This paper analyzes the problems and contradictions that occur when the traditional special theory of relativity which uses the speed of light in a vacuum as an invariant constant, studies the propagation of light in media. These problems are re-examined and discussed with the special theory of relativity of variable speed of light. The transformation relationship of the characteristic quantities describing light wave frequency ν, phase velocity w and the direction angle α of the wave normal between the two inertial coordinate systems in vacuum S and in medium S' were derived; combining the transformation of the light ray speed u which describes light granular motion, the de Broglie wave-particle velocity relationship in the vacuum u w = c2 is νextended to the medium to become u' w' = c'2. Corrected the approach of the traditional special theory of relativity when dealing with these problems, in which the transformation from the space-time coordinates to the relevant physical quantity is limited to the half-sided transformation of the media into the vacuum (not two sided transformation), so that the resulting contradictions and problems are all solved. Optical experiments that support the traditional special theory of relativity, such as the Fizeau experiment and the Michelson-Morley experiment, not only still support and agree with the generalized special theory of relativity with variable speed of light, but also obtain a more correct and satisfactory interpretation from it.

In this paper, the special theory of relativity in different media is established, based on the fundamental invariant of the space-time four-dimensional space x2 + y2 + z2 - c2 t2 = x'2 + y'2 + z'2 - c' t'2 . First of all, the inertial coordinate system is strictly defined in mathematical language. The inertial coordinate system that uses the actual measured different speeds of light as the limit speed still retains its most basic characteristics as an inertial coordinate system. Then, the space-time coordinate transformation and velocity transformation formulas between inertial coordinate systems with different light velocity are derived. These results not only break through the limitation of "vacuum", but also all are exactly the same as the conclusions of the traditional special theory of relativity when c = c' ; and when c ≠ c' give the new physical content. This all lifted the threat of the theory of relativity by the speed of light experiment, making c = c' ; and c ≠ c' both inclusively under the basic point of view of the theory of relativity; which will inevitably broaden the way of using relativity to deal with physics problems and clarify many problems left over in the study of relativity. The article discusses the problem of relativistic kinematics involving the measurement of time and space, correctly interprets the effects of “ruler contraction” and “clock retardation”, and uncovers and correctly answers the “clock paradox” that accompanied the birth of relativity. For two motion systems S and S', that are separated from each other by constant velocity, at any time and where, the product of the proper time elapsed evenly and uniformly and the speed of light in the respective system are equal, cτ = c' τ'; and the product of the coordinates time read out in observing and recognizing the other party's proper time and the speed of light in the respective system are also equal, ct = c' t' . It is confirmed that the product of any moving individual's uniform disappearance proper time and its measured speed of light remain unchanged; and the proper time cannot be determined purely by the individual's subjective way. Deduced the uncertain relationship between the proper time and the coordinate time for an inertial coordinate system which was not noticed by the traditional special theory of relativity. Remind the practical astronomy workers who do the time measurement and the time service work to understand that it is impossible to equate practical scientific coordinate time and the proper time of ideal uniform disappearance (the so-called “Ephemeris Time”). Thereby pay attention to the impact of this uncertain relationship on the time measurement and the time service work, and propose ways to verify. Subsequent work will use this expanded special theory of relativity to conduct a comprehensive review of related physics, which will inevitably extend to issues that have not been or cannot be examined by traditional special theory of relativity.

In the general theory of relativity the four-dimensional space-time described the accelerated motion or moves in a gravitational field of a mass body, from the perspective of integral geometry, although it is a curved Riemannian geometric space, but for any instantaneous position of the moving mass body, there is a local Flat Space of Riemannian geometric space. The local Flat Space is a Mincowski space in which the inertial coordinate system can be used in the local small area. Between the proper coordinate systems of two interacting moving masses, or between a series of follow-up proper coordinate systems experienced by a mass body moving in any way, there must be a coordinate transformation relationship similar to the traditional special theory of relativity. However, they have an important difference: In these instantaneous local inertial systems, the speed of light is no longer the constant c of vacuum, the effect of gravitational field or acceleration on the speed of light is the same as that of a medium with a dielectric constant of ε and a permeability of μ. Using the special theory of relativity with variable speed of light that the author has established can discuss relevant relativity physics issues in these instantaneous local inertial systems. This article uses the special theory of relativity with variable speed of light to infer the functional relationship between a moving mass and the change of speed. In addition to obtaining the traditional continuous increasing function relationship, a step function relationship with stepped discontinuous changes is also obtained. At the same speed, the mass can have two values, such as a ladder upgrade one level; the same mass can be matched with two different speeds, such as one step extension forward on the same step stair. From the perspective of the increase in speed, the mass is stagnant on the step platform (the speed increases, the mass does not change), and it jumps in the step up ladder (the speed does not change, the mass has a jump change). This obviously incorporates the main image of quantum theory into the theory of relativity, which is the result that all physics researchers care about and expect.

On the basis of establishing the special theory of relativity with variable speed of light and obtaining the step function relationship between mass and speed, this article further seeks the proper collocations of mass, energy and momentum allowed by the "ontology" of moving masses which are in various stages of motion properties or in different physical environments. Three ontology collocation types are obtained. If we consider the basic fact that the lower the energy, the more stable it is, the real physical world ranges from astrophysics issues such as white dwarfs, red giants, and celestial space speeds, to the various light and heavy elementary particles existence, combination and performance，which qualitative knowledge can all be derived from the "ontology collocation ". Two of these three types of collocations are derived from the mass-velocity step function relationship contented of quantum properties, so all the quantum phenomena of modern physics will not be obliterated. It is hoped that the modern physics knowledge accumulated in the laboratory and the scattered various theories will be explained under the dominance of a classic theory. The article also deduced the conversion relationship between the inertial system S and S’ of the three collocation types of mass, energy and momentum of the moving mass. Derive the upgrade and downgrade law of the complete special relativity system, this also greatly expands the way to understand modern physics from the theory of relativity.

In the general theory of relativity, the four-dimensional space-time of describing a mass body accelerated motion or in a gravitational field, although it is a curved Riemannian geometric space from the perspective of "integral geometry", but for any instantaneous position of the moving mass body, there is a local Flat Space of Riemannian geometric space. The local Flat Space is a Mincowski space in which the inertial coordinate system can be used in the local small area. Between the proper coordinate systems of two interacting moving masses, or between a series of following proper coordinate systems experienced by a mass body moving in any way, there should be a coordinate transformation relationship similar to the traditional special theory of relativity. However, they have an important difference: in these instantaneous local inertial systems, the speed of light is no longer the constant c of vacuum, the effect of gravitational field or acceleration on the speed of light is the same as that of a medium with a dielectric constant of ε and a magnetic permeability of μ. Using the special theory of relativity with variable speed of light that the author has established can discuss relevant relativity physics issues in these instantaneous local inertial systems. This article uses the special theory of relativity with variable speed of light to derive the functional relationship between a moving mass and the change of speed. In addition to obtain the traditional continuous increasing function relationship, a step function relationship with stepped discontinuous changes is also obtained. At the same speed, the mass can have two values, such as a ladder upgrade one level; the same mass can be matched with two different speeds, such as one step extension forward on the same step stair. From the perspective of the increase in speed, the mass is stagnant on the step platform (the speed increases, the mass does not change), and it jumps in the step up ladder (the speed does not change, the mass has a jump change). This obviously incorporates the main image of quantum theory into the theory of relativity, which is the result that all physics researchers care about and expect. K E Y W O R D Scoordinate transformation in the flat space of riemannian geometric space, instantaneous local inertial coordinate system, relationship between mass and speed, special and general relativity, step function relationship of mass with speedThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.