This is the 2nd edition as of 15 November 2018.
For the first edition see: Relativity of Simultaneity - Edition 1
• 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.
Physics has to do with processes and how processes each, can influence each other. Within each process specific events can happen and as such each events can influence many process in different ways. Observers in general have nothing to do with these. The same for the Observers reference frame.
If there are two observers and one observer declares two events (happening within a process) simultaneous and the other observer not the final outcome (state) of the process will be the same. The observation as such does not influence the process.

### 1. Description

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.
Their are two issues:
• Is it physical possible that two events are happening at the same time i.e. simultaneous in space?
• What is the position of any Observer related to this issue.
The answer on the first question is Yes. At the same time, at each instant, millions of events are happening in space.
The problem for any observer is the same: it is very difficult to decide which events are simultaneous or not. In fact each observer should give the same answer.
If one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events (the only exception being when motion is exactly perpendicular to the line connecting the locations of both events).
The sentence is not clear, because what means: Assigns a time. This requires a clock, which has its own problems.
The sentence should be changed as follows:
If one observer observes two events that are at different points in space simultaneous, than a second observer which is moving relative to the first observer, will generally speaking not observe these two events simultaneous. Simultaneous meaning at the same time.
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 because it lacks the details how it is actual performed
Observer 1 on earth can never observe the two crashes because the earth is not flat.
Observer 2 in the sky has a much better chance to observe both crashes
Let us assume that the earth is 'flat' and observer 1 on earth can observe both car crashes. His observation is that the crashes are simultaneous. The same for observer 2 in the sky. (You never know)
If this is the case observer 2 in the sky will see the two crashes later as observer 1 because the distances involved are larger. The whole physical issue is how lightsignals are propagated in space.
The question of whether the events are simultaneous is relative:
This line has been removed from the Wikipedia article. However in some way or an other you have to explain the word 'relative' in the subject. I agree this is difficult.
There are two issues:
1. There is a high chance that you observe two simultaneous events, not simultaneous.
2. There is a small chance that you observe two events which are not simultaneous, simultaneous.
The whole issue is to recognize that physical events can happen simultaneous. This has nothing to do with an observer.
Furthermore, if the two events cannot be causally connected (i.e. the time between event A and event B is less than the distance between them divided by the speed of light), depending on the state of motion, the crash in London may appear to occur first in a given frame, and the New York crash may appear to occur first in another.
A simpler version of this sentence is:
Furthermore, if the time between event A and event B is less than the distance between them divided by the speed of light, depending on the positions of the observers, observer 1 may observe event A first and observer 2 may observe event B first.
This is the case in the following example:
 ``` 1<. . . .A<. . . . . .X- - ->B- - - ->2 . . . . . . . . . . . . . > <- - - - - - - - - - - - - - Figure 1A ``` Figure 1 shows an initial event X which emits two light flashes. These two flashes causes two events A and B. Event B is first. The numbers 1 and 2 represent two observers. Event A causes two light flashes which are indicated as points. Event B causes also two light flashes which are indicated as dashes.
The time from X to A is 6 counts that means tA = 6. The time from X to B is 3 counts that means tB = 3.
The difference in time between the two events A and B = 3.
The time from A to 1 is 4. That means tA1 = 6+4 = 10. The time from A to 2 is 13. That means tA2 = 6+13 = 19.
The time from B to 1 is 14. That means tB1 = 3+14 = 17. The time from B to 2 is 4. That means tB2 = 3+4 = 7.

In Figure 1 the time between the two events A and B (6-3=3) is less than the distance between the two events (6+3=9).
Observer 1 will see event A first because tA1 is less than tB1. (10 < 17). Observer 2 will see event B first because tB2 is less than tA2. (7 < 19).

However, if the events are causally connected, precedence order is preserved in all frames of reference.
A simpler version of this sentence is:
However, if the time between event A and event B is larger than the distance between the two events divided by the speed of light, each oberver will always see them in the order they have happened.
That means the events can not be simultaneous.
 ``` 1<. . . .A<..............X-->B- - - ->2 . . . . . . . . . . . . . > <- - - - - - - - - - - - - - Figure 1B ``` Figure 2 shows an initial event X which emits two disturbances. These two disturbances causes two events A and B. Event B is first. It is important to observe that the speed of disturbances is much lower than the speed of light. The numbers 1 and 2 represent two observers. Event A causes two light flashes which are indicated as points. Event B causes also two light flashes which are indicated as dashes.
The time from X to A is 14 counts that means tA = 14. The time from X to B is 2 counts that means tB = 2.
The difference in time between the two events A and B = 12.
The time from A to 1 is 4. That means tA1 = 14+4 = 18. The time from A to 2 is 13. That means tA2 = 14+13 = 27.
The time from B to 1 is 14. That means tB1 = 2+14 = 16. The time from B to 2 is 4. That means tB2 = 2+4 = 6.

In Figure 2 the time between the two events A and B (14-2=12) is more than the distance/c between the two events (9 or 10), which is the same as in Figure 1A.
Observer 1 will see event B first because tB1 is less than tA1. (16 < 18). Observer 2 will also see event B first because tB2 is less than tA2. (6 < 27).

The question to ask is what is the purpose of this exercise.

Figure 2A and Figure 2B show the same information but now as a spacetime diagram

 ``` | | | | | | | | | | | | | | | | | | | .tA2 . tB1. . | . . | | . . | | . . | | . . | tA1. . . | . . . | | . . . | | . . . | tA | A. . .tB2 | . . . | | . . . | | . .B |tB | . . | . . Ob 1 X Ob 2 Figure 2A ``` ```26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 t ``` ``` | | | .tA2 | . | | . | | . | | . | | . | | . | tA1. . | | . . | | . . | tA | . A . | tB1. . | | . . | | . . | | . . | | . . | | . . | | . . | | .. | | . | | .. .tB2 | . . . | | . . . | | . .B |tB . . Ob 1 X Ob 2 Figure 2B ```

### 2. 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.

### 3. Thought experiments

It should be mentioned that generally speaking thought experiments are tricky. You can easily make the tought experiment too simple while the reality is more complex.

### 3.1 Einstein's train

Einstein's version of the experiment presumed that one observer was sitting midway inside a speeding traincar and another was standing on a platform as the train moved past.
In this case one observer is sitting inside the train and one observer is standing on a platform.
For more about Einstein's version See: Reflection 3 - More to read
The train is struck by two bolts of lightning simultaneously, but at different positions along the axis of train movement (back and front of the train car).
The question to ask is: How do you know that lightning stricks simultaneous.. Neither observer can actual see the two events directly.

For a better description of an almost identical experiment select this: Reflection 1 - Train Experiment

In the inertial frame of the standing observer, there are three events which are spatially dislocated, but simultaneous: standing observer facing the moving observer (i.e., the center of the train), lightning striking the front of the train car, and lightning striking the back of the car.
All of this seems easy but it is not that simple.
Since the events are placed along the axis of train movement, their time coordinates become projected to different time coordinates in the moving train's inertial frame.
That maybe true but it is very difficult to understand.
Events which occurred at space coordinates in the direction of train movement, happen earlier than events at coordinates opposite to the direction of train movement.
That maybe true but it is very difficult to understand.
In the moving train's inertial frame, this means that lightning will strike the front of the train car before the two observers align (face each other).
All of this is not as simple as it looks.
Generally speaking if you want to compare the time of different events you need a clock at the position of each event.

### 3.2. The train-and-platform

A popular picture for understanding this idea is provided by a thought experiment etc. and Einstein in 1917. It also consists of one observer midway inside a speeding traincar and another observer standing on a platform as the train moves past.
So far so good

A flash of light is given off at the center of the traincar just as the two observers pass each other. For the observer on board the train, the front and back of the traincar are at fixed distances from the light source and as such, according to this observer, the light will reach the front and back of the traincar at the same time.
How do you know that?
Ofcourse the observer on the moving traincar can think that, but the same is true for every observer on any moving train, independent of the speed of the train. Even if the speed is zero.

For the observer standing on the platform, on the other hand, the rear of the traincar is moving (catching up) toward the point at which the flash was given off, and the front of the traincar is moving away from it.
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 (more or less) the same in all directions in local space and has nothing to do with any observer.
To measure the speed of light in one direction is very difficult. See also Reference 1

### 3.2.1 Space Diagrams

This paragraph starts with the following sentence:
It may be helpful to visualize this situation using spacetime diagrams.
That is correct.
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 point of view of an observer on the moving train with a speed v>0
However this is also the case of an observer on a train with a speed v=0.
What this means is that every observer, within his own train, at any speed, considers himself at rest and the two events are simultaneous.
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 from the point of view of an observer with speed v=0 which observes a train with a speed of v>0.
This is almost the same of the situation from the point of view of an observer with speed v>0 which observes a train with a speed of v=0: the events are not simultaneous.

### 4. 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 - LC simple
Now suppose that the first observer sees the second Observer moving in the x-direction at a velocity v.
The word suppose is simple. The word sees is also simple
The question is: How do you know that the speed in the x-direction is v? How is v calculated? This is tricky because you do not know when the speed v=0.
In order to calculate the speed you need a clock, maybe two clocks.
Secondly you must know the time at two different positions of observer 2, measured by observer 1 (in the frame of observer 1)
If two events happen at the same time in the frame of the first observer, they will have identical values of the t-coordinate.
The second half of the sentence is is true. The question is how do you know that the two events happened at the same time. Normally there is a distance between the two.
However, if they have different values of the x-coordinate (different positions in the x-direction), they will have different values of the t' coordinate, so they will happen at different times in that frame.
The real question to ask is if two events can be simulatenous in one frame (as observed by one observer) and not be simulataneous in an other frame (as observed by an other observer) The difference is the speed between the observers.
In the paragraph Reflection 2 - General Reflection I expres that this cannot be the case.
What is the cause of this conflict. IMO that both observers use there own clock. The problem is that moving clocks tick at their own rate. The clock which moves the fastest ticks the slowest. As such both observers can attach different clock readings to each event. This is physical not practical why the events are the same.
The solution is to use only one clock
The equation t' = constant defines a "line of simultaneity" in the (x', t' ) coordinate system for the second (moving) observer, just as the equation t = constant defines the "line of simultaneity" for the first (stationary) observer in the (x, t) coordinate system.
This sentence describes the same problem. The solution is the same.

### 5. 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 too complex. What is it purpose?
One caveat of this approach is that the time and place of remote events are not fully defined until light from such an event is able to reach our traveler.
One caveat is that in reality all experiments require acceleration i.e. that all observers are accelerated observers.

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

### Reflection 1 - Train experiment

The following sketch shows the two possibilities.
Figure 3A and Figure 3B show the same events.
At t0 the position of the Observer at the platform (Observer X) is the same as the observer in the train (Observer Y) Figure 3A describes the situation for the train at rest
Figure 3B describes the situation for a moving train with a speed v to the right
Figure 3C describes the situation where both Observers are moving towards the right
 ``` | | | | | | | | | | | | | | | | | t3. | | . . | | . . | | . . | | . . | t1. t2. | . . | | . . | | . . | | . . | | t0 Y | ------------------------- X Figure 3A v=0 ``` ``` t5. | . | . | |. | | . | | . | | t3. | | . . | | . . | | . t2.| | . . | | . . | | t4. . | | . . | | . . | t1|. . | | . . | | . . | | t0 Y | ------------------------------ X Figure 3B v>0 ``` ``` . | . | . | |. | | . | | . | | t3. | | . . | | . . | | . t2.| | . . | | . . | | . . | | . . | | . . | t1|. . | | . . | | . . | | t0 Y | ------------------------------ X Figure 3C v>0 ```
• Figure 3A 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 t3.
The two reflections at t1 and t2 are simultaneous.
• Figure 3B shows the situation for a moving train.
The observer X on the platform will see the two reflections at t4 and t5. The observer Y on the train will see the two reflections simultaneous at t3. The question to ask is are the two reflections t1 and t2 simultaneous.
• Figure 3C describes the situation where the platform and the train are not considered at rest. They have both the same speed. The train stands stil at the platform.
Both observers will see the two reflections simultaneous at t3.
What is important:
• That the two events t1 and t2 (in Figure 3B/3C) are not simultaneous.
• That t1 (in Figure 3B/3C) is earlier than t1 in Figure 3A and that t2 (in Figure 3B/3C) is later than t2 in Figure 3A
• Finally that t3 (in Figure 3B/3C) is earlier than t3 in Figure 3A
This means when you consider each meeting point of the two flashes as a tick of a clock that the moving clock runs slower..

### Reflection 2 - General Reflection

What means "Relativity of simultaneity"?
Starting point of any discussion that in the Universe at any moment millions of events happen simultaneous. For simplicity this set of simultaneous events is called events A. The definition of simultaneous implies that none of these events A can influence each other. They are all caused by previous (earlier) events and in turn be the cause of other events (in the future)
The problem is how to decide which events belong to this simultaneous set of events A.

Accordingly to that same rule you can also define a different set of simultaneous events B. These events happened either before or after the events A.

This leads to the rule: any event can only belong to one set of simultaneous events.

A different "thought": Let us start with a set of observers. Each observer is at the center of two equally spaced mirrors. Suppose each of the observers emits a flash of light towards both mirrors. What the figures 3A, 3B and 3C tell you is that each observer will receive both flashes simultaneous assuming the observer is moving with a constant speed v in straight line. However none of the observers can claim that the two events, when the flash hits the mirror, happened simultaneous (assuming that they all have a different speed - in absolute sense). At the most only one observer. That means only one observer can call himself or herself at rest. (as depicted in Figure 3A).

### Reflection 3 - More to read

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.
3. To investigate how a clock behaves in linear accelerator and centrifuge read this: Clock and Centrifuge. In this document is explained that the behaviour of a clock using light signals behaves differently depending how the clock is build. In one case the clock behaves accordingly to the Lorentz tranformation. In a second case not.
In this document also the train experiment is discussed, including a possible test how to detect if a system is at rest.
For more complex examples about the behaviour of a clock select this: Clock and Centrifuge - part 2

### Reflection 4 - A philosophical question.

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, based on what you see/observe.
• Accordingly to SR the answer is "Yes".
SR takes length contraction into account. The issue what does this physical means.

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Created: 7 January 2008
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