Yucong,

   Let me to try to answer your questions

   Question 1.

   Given that the velocity of the light is independent of the state of motion, this means that time and space are always related by a constant for inertial observers. The relationship among all the inertial observers are the Lorentz transformations, which form a continous group SO(3,1) and keeps the Minkowski metric invariant. There is a strict proof of this, once the above concepts are assumed.

   Question 2.

   Time is a fourth coordinate because you interpret geometrically the wave-front of light as

     c2dt2-dx2-dy2-dz2=gabdxadxb

being gab the Minkowski's metric with diagonal principal (1,-1,-1,-1) and the other elements zero.

    It is not necessary to write the time as a coordinate (actually Einstein wrote his paper without using it ), but it is convenient for having a geometrical and intuitive form of how the physical magnitudes are related by means of tensors and their associated invariants.

   I hope that this can helps.

The Lorentz transformation was presented first, as i understand it, as a way to relate electromagnetic phenomena in one frame of reference to that in another. The classic example is that of a charge moving in a magnetic field. It acts as if an electric field is accelerating it. Only the Lorentz transformation (or some similar transformation) would explain why the charge experienced the effect of an electric field. Previously, the Galilean transformation was thought to apply to all physics. Einstein said the Galilean transformation was wrong and the Lorentz transformation was the correct way to relate physics in one frame of reference to physics in another (moving) frame of reference. So, this is a theory, not a mathematical proof.

The time coordinate is not the same as the space coordinates. You can go backward and forward in space (just walk backward) but you can't in time. Minkowski made an important contribution to relativity theory by coming up with the concept of spacetime. Again, this is more like a theory than some sort of mathematical proof. That's the way science is. We can't prove nature, we can only propose ideas that we think nature lives by. Minkowski space has been incredibly fruitful in the physics of special and general relativity.

Please note that their time of publication

Relativity special is 1905.

Space Minkowski is 1909.

  Actually the paper where Minkowski published the tensorial formalism of special relativity applied to electrodynamics was in 1908:

Minkowski, Hermann "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 53 (1908)

  But in fact he gave many talks about it in different public places (Sommerfeld summarized some of them). The mathematical formalism is necessary and natural for understanding properly the physics behind special relativity theory. And much more if a generalization is done as in general relativity.

My answers to two right questions:

1. Einstein chose Lorentz' s Transformations because in those years (in the beginning of the 20th century) those transformations seemed the only solution in order to explain the result of Michelson-Morley's experiment. Already Poincaré, Lorentz and others had demonstrated those transformations modifying properties of ether and assuming ether caused the length contraction and the local time for moving objects. Einstein demonstrated instead those transformations renouncing the concept of ether and assuming a new metric of spacetime. Minkowski reached Einstein's same conclusions with a more elegant demonstration. Today we know Lorentz's Transformations aren't the only solution to problem.

2. Einstein's spacetime and Minkowski's spacetime has some differences but substantially both represent a virtual and imaginary mathematical model of space and of time. That representation seemed necessary in order to explain the result of Michelson-Morley's experiment starting from the postulate of the constancy of the speed of light. Today we know that postulate isn't the only solution to problem.

If we are inclined to accept Galileo and Newton mistaked relative to their interpretation of the physical reality, I don't understand why it is impossible to accept also Einstein and Minkowski can have mistaked and to think their interpretation of the physical reality isn't infallible. Physics is a science in progress and no theory can be considered final and definitive.

Christian,

it isn't possible to affirm there are differences between classic physics and modern physics only when v

Christian,

your question: by what? is fruit of ignorance in the positive meaning of the word. Einstein didn't accept QM and many quantum physicists don't accept SR and GR. Many present physicists don't accept SM. What is strange if a researcher, as me, has a critical position towards those theories. I will be very grateful if you will assume a critical position towards my research.

Dear Christian,

I am obliged to insist because you talk knowing SR and I talk knowing SR and TR. TR is the Theory of Reference Frames in which starting from a critical viewpoint the author reaches a few important results. When you claim I make use of a prerelativistic physics you again demonstrate to not know prerelativistic physics is the Aristotelian physics because the Principle of Relativity was defined by Galileo and used by Newton. Criticism towards modern and postmodern physics in order to get back to classic physics (Galleo, Newton, etc..) represents the neoclassic viewpoint. My criticism is different, it starts from SR in order to go ahead and this represents the contemporary viewpoint. You cite particle accelerators that are no doubt an important laboratory for testing the behavior of elementary particles. The working problem of accelerators concerns time synchronizations and Newtonian mechanics isn' t able to solve this problem. You mistake because you think I am a neonewtonian physicist. I have great admiration for Newton, Galileo, Einstein, Planck, etc.. but it doesn't mean I cannot assume a critical position towards their very important scientific viewpoints. Returning to problems of particle accelerators I think now the best thing to do is that you raise me one problem and I will see if it is surmountable in the order of TR. On the other hand I was always available to give explanations about the Theory of Reference Frames.

Greetings.

Yucong,

It is very simple.

There is three spatial coordinate and one time scalar parameter independent of each other. You can have a neat mathematical representation if you add one extra spatial coordinate using scalar parameter t as w=c*t where t represents the state of a reference clock.

You can use Galilean transformations for small velocities in this form easily using matrix notation this way and works fine without claiming existence of mysterious time space .

It appears that on the round trip of a light beam any inertial system may give the same average speed which could  be explained assuming the speed of light being the same  in both direction in any inertial system.

From that Lorentz transformations were derived. They show that the length of objects in the direction of motion may change and clock rates must differ accordingly. Everything else is all dressing it up in different clothing.

Christian, I did and I am searching for doing what you suggest me. Thanks for your care.

Best wishes for your research.

WHAT ABOUT THIS TRANSFORMATION x=R^2(x'-vt') t=R^2(t'-vx'/c^2) y=Ry' and z=Rz' It expresses about the wave-particle duality and the uncertainty principle. Space is invariant and it is only time related to time dilation. My transformation is vacuum energy dependent, and by my equivalence principle gravity is gauge theory. Now, according to my transformation

x=R^2(x'-vt') t=R^2(t'-vx'/c^2) y=Ry' and z=Rz' Now this transformation express about the wave-particle duality and the uncertainty principle and it is vacuum energy dependent. Now if we consider t-term in my transformation we get: t-=R^2(t'-vx'/c^2) and t+=R^2(t'+vx'/c^2). Now if we consider x'=L which space is invariant in my transformation we get delta t=2Lv/c^2 R^2 where R is the Lorentz factor. That is exactly the same explaining the Sagnac effect in the framework of the ether theory. Review this paper http://arxiv.org/pdf/1404.4075v1.pdf