At the crossections of the rods there should be a clock and an observer.
  The simplest case to use a single dimension in the x direction. That defines the speed of a moving rod along the x axis as: (x(t2)-x(t1))/(t2-t1). with x(t1) the front of the moving rod at t1 and x(t2) the the front of the moving rod at t2.
  The same can be done for the back of the moving rod. The same can be done for rod which moves in the opposite direction.
  However and that is very important: these speeds should never be added because such a speed does not represent a parameter of the physical reality.