• 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.
Einstein's theory was framed in terms of kinematics (the study of moving bodies), and showed how quantification of distances and times varied for measurements made in different reference frames.
In this sentence both distances and times are discussed as if they are the 'same'. In reality they are very different and each should be discussed separately. Distances in general are measured with rods and times are measured with clocks (generally assuming that its inner workings is based on the speed of light). If distances are measured with clocks you have a problem.
In 1908, Hermann Minkowski�once one of the math professors of a young Einstein in Zurich�presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space.
Yes, but what he calls Minkowski space is not something that exists in the reality. It is much more a tool to make movement in 2D vissible in 3D.
A key feature of this interpretation is the definition of a spacetime interval that combines distance and time.
That is correct. But this is much more a mathematical construct and very difficult in its pratical usage.
Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.
This raises the question: why using different inertial frames? Newton only used one.

### 1.1 Definitions

Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.
Classical mechanics (as understood by Newton) does not claim that time a rate. In any way time has nothing to do with any observer.
Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.
Any way space has nothing to do with common sense. Newton does not discuss space as such. He discusses the movement of objects through space, mainly our solar system.
In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer.
This sentence is vaque, specific the difference between 'time' and the 'rate of time'. What is clear is a clock as a tool to measure the time.
General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.
Very unclear sentence. What has an observer to do with all of this.
Skip a large part. Goto the text near Fig 1-1.
In Fig. 1-1, imagine that a scientist is in control of a dense lattice of clocks, synchronized within her reference frame, that extends indefinitely throughout the three dimensions of space.
Yes you create this concept of a scientist. In following text I call this a 'Virtual Observer'. Specific the terminology 3D space is correct.
Her location within the lattice is not important. She uses her latticework of clocks to determine the time and position of events taking place within its reach.
Okay.
The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference. In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording.
In some sense this is not an idealized case. This is the real physical reality. The 'Virtual observer' becomes much more like an omnipresent human being which is somewhere and nowhere in the real Universe simultaneous.
A real observer, however, will see a delays between the emission of a signal and its detection due to the speed of light.
That is correct, but (the real observer) has nothing to do with the laws of physics. In relation to the movement of objects only garavity is important, specific the speed of gravity.
In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.
It will be interesting what Einstein's opinion is about this. IMO he only uses the concept 'real observer', specific in relation to 'Relativity of Simultaneity'

### Figure 1.1

Figure 1-1. Each location in spacetime is marked by four numbers defined by a frame of reference: the position in space, and the time (which can be visualized as the reading of a clock located at each position in space). The "observer" synchronizes the clocks according to their own reference frame.
In reality what you should do is to consider only one reference frame. i.e. the universe in its totality. All the clocks are synchronised and all the clocks are at rest.

Figure 1-1 comes from the book: Spacetime Physics: Introduction to Special Relativity (1st ed.). San Francisco: Freeman. by Edwin F. Taylor and John Archibald Wheeler, (1966). ISBN 071670336X.

### 1.2 History

Minkowski's talk 'Space and Time' included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.
spacetime diagrams and the observation and or postulate that speed of light is finite and or constant are two completely different subjects.

### 2.1 Spacetime interval

In three-dimensions, the distance between two points can be defined using the Pythagorean theorem:
dd^2 = dx^2 + dy^2 + dz^2
The emphasis is on the word defined. In general the relation is only true points as part of rigid objects.
Although two viewers may measure the x,y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both (assuming that they are measuring using the same units).
If two observers measure the coordinates of the same points of course they should measure the same distance. The issue is that this is a rigid object.
In special relativity, however, the distance between two points is no longer the same if it measured by two different observers when one of the observers is moving, because of the Lorentz contraction.
If one observer measures the dimensions of a rigid object on a platform and an other observer on a train they should measure the same distance. THere is no Lorentz contraction involved, (At first sight)
The situation is ever more complicated if the two points are separated in time as well as in space.
The situation becomes more complex if you want to measure the distances on a non fixed object, for example on your ticking heart. The PQRS chart gives you an impression.
For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because (from their point of view) they are stationary, and the position of the event is receding or approaching.
That is totally correct. My advice keep it simple.
The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction).
The fact 'that space and time separately are not invariant' is that realy a problem? Does that realy have anything to do with how objects move?
But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time.
Okay
All observers who measure time and distance carefully will find the same spacetime interval between any two events.
That maybe true. The problem is how does each observer measure the distance and time between two events which are not simultaneous in his frame.
Suppose an observer measures two events as being separated by a time delta t or dt and a spatial distance delta x or dx.
When you know dt and dx then to calculate ds is simple. The issue is how do you calculate ds and dt.

### 2.4 Relativity of simultaneity

The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.
Almost by defination anything that an observer observes is relatif. The true issue how to combine the different observations of different observers (in time) in one reference frame.
The three events (A, B, C) are simultaneous from the reference frame of an observer moving at v = 0.
That is correct in Figure 2.6. But how do you know that in reality? How do you know that the speed = 0?
From the reference frame of an observer moving at v = 0.3 c, the events appear to occur in the order C, B, A.
That is correct in Figure 2.6. But how do you know that in reality? How do you know that the speed = 0.3 c?
From the reference frame of an observer moving at v = -0.5 c, the events appear to occur in the order A, B, C.
That is correct in Figure 2.6. But how do you know that in reality? How do you know that the speed = -0.5 c?

### 2.6 Time dilation and length contraction

Different world lines represent clocks moving at different speeds.
in the x direction. Okay
A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time.
This clock is identified as clock C0. And what about all the other observers for each clock with a different speed?
They are also at rest in their respectivily frames?
For a clock traveling at 0.3c, the elapsed time measured by the observer is 5.24 meters
This clock is identified as clock C0.3. That is mathematically correct, but how do you measure this speed of 0.3c?
And what is the situation for clock C0 in the frame of C0.3?
Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x�axis than they would have without time dilation.
This sentence is not very clear.
The issue is that it takes longer for a moving clock to count the same number of ticks compared to a clock at rest and the faster you move the longer the distance travelled for the moving clock to measure the same ticks.
The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O' as running slower in his frame, observer O' in turn will measure the clocks of observer O as running slower.
what should be added: is his frame.
This cannot be true Each experiment that you perform has physical nothing to do with the frame within you perform the experiment.
Each experiment has only one outcome and all the observers have to agree with this result. If an observer claims that the outcome of the (same) experiment in his frame is different then she is wrong.

### 2.7.1 Mutual time dilation

Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts.
Both concepts should be treated separately.
The worry is that if observer A measures observer B's clocks as running slowly, simply because B is moving at speed v relative to A, then the principle of relativity requires that observer B likewise measures A's clocks as running slowly.
The only thing that is important is to describe the experiment in detail and the result of an actual experiment.
What the principle of relativity predicts is in some sense unimportant. The most important issue is how does A measures B's clock and B measures A's clock.
Basically, A and B are performing two different measurements.
Why this sentence?
Why not write: A and B (each) are performing two measurements.
In order to measure the rate of ticking of one of B's clocks, A must use two of his own clocks, the first to record the time where B's clock first ticked at the first location of B, and second to record the time where B's clock emitted its second tick at the next location of B.
How does A do this
Observer A needs two clocks because B is moving, so a grand total of three clocks are involved in the measurement.
What is the reason of this third clock? Used by A?
A's two clocks must be synchronized in A's frame.
Why not A's three clocks must be synchronized in A's frame.
This whole description of the experiment is not very clear.

The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more.
You cannot discuss the twin paradox as a thought experiement. What you can do is to design and describe a certain experiment and predict the result of the experiment based on certain reasoning or logic. The prove is to perform the experiment and test if the outcome is according to the predictions.
This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less.
This logical reasoning is wrong when the result of an actual experiment shows that only one is older and the other one is younger.
In actual experiments no observers but clocks are used. The result shows that the clock readings are different, implying a non-symmetrical experiment.

### 3.2 Relativistic composition of velocities

To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
Beta = v / c
The problem is how do you calculate Beta. To be more specific how do you measure v (relatif to c). This is a very difficult problem. One example is when do you know that v=0. Ofcourse you can claim that you are at rest in your reference frame, but the same can all the other observers claim in their reference frame.

### 3.3 Time dilation and length contraction revisited

and the red line representing the world line of a particle in motion has the equation w = x/Beta = xc/v.
Again here the relation Beta = v/c is used, which causes the same comments as above.

### 4.3 Acceleration

It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required
This is the case for all applications were moving objects, like our solar system, are involved.

The Dewan�Beran�Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.
Also here the issue is: is this a real experiment or a thought experiment.
The main article for this section recounts how, when the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution.
The only way to find the solution is by performing actual experiments.

### 5.1 Basic propositions

In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun.
That is a very strong pronunciation. This seems to indicate that the path is the same without the Earth, Moon and Sun etc
Instead, the satellite moves through space only in response to local conditions.
And what are these local conditions?
In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime.
What are these local inhomogeneities? Why not global?
Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime.
That again requires a discussion about what causes such a curvature.

### 5.2 Curvature of time

The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower.
Gravity makes a clock run slower. A clock is a physical process.
Clocks in a gravitational field do not all run at the same rate.
This is IMO the same as saying: Acceleration causes clocks to run at a different rate. A whole different issue is what is a gravitational field

### Reflection 1 - Worldline in SR versus Newton's Law

The purpose of this reflection is to compare the behaviour of a moving clock using classical mechanics (Newton's Law) versus the concept of worldline (SR or Einstein)

To explain the concept worldline we use the book GRAVITATION by MTW at the pages 21 and page 315.
The equation that describes the situation is the same (tau = sqrt(T^2 - R^2) but the pictures are different. It should be mentioned that the equation is confusing. IMO this should be (tau = sqrt(T^2 - (R/c)^2). However is c=1 then ....
Figure 5A shows the picture of page 21. Figure 5B shows the picture of page 315. For more detail See below.

To explain the behaviour of a moving we use Reflection 1 - Lorentz Contraction
Reflection 1 starts with the following text:
In this experiment the train travels in the x direction with a speed v. In the train are two "mirrors". One at the bottom of the train and one at the ceiling. The two mirrors are used as a clock. The light signal is supposed to go in the y direction.
 ``` B1------B-------B2 /|\ / | \ t1/ | \ tau / | \ D L D / |t0 \ v--> / t2 | \ / -.A-.-.-.-E-.-.-.-C.-. R Figure 3 ``` The time for a light signal to travel (v=0) from E to B and back is 2*t0 In this case t0 = BE / c = L /c The time for a light signal to travel from A to B and back to C is 2 * t1 The time for a "train" to travel with speed v, from A to E to C is 2 * t2 If both arrive at the same time than: t1 = t2 = t AB = D = c*t1 = c*t and AE = EC = v*t2 = v*t Using some arithmatic we get: AB^2= AE^2 + BE^2 = c^2*t^2 = v^2*t^2+L^2 c^2*t^2 - v^2*t^2 = L^2 : t^2 = L^2/(c^2-v^2) : t^2 = t0^2*c^2/(c^2-v^2) Now we get t^2 = t0^2/(c^2-v^2)/c^2 or t1 = t0/ SQR(1-v^2/c^2)). The factor SQR(1-v^2/c^2)) with v > 0 is smaller than 1. You can also rewrite the last two lines and then you get: c^2*t^2 - v^2*t^2 = L^2 or c^2*t^2 - R^2 = c^2*t0^2 That gives c2*t2 = c2*t0^2 + R^2 or tau^2 = T^2 + R^2 or tau = sqrt(T^2 + R^2)
As mentioned above the concept of worldline is discussed in the book GRAVITATION by MTW at the pages 21 and page 315.
The equation that describes the situation is the same (tau = sqrt(T^2 - R^2) or (tau = sqrt(C^2*T^2 - R^2) with T being the time of the observer at rest and tau being the proper time of the moving particle.
The two pictures are different, which can raise some problems. Figure 5A shows the picture of page 21. Figure 5B shows the picture of page 315.
 ``` Z | t+x .Z |\ | \ | \ | \ | \ t |-----.B | /. | / . | / . | / . |/ . t-x .P . | . | . A----------> x Figure 5A (p21) ``` Figure 5A page 21 shows two worldlines: the wordline of a particle at rest. This can also be a clock at rest. B is an event outside this worldline. the worldline of one photon. It is important to mention this. The event B creates a light ray which strike the worldline at Z. P is an earlier event that creates a lightray which coincides with B. The angle PBZ is 90 degrees. It is important to define the units of the z axis. The z axis can represent time, In that case the units are in seconds or years. The z axis can also represent distance = ct. In that case the units are in lightseconds or lightyears. The lines PB and ZB have the same length. The length in that case is sqrt(X^2 + C^2*t^2) The z axis can also represented as an imaginary axis ict. The length in that case of DB and ZB is sqrt(X^2 - C^2*t^2) See also the book "Introducing Einstein's Relativity by Ray d'Inverno" at page 30. In reality (in 3D space) the lightray moves along the x axis to a point at distance x, is reflected and returns back to the origin at 0. The time (duration) t is 2x/c in seconds or years.
There is a problem with this figure. What is its real purpose? IMO this figure is too simple. It does not make sense to discuss a particle with has a speed v from such an 'angle' that its speed is zero.
 ``` ^ t | T B . |\ | . | \ | . | \ | . | \ 0.5T|----.P | /. | .. | / . | . . | / . | . . |/ . . . 0 A--------->Z 0.5R Figure 5B (p315) ``` Figure 5B page 315 demonstrates a non straight worldline. In this case a particle moves in spacetime with uniform velocity v from A to P and back to B. The angle APB is larger than 90 degrees because v In reality (in 3D space) the particle moves along alongs the Z axis to a point at 0.5R, is reflected and returns back to the origin at 0. The lapse of proper time from start to finish ("length of the worldline") is : tau = sqrt(T^2-R^2) Thus the length of proper time is diminished from its stright-line value and diminished moreover for any choice of R whatsoever. In this figure the worldline of a photon is not indicated. When you do that you can see that when the particle is a clock that the moving clock ticks slower than a clock at rest.
The biggest problem in this whole discussion is what means: "length of the worldline". It is not a physical length because the line does not connect two points in 3D.
In Figure 3 there are two light signals involved. There is one lightsignal which represents the clock at rest (t0). There is a second lightsignal which represents the the moving clock (t1,t2 and t). All the lines drawn are in 3D. As such you can calculate the direct relation between a clock and rest and a moving clock.

### Reflection 2 - Real Observer versus Virtual Observer

In 1.1 Definitions the concepts of 'Virtual Observer' and 'Real Observer' are defined.
To explain the two concepts Figure 1.1 is used. Figure 1.1 shows an inertial reference frame of a 2D or 3D grid of equally spaced clocks. A 'Virtual Observer' can instantaneous observe all these clocks simultaneous, which will all show the same time. A 'real observer' can also observe all these clocks simultaneous but the time will be different as a function of distance.
The concept 'real observer' comes from the book Spacetime Physics: Introduction to Special Relativity (1st ed.). San Francisco: Freeman. by Edwin F. Taylor and John Archibald Wheeler, (1966). The concept 'Virtual Observer' is the scientist mentioned in the text. I expect also Figure 1.1 comes from the same book.

Both concepts 'Real Observer' versus 'Virtual Observer' are important concepts to study the evolution of the Universe. The concept 'Virtual Observer' is important because it allows you to study the Laws of Nature i.e. the trajectories of the stars and planets not using the speed of light. Classical Mechanics (Newton's Law) does the same. The major problem with Classical Mechanics is that forces do not act instantaneous but propagate with a constant speed.

### Spacetime Physics

To unravel the laws of nature consider the next picture of the Local Group i.e. https://en.wikipedia.org/wiki/File:Local_Group.svg What this picture shows is an image of the most important Galaxies in our neighbourhood. In fact near each object you should indicate the nearest clock that represent the position as observed by the 'virtual observer'. What the image should show is that all the clock readings are identical. For example all at 12 o'clock to day. The 'virtual Observer' is the ideal person to perform this task. In fact she should do that for one whole year. As such you get for all the objects considered 365 positions equally spaced in time.

What makes this approach IMO so powerfull is because no moving clocks, no real observers and no speed of light issues (including the bending of light) are involved. The attention is towards the physical issues directly related to how objects behave i.e. specific the forces and retarded issues.

### Reflection 3 - 3 objects (points) in space.

Consider 3 objects(points) in space. All the 3 points are moving against a fixed background. What is the best strategy to describe the trajectories of each object? It is also possible that there are 4 objects.

IMO it is the best strategy to use the fixed background as your reference frame or coordinate system.

A different strategy is what SR does. Each object becomes its own inertial reference frame and the object is considered at rest in that frame. The problem is where do you draw the other objects in the reference frame of object n. The problem is horizontal lines in frame n define simultaneous events in frame n. Simultaneous events are the positions of the other objects at the same time in frame n. However these same events are not simultaneous in any other frame, which makes evertything complicated.

### Reflection 4 - Twin paradox

 ``` | 4 | 0 C 0 | . . | a b |. .| 0 a 0 . . a b |. .| 0a b0 | . . | a b | 3 | 0 b 0 | . . | a b |. .| 0 B 0 . . a b |. . 0 a 0 | . . | a b | 2 0a b0 | . . a b |. 0| b 0 . a b |. 0.|A 0 | . . a b | 1 a| 0 | . . | b |.0a .|b0 . a b |o b|0 | a b | ----X----------- L1 L2 Figure 6A ``` ``` | 4 | 0 C 0 | B | 2 | B| 0 2 x . . 2 | |B 02 0 | | B 2 | 3 | x 0 | | 2 B | |20 B 0 . . | 2 0 x | 2 | | 2 0 1 0 | A | 1 | A|1 0 . 1 | 01|A 0 | 1 | | 1 | x | | | 0 | 0 . . |x |0 | | ----X----------- L1 L2 Figure 6B ``` Figure 6A shows one clock at rest and a moving clock with a speed v towards the right. Each clock uses two mirrors, perpendicular to the direction of movement. The two mirrors are identified with the letters L1 and L2. The clock at rest shows 4 ticks. The ticks are identified with the numbers 1, 2, 3 and 4. The path of the two lightbeams is identified with a dot. This becomes clearer after the second tick. The moving clock shows 3 ticks. The position of the mirrors is identified with the letter o. The ticks are identified with the letters A, B and C. The path of the two lightbeams is identified with the letters a and b. This becomes clearer after the second tick. Figure 6B shows the same as Figure 6A. Deleted are the trajectories of the light signals. The clock at rest shows the same for 4 ticks. The ticks are identified with the numbers 1, 2, 3 and 4. The moving clock shows 3 ticks. The position of the mirrors is identified with the letter o. The ticks are identified with the letters A, B and C. What is added for two the clock at rest are two communication signals towards the moving clock. The first one is identified with the letter 1 and is issued after the first tick of the clock at rest. The same for tick 2. The moving clock receives the first signal in between the first and second tick. The second signal at the same time with the third tick. The moving clock also sends a communication signal to the clock at rest.The first one is identified with the letter A and is emitted after the first tick of the moving clock. The second one is identified with the letter B. The clock at rest receives the first signal when its clocks strikes 2 and the second signal when the clock at rest strikes 4.
What are the results of this experiment?
• What figure 6A and 6B show that when the clock at rest shows 4 ticks the moving clock shows 3 ticks. That means when the moving clock stops and returns back to the clock at rest the final result will be that the clock at rest shows 8 ticks and the moving clock 6. That means the moving clock runs slower. Both observers will agree about this.
• Figure 6B shows also shows "internal communication" between the two spaceships. As such Figure 6B shows that the clock at rest shows 4 ticks and at that moment it receieves the signal that the moving clock has performed 2 ticks. Figure 6B also shows that the moving clock shows 3 ticks and at that moment it receieves the signal that the clock at rest has performed 2 ticks. Also this demonstrates that the moving clock runs slower.
• What Figure 6 does not show if the clock at rest is the fastest clock. What the experiment demonstrates is that there are two spaceships which each has one clock. A the beginning of the experiment both clocks are reset and one spaceship is started and moves with a certain speed in one straight direction and back. This space ship contains the moving clock and the other one is the clock at rest. The issue is if this physical true in the sense that is the clock at rest trully the fastest moving clock? I have my doubts. What makes this whole experiment tricky is that it is not symetrical.
• You can perform a twin type experiment starting from earth and decide that the spaceship travelling towards Sigitarius has the slowest clock.
• You can also start from jupiter and decide that the spaceship travelling towards Sigitarius has the slowest clock.
• But that does not validate any claim to decide which clock on either earth or jupiter is the fastest.

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