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This question is related to this answer of John Rennie.

He says:

The length of the red line is the same in both figure 1 and figure 2

I guess his meaning of red line is the space-time distance traveled by the same particle between points (space-time states) A and B. It makes sense.

Then he adds:

For special relativity we need to extend this idea to include all three spatial dimensions plus time. There are various ways to write the line element for special relativity and for the purposes of this article I’m going to write it as:

$$\mathrm ds^2 = - c^2\mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 +\mathrm dz^2$$...

note that we can’t just add time to distance because they have different units - seconds and metres - so we multiply time by the speed of light $c$ so the product $ct$ has units of metres.

  1. I think this (they have different units) is not a good reason. Because he is talking about fundamental basic concept of the physics. He could easily say that "so, we discovered that our understanding about space and time was wrong and now we have space-time with four new coordinates $(t,x,y,z)$ and the unit of them are the new unit (for example) we name it $\mathrm {st}$. $\mathrm {st}$ is a new unit not meter nor second and after now meter and second are meaningless".

And then, he could write: $$\mathrm ds^2 = \mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 +\mathrm dz^2$$

But I guess his true reason of that formula is the results of the experiments not units difference.

  1. Why does he use the speed of light $c$ in that formula? Is this obtained from his strong (at least I think that principle is very strong) principle (The length of the red line is the same in both figure 1 and figure 2)?

  2. Why do we need to improve our physics?

I think because we want our physics matches with the results of the experiments. But, there are some problems here:

3.1. How are we sure that what we measure in the experiments, is the same thing that we want to (or we must) measure? For example, consider to speed. How are we sure that we exactly measure the $\frac{\mathrm dx}{\mathrm dt}$. Because as far as I know measurement needs to a time interval and the speed $v=\frac{\mathrm dx}{\mathrm dt}$ is defined for a single time instance.

3.2. Assuming we must improve the physics, why should we change the definition of an undefined concept? (As far as I know, time is an undefined concept like point in geometry.) Why do we not change definitions those are defined by ourselves like velocity, kinetic energy, etc.? I think talking about time dilation is completely similar to talking about size of points on the plane. It is similar to that we say some points on the plane are bigger than the others! enter image description here Maybe it is true (maybe some points are bigger than the others in fact) but I think we cannot discover it because we don't know what point is (it is an undefined concept). As far as I remember, I have learnt that we have some undefined concepts and we define other concepts by getting help of them but we never can define them because if we could, they weren't called undefined. Can someone please define point for me? If he/she can, I will prove that there is no time dilation!

Note that this is not a definition for point: "A plane is created by points" because I will immediately ask "What is the plane?" And so on.

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    $\begingroup$ Worth reading Taylor and wheeler - spacetime physics, he treats both space and time with the dimension of meters, and completely ignores the constant $c$, reading this book should probably answer all your questions $\endgroup$ – Courage Jul 10 '16 at 10:39
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    $\begingroup$ Definition: A "point" in geometry is a location. It has no width, length, or depth. It has no volume; only coordinates of the location. Now, please prove to me there is no time dilation. $\endgroup$ – Jim Jul 11 '16 at 12:33
  • $\begingroup$ @Jim OK, so what is the location? What is the width? What is the length? What is the depth? What is the volume? Is location equal to coordinates of the location? And finally, what is the coordinates? $\endgroup$ – lucas Jul 11 '16 at 12:37
  • $\begingroup$ @lucas There is NO length, width, depth, or volume. None. Nada. You can't ask what it's length is because it has none; that is not a property of a point to which a value can be ascribed. The location is whatever you choose it to be. I haven't specified which point, only what a point is. The coordinates specify the location. Example: (0,0,0) specifies the origin. That is a point. Literally anywhere else is also a point. The coordinates of the point are like its name. And that is literally the only thing there is to a point. $\endgroup$ – Jim Jul 11 '16 at 12:47
  • $\begingroup$ @Jim "There is NO ..." So, you have defined point by nothing! Sorry, but I don't agree with you. You can get useful guide from mathematicians about undefined concepts. $\endgroup$ – lucas Jul 11 '16 at 12:52
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He says: The length of the red line is the same in both figure 1 and figure 2. I guess his meaning of red line is the space-time distance travelled by the same particle between points (space-time states) A and B. It makes sense.

Actually, it doesn't. Because there is no motion in spacetime. See relativist Ben Crowell saying so here: "Objects don't move through spacetime. Objects move through space". The spacetime interval is the same because time is a cumulative measure of local motion, and when you move fast through space your local motion is of necessity reduced. Otherwise the local motion plus macroscopic motion would exceed c, which can't happen because of the wave nature of matter.

Then he adds: "For special relativity we need to extend this idea to include all three spatial dimensions plus time. There are various ways to write the line element for special relativity and for the purposes of this article I’m going to write it as: $\mathrm ds^2 = - c^2\mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 +\mathrm dz^2$.

Yes, note the minus sign, and see Einstein's derivation.

I think this (they have different units) is not a good reason. Because he is talking about fundamental basic concept of the physics. He could easily say that "so, we discovered that our understanding about space and time was wrong and now we have space-time with four new coordinates $(t,x,y,z)$ and the unit of them are the new unit (for example) we name it $\mathrm {st}$. $\mathrm {st}$ is a new unit not meter nor second and after now meter and second are meaningless".

You're right. Because what we're really dealing with isn't length or time, it's motion. We define our second and our metre using the motion of light.

And then, he could write: $\mathrm ds^2 = \mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 +\mathrm dz^2$

You missed the minus sign out. That apart the expression is correct, but it's only really a restatement of Pythagoras's theorem. See the simple inference of time dilation on Wikipedia.

Why does he use the speed of light $c$ in that formula? Is this obtained from his strong (at least I think that principle is very strong) principle (The length of the red line is the same in both figure 1 and figure 2)?

The speed of light is in the expression because "A light-signal, which is proceeding along the positive axis of x, is transmitted according to the equation x = ct". See Einstein's derivation.

How are we sure that what we measure in the experiments, is the same thing that we want to (or we must) measure?

Because of pair production and electron diffraction and the wave nature of matter. Hence matter behaves like the light between the parallel mirrors.

Assuming we must improve the physics, why should we change the definition of an undefined concept? (As far as I know, time is an undefined concept

It isn't. See what I said about it here. As Einstein said, time is what clocks measure. And if you take a look at what a clock actually does, if you open up a clock and take a cold scientific look at the empirical evidence, you will see cogs turning or a crystal oscillating. You will see that the clock features some kind of regular cyclical motion along with something like gears or a counting device, and it gives some kind of cumulative display of the thing we call "the time". It's that simple.

I think talking about time dilation is completely similar to talking about size of points on the plane. It is similar to that we say some points on the plane are bigger than the others!

It isn't I'm afraid Lucas. Time dilation is just a reduced rate of local motion. Again, it's very simple.

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  1. We are only allowed to add quantities with the same units. Time and distance can't be added just like apples and oranges shouldn't be mixed. To add them, one must first convert them to quantities of the same unit, and a speed is needed.

  2. The speed of light in the vacuum $c$ is the right speed that produces geometry of space and time as seen in the experiments. In particular, it's the limiting speed that can't be surpassed. For example, the protons at the LHC move at the speed 99.9999% of the speed of light or so. For all other speeds $v$, it's not true that $v$ is the maximum speed that massive objects may approach.

  3. Physics needs to be improved all the time for tons of reasons. In particular, physics needed to be improved by switching to special relativity because Newton's theory of motion was ruled out by the Morley-Michelson experiment - it died – and science cannot work with theories that are dead i.e. known to be wrong. In Einstein's theoretical viewpoint, the physics needed to be fixed because Newton's mechanics and Maxwell's theory disagreed what should happen when a source of electromagnetic radiation approached (or surpassed) the speed of light.

The difference between what you wrote and what John Rennie wrote is that the latter is the verifiable foundation of modern science that has been verified in all the experiments and agrees with everything that people have ever observed, while your changes disagree with the experiments that may be done in these conditions – speeds close to the speed of light.

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You sound confused but it cannot be otherwise - you are trying to make sense of the absurd consequences of Einstein's 1905 false constant-speed-of-light postulate. I am not going to continue (gatekeepers will delete my answer anyway) - just to note that Motl's "Newton's theory of motion was ruled out by the Morley-Michelson experiment" is both silly and wrong. The experiment CONFIRMED Newton's emission theory of light:

http://philsci-archive.pitt.edu/1743/2/Norton.pdf John Norton: "In addition to his work as editor of the Einstein papers in finding source material, Stachel assembled the many small clues that reveal Einstein's serious consideration of an emission theory of light; and he gave us the crucial insight that Einstein regarded the Michelson-Morley experiment as evidence for the principle of relativity, whereas later writers almost universally use it as support for the light postulate of special relativity. Even today, this point needs emphasis. The Michelson-Morley experiment is fully compatible with an emission theory of light that contradicts the light postulate."

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  • $\begingroup$ @Pentcho Valev, your statement"Einstein's 1905 false constant-speed-of-light postulate" isn't true; this postulate of Einstein's has been proven time and time again in experiments. $\endgroup$ – heather Jul 10 '16 at 12:04
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    $\begingroup$ "this postulate of Einstein's has been proven time and time again in experiments" There are no such experiments. At least the Michelson-Morley experiment is not one of them. Others - e.g. the Alväger experiment - are inconclusive. $\endgroup$ – Pentcho Valev Jul 10 '16 at 12:09
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    $\begingroup$ @heather : you should read the Einstein digital papers: "the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned." "_ $\endgroup$ – John Duffield Jul 10 '16 at 14:29

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