The Space-evolution Frame as Alternative to Spacetime

—As an alternative to Minkowski spacetime frame, this paper propose a four-dimensional Euclidean space that com- bines three spatial dimension with proper time instead of time. We call it space-evolution, in which proper time is interpreted as evolutionary position, time is considered as world line length and is absolute. Space-evolution frame provide a new perspective for our understanding of time, space and special relativity. The new frame is self-consistent without losing compatibility to special relativity, the Lorentz transform and its predictions could be derived geometrically by simple coordinate rotation.


I. INTRODUCTION
In 1905, Einstein introduced special relativity in its modern understanding as a theory of space and time [1]. Around 1907, Minkowski recognized that the work of Hendrik Antoon Lorentz (1904) and Einstein on the theory of relativity can be understood in a non-Euclidean space. In 1908, Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. In the publication [2] Hermann Minkowski introduced the the concepts of spacetime interval, proper time and worldline. Subsequent work of Hermann Minkowski, in which he introduced a 4-dimensional geometric "spacetime" model for Einstein's version of special relativity, paved the way for Einstein's later development of his general theory of relativity and laid the foundations of relativistic field theories. Though Minkowski took an important step for physics, spacetime is, in particular, not a metric space and not a Riemannian manifold with a Riemannian metric. In Minkowski spacetime, the position of an event is given by x, y, z and time t. Unfortunately, space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events.But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure time and distance carefully will find the same spacetime interval between any two events. Then the spacetime interval(∆s) 2 between the two events that are separated by a distance ∆x in space and by ∆ct in the time coordinate is: (∆s) 2 = (∆ct) 2 − (∆x) 2 (1) It seems mathematically feasible to write the equation as (∆ct) 2 = (∆s) 2 + (∆x) 2 (2) to make the equation more elegant, but such rewrite lack of motivation from perspective of physics. This paper propose a brand new reinterpretation of time, space and event interval, so that they could be described with a standard Euclidean space. We call it "space-evolution". In section 2, we clarify the concept of evolution of object, in further, evolutionary position and evolutionary speed are introduced as physical quantities that similar to spatial position and spatial speed. In section 3 we integral evolutionary position with three spatial positions to establish the four-dimensional Euclidean space-evolution frame. Section 4 discusses the coordinate transformation between different observers, with which the Relativistic effects are explained. Section 5 derives the Lorentz transformation from the rotational transformation of spaceevolution frame.

II. EVOLUTIONARY POSITION AND EVOLUTIONARY
SPEED.
Evolution in this paper refers to progress of observed rest subject, e.g ageing of person, timing of clock, stellar evolution, decay of element. we exclusively discuss isolated subject whose internal process is not affected by any field. Clock is perfect model to explain some important concepts in this paper, its reading at any time is uniquely determined by inner structure and law of physics, and the ticking is independent to spatial position. Evolutionary position, correspondingly, is a physical quantity which determines status of subject's evolution process. For a rest clock, the convenient way to coordinate/calibrate evolutionary position is by its reading; For any other subject the evolutionary position can be calibrated by the reading of a mass-less clock stick with it. In some sense, evolutionary position is reinterpretation of proper time, we represent it with τ . Though such reinterpretation is quite essential, the term "evolution" is preserved; it also helping to avoid confusion with coordinate time. The term "time" specifically refers to coordinate time(denoted by t) in this paper.
The original idea was inspired by the time dilation in special relativity, which saying that a moving clock with spatial speed u ticks slower in perspective of a rest observer. The time between two ticks for moving clock(∆t ′ ) and rest clock(∆t) has relation ∆t ′ = 1 We define evolutionary speed v τ = ∆t/∆t ′ to quantify raletive ticking rate of the moving clock, or progressing rate of any other process. Denote spatial position with x, define normalized spatial speed as Equation (5) put a constrain to evolutionary speed: (0 ≤ v τ ≤ 1); and according to general relativity, the ticking rate of a clock can be slowed down by a gravitational field, though there is no way to speed up ticking rate of a clock. The above facts indicate that evolutionary speed is capt at 1 just like spatial speed. The upper limit for spatial speed has been noticed and widely discussed. However, the fact that has been neglected for long time is, aging rate of any evolvable subject also have a upper limit.
According to Equation(5), any subject with spatial speed v x is evolving with speed v τ , obviously v τ will drop to zero if v x approach to 1c. Equation(6) strongly suggests that v τ and special speed v x are components of a unit velocity vector, thus we consider the evolutionary position as a component of coordinate system in next section.
Besides, the interpretation of evolutionary position suggest that it is more qualified than coordinate time to unite with three spatial positions, to form a four-dimensional coordinate system. Evolutionary position describes status the subject itself like spacial position did; τ are also independent/orthogonal to spacial position: an astronaut would stay at same position if he is aging with full speed, and wouldn't age if he is traveling with speed of light. Though time is reading of a rest clock that irrelevant to the subject, and it is quite distinguishable to spatial position components.

III. THE SPACE-EVOLUTION FRAME
Minkowski space differs from four-dimensional Euclidean space, because time is, unlike 3 spacial coordinate, reading of lab clock rather than description of the subject itself. In this section we fuse the evolutionary position τ space and the three spatial position x, y, z into a single fourdimensional manifold. We call such new coordinate system "space-evolution", which is distinguishable to "spacetime" it can potentially be a promising alternative to the well known Minkowski spacetime.
Status of a subject can coordinated by a raw vector we refer to as four-position space-evolution is the collection of such points. First we assume that such space-evolution is Euclidean. Similar to spacetime, we define worldline of an subject as its path traces in 4-dimensional space-evolution. Accordingly, define an infinitesimal interval between two status as line element dl, which is coordinate-independent, written as Equation (8) can be rewrite as a differential equation By comparing Equation (9) with Equation (6), we speculate that the worldline element and time may related as dl = c dt (10) In space-evolution diagram, the stretch of worldline drives variation of 4-position of the Endpoint; in physics point of view, time drives succession of subject's evolutionary position and spatial position. Therefore this paper accept worldline length as geometric representation of time, in other words, the stretch of world line is driving only by growth of time. In further we rewrite Equation (9) as is a four-dimensional unit vector called four-velocity, the tangent vector of the world line at a point in space-evolution. It is quite differ from that of in special relativity. A infinitesimal stretch of world line (or time interval) could be expressed as The world line will be curved if the subject experienced acceleration, the extended discussion about this will be carried out in other papers. Assume the coordinate of a subject when time l = a is P (a) = τ a x a y a z a , then its coordinate when time l = b can be calculated by accordingly, the time spent on the process is equal to stretched arc length Or if the entire world line is known to us, the coordinate of the subject could be written as a set of univariate time series For the sake of simplicity and two dimensional display, from now on we assume y = z = 0 and v y = v z = 0. Figure 1 drawing a space-evolution diagram with worldline to illustrate the geometric relationship between time, evolutionary coordinate, and spatial coordinates. The worldline is the trace of a subject in the time period 0 ≤ l ≤ L. The subject was accelerated, thus the worldline is curved. It should be pointed out that the limitation of spatial speed is implemented in a geometric fact, that arc length dl is always not shorther than its spacial projection dx, thus v x = dx/dl ≤ 1.
One may argue that this paper just simply switches the role of coordinate time and proper time in spacetime diagram, but in space-evolution, the concept of a point and a world line is quite different. The spacetime diagram describe events, though the space-evolution diagram describe evolve-able subjects. An event is something that happens instantaneously at single point in spacetime, represented by a set of coordinates x, y, z and t. The spacetime observer wait until subject reaches specific evolutionary stage, then record the time of rest clock(length of stretched world line) and the spacial position of subject. But in perspective of space-evolution, a event is considered to be subject that evolve to the evolutionary position that specified by the event. Take a time bomb as example, spacetime describe explosion, track its location and time when it happen. But space-evolution is description of the bomb, track its timer reading and spacial coordinate, explosion is a specified evolutionary position(stage) marked on τ axis, occurs when .
Before we proceed, a fundamental postulate must be posited: Synchronous World Line Postulate: In flat spaceevolution frame, all subject's world line stretch the same length between two observations. This postulate is based on some common sense, that in different location of flat space, rest clocks of identical structure tick with same rate, and upper limit of spacial speed for all subjects are the same. The postulate saying that the universe has a unified time, and the world lines of all matters in the universe stretch synchronously over this time, we call it 'universal time'. In rest of this paper, time, l and ct are all represent the universal time. We would speculate that the postulate also holds in gravitational field, as long as the coordinates system is established in a proper way. The postulate allows us to measure the worldline length of all other subject by checking the reading of a rest clock. A time interval (c△t) that recorded by resting clock is, actually, mutual geometrical length(△l) that all subject's world line stretched. Fig. 2 demonstrate events in space-evolution diagram. We define the event-auxiliary-line as a auxiliary line that perpendicular to evolution axis and through the evolutionary position that determines the event occur. An event is the cross point of a world line and event-auxiliary-line. An example for understanding, the expression "the observer sees the event happen on the subject" is equal to "the observer sees the subject reach the evolutionary position that defined by the event".
Two events may or may not occur on same subject, if dose, may or may not on a stationary subject. Space time diagram doesn't specify those conditions. But there is a hidden premises, spacetime observer always presume that the event happens on rest object before coordinate transformation, so in space-evolution diagram that preparing for Lorentz transformation, the world line of this subject should parallel to τ axis.
In frame of space-evolution, a rest clock record a evolution interval △τ = c△t = △l but zero spacial interval, thou a light speed clock shows zero evolution interval to observer but its spacial interval △x = △l. The speed of light c is treated as conversion constant used to normalize spatial distance and evolutionary distance. Though we haven't find a way to properly describe photon and EM field with space-evolution as they have no evolution component, treating photon as light speed subject will cause trouble in coordinate transformation.

IV. ROTATION OF SPACE-EVOLUTION FRAME
Consider a space-evolution frames S, define a subject B to have coordinate [τ x] and velocity ? Given that v is a coordinate invariant vector in a Euclidean space, we use rotation instead of addition and subtraction to describe changes of v. First we aim to find S ′ , in which C is stationary. Differ from space-time, in space-evolution it is not necessary to stick the origin of S ′ to the moving subject so that the subject is relatively stationary. Instead, we just need to rotate S with angle θ, which satisfy sin(θ) = µ x and cos(θ) = µ τ .
Where [1,0] is the velocity of subject C in frame S ′ , indicating the subject C have zero spatial velocity but evolve with speed of light, we identify S ′ as frame of reference for the subject C. Accordingly, B's coordinate and velocity in new frame S ′ could be calculated with the same transformation Naturally, in such rotational transformation: • distance between any two subject points is preserved, including the origin. Therefore infinitesimal world line length dl is preserved, even the shape of all world lines are preserved.
• angle between any pair of velocity vector is preserved.
• the origin is preserved and fixed, rather than be a regular evolvable subject. Such transformation is self-consistent from a geometric point of view, but quite differ from Lorentz transformation of velocities. Though the velocities in Lorentz transform refer to "speed of event" rather than "speed of subject". The question is, dose Eqn.17,18 explain physical phenomena properly, especially those in special relativity? Fig. 3 demonstrates that the frame rotation expressed by Eqn. 17 is able to explain the typical consequences derived from the Lorentz transformation. Frame S is represented by solid axis and frmae S ′ by dotted axis. S ′ is the frame of reference for spacial speed v x = sin(π/6) in the figure. The Lorentz factor γ is introduced so that the result is comparable to spacetime diagram Fig. 3(a) show a subject in frame S with velocity vector v = [v τ v x ]. But in transformed frame S ′ , the velocity vector is [1 0], parallel to evolution axis. Thus the subject is identified as a spacial stationary subject with coordinates[τ ′ x ′ ], the frame S ′ is considered as the frame of reference for the subject. Fig. 3(b) demonstrate length contraction of a measuring rod. The rod is draw with dots so that it distinguishable to world line. The rod is at rest and aligned along the x-axis in the frame S. In this frame, the length of this rod is written as ∆x, but in frame S ′ the rod is moving towards the origin with spacial velocity −v x , the spacial length projection ∆x ′ = ∆x 1 − v 2 x = ∆x/γ One should also notice that two synchronized clocks in frame S, placed at the two ends of the rod, is not synchronized in frame S ′ since they have different evolutionary position. Though ∆x ′2 + ∆τ ′2 = ∆x 2 is invariant under coordinate transformation, indicating that the 4-distance between two subject is also preserved. Fig. 3(c) suppose a clock is at rest in the unprimed system S, its world line is the blue strate line. The clock ticks when the subject evolve to specific evolutionary position. The two ticks are intercepted by two event-auxiliary-lines. The world line length that intercepted by two event-auxiliary-lines is ∆l, which is time interval between two ticks seems to observer. Though in frame S ′ , the intercepted world line length is It should be pointed out that at a specific moment of universal time, an event that already happened to one observer may hasn't happened yet for another observer, depend on how dose stretching worldline hit the event auxiliary line. The two blue lines in Fig. 3(d) is world line of two subjects with different spacial locations. In perspective of spacetime, the two event occur simultaneously when two wrold lines cross the same-evolution-plane that denote the occurring of event. The event plane(sam-evolution-plane) of frame S ′ is tilt, the world lines that stretched before the event occur in S ′ is represented by red dashed lines. The length differences between them could be calculated geometrically as which is time interval of two events occur in frame S ′ .
When we looking at a event located in the space-evolution coordinate, rotation of coordinate works very well on explaining the Time dilation,Length contraction and Relativity of simultaneity. Geometrical interpretation of Lorentz transformation in spaceevolution configuration. The moving spacetime observer measured time ct ′ and position x ′ are parameters to be solved,highlighted with red color. The blue line is world line with start point A, K is when stationary observer confirms the event occurring, K ′ is when moving observer confirms the event occurring.
As mentioned abouve, photon,which is unevolvable, can not be described by the space-evolution frame and it's transformation. But particle with mass is possible to treat as a evolvable subject and draw it's world line, though the meaning of evolutionary position for stable particles are unknown.
V. LORENTZ TRANSFORMATION Lorentz covariance is considered to be the fundamental postulate of special relativity. In this section, we try to drive the Lorentz transform from the rotation of space-evolution coordinate. Consider two space-evolution frame S and S ′ , S ′ is frame of reference for a moving spacetime observer, whose spacial velocity is µ x in S. As a example, in frame S we define a world line for a rest subject at spacial position d as follows: According to Eqn. 17, the function of the same world line in S ′ is in S ′ be calculate geometrically. The world line and its start point is invariant of coordinate transformation , but the eventauxiliary-line is rotated with the frame thanks to its definition. Thus the cross point with world line is differ from which in frame S. Based on Fig. 4, by inspecting the geometric relation between τ, ct, x, the spacetime coordinate of the event in S ′ is calculated as The Lorentz transform is obtained.

VI. DISCUSSION
The coordinate in-dependency of worldlines and causality becomes clear by carrying out a thought experiment. As shown in Fig. 5, consider at the moment l = 0, a rest target is hidden in a bunker that located at x = √ 3/2 L. The target pop out the bunker for a very short time when l = L. A time bomb is located at x = 0, moving towards the target with spatial speed v x = √ 3/2 , its explosion radius is much smaller than L and will not be blocked by the bunker. Bomb's timer initial countdown is set to 0.5L. The question is, will the target be destroyed? According to special relativity the answer is clear, the target can be destroyed only when timer initial is set to 0.5L. Now we try to inspect the process in space-evolution frame both for target and bomb. Fig. 5 (a) shows the worldline of the target and the bomb, both stretch from 0 to L synchronously. when l = L, the bomb and target encounter each other at x = √ 3/2, the evolutionary position of bomb is τ = 0.5L thus it explodes; the evolutionary position of target is L thus is pop out of the bunker. The outcome is that the target get destroyed. Fig. 5 (b) shows the same process but in perspective of the bomb. the space-evolution frame is established as bomb's frame of reference. The target move towards the bomb with spatial speed v x = − √ 3/2, its location coincide with the rest bomb at x = 0. When l = 0.5L, the world line of the bomb stretches to [0, 0.5L] so it explodes; the target's worldline stretch to [0, L] so it pop out of bunker. The target is destroyed l = 0.5L What the bomb and the target can reach a consensus is the physical fact "bomb destroys the target", which is supposed to be coordinate-independent, despite they may interpret the process in different way. We conclude that the coordinate transformation of space-evolution dose not violate causality. The Synchronous-World-Line-Postulate is necessary for the causality, otherwise explosion of the bomb and exposure of the target may happen at different places, the target escape from explosion and the causality breaks.

VII. CONCLUSION
By introduce the evolution axis, we successfully established the space-evolution as an Euclidean othoughnorm coordinate system, without losing compatibility to special relativity.
The coordinate transformation for different observers is achieved by rotation, which is typical property of Euclidean space. In such frame, the speed of light c is nothing but a constant used to normalize space-evolution coordinate system. The evolutionary and spacial speed naturally have a upper limit c as a geometric fact. Lorentz transformation is not founding principle but rather a simple consequence of the geometrical nature of the theory, and it's consequences such as length contraction and time dilation could be obtained without much mathematical effort. The good geometric property of spaceevolution suggests that the evolution axis is more qualified than time to combine with spatial axis. The space-evolution frame is likely to further improve our understanding of time and space. Though difficulties encountered in explaining photons. Further attention is expected to discover its potential value in physics.