PACS Numbers: 03.30.+p
1 Introduction
A century ago, Newton and Galileo's absolute view of time and space was replaced by Einstein’s special relativity, in which the Galileo transformation formula was substituted by the Lorentz transformation formula. Special relativity based on the principle of relativity and the principle of the constant velocity of light and space-time transformation between inertial system observers are characterized by the observer-independent velocity scale c (i.e., the velocity of light). Twenty years ago, a modified theory of special relativity was postulated by Amelino-Camelia as doubly special relativity[1] (also referred to as deformed special relativity), which is based on quantum-gravity arguments. Doubly special relativity, the new relativistic theory in which the space-time transformations between inertial system observers are characterized by two observer-independent scales (in addition to the light velocity scale, there is a second new observer-independent length/momentum scale, the Planck length/momentum). Further, doubly special relativity predicts that a value of Ep ≈ 1028 eV can be regarded as the maximum value of energy and momentum for fundamental particles, while length/momentum remains unchanged in the space-time inertial frame under the Planck scale.
In mathematics, the mathematical basis for further analysis of relativity has been provided by new mathematical models, such as boundary value problems[2][3] and discrete mathematics[4][5] ( temporal and spatial discontinuity).
There are two concepts that are logically debatable in relativity. Firstly, the velocity of light is a finite speed, but it is also a limited speed, which indicates that there are infinitely many different speed values to choose from between the velocity of light and a velocity of 0. This concept can be seen in the Lorentz space-time transformation formula:
X’1 =X1-vl/{1-(v/c)2}1/2 and l’1 =l-vx1/{1-(v/c)2}1/2
Secondly, the physical quantities remain unchanged in different inertial systems; in other words, if we do not assume the two inertial systems with different motion states, then the two inertial systems cannot be distinguished. This is called the principle of relativity.
In this paper, based on Axioms 1[6] and 3[7], four perspectives of inertial systems that differ from Einstein’s special relativity are proposed: ① the principle of relativity can be replaced by a concept in which an inertial system is only a specific quantity, ② relative velocity is meaningless; any velocity is constant with respect to any other velocity, ③ a definition of velocity that includes two dimensions (space/time) is not valid and there is only one-dimensional space or time, and ④ unlike two dimensions, where the magnitude is determined by the ratio, with one dimension, it is determined by unit values.
2 There is no instantaneous or two-dimensional velocity
The relationship between space and time can be expressed in terms of velocity: v = s/t, where v is the velocity, s is the length of space, and t is time. There are two implications regarding the relationship between space and time arising from this formula: ① s and t are dimensions that can be compared, and ② s and t are equivalent. For example, for two velocities of 3 m/s and 4 m/s, 1 second should be equivalent to 3 meters for the first velocity, while 1 second should be equivalent to 4 meters in the second. From Axiom 1, it is known that each length value is specific (because the unit value differs, each length value can only be itself), thus the unit second of time is also specific. If 1 second is equivalent to 3 meters, then 1 second is not equivalent to 4 meters, so the definition of velocity is meaningless in Axiom 1. Another example is that 1 second is a finite number in Axiom 1, so an infinitely great velocity (one second goes into the infinite distance) is not valid in Axiom 1, because 1 second cannot be equivalent to an infinite length. Therefore, in Axiom 1, because the conditions for the two-layered meaning of v = s/t cannot be satisfied simultaneously (that is, the size can be compared and the equivalence can be met simultaneously), a definition of velocity based on two dimensions representing the relationship between space and time cannot be established.
If the two-dimensional property of permissible velocity is true, then certain conditions must be met. As can be seen in Figure 1, the properties that can be compared are eliminated. For a velocity composed of two dimensions, time and space are reduced to dimensions that cannot be compared, that is, the finite and infinite quantities cannot be distinguished, nor can the sizes be compared with each other. For example, 1 meter or 1 second in a general sense can represent any quantity. For the convenience of the following description, this concept is defined as two-dimensions-without-size Axiom 1. Two-dimensions-without-size Axiom 1 is a paradox and meaningless, so the conclusion can be drawn that, in Axiom 1, only one dimension exists and space and time are independent of each other. It is now important to look at some basic properties of Axiom1.