• The text in italics is copied from the article
• Immediate followed by some comments
In the last paragraph Reflection I explain my own opinion.

### Introduction

The article starts with the text:
In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame.
The first question to answer, without thinking about clocks, is: Does it make sense to call, the events which all happen at the same moment (now) somewhere in the Universe, simultaneous?
The property of such simultaneous events is that one can never be the cause of the other and vice versa. The cause of both can be the same, but previous, event.
This definition of simultaneity has nothing to do with any observer. The question remains how do humans decide that events are simultaneous.
According to the special theory of relativity, it is impossible to say in an absolute sense that two distinct events occur at the same time if those events are separated in space.
• Simultaneity is not specific related to Special Relativity.
• This is not a denial that two events physically can happen at the same moment.
• The problem is how.
For example, a car crash in London and another in New York, which appear to happen at the same time to an observer on the earth, will appear to have occurred at slightly different times to an observer on an airplane flying between London and New York.
This example is too complex, partly because the earth is not flat.
There is a small chance that the observer in the airplane also observes these two crashes simultaneous. However because this observer is high in the sky he always will observe (in this specific case) both crashes later.
The question of whether the events are simultaneous is relative:
This is not the issue. There are two issues:
1. There is a high chance that when you observe two simultaneous events not simultaneous.
2. There is a high chance that when you observe two events simultaneous that they are not simultaneous.
immediate next
in the stationary earth reference frame the two accidents may happen at the same time but in other frames (in a different state of motion relative to the events) the crash in London may occur first, and in still other frames the New York crash may occur first.
First of all it is a question if the earth is a stationary reference frame. Secondly within this same frame dependent about the position of the observer, both events can be observed simultaneous, the Londeon crash first or the New York crash.

### 1. The train-and-platform thought experiment

A popular picture for understanding this idea is provided by Einstein's thought experiment consisting of a moving train with one observer midway in the train, and another observer midway on the platform as the train moves past.
Thought experiments in science are always tricky.
You can discuss experiments in your mind before you actual build and perform them.
• For example you can decide first to make a model.
What is dangerous to discus the actual outcome of an experiment without performing the experiment.
• For example it is tricky to discuss experiments with objects travelling close to the speed of light

(1) A flash of light is given off at the center of the train just when the two observers pass each other.
(2) The observer on the train sees the front and back of the train at fixed distances away from the source of the light
(3) and as such according to this observer, the light flashes reach the front and back of the traincar at the same time.
This scenario is wrong and it demonstrates clearly the danger of a thought experiment!
What you need are mirrors at both ends. What the observer at the center of the train will see are the two reflections from the light flash.
see 1.1 Spacetime diagrams for a better explanation.

The observer standing on the platform, on the other hand, sees the rear of the traincar moving (catching up) toward the point at which the flash was given off and the front of the traincar moving away from it.
Clearly also this text is very misleading.
What you need are mirrors at both ends of the train. What the observer at the platform will see are the two reflections from the light flash. However not simultaneous.
see 1.1 Spacetime diagrams for a better explanation.
As the speed of light is finite and the same in all directions for all observers, the light headed for the back of the train will have less distance to cover than the light headed for the front.
The movement of photons is physical the same in all directions only for one reference frame.
The calculated speed of light in that reference frame is c and the same in all directions
In all other reference frames, based on moving observers the speed of light is physical not the same all directions, because the speed of the reference frame has to be taken into account.

### 1.1 Space Diagrams

This paragraph starts with the following sentence:
It is sometimes helpful to visualize this situation with spacetime diagrams.
The following sketch shows the two possibilities.
The left picture is describes the situation for the train at rest
The right picture is describes the situation for a moving train with a speed v to the right
In both cases the speed of light is the same.
 ``` | | | | | | | | | | | | | | | | | t2 . | | . . | | . . | | . . | | . . | t1. .t3 | . . | | . . | | . . | | . . | t0| Y | ------------------------- X Figure 1A v=0 ``` ``` t5 . | . | . | |. | | . | | . | | .t4 | | . . | | . . | | . .|t3 | . . | | . . | | .t2 . | | . . | | . . | t1|. . | | . . | | . . | t0| Y | ------------------------------ X Figure 1B v > 0 ``` ``` . | | . | | . | | .t5 | | . | | .| | | . | | . | | .t4 | | . . | | . .t3 | . .| | . . | | .t2 . | | . . | t1|. . | | . . | t0| Y | ------------------------------ X Figure 1C v > 0 ```
• Figure 1A shows the situation for a train at rest.
The observer X on the platform and the observer Y on the train will both see the two reflections simultaneous at t2.
The two reflections at t1 and t3 are simultaneous.
• Figure 1B shows the situation for a moving train.
The observer X on the platform will see the two reflections at t2 and t5. The observer Y on the train will see the two reflections simultaneous at t4.
• Figure 1C shows the situation for a moving train as Figure 1B, but with length contraction.
The observer X on the platform will see the two reflections at t2 and t5. The observer Y on the train will see the two reflections simultaneous at t4.
What is important that the two events t1 and t3 (in Figure 1B and 1C) are not simultaneous and that t4 is larger than t2
This means when you consider each meeting point of the two flashes as a tick of a clock that the moving clock runs slower. To take care that both the clock at rest and the moving clock show the same time you have to introduce length contraction. This is depicted at Figure 1C.
In the first diagram, we see the two ends of the train drawn as grey lines. Because the ends of the train are stationary with respect to the observer on the train, these lines are just vertical lines, showing their motion through time but not space. The flash of light is shown as the 45° red lines. We see that the points at which the two light flashes hit the ends of the train are at the same level in the diagram. This means that the events are simultaneous.
The first diagram shows the bottom part of the left picture. This is situation for a train at rest. It is obvious that the first diagram is not complete.
In the second diagram, we see the two ends of the train moving to the right, shown by parallel lines. The flash of light is given off at a point exactly halfway between the two ends of the train, and again form two 45° lines, expressing the constancy of the speed of light. In this picture, however, the points at which the light flashes hit the ends of the train are not at the same level; they are not simultaneous.
The second diagram shows the situation for a moving train. Also here the picture is not complete because it does not show what any observer actual observes.

### 2. Lorentz transformations

The relativity of simultaneity can be calculated using Lorentz transformations, which relate the coordinates used by one observer to coordinates used by another in uniform relative motion with respect to the first.
The problem with this paragraph is that the actual derivations are not shown. For a simple derivation see: Reflection 3

### 3. Accelerated observers

The Lorentz-transform calculation above uses a definition of extended-simultaneity (i.e. of when & where events occur at which you were not present) that might be referred to as the co-moving or "tangent free-float-frame" definition.
That may be true. However it is far to complex. What is it purpose?

### 4. History

This was done by Henri Poincaré who already in 1898 emphasized the conventional nature of simultaneity and who argued that it is convenient to postulate the constancy of the speed of light in all directions.
When you are at rest and when you assume that the speed is constant in all directions the concept of simultaneity becomes easy. That means which events are simultaneous and which events are not.
However, this paper does not contain any discussion of Lorentz's theory or the possible difference in defining simultaneity for observers in different states of motion
The issue is if motions of observers have any thing to do with the question if two events are actual simultaneous or not. IMO it does not.
This was done in 1900, when he derived local time by assuming that within the aether the speed of light is invariant.
When you assume that the speed of light is constant (locally) relative to eather this is easy, but that does not mean it is correct.
Due to the "Principle of relative motion" also moving observers within the aether assume that they are at rest and that the speed of light is constant in all directions (only to first order in v/c).
Again here the word assume is used.
If you want to compare a situation with an observer at rest (See Figure 1A) with the same situation assuming an eather than you have to give the observer a speed opposite the eather drift in order to compare the same.
A much simpler situation is when the eather drift is zero but that does not mean that the observer is at rest.

### 4.1 Einstein's train thought experiment

Einstein's version of the experiment presumed slightly different conditions, where a train moving past the standing observer is struck by two bolts of lightning simultaneously, but at different positions along the axis of train movement (back and front of the traincar).
 ``` | | |Y | | | Yt3 | | | .Y| | | | . Y | | | | . Y | | | | . Y | | | | t2 . Y | | | | . . Y | | | | . Y.t1 | | | | . Y . | | ||. Y . || t0. Y .t0 | . . | | . . | | . . | | . . | | Y | ------------------------- X Figure 2 v ---> ``` The events are simulataneous assuming an observer at rest if the distance between the observer and the two markers (at rest) is the same. In figure 2 the two bolts of lightning are the two events t0. The observer on the platform observes them simultaneous at t2. The observer Y on the moving train observes these two events at t1 and t3 not simultaneous. This whole effect is independent of the length of the train. As such this issue is not special for SR or GR because no Time dilation and Length contraction in this example is involved.
Events which occurred at space coordinates in the direction of train movement (in the stationary frame), happen earlier than events at coordinates opposite to the direction of train movement. In the moving train's inertial frame, this means that lightning will strike the front of the traincar before two observers align (face each other).
The issue is that an observer at the center of the moving train will observe the ligtning in the direction first at t1 in the forward direction of the moving train and secondly at t3.

Comments on the article Quantum_and_classical_clocks.htm "Einstein’s quantum clocks and Poincaré’s classical clocks in SR" by Yves Pierseaux

### Reflection

When you study the concept of simultaneous it is very important to indicate what we mean.
• The first question to answer is are there at each instant during the evolution of the Universe simultaneous events happening. IMO the answer is clearly Yes. That does not mean that I know which these simultaneous events are, but they occur. In fact it does not matter if I know, these events 99.99 % are happening without my involvement or any human involvement.
• The next question is how do we know if two events are simultaneous.
That is a much more difficult question to answer.
The problem is if I observe two events as simultaneous it does not mean that they happened simultaneous. They can be millions of kilometers appart.
 ``` /| / | / | ^ / | | / O | | / | | | | E1...|..X......E2 | | \ | / _\| | / | / | / |/ Figure 3 ```
The problem is also when two events happen simultaneous I have to be at specific places to observe them simultaneous. To be more precisely I have to be in the plane half way between the points where the two simultaneous events happened at the moment when the light signals hit that plane (simultaneous).

Figure 3 shows two events E1 and E2. After they happend they can move away.
The plane is drawn perpendicular through the point X half way between the two events.
The observer at O has there be exactly at the right moment when the two light flashes reach that point, assuming that the light rays are not bendend and that the speed of light is the same throughout the Universe in all directions.

This is a very difficult exercise in practice.
• A different strategy is to start from simultaneous events and to see what the results are in different reference frames. Events in this case are events which happen at the same location. For example when a the front (or the back) of moving train meets the front (or the back) of a train at rest. In this particular case we study when clocks meet.
 ``` c1 c2 /|C2 /|C3 / | / | / / x2 / / , , / / , | , / x1 / , | t3 | / X1 , , / X2 | , / | / . . | . / , | , / | . , . | t2 , / | . / , . | . / , , | / T1 / , T2 / , , T3 / | t1 Z | . s2 . | . | / . . | / . . | / . |/ . . |/ . . |/ . C1 S1 C2 S2 C3 S3 c1 s1 c2 c3 Figure 4 ```
Figure 4 depicts 2 frames: One in rest and one moving. The vertical axis is the time and the horizontal axis is the X axis.
In each frame there are three clock marked C(c) and three light sources sources S(s) to synchronise the clocks. The small letters are used to indicate refernce frame 2.
In this particular Figure the distance between the Clocks in both frames is identical.
This means in the frame at rest there are simultaneous events when the clocks meet. The question is what is observed in the moving frame. In Figure 5 the distance is not the same.
At each Clock there are also mirrors which are used to reflect lightsignals and to synchronise the clocks.
Now we are performing a number of experiments and observe what is observed in each reference frame.
1. First we are going to synchronise the clocks in each reference frame.
• In frame 1 this means we use source S1 which emits two signals. They will be reflected at time T1 and T2 at respectivily clock C1 and C2. The two light signals will combine at point X1.
The light signals are directly used to start the clocks.
• In frame 2 this means we use source s1 which emits two signals. They will be reflected at time t1 and t2 at respectivily clock c1 and c2. The two light signals will combine at point x1.
The light signals are directly used to start the clocks c1 and c2
What Figure 4 shows is that the time that the light signal reaches c1 at t1 is earlier than T1 and the time that the light signal reaches c2 at t2 is later than T2.
The condition around S1 shows the synchronisation condition.
In reality what we will do we will try to synchronise the clocks that T1 and T2 coincide with C1 and C2 and that t2 and T2 also coincide. That means t1 happens the first. At the point of synchronistaion three clocks C1, C2 and c2 all show the same time i.e 0 counts. The clock c1 will show different times. For example c1 shows 30 counts.
2. The condition around S2 shows the situation when clock's c1,c2 (frame 2) have reached the clock's C2,C3 in frame 1.
The two clocks C2 and C3 will show the same time for example 100 counts.
the two clocks c1 and c2 will show different times. For example c1 shows 110 counts and c2 80 counts. The important point is that the moving clocks have increased less.
3. The condition around S3 should show the situation when clock's c1,c2 (frame 2) have reached the clock's C3,C4 in frame 1.
The two clocks C2 and C3 will again show the same time for example 200 counts.
the two clocks c1 and c2 will show different times. For example c1 shows 190 counts and c2 160 counts.
The moving clocks have increased less.
4. What the above shows is that not all clocks tick at the same rate.
 ``` t5 / |\ . | / | \ | / / | \ . | / / | \ | / / | \ | . / / | \ | /t4 / | \ | / . / | \ |/ / |t1 \|t2 .t3 | /| /| | / | . / | | / | / | | / | . / | | / | / | | / . | / | | / | / | | / . | / | |/ |/ | A B C Figure 5 ```
But the question is: is this correct?
Figure 5 shows 5 examples assuming that moving clocks tick slower. The distance AB and BC are equal.
1. The first case is a clock at rest in A.
The time at the clock will be: t0 = 0, t1 = 100 and t5 = 200 counts.
2. The second case is a clock which moves from A to B and back to A
The counts are respectivily at t2 = 80 and at t5 = 160 counts
3. The third case is a clock which moves from A to C and back to A. (dotted line)
The counts are respectivily at t3 = 60 and at t5 = 120 counts
4. The fourth case is a clock which moves from A to C and back to B.(dotted line)
The counts are respectivily at t3 = 60 counts and at t4 = 72 counts.
5. The fifth case is a clock which moves from A to B and continues.
The counts are respectivily at t2 = 80 counts and at t4 = 96 counts.
What this shows that a moving clock ticks slower than in a frame at rest, but that a moving clock in a moving frame ticks even slower (represented by the dotted line)
5. The following two tables show the outcome of two possible experiments:
 ``` C1 C2 C3 C3 Case 1 0 0 0 0 --> 30 0 Case 2 100 100 100 100 --> 140 110 80 Case 3 200 200 200 200 --> 220 190 160 ```
Table 1
 ``` C1 C2 C3 C3 Case 1 0 0 0 0 --> 30 0 Case 2 100 100 100 100 --> 180 150 120 Case 3 200 200 200 200 --> 300 270 240 ```
Table 2
• Each table is a summary table showing the three cases of simultaneous events in frame 1 and frame 2
• The first line in each case is for frame 1
• The second line is for frame 2 (moving)
• The importance of this table is to demonstrate that clocks in a moving frame are running slower and as such can be used to demonstrate which frame is not at rest (assuming that clocks in a moving frame run slower).
However:
• In Table 1 the distance in frame 2 is 80 counts.
• In Table 2 the distance in frame 2 is 120 counts.
• That means in Table 1 frame 1 is at rest and in Table 2 it is frame 2.
What this implies is that the result of practical experiments do not have to be symetric.
• Now we observe what happens in case the length between the clocks in the two frames is different.
 ``` |/ / / | / / | / / | / | / / | / /| / | t5|/ / | / t6|/ / | / | / / | | / | / /| / | | / | / / | / | | / | / / | / | | / |/ / | / |/ | / x |t3 / | / /|t4 | / , .t2 / | / / | | / , ./ | / | / / | |t1 . / | / | / / | C1 s1 c2 C2 c3 C3 c5 C4 c1 c4 Figure 6 ```
Figure 6 shows the situation were the distance between the clocks in frame 2 is shorter than the distance in frame 1. The distance in frame 1 is 15 units and in frame 2 10 units.
• s1 is a lightsignal to synchronise the clocks in the moving frame. t1(c1) and t2(c2) are the synchronisation events in frame 2. They are not synchroneous in respect to frame 1.
• t3 and t4 are two synchroneous events in frame 1. That is because c2 meets C2 and c5 meets C4.
• t5 and t6 are two synchroneous events in frame 1. That is because c1 meets C2 and c4 meets C4.
• In both cases the reason is because the length between the clocks in each frame is the same and the length between the clocks in the two frames has the relation 3 to 2.
• The final lesson of all of this is that if you are equal distance between two simultaneous observed events:
• only the events happening in the frame with the lowest speed can be truelly called simultaneous.
• this is the frame with the highest clock rate (as demonstrated by real experiments?)
• This is not the frame of the Earth, not of the Sun, not of the Milky Way Galaxy

### Reflection part 1

There are two other documents which explain the same:
1. The original version by Albert Einstein: Albert Einstein (1879–1955). IX. The Relativity of Simultaneity (1920). It is simple.
(1) Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also are simultaneous relatively to the train? (2) We shall show directly that the answer must be in the negative.
The second sentence is easy. Both observers will never both claim, that each sees the two events simultaneous.
The first sentence is much more difficult to evaluate. How do you know that the events are simultaneous with reference to the embankment and not to the train ? Why not the reverse ? Or why not neither one ?
2. A different version is from the book "Introducing Einstein's Relativity" by Ray d'Inverno. See for more detail: Introducing Einstein's Relativity The most serious problem here is that length contraction is involved.

### Reflection part2

The central issue of the discussion is the following:
Suppose two events happen along a straight line, and I'am standing at the center of those two events. Suppose I see those two events simultaneous. The question is now: Are those two events simultaneous ?
• Newton will say "No": Because in his opinion you have to take the movement of the Observer into account. Only if the movement of the observer is zero than the events are simultaneous.
• Accordingly to SR the answer is "Yes".
SR takes length contraction into account. The issue is how much.

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