Figure 1. Ability to compare sizes is eliminated and space and time turn into one dimension that cannot be compared in size for a velocity consisting of two dimensions. Further, the finite and infinite quantities cannot be differentiated and sizes cannot be compared with each other. This property is achieved by s=t., where s is space and t is time. For convenience, this concept is referred to as two-dimensions-without-size Axiom 1.
Property 1 There is only one dimension, space or time, and they are independent of each other. For example, for an event moving at an infinite distance of 1 second, 1 second is a finite quantity, and space at an infinite distance is an infinite quantity. The two quantities are neither equivalent nor dependent on each other. In any other velocity-describing event, the magnitude of space or time is neither equivalent (except for each magnitude itself) nor correlated.
Property 2 There is no instantaneous velocity at infinity. Instantaneous velocity is defined as moving to any point of length in space without time, that is, point 0 corresponds to any point of length in space, a moment of 1 second corresponds to any point of length in space, a certain distance in space (for example, 1 meter) corresponds to any moment in time, and so on.
In Axiom 1, the absence of instantaneous velocity has two implications. First, as mentioned earlier, a single dimension means that there is no velocity with two dimensions that can be compared. Second, the independent existence of space and time does not mean that a certain moment of the clock only corresponds to any distance or position in space; rather, it means that there is only one dimension, space or time. Each value of space corresponds only to itself, not to other quantities, and each value of time corresponds only to itself, not to other quantities; thus, space or time are independent of each other. For example, point 0 only corresponds to point 0 and does not correspond to other quantities (including infinite quantities), while 1 meter only corresponds to 1 meter and does not correspond to other quantities. Unlike the concept of simultaneity/non-simultaneity in relativity, this independence is given a new definition.
The independence of the relationship between space and time can also be illustrated as follows. If we talk about space, it makes no sense for us to talk about time, and if we talk about time, it makes no sense for us to talk about space. A given interval of time does not correspond to any length of space, and a given distance of space does not correspond to any interval of time. Thus, it can be said that, for two different locations in space, whether they exhibit simultaneity or non-simultaneity in time is of no significance; similarly, for two different intervals in time, whether they are in the same or different locations in space is also of no significance.
The absence of instantaneous velocity does not mean that infinite space and infinite time do not exist, just that they exist independently. The absence of instantaneous velocity does not mean that infinite velocity does not exist, nor does it mean that there is only a finite velocity, such as the velocity of light. In Axiom 1, the velocity of light is only a finite speed (300,000 kilometers and 1 second are both finite), thus it is neither an infinite velocity nor a limit velocity. In Axiom 1, the single dimension dictates that each value corresponds to itself and does not correspond to other values, explaining why a clock at some point in the theory of relativity only corresponds to a certain space with an equal distance or the position itself, not to the concept of the distance or the position of others. However, unlike the description of the theory of relativity, the concept of a single dimension described does not deny that infinite values exist, and there is aslo no concept of time shortening or space lengthening here. Details on this will be described in section 6.
3 Principle of special relativity and the principle of the constant velocity of light
Concrete descriptions of the single-dimensional properties of Axiom 1 are provided in this section. By comparing the concept of time and spacein Einstein’s special relativity, the properties of a single dimension can be more clearly understood.
Principle of relativity Here, inertial frames and relativity principles are discussed. If K is defined as a Cartesian frame-of-reference system (i.e., an inertial frame), then another Cartesian frame of reference K’, which is moving uniformly in a straight line with respect to K, is also an inertial system(A rotating Cartesian inertial frame is classified as a non-inertial frame and is beyond the scope of this paper).There are three implications here. First, for any coordinate system K’, all space-time quantities (i.e., spatiotemporal variables) can be expressed in this coordinate system, and all quantities are static relative to K’. For example, consider two velocity events s=ct or s=vt, both of which can be expressed in K’, where c is the velocity of light and v is any velocity. If K and K’ without comparison, then the spatiotemporal variables relative to K at rest cannot be used to distinguish the motion state from the spatiotemporal variables relative to K’ at rest. This is known as the relativity principle. Second, the coordinate system itself and the quantity expressed in the coordinate system can be described by different quantitative terms, such as K’ moving with velocity v1. Any number of values that differ from v1 can be described along the x, y, and z axes of the coordinate system, such as s1=ct1 or s2=v2t2, where c is the velocity of light and v2 is any velocity. Third, in static coordinate system K with a velocity of 0, the velocity at all points is 0. In coordinate system K’ with a uniform velocity of v, the velocity at all points is v. The difference between K’ and K is quantitative, that is, the difference between v and 0. These concepts apply to Axiom 2[6].
Principle of the constant velocity of light It has been proven by Michelson's experiment that the speed of light remains constant in Cartesian coordinates with uniform linear motion at any velocity. A moment of a clock corresponds only to a certain distance or position in space equal to itself and does not correspond to any other distances or positions. For example, 1 second only corresponds to 300,000 kilometers (i.e., 1 second is equivalent to 300,000 kilometers) and does not correspond to other distances.
The implications of the transformation of Cartesian coordinates based on these two principles are as follows:
⑴ In a Cartesian coordinate system that allows instantaneous velocity, relative velocity is meaningful, which indicates that the quantity of velocity for given Cartesian coordinates will vary for Cartesian coordinates with different velocities; that is, the quantity of a given velocity depends on the motion velocity of the Cartesian coordinates. Because a certain moment of a clock corresponds to an arbitrary distance in space, and a certain distance in space corresponds to an arbitrary time of the clock, the transformation between the two Cartesian coordinate systems K’ and K is arbitrary. In fact, this concept is two-dimensions-without-size Axiom 1.
⑵ In a Cartesian coordinate system with a constant velocity of light, the velocity of light is used as the basis for defining space and time (i.e., light time and light space). For optical space coordinate X1 in frame K (stationary coordinates with velocity 0), the corresponding optical space coordinates in frame K’ (a coordinate system with velocity v) is
X’1= 1/(1-v/c)X1(X’1> X1)
This formula is the revised version of the Lorentz transformation. Unlike the relativistic principle, which holds the coordinates of K’ and K to be identical, here the coordinates for K’ and K differ due to the fact that all quantities within frame K are stationary with respect to frame K, but they are not stationary with respect to frame K’.
⑶ The formula X’1 =X1-ct cannot be established for a comparison of the coordinates between the two frames K’ and K because relative velocity is non-existent in relativity; in other words, a minus sign in the formula does not exist.
⑷ It is known from ⑵ that the same proportional extension of K’ and K coordinates for the two coordinate systems is carried out as
X’1:1/(1-v/c)X1 ,
where X’1= 1/(1-v/c)X1
The purpose of this formula is to facilitate a comparison of the coordinate transformation of the two coordinate systems so that the two coordinates are compared at the same length value and the same scale of time.
⑸ According to ⑵, X’1 =1/{1-(v/c)2}1/2(X1-ct)[8]
The Lorentz transformation is meaningless; instead, X’1 is given by the formula
X’1= 1/(1-v/c)X1.
Therefore, the notion that frames K’ and K coincide at origin 0 is meaningless and frame K’ does not start at origin 0.
⑹ From ⑵, because the K’ and K coordinates are different, the two Lorentz transformation equations
X’1 =1/{1-(v/c)2}1/2(X1-ct)
and X’1 =1/{1-(v/c)2}1/2(X1-ct’)
are not valid, and they are replaced by the following two equations:
X’1= ct’1 and X1=ct1
Here X’1= 1/(1-v/c) X1 and t’1 =1/(1-v/c)t1 .
The main characteristics of these two equations that differ from the Lorentz transformation are that their coordinates are given by X’2-X’1>X2-X1 (Figure 2).