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

### Introduction

The article starts with the following sentence.
In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and a Lorentz scalar.
In short a moving clock shows proper time. Why this distinction is not clear.
The proper time between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events.
How do you know the shape of the world line that connects two events?
IMO this whole sentence is purely "academic".
The most important part is that the readings of identical clocks, which are started at a primary event, are moved to a different position and stopped at the occurence of a secondary event, can be different. Each clock shows its proper time and the readings all can be different. This difference depents about the length of the path.
An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events.
It is very important what is meant by events. For example it should be clear if each clock should be at the same position as were the event happened. If this is not the case, then the clock will observe the event later and that is maybe not what is expected.
By contrast, coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity.
This definition is wrong. In fact the actions involved to measure the time between two events should be the same for all parties involved. This symmetry is important because both parties have the same opinion: they other one is running slow (which ofcourse physical is impossible)

### 2.1 Example 1: The twin "paradox"

For a twin "paradox" scenario, let there be an observer A who moves between the A-coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially.
The question is how does observer A knows that he or she has travelled for 10 years.
IMO the only way is to use a clock (something that oscillates) which indicates based on the number of oscillations that 10 years have passed.
Let there now be another observer B who travels in the x direction from (0,0,0,0) for 5 years of "A-coordinate time" at 0.866c to (5 years, 4.33 light-years, 0, 0).
Again how does observer B that he or she has travelled for 5 years ("A-coordinate time") and that his speed is 0.866c.
One solution in priniple is to have clocks troughtout the Universe which all run synchronuous with A's clock and which keep the same distance between each other.
For more detail to this problem select Reflection 2 - Proper Time and coordinate time

### Reflection 1 - Proper Time

The concept of "Proper Time" is very tricky.
In fact the whole concept of Time is tricky. The only way to realise that there exist something like time is when we observe a moving object. More specific a moving object that oscilates i.e. an object that moves at a regular rate (more or less) between the same positions. A clock is such an object. What you want is an accurate clock i.e. a clock that "ticks" at a constant regular rate. If you have such a clock you can compare times between different events.

### Reflection 2 - Proper Time and coordinate time

This reflection discusses the s similar example as discussed in: 2.1 Example 1: The twin "paradox". In order to explain the difference between "proper time" and "coordinate time" we study two experiments: 1) First a "real" experiment and 2) a thought experiment.
In the real experiment we try as good as possible to actual perform the same experiment as depicted in paragraph 2.1.
 ``` B 8(4) 6 3 . . 4 2C . | . 2 1 . | . 0 --A----------> distance ```
The picture at the left shows two observers.
• Observer 1 stays at home. He starts his clock and waits. On his clock he observes the ticks of his clock 1,2,3,4,5,6,7 and 8. These ticks are depicted in the line A-B.
• Observer 2 also starts his clock and at the same time he moves away as fast as he can in a certain direction. He observes the ticks 1,2 and then he stops and travel backs as fast he can. His clocks ticks 3 and 4 and he is back at base.
• Immediate they compare clock readings. Observer 1 his clock shows 8 ticks and clock 2 shows 4 ticks. That means observers 2 his clock runs behind. This is the same result as in paragraph 2.1 An identical result would be 1000 ticks versus 500 ticks.
• The result 1000 versus 990 means that the speed of observer 2 is much slower.
Now we are going to do a thought experiment to simulate this experiment. Observer 1 and 2 are sitting near each other.
1. Both say start and the experiment starts. Observer 2 starts his enigine and moves away
2. After a certain time Observer 2 says: My clock reads 1. Immediate observer 1 says: My clock reads 2.
3. After the same interval observer 2 says: My clock reads 2. Immediate observer 1 says: My clock reads 4.
4. After the same interval observer 2 says: My clock reads 3. Immediate observer 1 says: My clock reads 6.
5. After the same interval observer 2 says: My clock reads 4. Immediate observer 1 says: My clock reads 8.

What Observer 1 does, because he knows the final outcome, whenever Observer 2 shows a clock reading he multiplies this number with 2 such that when the two meet again there clock readings agree.

What you can learn from this experiment that the moving clock runs slower than the clock at rest. You can also say this different that the clock which travels the longest path runs the slowest.
Related to the concepts "proper time" versus "coordinate time" this means that the clock at rest shows coordinate time and the moving clock shows "proper time".

The question is is that always the case. To challenge that we are first performing a thought experiment. However slightly different. Observer 1 and 2 agree that before the experiment start they both will start there engines. They also agree that Observer 2 will not return when his clock reading is 2 ticks.

1. Both say start and the experiment starts. Observer 1 will immediate stop his engine (speed). Observer 1 will move away from Observer 2
2. After a certain time Observer 2 says: My clock reads 1. Immediate observer 1 says: My clock reads 2.
3. After the same interval observer 2 says: My clock reads 2. Immediate observer 1 says: My clock reads 4.
At that same moment Observer 1 will start his engine in turbo speed (much faster than Observer 2) in the direction of Observer 2.
4. After the same interval observer 2 says: My clock reads 3. Immediate observer 1 says: My clock reads 5. And they meet each other. (observer 1 reduces his speed)
What this thought experiment demonstrates is that Observer 2 is now the observer at rest and Observer 1 is the moving observer which travels a certain path relatif to observer 2. However the clock of Observer 1 runs the fastest and his path travelled is the longest compared to observer 2 which stayed at rest. This is in conflict with the first experiment where the path of observer 2 was the longest and his clock runs the lowest.

What this shows that when you consider yourself at rest it is possible that you can perform experiments which demonstrates that your clock runs the fastest and moving clocks slower, but there is no garantee that this is always the case. It is possible that there are moving clocks (relatif to you), which run faster.

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Created: 2 February 2017

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