On New Space-Time Theory (Part I)

Set New Principle of Relativity

The first postulate of Einstein special relativity i.e. the principle of relativity: “The laws of physics apply in all inertial reference systems” [28, 29]. It can be checked even with the event of everyday life. It makes people firmly believe “no inertial reference system is special, any two reference systems in uniform relative motion are identical for the laws of physics”. It seems to be absolutely right.

However, John C. Mather and George F. Smoot’s discovery of the blackbody form and anisotropy of the cosmic microwave background radiation (2006 Nobel prize) distinctly tell us: “Two reference systems in uniform relative motion are different”. Therefore, we have no choice but to amend the principle of relativity to new principle of relativity: “The laws of physics apply in all inertial reference systems, while any two reference systems in uniform relative motion are different” (the different is the data of the two reference systems taking simultaneous measurement of the same physical quantity of the same body are different, please see later in 2.2 and 2.4, while the identical is their using his own measurement data of the physical quantities to build laws of physics the two reference systems are identical). Two reference systems in uniform relative motion are different, in different reference system taking measure of the anisotropy of the 3K background radiation’s radiation temperature is different, being in accord with John C. Mather and George F. Smoot’s discovery. Although as formerly theory when the reference system’s speed relative to the 3K background radiation field is v, because of Doppler effect, this reference system’s measurement data of the background radiation temperature will be T = [1–(v/C)2]1/2 ╱[1–(v/C)cosθ] [30]. Only in the 3K background radiation field reference system v=0, the radiation temperature’s anisotropy disappears. However, does this mean that we can take the 3K background radiation field reference system v=0 as an absolute rest system in violation of relativity? Of course not! Because the movement still must be one relative to the other.

Set New Postulate of Light Speed

The second postulate of Einstein special relativity i.e. the universal speed of light: “The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source, the observer, or any assumed of propagation” [28, 29]. The speed is the same regardless of the motion of the source or the observer etc is contrary to the common practice. Many studious persons have proposed many amended means before [9, 10]. This paper different from Einstein, also different from any persons had made before, we fix the light source on to a reference system. Because “the average speed measured over a closed path is constant C” is a conclusion on a large numbers of experiments [31], and is the reasonable part in all the studious person’s efforts had made to amend the special relativity more perfect, we amend the universal speed of light to new postulate of light speed: “The average speed of any light ray from a stationary light source measured over a closed path in vacuum is always constant C≈3×108 ms-1. Our amendment is either obeying with the result of the experiments or able to give the light speed more freedom: “the average over a closed path is constant C” allows the local speed of light over each short line segment component which built the closed path may not be C. The “light ray from a stationary light source” and cast off “regardless of the motion of the source, the observer, or any assumed of propagation” set our heart at rest. Our light source is fixed in the reference system, “the light ray come from the source” will be more clear, more unassailable.

Here new postulate of light speed’s “the average speed over a closed path is constant C” while the local speed of light over each short line segment component which build the closed path may not be C, seems to be similar as new principle of relativity’s “using his own measurement data of the physical quantities to build laws of physics the two reference systems in uniform relative motion are identical” while the data of measurement of the two reference systems taking simultaneous measurement of the same physical quantity of the same body may not identical.

Set New Added Postulate

The new physics experiments were performed and analyzed at CERN since 1998 relating this paper are: 1)Direct test of wave-particle duality (complementarity) by a “which-way” experiment in an atom interferometer [32, 33]. 2) Einstein-Podolsky-Rosen (EPR) experiments were performed in two-photon entangled state to show the violation of Bell inequality under strict Einstein locality conditions [34] or to show [35, 36] the quantum correlation over long distance (>10km). Also an EPR experiment was achieved at CERN to test the non- separability of entangled neutral-kaon wave function [37]. 3) First direct observation of time-reversal non- invariance in the neutral-kaon system [38, 39]. The experiments 1) and 2) are directly related to reveals the essence of the measurement which can be summarized as three propositions: a) The measurement is founded to change the state of the object. b) The measurement is also quantum in essence. The quantum correlation (i.e. entanglement) between the measurement apparatus (with its reference system) and the object (with its environment) is founded to destroy the quantum correlation (quantum coherence) originally existing in the object and its environment. c) There is not any information (experimental data) existed before the measurement is taken. Now, the “measurement is founded to change the object by destroying the original quantum coherence between the object and object’s environment” is already general knowledge in physics circles [40]. So, this paper set it as one of the basic postulates: the third postulate (new added postulate after the two amended postulates of the Einstein special relativity).

It must be pointed out that: 1) The “measurement is founded to change” in the third postulate is on both sides. Not only the being measured object been changed by the reference system’s taking measurement, but the reference system in taking measurement also been changed by the being measured object. Because it is the quantum correlation (i.e. entanglement) between the measurement apparatus (with its reference system) and the object (with its environment) been founded that destroies the quantum correlation (quantum coherence) originally existing between the object and its environment, of course it also destroies the quantum correlation (quantum coherence) originally (before the measurement is taken) existing between the reference system and the reference system's measurement apparatus. 2)If two reference systems (for example Σ and Σa) simultaneously take measure of the same object, because of the third postulate, the simultaneous measurements of Σ and Σa will disturb each other, leading that both the measurement data of Σ and of Σa contain the interactional impact of simultaneous measurement come one from the other (more precisely the interactional impact is among Σ and Σa and the being measured object three sides come one from the other instead of only between Σ and Σa two sides). 3)If many reference systems (for example Σ, Σa, Σb, Σc et al) joining in simultaneously taking measure of the same object, each reference system’s measurement data will contain all of the interactional impact come from all of the other reference systems’ simultaneous measurement, it is very complex. Of course the first simple case is only one reference system (for example Σ) in measuring, the interactional impact is only between Σ and the being measured object. The second simple case is only two reference systems (for example Σ and Σa) in simultaneously measuring, only two simultaneous measurements of Σ and Σa disturb each other ——or perhaps there are other reference system Σb, Σc et al while Σb, Σc et al do not join to measure with Σ and Σa, or there are Σb, Σc et al joining in simultaneous measurement with Σ and Σa while Σb, Σc et al are far (>>10km) off the place so that the interactional impact of simultaneous measurement from Σb, Σc et al are too weak to be neglected. 4)If two (or more) reference systems are not simultaneously taking measure of, the joining measurement of new reference system (or in simultaneous measurement reference system’s stopping measurement) will change the reference system(s) being in taking measurement and the being measured object by destroying the original quantum coherence between the reference system(s) and the being measured object.

To express simply, in the following Σ and Σa will be always in this case: Σ is moving along the positive direction of the x-axis of itself relative to the Σa, the Σ’s moving speed measured by Σa is constant v (of course the v is not limited i.e. it may be v→0 or ＞C or＞＞C), both the x-axis of Σ and the xa-axis of Σa are on the same horizontal line and the positive directions are from left to right, both the y-axis of Σ and the ya-axis of Σa are horizontal lines and the positive directions are from the book point to the reader, both the z-axis of Σ and the za- axis of Σa are vertical line and the positive directions are from below to above.

The Second Step: See New Things Certainly Come

See New Things Certainly Come from New Postulate of Light Speed

Considering Σ and Σa are simultaneously measuring the same a horizontal photon from the light source fixed at the Σ’s origin, we fix a glass plate on to the Σ’s x-axis to reflect the photon come from the Σ’s origin back to the Σ’s origin. How long time does it take that a photon to make this trip? The light source is in stationary relative to the Σ and the glass plate on to the Σ’s x-axis is also in stationary relative to the Σ so the light source’s mirror image is also in stationary relative to the Σ. Because the light ray pass to and fro through the same path on the Σ’s x-axis is a special case over a closed path, as new postulate of light speed, in Σ the average speed of the light ray should be the constant C. Using the absolute value to list the time equation in Σ is x/C-x+x/Cx =2x/C (assume the Cx is a constant and the C-x may be another constant), Reduced the x it becomes 1/C-x+1/Cx =2/C. While in mathematics it always is (C–x1/2 – Cx1/2)2≥0 combine it with 1/C-x+1/Cx =2/C we get (C–x1/2∙Cx1/2)≥C bring into 1/C-x+ 1/Cx =2/C we can get (C-x+Cx)≥2C. It tells us: at least C-x or Cx is higher than C, then we can guess: if nobody nearby Σ and Σa, it must be that the C-x and Cx just are C-x≥C and Cx≤C, in mathematics if and only if C-x=Cx can we get they are C- x=Cx=C. Of course when Σ and Σa are simultaneously measuring the same a horizontal photon from the light source fixed at the Σa’s origin the Σa’s measurement data of light speed must be C-ax≤C and Cax≥C (just opposite to C- x≥C and Cx≤C). While the C-x≥C (or Cax≥C) breaks “C is the maximal and unsurpassable speed”.

Of course in 1/C-x+1/Cx =2/C the two speed of light C-x and Cx must be: the more the one, the small the other. For example C-x at maximal is C-x→∞, and then the Cx must be at lowest Cx→C/2. i.e. when light source is “in stationary” the photon’s speed will always between (C/2, ∞).

See New Things Certainly Come from New Added Postulate

In 2.1 above, in Σ and Σa’s measuring the same a horizontal photon from the light source fixed at the Σ’s origin when no other body nearby, the Σ’s measurement data of the horizontal photon’s speed along the positive direction of the x-axis would be Cx≤C and along the opposite direction be C-x≥C. As new added postulate, besides experienced the Newtonian universal gravitation and other actions it is originally because the measurement data of Σ is disturbed by the simultaneous measurement of Σa. It is the reference system’s taking measurement (more precisely the quantum correlation (i.e. entanglement) between the measurement apparatus (with its reference system) and the being measured object been founded) instead of the ether or the being measured object’s motion that changes both the being measured object and the reference systemself. It is evident that different Σa will bring different disturbing then result in different C-x and Cx, only 1/C-x+1/Cx =2/C being unchanged in form. It is in accord with the new principle of relativity: “The laws of physics apply in all inertial reference systems, while any two reference systems in uniform relative motion are different”.

“Two reference systems in uniform relative motion are different”. Of course the most acceptable difference between two reference systems is the mass rest in the reference systems (more precisely the mass joining in the quantum correlation of taking the measurement). Therefore, we define the mass rest in the reference system (joining in the quantum correlation) as the reference system’s referenced weight, define the center of the mass as the reference system’s origin. Then, as the new added postulate and “the measurement is founded to change” in the new added postulate actually is on both sides, we can consequently get: In taking measure of, the greater the referenced weight a)the stronger the reference system destroies the original (before the measurement is taken) quantum coherence between the being measured object and its environment, b)the less the reference systemself being changed by the being measured object, c)the stronger the reference system disturbs the other reference system’s measurement data of taking simultaneously measure of the same object, d)the less the reference systemself’s measurement data been disturbed by other reference system’s taking simultaneously measure of the same object; on the opposite, the less the referenced weight, it is just the reversed case of a), of b), of c), of d). Then we can guess: It perhaps that space is not empty the reference system’s space is something around the referenced weight, if there is not referenced weight then saying nothing of the space around the referenced weight, so do the reference system’s time.

As above, for example we (on earth) take measure of a micro-particle, Σa is our earth’s reference system, while Σ is the particle’s reference system (the particle is “stationary” in it and its moving speed measured by Σa is constant v). Compared with our Σa earth’s mass the Σ particle’s mass is infinitely small. Therefore, the Σ particle’s taking measure of us disturbs our earth Σa infinitely small. However, our earth Σa’s taking measure of the particle disturbs the Σ particle infinitely great, almost type of deciding the particle’s there be or there not be. Then we educe: In taking measurement of a micro- particle, because the micro-particle’s mass is too small, the “on” or “off” of the quantum correlations (i.e. entanglements) between the micro-particle and the other objects in the environment make the micro-particle’s behaviour uncertainty. Perhaps it is the essence of the uncertainty in the Heisenberg Uncertainty Principle.

See New Physics Meanings Certainly Come from the New Added Postulate

About the physical meanings of the Lorentz transformation even Einstein and Lorentz himself each stucks to his own opinion until they past away [30]. In fact, off a physical quantity’s measurement process to discuss its physics meanings is a thing cannot exist without its basis. Therefore, as the new added postulate’s suggestion, we establish the coordinates relation of the two inertial reference systems Σ and Σa from Σ and Σa (there may be other reference systems Σb, Σc et al and perhaps some of the Σb, Σc et al are joining in) simultaneously take measure of the same object’s process of a physics event taking place.

First of all, we stipulate “the definition of measurement unit of Σ and of Σa is the same”. For example one second is the time in which there occur 9192631770 oscillations of the cesium atom “stationary” in the reference system, one centimeter is 165076373 wavelengths of red light from Kr86 “stationary” in the reference system etc. We suppose the mass rest in the Σ is M0, the center of M0 is Σ’s origin; the mass rest in the Σa is M${a0}$, the center of $M{a0}$ is the $\Sigma_a$'s origin. Here we must remind you: A) The "actual length" of time of the same cesium atom "stationary" in $\Sigma$ being measured alone by $\Sigma$ occur 9192631770 oscillations is not equal to "stationary" in $\Sigma_a$ being measured alone by $\Sigma_a$ occur 9192631770 oscillations, because $M_0 \neq M_{a0}$ they are different quantum correlation (i.e. entanglement). As we known, different quantum correlation (i.e. entanglement) is corresponding to different quantum energy level. B) Also because they are different quantum correlation (i.e. entanglement), the "actual length" of time of the same cesium atom "stationary" in $\Sigma$ being measured alone by $\Sigma$ occur 9192631770 oscillations is not equal to being simultaneously measured by $\Sigma$ and $\Sigma_a$ occur 9192631770 oscillations. So do other unit only the measurement unit's "definition" is unchanged, while the measurement unit's "actual length" can change or be changed in different quantum correlation (i.e. entanglement) is different. C) Although the "actual length" of the same unit in different case (for example as above in A) and in B)) is different, while the reference system himself is not aware of it using his own unit taking measure of himself cannot obtain his own change, he thinks the "actual length" of his unit is always the same and being unchanged in different case.

Now, we set $\Sigma$ and $\Sigma_a$ "start his own clock" (t =0 and ta=0) at the moment $\Sigma$'s coordinate axis frames and $\Sigma_a$'s coordinate axis frames coincide. It must be especially pointed out: Now that the reference systems "start his own clock" and the stipulation "the definition of measurement unit of the two reference systems is the same", time coordinate should be something as space coordinate, each the reference systems is severally using his own clock to determine his own time coordinate in simultaneously measuring the same object's physics process taking place, it must be that there is not the problem to have to synchronize the clocks of the two reference systems before simultaneous time measurement (do you think you need to synchronize the x and x$_a$ to y, or y and y$_a$ to z, or z and z$_a$ to x etc before space coordinate measurement?).

We suppose $\Sigma$ and $\Sigma_a$ are in the simultaneous measurement of the same object's process of a physics event taking place, the $\Sigma$'s measurement data are from (0,0,0,0) to (x, y, z, t) and the $\Sigma_a$'s are from (0,0,0,0) to (x$a$, y$_a$, z$_a$, t$_a$). Namely, $\Sigma$'s time t =0 and $\Sigma_a$'s time t$_a$ =0 are at the same instant of time, while $\Sigma$'s time t and the $\Sigma_a$'s time t$_a$ are at the another same instant of time. It seems that with unit it must be "$\Sigma$'s t second (i.e. t$\Sigma$'s one second)=\Sigma_a$'s t$_a$ second (i.e. t$\Sigma$'s one second)" because $\Sigma$ and $\Sigma_a$ are simultaneously measuring the same object's process taking place; while the numerical value before the unit is "t≠t$_a$" because $M_0 \neq M{a0}$ and v≠0, from the new added postulate, the disturbing of $\Sigma$ and of $\Sigma_a$ in simultaneous measurement are different, leading the "actual length" of time unit to be "$\Sigma$'s one second ≠$\Sigma_a$'s one second", though both the definition of "one second" of $\Sigma$ and of $\Sigma_a$ is the same. No. That is not the case. In fact, even with unit it still is "$\Sigma$'s t second (i.e. t$\Sigma$'s one second) ≠$\Sigma_a$'s t$a$ second (i.e. t$_a$$\Sigma_a$'s one second)", the "actual length" of the same definition of time's unit in $\Sigma$ ≠ in $\Sigma_a$ because $M_0 \neq M{a0}$ the same definition of time's unit in $\Sigma$ and in $\Sigma_a$ are different quantum correlation (i.e. entanglement). Even v =0 it still will be that the "actual length" of the same definition of time's unit in $\Sigma$ ≠ in $\Sigma_a$ because $M_0 \neq M_{a0}$ the same definition of time's unit in $\Sigma$ and in $\Sigma_a$ still are different quantum correlation (i.e. entanglement). For example, when v =0, $\Sigma$ and $\Sigma_a$ taking simultaneous measurement of the same time of an object's process of a physics event taking place, it must be that both $\Sigma$'s measurement data and $\Sigma_a$'s measurement data are $\tau$ second, however, $\Sigma$'s one second ≠$\Sigma_a$'s one second (only the numerical value before the $\Sigma$'s time unit and the numerical value before the $\Sigma_a$'s time unit are the same $\tau$) because $M_0 \neq M_{a0}$ the same definition of time's unit in $\Sigma$ and in $\Sigma_a$ are different quantum correlation (i.e. entanglement). Of course the space length unit is also something as time unit: the "actual length" of space length unit of $\Sigma$'s one meter ≠ of $\Sigma_a$'s one meter, so do other physical quantities' unit.

In fact, from the C) of "we must remind you" before, we can see: We need not to pay attention to the "actual length" of the same definition of an unit in $\Sigma$ ≠ in $\Sigma_a$ and in $\Sigma$'s alone measuring ≠ in $\Sigma$ and $\Sigma_a$'s simultaneous measuring etc (we have stipulated the "definition of measurement's unit of $\Sigma$ and of $\Sigma_a$ is the same") enough. What we have interest in are: in $\Sigma$ and $\Sigma_a$'s simultaneous measuring the same physical quantity of the same object, the numerical value before unit of $\Sigma$ ≠ of $\Sigma_a$, and what the relation between the numerical value before unit of $\Sigma$ and of $\Sigma_a$ is? Generally, we suppose the relation about (x, y, z, t) and (x$_a$, y$_a$, z$_a$, t$_a$) is linearity as below:

$$\begin{align*} (x) & = a_{11} a_{12} a_{13} a_{14} \\ (y) & = a_{21} a_{22} a_{23} a_{24} \\ (z) & = a_{31} a_{32} a_{33} a_{34} \\ (t) & = a_{41} a_{42} a_{43} a_{44} \end{align*}$$

(1)a

(why it is linearity? Fuke BA [41] though our definiens and postulates are different from them, while the reasons or principles are analogous). The $a_{ij}(i, j =1,2,3,4)$ of (1)a

actually is function $a_{ij}(M_{a0}, M_{0}, m_{0}, m_{10}, m_{20}, \dots, v, u_{w}, u_{a1}, u_{a2}, \dots, \omega, \xi, \psi, \dots)$, where $m_{0}$ is the rest mass of the being measured object and $u_{a}$ is its speed measurement data of $\Sigma_{a}$ while $m_{10}, m_{20}, \dots$ are the other objects’ rest mass (including the rest mass of referenced weight of other reference systems joining simultaneous measurement with $\Sigma$ and $\Sigma_{a}$) and not joining simultaneous measurement but joining quantum correlation (i.e. entanglement) with $\Sigma$ and $\Sigma_{a}$), and $u_{a1}, u_{a2}, \dots$ are the corresponding speed of $m_{10}, m_{20}, \dots$ — measurement data of $\Sigma_{a}$, and $\omega, \xi, \psi, \dots$ are variable representing the simultaneous measurements’ disturbance and the other actions, we denote $a_{ij}(i, j = 1, 2, 3, 4)$ for shot. Of course different being measured object will result in different $a_{ij}(i, j = 1, 2, 3, 4)$, different reference systems joining to measure simultaneously with $\Sigma$ and $\Sigma_{a}$ will also result in different $a_{ij}(i, j = 1, 2, 3, 4)$, different $(x, y, z, t)$, different $(x_{a}, y_{a}, z_{a}, t_{a})$ for simultaneous measurement’s interaction impact.

In taking measure of the same a stationary space-length or a time-length or a mass when $v = 0$: 1) $\Sigma$ alone ($\Sigma_{a}$ and other reference system do not join) in taking measure of, the $\Sigma$’s measurement changes $\Sigma$ self and the being measured object in the same scale (for the being measured object is “stationary” in $\Sigma$), $\Sigma$’s measurement data is l0 or $\tau$ or m0. 2) $\Sigma_{a}$ alone ($\Sigma$ and other reference system do not join) in taking measure of, the $\Sigma$’s measurement changes $\Sigma_{a}$ self and the being measured object in the same another scale (for M0≠Ma0). Because a) the definition of measurement unit in $\Sigma$ and in $\Sigma_{a}$ is the same, b) the being measured object’s atoms number is the same, although “$\Sigma_{a}$’s measurement changes both $\Sigma_{a}$ self and $\Sigma_{a}$’s being measured object” in 2) is different from “$\Sigma$’s measurement changes both $\Sigma_{a}$ self and the $\Sigma$’s being measured object” in 1) (for M0≠Ma0), while the $\Sigma_{a}$’s measurement data is the same l0 or $\tau$ or m0 as $\Sigma$’s in 1) (Please note: only the numerical values before $\Sigma_{a}$’s unit in 2)= the numerical values before $\Sigma_{a}$’s unit in 1) while the actual length of $\Sigma_{a}$’s unit in 2)≠ the actual length of $\Sigma_{a}$’s unit in 1) for M0≠Ma0). 3) $\Sigma_{a}$ joins simultaneous measurement with $\Sigma$ in $\Sigma$’s taking measure of, the $\Sigma$’s measurement data will still be the same l0 or $\tau$ or m0 as in 1), although here “$\Sigma$’s measurement changes $\Sigma$ self and the being measured object in the same scale” is different from in 1) for $\Sigma_{a}$’s measurement disturbing. It is because $\Sigma$ and the $\Sigma$’s being measured object are equally changed i.e. are changed in the same scale (because the object being “stationary” in $\Sigma$) by the disturbing of $\Sigma_{a}$’s simultaneous measurement and by the $\Sigma$’s self’s measurement. Therefore, although here $\Sigma_{a}$’s unit and $\Sigma_{a}$’s being measured object have been changed into not the same as $\Sigma_{a}$’s alone taking measure of in 1), however, the $\Sigma_{a}$’s measurement data is still the same as in 1) (please note: only the numerical values before $\Sigma_{a}$’s unit in 3) = the numerical values before $\Sigma_{a}$’s unit in 1), while the actual length of $\Sigma_{a}$’s unit in 3)≠ the actual length of $\Sigma_{a}$’s unit in 1), because “$\Sigma_{a}$ joins simultaneous measurement with $\Sigma$ “changes $\Sigma$ and $\Sigma_{a}$’s being measured object). 4)In 3) on $\Sigma_{a}$ hand, because of $v=0$, the being measured object also is “stationary” in $\Sigma_{a}$ as in $\Sigma$, “$\Sigma_{a}$ and $\Sigma_{a}$’s being measured object stationary in $\Sigma$” are equally changed (i.e. changed in the same scale) by the disturbing from $\Sigma_{a}$’s simultaneous measurement and by the $\Sigma_{a}$’s self’s measurement, therefore, although both $\Sigma_{a}$’s unit and the $\Sigma_{a}$’s being measured object have been changed into not the same as in 2), however, the $\Sigma_{a}$’s measurement data is still the same l0 or $\tau$ or m0 as $\Sigma_{a}$’s alone taking measure of in 2). Namely, when $v=0$, whether simultaneous measurement or alone measurement, both $\Sigma$ and $\Sigma_{a}$ can accurately get that not only origins be coincident but also any other corresponding points on axis frames be coincident as well, although the “actual length” of the reference system’s unit and the being measured object have been changed (dilate or contract) by his own measurement or both by his own and by the disturbing from another reference system’s simultaneous measurement, the $\Sigma$’s measurement data of the being measured object is always the same and not different from the $\Sigma_{a}$’s. Represented with (1)a (of course (1)a only represents the measurement data of space and time) it will be stipulation: When $v=0$, the coefficient matrix of (1)a becomes identity matrix (the coefficient matrix’s element becomes Kronecker symbol $a_{ij}|v=0=\delta_{ij}$ $=0$ for Inj and $\delta_{ij}=1$ for $i=j$) i.e.

$$\begin{bmatrix} x \\ y \\ z \\ t \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_{a} \\ y_{a} \\ z_{a} \\ t_{a} \end{bmatrix}$$

The $l_{0}$ or $\tau_{0}$ or $m_{0}$ etc above should be something as the so called “proper distance” or “proper time” or “proper mass” in the Special Relativity. We would remind you again: a) Though “both $\Sigma$ and $\Sigma_{a}$ can accurately get that not only origins to be coincident but also any other corresponding points on axis frames to be coincident as well”, however, “the actual length” of the same unit of $\Sigma$ and of $\Sigma_{a}$ are not identical (in taking measure of the same object $\Sigma$’s one second $\neq \Sigma_{a}$’s one second, $\Sigma$’s one meter $\neq \Sigma_{a}$’s one meter etc) because of $M_{0} \neq M_{a0}$. b) The “actual length” of the same unit of the same reference system is different in different case (for example one $\Sigma$’s second in $\Sigma$’s alone measurement $\neq$ one $\Sigma$’s second in $\Sigma$ and $\Sigma_{a}$’s simultaneous measurement, one $\Sigma$’s second in $\Sigma$ alone taking measure of m1 ≠one Σ’s second in Σ alone taking measure of m2 if m1≠m2 etc). Different measurement will result in different change (dilation or contraction) of the reference system and the being measured object, only “the numerical values before Σ’s unit” = “the numerical values before Σa’s unit” when v =0, namely the axis frames of Σ and of Σa only are number axis (not with unit). Of course (1)a is only the space-time coordinates numerical values relation not with unit, (only we have stipulated “the definition of measurement unit of Σ and of Σa is the same”). Of course so do other physical quantities. However, it just is the true fact what we see in our real- world. Do you understand? If not, you need to re-think the C) of “we must remind you” before, till understand.

In taking measure of the same a stationary space- length or a time-length or a mass when v≠0: 5) The being measured object is stationary in Σ and Σ alone (Σa and other reference system do not join) in taking measure of, the Σ’s measurement data also will be the same l0 or τ0 or m0 as in 1) (there v =0). Because Σa and other reference system do not join, the quantum correlation (i.e. entanglement) between Σ and the Σ’s being measured object is the same as in 1). 6)The being measured object is stationary in Σa, and Σa alone (Σ and other reference system do not join) taking measure of, the Σa’s measurement data will be the same l0 or τ0 or m0 as in 2) (there v =0), the same data as the Σ’s in 5) and the Σ’s in 1), also because Σ and other reference system do not join, the quantum correlation (i.e. entanglement) between Σa and the Σa’s being measured object is the same as in 2). 7) The being measured object is “stationary” in Σ, and Σa join simultaneous measurement with Σ in Σ’s taking measure of, the Σ’s measurement data will still be the same l0 or τ0 or m0, although this time both the Σ’s measurement unit and Σ’s being measured object have been changed into not as the same as in 3) (the quantum correlation (i.e. entanglement) is not the same as in 3) for there v =0 here v≠0). It is because the Σ’s measurement unit and the Σ’s being measured object are equally changed i.e. are changed in the same scale (for the object “stationary” in Σ) by the Σ self’s measurement and by the disturbing from Σa’s (or as well as and other reference system’s) simultaneous measurement. However, this time on Σa hand, because of v≠0 the Σa’s being measured object is “in motion” (with Σ) in Σa, therefore, this time the Σa’s unit and the Σa’s “in motion” being measured object in Σa are not equally changed (i.e. are changed not in the same scale) by the Σa self’s measurement and by the disturbing from Σ’s (or Σ and other reference systems’ simultaneous) measurement in Σa’s angle of view! Therefore, this time the Σa’s measurement data will not be the same l0 or τ0 or m0 as Σa’s in 6)! 8)The object “stationary” in Σa, and Σ (or Σ and other reference systems) join(s) simultaneous measurement with Σa in Σa’s taking measure of, the Σa’s measurement data is the same l0 or τ0 or m0 as in 3) on Σa hand (there v=0 here v≠0), although different measurement state result in different change (dilation or contraction) of the reference system and the reference system’s being measured object, however, here Σa’s measurement unit and Σa’s being measured object are equally changed or changed in the same scale (for the object “stationary” in Σa), therefore, Σa’s measurement data still be the same l0 or τ0 or m0 as in 3), in 2), in 6) on Σa hand. However, this time on Σ hand, because Σ’s being measured object is “in motion” in Σ, the Σ’s measurement unit and the Σ’s being measured object are not equally changed (i.e. are changed not in the same scale) by the Σ self’s measurement and by the disturbing from Σa’s (or Σa and other reference systems’ simultaneous) measurement in Σ’s angle of view, therefore, this time the Σ’s measurement data will not be the same l0 or τ0 or m0 as Σ’s in 3) (there v =0) or in 7). Both 7) and 8) show us: in a reference system the measurement data of the same being measured object’s physical quantity, the object “in motion” is different from “in stationary”, two reference systems’ simultaneous measurement is different from one reference system’s alone measurement. Namely when v≠0 in Σ and Σa taking simultaneously measure of the same a physical quantity, the numerical value before Σ’s unit ≠ the numerical value before Σa’s unit, both Σ and Σa can accurately get that at the two axis frames coincide moment (t =0 and ta=0 ) only Σ’s origin and Σa’s origin coincide while any other corresponding points on axis frames do not coincide (though the two axis frames coincide)! Namely “when v≠0 the coefficient matrix of (1)a is not identity matrix”.

From above we can see: When v =0, the reference system taking measure of a stationary object cannot see that his measurement have changed both himself and the being measured object, cannot see that he and another reference system’s simultaneous measurement disturb each other, for the numerical value before Σ’s unit = the numerical value before Σa’s unit (though the measurement data of the same being measured object “in motion” is different from “in stationary”). If and only if v≠0, can the simultaneous measurement of Σ and Σa disturbing each other be seen by Σ and Σa themselves —— the Σ’s measurement data different from the Σa’s, the difference on the space-time coordinates between Σ and Σa (there may be other reference systems Σb, Σc et al and perhaps some of the Σb, Σc et al are joining) in simultaneously taking measure of the same object’s process of a physics event taking place as shown in (1)a, the coefficient matrix of (1)a is not identity matrix when $v \neq 0$.

Now, we can see: In explaining the Heisenberg Uncertainty Principle in ending of 2.2, the uncertainty must occur and only occurs in taking measure of the “in motion” micro-particle. We also can see: In taking measure of an “in motion” micro-particle, our simultaneous measurement disturb the micro-particle infinitely great is of course measurable phenomenon ourselves, while the micro-particle himself is not aware of it (using his own unit taking measure of itself cannot get his own change). We still can see: When $v \neq 0$, only if $v \rightarrow 0$ is the difference between the simultaneous measurement data (numerical value before unit) of $\Sigma$ and of $\Sigma_a$ close to zero (actually only $v = 0$ can we get “the measurement data (numerical value before unit) of $\Sigma =$ of $\Sigma_a$”).

Of course whether or not $v = 0$ and $v \neq 0$, one can compare the same physical quantities of his own reference system, for example, compare the speed of a light ray from a stationary source to some direction with to the opposite direction. If the light speed in this direction is greater than in the opposite direction, he can guess: from the stationary light source to some not far away place in this direction there may be a big mass object. Or comparing the speed of a light ray from a stationary source to the same direction in different time, if the speed is increscent (more and more great), he can guess: in this direction from the stationary light source to some not far away place, there may be a big mass object is closing to the stationary light source.

Of course usually (1)a must be

$$\begin{align*} x & = a_{11} x_a - v_t a_11 \\ y & = a_{11} y_a - v_t y_a \\ z & = a_{11} z_a - v_t z_a \\ t & = a_{41} t_a - v_t t_a \end{align*}$$

Then (1)a$^{(-1)}$ goes to (i.e. we solve the equation (1)b)

$$\begin{align*} x & = a_{44} / \eta \\ y & = a_{22} / \eta \\ z & = a_{33} / \eta \\ t & = a_{41} / \eta \end{align*}$$

(where $\eta = a_{11} a_{44} - a_{41} a_{14} = a_{11} (a_{44} + v_a_41)$). Of course we also can solve (1)b$^{-1}$ back to (1)b, here (1)b$^{-1}$ and (1)b$^{-1}$ being analogous as $\alpha = \lambda \beta$ and $\beta = (1/\lambda) a$ actually are only one equation's two different forms. Compared the first row of (1)b i.e. $x = a_{11} (x_a - v_t a_11)$ with the first row of (1)b$^{-1}$ i.e. $x_a = (a_{44} / \eta) x + (v_a_11 / \eta) t = (a_{44} / \eta) x - (-v_a_11 / a_44) t$ we can see: The speed of $\Sigma$'s moving along $x_a$-axes measured by $\Sigma_a$ is $v$ while the same speed of $\Sigma_a$'s moving along $x$-axes measured by $\Sigma$ is $(-v_a_11 / a_44)$ instead of $(-v)$, although it is $\Sigma$ and $\Sigma_a$ take simultaneous measure of the same speed of relative motion. Actually, it reminds us again: $\Sigma$ and $\Sigma_a$ are different (something as $v_a_11 / a_44$ and $v$ are different) it proofs $\Sigma$ and $\Sigma_a$ are different on the hand of the same proofs's measurement data. As light speed's “the average over a closed path is constant $C$” while the local speed of light over each short line segment component which build the closed path may not equal to $C$, “the $\Sigma$ and $\Sigma_a$ are identical for taking his own measurement data to build laws of physics” while may not identical for each the component physical quantity which build the laws of physics (of course physics law is built by the physical quantities such as distance, time interval, speed etc.) in $\Sigma$ and $\Sigma_a$ simultaneously taking measure of the same physical quantity of the same object, the measurement data of $\Sigma$ and of $\Sigma_a$ are different because $\Sigma$ and $\Sigma_a$ are different for $M_0 \neq M_{a0}$ and $v \neq 0$, being in accord with John C. Mather and George F.'s discovery, while in using his own measurement data of the physical quantities to build laws of physics $\Sigma$ and $\Sigma_a$ are identical (therefore the laws of physics apply in all inertial reference systems, while any two reference systems in uniform relative motion are different).

Determine the (1)b Coefficient Matrix's Element when we Re-Reduce the Case

A little generally it may be $M_0 \neq M_a_0$: on right side of (2)'s $\tau$ and of (3)'s $\tau_a$ are the same $\tau_0$ but on left side of (2)'s $\tau_a^*$ and of (3)'s $\tau^*$ it may be $\tau_a^* \neq \tau^*$ we cannot get $1 = \frac{1}{[a_{44}(a_{44} + va_{41})]}$; on right side of (4), (6), (8)'s $l$ and (5), (7), (9)'s $l_a$ are the same $l_0$ but on left side of (4)'s $l_a^*$ and (5)'s $l^*$ it may be $l_a^* \neq l^*$ we cannot get $1 = a_{44}/[(a_{44} + va_{41})a_{11}^2]$, (6)'s $l_a^*$ and (7)'s $l_y^*$ may be $l_a^* \neq l_y^*$ we cannot get $1 = \frac{1}{a_{22}^2}$, (8)'s $l_a^*$ and (9)'s $l_y^*$ may be $l_a^* \neq l_y^*$ we cannot get $1 = \frac{1}{a_{33}^2}$, i.e. a little generally (need not to say generally) we cannot get (1)c.

It must be pointed out that $l^*$ and $l_a^*$ is not analogous to $\Sigma$ and $\Sigma_a$'s taking simultaneous measurement of the same distance between two reference system's origins i.e. measurement data of $tv(a_{11}/a_{44})$ and $(t_a v)$. Because the rest length of $l^*$ and $l_a^*$ are the same "stationary" length $l_0$ ($l = l_a = l_0$), while $tv(a_{11}/a_{44})$ and $(t_a v)$ are the same "in motion" length — of course the "in stationary" length of $tv(a_{11}/a_{44})$ and of $(t_a v)$ are impossible the same. Therefore, $l^* / l_a^*$ is not equal to $[tv(a_{11}/a_{44})]/(t_a v)$. So, taking $l^* / l_a^* = [tv(a_{11}/a_{44})]/(t_a v)$ as an added equation is a wrong idea. So, generally the equations (2)~(9) are useless to determine the element of (1)b's coefficient matrix. Generally determining the element of the (1)b's coefficient matrix is very very difficult or impossible. We have no more better choice but to re-reduce the case (we have reduced the case since 3): If nobody nearby $\Sigma$ and $\Sigma_a$ (i.e. except $\Sigma$ and $\Sigma_a$ any other Newtonian universal gravitation and simultaneous measurements' disturbing being neglected), the object being measured only are photon coming from the light source stationary in $\Sigma$ or $\Sigma_a$ both $M_0$ and $M_a_0$ close to zero only keeping $\sigma = M_0/M_a_0$ as an arbitrary constant had been determined like $v$, i.e. all the Newtonian universal gravitation even from $M_0$ and $M_a_0$ would be reduced, it seems that only the interactional impact of simultaneous measurement of the reference systems $\Sigma$ and $\Sigma_a$ becomes main part. It must be: at anywhere except the two infinitely small regions (one infinitely small region contains the $\Sigma$'s origin, the other contains the $\Sigma_a$'s origin) the $C_a$ is a constant and the $C_a_x$ is another constant; $C_y = C_y = C_x = C_z$ and $C_y = C_y = C_z = C_z$. Therefore, it must be $a_{22} = a_{33} = a$ in this case (whether $C_y = C_y = C_z = C_z$ or not $C_y = C_y = C_z = C_z$ and whether $a_{22} = a_{33} = a = 1$ or not $1$ we can only answer that we don't know, but we can confirm that when $\sigma = M_0/M_a_0 = 1$ it must be $C_x = C_x = C_x = C_x$ and $C_y = C_y = C_x = C_x = C_x$ and $C_y = C_y = C_x = C_x = C_x$ at anywhere except the two infinitely small regions (of course only when $M_0$ or $M_a_0$, or both $M_0$ and $M_a_0$ not close to zero and then $C_x = C_x$, $C_x = C_x$ are not constants). Can we find out the element of the (1)b's coefficient matrix in this case?

$\Sigma$ And $\Sigma_a$ Simultaneously Taking Measure of the Same Wave Front Surface of Light Emitted from $\Sigma$'s Origin:

At first let us take simultaneous measure of the same wave front surface of light emitted from the $\Sigma$'s origin. In $\Sigma$ we set some stationary glass plates on to the points at appropriate angle to reflect the light ray come from the $\Sigma$'s origin back to the $\Sigma$'s origin, it can change the light ray's direction from $\Sigma$'s origin into from any other point of $\Sigma$'s stationary light source (we neglect the two infinitely small regions one contains the $\Sigma$'s origin and the other contains the $\Sigma_a$'s origin because in the two infinitely small regions the Newtonian universal gravitation cannot be neglected): From the $\Sigma$'s origin along the $x$-axis' positive direction to the stationary point $x$ and then along the opposite direction back to the $\Sigma$'s origin (as in 2.1 the light source fixed at the $\Sigma$'s origin), on this a closed path, as new postulate of light speed, in $\Sigma$ the average speed of the light ray should be the constant $C$. Using the absolute value to list the time equation in $\Sigma$ we get $x/C_x + x/C_x = 2x/C$, reduced the $x$ it goes to

$$\frac{1}{C_x} + \frac{1}{C_{-x}} = \frac{2}{C}$$

As known in the ending of 2.1 the two speed of light $C_x$ and $C_x$ in (19) must be: the more the one, the small the other e.g. $C_x$ at maximal is $C_x \rightarrow \infty$, and then the $C_x$ must be at lowest $C_x \rightarrow C/2$. However, does this mean that the photon’s speed will always between $(C/2, \infty)$ cannot be zero? Of course not! Here photon’s speed is always between $(C/2, \infty)$ is because the light source is “in stationary”. When the light source is “in motion” it will be not the case. Please see later the discussion after (23) and (24) under (25), (27) and (28) under (29): Although when $v < C$ the $1/C_x + 1/C_x$ (or $1/C_x + 1/C_x$) always is equal to $2/(C_x) \approx 2/C$ almost as $1/C_x + 1/C_x = 2/C$ (or $1/C_x + 1/C_x = 2/C$) because it always is $\lambda \approx 1$, while when $v$ is sufficiently great, in (24) the $C_x \rightarrow C_x$ may be $>0$ or $=0$ or $<0$ (meanwhile $C_x \rightarrow C_x$ may be $>C$ or $>>C$ when $\sigma = M_0/M_a0 > 1$ or $\rightarrow \infty$ in taking measure of the light come from a quasar please see later in (34) $|{\sigma \rightarrow \infty}$); in (28) the $C_x \rightarrow C_x$ may be $>0$ or $=0$ or $<0$ (meanwhile $C_x \rightarrow C_x$ may be $>C$ or $>>C$ when $\sigma = M_0/M_a0 < 1$ or $\rightarrow \infty$ in our (on earth) taking measure of the light come from a small mass particle please see later in (34) $|{\sigma \rightarrow 0}$).

From the $\Sigma$’s origin along the $r$’s positive direction to the stationary point $p$ and then along the opposite direction back to the $\Sigma$’s origin on this a closed path, as new postulate of light speed, in $\Sigma$ the average speed of the light ray should be the constant $C$. Using the absolute value to list the time equation in $\Sigma$ we get

$$\frac{r}{C_r} + \frac{r}{C_r} = \frac{2r}{C} \tag{20}$$

From the $\Sigma$’s origin along the $r$’s positive direction to the stationary point $p$ and then along the $z$-axis ‘opposite direction, the $y$-axis ‘opposite direction, the $x$-axis ‘opposite direction back to the $\Sigma$’s origin. As new postulate of light speed, using the absolute value to list the time equation in $\Sigma$ we get

$$\frac{r}{C_r} + \frac{z}{C_z} + \frac{y}{C_y} + \frac{x}{C_x} = \frac{r + z + y + x}{C} \tag{21}$$

From the $\Sigma$’s origin along the $x$-axis’ positive direction, the $y$-axis’ positive direction, the $z$-axis’ positive direction to the stationary point $p$ and then along the $r$’s opposite direction back to the $\Sigma$’s origin. As new postulate of light speed, using the absolute value to list the time equation in $\Sigma$ we get

$$\frac{x}{C_x} + \frac{y}{C_y} + \frac{z}{C_z} + \frac{r}{C_r} = \frac{x + y + z + r}{C} \tag{22}$$

Now, from (21) minus (22) plus (20) (please note $C_y = C_y$ and $C_z = C_z$) we get

$$2 \frac{r}{C_r} + x \left( \frac{1}{C_x} - \frac{1}{C_z} \right) = \frac{2r}{C} \tag{23}$$

Although in different octant (21) and (22) will change while (23)” is always unchanged in form (please see appendix II). Bring (19) into (23)” (please see appendix III) the (23)” will go into

$$C_r = \frac{C}{1 + \frac{C_x - C_z}{C_x + C_z}} \cos \alpha \tag{23}'$$

Here (23)’ appears: If $C_x$ and $C_z$ have been determined, $C_r$ will have been determined, and on the $x$-y plane ($\alpha = \pi/2$) $C_r$ will be $C_r|_{\alpha = \pi/2} = C$.

If we do not only bring (19) as above but bring (19) and $r = (x^2 + y^2 + z^2)^{1/2}$, $r/C_r = t$ into (23)’”, the (23)’” will turn not into (23)’’, but into (please see appendix IV) (23):

$$\frac{x - t \left( \frac{C_x - C_z}{C_x} \right)^2}{t \left( \frac{C_x + C_z}{C_x} \right)^2} = 1 \tag{23}$$

Of course (23)’ and (23)’ all are $\Sigma$’s result of taking measure of the same wave front surface of light emitted from the $\Sigma$’s origin at $t$ instant of time of $\Sigma$ in three different angle of view. The (23) is an ellipsoid and as known in 2.1 here $(C_x - C_z) < 0$. Analytic geometry tells us: the $\Sigma$’s origin (light source) is just on the right focus of the ellipsoid (23). While the $\Sigma$’s origin, as 2.2, $\textcircled{1}$ may on the ellipsoid’s two focus join-line (when $M_a > M_0$ (the $M_a$ is “$M_a0$ being in motion” in $\Sigma$ with speed $-va_{11}/a_{44}$ please see 2.4), $\Sigma$’s light be disturbed greater by $M_a$ than by $M_0$ self so that two focus join-line longer than two origins join-line, great mass object $M_a$ being wrapped in (23)); $\textcircled{2}$ may on the ellipsoid’s two focus join-line’s leftward extended line out of the two focus join-line but still being wrapped in (23) (when $M_a < M_0$ and $M_a$ is with not great enough speed $-va_{11}/a_{44}$ $\Sigma$’s light be disturbed lighter by $M_a$ than by $M_0$ self so that two focus join-line shorter than two origins join-line i.e. with the speed not great enough the small mass object at $\Sigma_a$'s origin is still being wrapped in (23)); $\textcircled{3}$may on the (23)'s two focus join-line's leftward extended re-extended even out of the ellipsoid (when $M_a \rightarrow < M_0$ and $M_a \rightarrow v$'s speed $(-v a_{11}/a_{44})$ is great enough i.e. with high speed small mass object $\Sigma_a$ may not being wrapped in (23)). —In $\Sigma$ we can see: If the speed (or kinetic energy) is great enough, a small mass object can go beyond the light which coming from the light source being stationary in a big referenced weight reference system.

Since it is $\Sigma$ and $\Sigma_a$ take simultaneous measurement of the same wave front surface of light emitted from the $\Sigma$'s origin, the $\Sigma$'s measurement result is (23). What result the $\Sigma_a$'s is? (Note in $\Sigma_a$ the light source is "in motion"). Bring (1)b and $a_{22}=a_{33}=a$ into (23) we get (please see appendix V and VI)

$$\frac{x_a - t_a \left( \frac{C_{\vec{ax}} - C_{\vec{-ax}}}{2} \right)^2}{t_a \left( \frac{C_{\vec{ax}} + C_{\vec{-ax}}}{2} \right)^2} + \frac{y_a^2 + z_a^2}{t_a^2 \left( C_{\vec{ax}} - v \right) \left( C_{\vec{-ax}} + v \right)} \frac{a_{11}}{a^2} = 1$$

(where $C_{\vec{ax}}$ is shown in (12) and $C_{\vec{-ax}}$ is shown in (14)). Since (23) and (24) are $\Sigma$ and $\Sigma_a$ take simultaneous measurement of the same wave front surface of light emitted from the $\Sigma$'s origin, the $\Sigma$'s measurement result (23) is that the light source is on the right focus of the ellipsoid (23), the principle of relativity pledge: it must be that the $\Sigma_a$'s measurement result (24) is also that the light source is on the right focus of the ellipsoid (24)! Consequently it must be

$$\frac{a_{11}^2}{a^2} = 1$$

Because only (25) can let the half minor axis' square of (24) become $t_a^2 \left( C_{\vec{ax}} - v \right) \left( C_{\vec{-ax}} + v \right)$, let the (24) become

$$\frac{x_a - t_a \left( \frac{C_{\vec{ax}} - C_{\vec{-ax}}}{2} \right)^2}{t_a \left( \frac{C_{\vec{ax}} + C_{\vec{-ax}}}{2} \right)^2} + \frac{y_a^2 + z_a^2}{t_a^2 \left( C_{\vec{ax}} - v \right) \left( C_{\vec{-ax}} + v \right)} \frac{a_{11}}{a^2} = 1$$

(24)]$$\text{under}(25)$$

then from the $t_a \left( C_{\vec{ax}} + C_{\vec{-ax}} \right)/2 = t_a \left( \frac{C_{\vec{ax}} - v}{v} + \frac{C_{\vec{-ax}} + v}{v} \right)/2$ can the analytic geometry pledge: The $\Sigma_a$'s measurement result also is that the $\Sigma$'s origin (light source) is just on the right focus of the ellipsoid (24) as (23); while the $\Sigma_a$'s origin, $\textcircled{1}$may on the (24)'s two focus join-line, $\textcircled{2}$may on the left drawn-out line out of the two focus join-line but still being wrapped in the wave front surface (24) of the light emitted from the $\Sigma$'s origin, $\textcircled{3}$may on the left re-drawn-out line even out of the (24), analogically as the $\Sigma$'s measurement result (23).

Please note that the (24)]$$\text{under}(25)$$ is still different from (23) for $v \neq 0$, being in accord with the new principle of relativity"The laws of physics apply in all inertial reference systems, while any two reference systems in uniform relative motion are different". Special remind —as Einstein special relativity $\Sigma = \Sigma_a$ the (24)]$$\text{under}(25)$$ covariant into (23) i.e. besides $a_{11}^2/a^2=1$ it must be $(C_{\vec{ax}} - v) = C_{\vec{-ax}}$ and $C_{\vec{ax}} = (C_{\vec{-ax}} + v)$ i.e. $\Sigma$'s origin and $\Sigma_a$'s origin just on each one of the same ellipsoid's two focus. It only is in a very special case: not only $M_{a0}/M_0=1$ and nobody nearby $\Sigma$ and $\Sigma_a$, but also the simultaneous measurements of $\Sigma$ and $\Sigma_a$ disturb each other just right. Generally we can only conclude: Taking simultaneous measure of the same wave front surface of light emitted from the $\Sigma$'s origin, both the measurement results of $\Sigma$'s and of $\Sigma_a$'s are that the light source ($\Sigma$'s origin) is just on the right focus of the ellipsoid; while the $\Sigma_a$'s origin, $\textcircled{1}$may on the ellipsoid's two focus join-line, $\textcircled{2}$may on the left drawn-out line out of the two focus join-line, $\textcircled{3}$may on the left re-drawn-out line even out of the ellipsoid.

From (25) we do get that analytic geometry pledges: both (23) and (24), $\Sigma$ and $\Sigma_a$ taking simultaneous measurement of the same wave front surface of light emitted from the $\Sigma$'s origin, are that the $\Sigma$'s origin (light source) is on the right focus of the ellipsoid. However, from (25) we also do get $a=a_{11}$ (abnegate the negative root) and hence $a_{11}=a_{22}=a_{33}=a$ and then we know: If the space length contract it must contract in all directions (instead of Einstein special relativity's only contract in the direction of motion)! Of course if dilate it will dilate in all directions, as quasars' apparent superluminal expansion observed in astrophysics (Blandford RD [3], Cohen MH [4]).

$\Sigma$ And $\Sigma_a$ Simultaneously Taking Measure of the Same Wave Front Surface of Light Emitted From $\Sigma_a$'s Origin: In taking simultaneous measurement of the same wave front surface of light emitted from the $\Sigma_a$'s origin, analogically as in 4.3.1 we install some stationary glass plates on to the points at appropriate angle to reflect the light ray come from the $\Sigma_a$'s origin back to the $\Sigma_a$'s origin, with the absolute value we list the time equation in $\Sigma_a$ we can get:

$\textcircled{1}$Being analogous as" .....from $x/C_x+x/C_x=2x/C$ reduced the $x$ we get (19)" in 4.3.1, in $\Sigma_a$ we can get

$$\frac{1}{C_{ax}} + \frac{1}{C_{-ax}} = \frac{2}{C} \tag{26}$$

$\textcircled{2}$Being analogous as" .....from (21) minus (22) plus (20) we get (23)" in 4.3.1, in $\Sigma_a$ we get

$$2 \frac{r_a}{C_{ar}} + x_a \left( \frac{1}{C_{-ax}} - \frac{1}{C_{ax}} \right) = \frac{2r_a}{C} \tag{27}$$

$\textcircled{3}$Being analogous in 4.3.1, bring (26) into (27)", the (27)" can be turned into

$$C_{ar} = \frac{C}{1 + \left( \frac{C_{-ax} - C_{ax}}{C_{-ax} + C_{ax}} \right) \cos \alpha} \tag{27}'$$

(Special remind: as known in 2.1, here $(C_x - C_{ax}) > 0$ is just opposite to $(C_x - C_{ax}) < 0$ of (23)'). Analogously, here (27)' appears: If $C_{ax}$ and $C_{ax}$ have been determined, it must be any $C_{ar}$ will have been determined; and on the $y_a$-$z_a$ plane $(\alpha = \pi/2)$ the $C_{ar}$ will be $C_{ar}$ at $\pi/2 = C$. Analogously if we do not only bring (26), but bring (26) and $r_a = (x_a^2 + y_a^2 + z_a^2)^{1/2}$ and $r_a/C_{ar} = t_a$ into (27)", the (27)" will not go to (27'), but go to

$$\left[ \frac{x_a - t_a \left( \frac{C_{ax} - C_{ax}}{2} \right)}{t_a \left( \frac{C_{ax} + C_{ax}}{2} \right)} \right]^2 + \frac{y_a^2 + z_a^2}{t_a^2 C_{ax} C_{-ax}} = 1 \tag{27}$$

Of course (27)" and (27)' and (27) all are $\Sigma_a$'s result of taking measure of the same wave front surface of light emitted from the $\Sigma_a$'s origin at $t_a$ instant of time of $\Sigma_a$ in three different angle of view. Being analogous (23) in 4.3.1, here (27) is an ellipsoid and analytic geometry tell us: The $\Sigma_a$'s origin (light source) is just on the left focus of the ellipsoid (27). While the $\Sigma_a$'s origin, $\textcircled{1}$may on the ellipsoid's two focus join-line (when $M_{a0} < M$) (the $M$ is "$M_0$ being in motion" in $\Sigma_a$ with speed $v$), $\Sigma_a$'s light be disturbed greatly by $M$ so that two focus join-line longer than two origins join-line), $\textcircled{2}$may on the rightward extended line out of the two focus join-line (when $M_{a0} > M$ and $M$ is in not great enough speed $v$, $\Sigma_a$'s light be disturbed lighter by $M$ than by $M_{a0}$ self so that two focus' join-line is shorter than two origins join-line i.e. with the speed $v$ not great enough the small mass object at $\Sigma_a$'s origin is still being wrapped in (27)). $\textcircled{3}$may on the rightward extended line re-extended even out of the ellipsoid (when $M_{a0} > M$ and $M$ is in great enough speed $v$, i.e. with high speed small mass object $\Sigma$ may not being wrapped in (27)) —In $\Sigma_a$ we also can see: If the speed (or kinetic energy) is great enough, a small mass object can go beyond the light which coming from the light source being stationary in a big referenced weight reference system, it is as similar as in 4.3.1 in $\Sigma_a$, while it just explain the reports on superluminal photonic tunneling experiments [5, 6, 7, 8] since 1993.

Being analogous in 4.3.1, since it is $\Sigma$ and $\Sigma_a$ taking simultaneous measurement of the same wave front surface of light emitted from the $\Sigma_a$'s origin, the $\Sigma_a$'s result is (27). What result the $\Sigma$'s is? (please note in $\Sigma$ the light source is "in motion"). Bring (1)b$^{-1}$ in 2.4 and $a_{22} = a_{33} = a$ into (27) we get (please see appendix VII and VIII)

$$\frac{x - t \left( \frac{C_x - C_{ax}}{2} \right)}{t \left( \frac{C_x + C_{ax}}{2} \right)} + \frac{y^2 + z^2}{t^2 \left( a_{44} - a_{41} C_{ax} \right) \left( a_{44} - a_{41} C_{ax} \right) \left( a_{11} + a_{41} C_{ax} \right) \left( a_{11} - a_{41} C_{ax} \right)} = 1 \tag{28}$$

(where $C_x$ is shown in (13) and $C_x$ is shown in (15)). Analogously as 4.3.1, since (27) and (28) are $\Sigma$ and $\Sigma_a$ taking simultaneous measurement of the same wave front surface of light emitted from the $\Sigma_a$'s origin, the $\Sigma_a$'s measurement result (27) is that the light source ($\Sigma_a$'s origin) is on the left focus of ellipsoid (27), the principle of relativity pledge: It must be that the light source ($\Sigma_a$'s origin) is also on the left focus of ellipsoid (28). Consequently it must be

$$\frac{(a_{44} + a_{41} C_{ax}) \left( a_{44} - a_{41} C_{ax} \right) \left( a_{11} + a_{41} C_{ax} \right) \left( a_{11} - a_{41} C_{ax} \right)}{a_{44} - a_{41} C_{ax} \left( a_{44} - a_{41} C_{ax} \right) \left( a_{11} + a_{41} C_{ax} \right) \left( a_{11} - a_{41} C_{ax} \right)} = 1 \tag{29}$$

Because only (29) can let the half minor axis' square of (28) be $t^2 \left( C_x \rightarrow \nu a_{11} / a_{44} \right)$. ( $C_x \rightarrow \nu a_{11} / a_{44} $), and then let the (28) become then from $t(C_x^+ + C_{-x}^+)/2 = t[(C_x^+ + va_{11}/a_{44}) + (C_{-x}^+ - va_{11}/a_{44})]/2$, can analytic geometry pledge: The $\Sigma$'s measurement result also is that the light source ($\Sigma_a$'s origin) is on the left focus of ellipsoid (28). While the $\Sigma$'s origin, $\textcircled{1}$ may on the ellipsoid two focus join-line, $\textcircled{2}$ may on the rightward extended line out of the two focus join-line but still being wrapped in the wave front surface (28) of the light emitted from the $\Sigma_a$'s origin, $\textcircled{3}$ may on the rightward extended line re-extended even out of the ellipsoid (28), analogously as the $\Sigma_a$'s measurement result (27).

Please not that the (28) is still different from (27) for $v=0$, being in accord with the new principle of relativity: "The laws of physics apply in all inertial reference systems, while any two reference systems in uniform relative motion are different".

Bring (13), (15) into (29), we solve the equation get $a_{41} = (a_{44}/v)[(a/a_{11})-1]$ (please see appendix IX). As it approximately is $\alpha_{ij} = \beta_{ij} = \gamma_{ij} = a_{ij}$, we can bring $a=a_{11}$ in 4.3.1 to $a_{41} = (a_{44}/v)[(a/a_{11})-1]$ in 4.3.2, leading $a_{41}=0$. Bring $a_{41}=0$ and $a_{11}=a_{22}=a_{33}=a$ into (1)b we get (1)b become

$$\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix} = \begin{pmatrix} a_{11} & 0 & 0 & (-v)a_{11} \\ 0 & a_{11} & 0 & 0 \\ 0 & 0 & a_{11} & 0 \\ 0 & 0 & 0 & a_{44} \end{pmatrix}$$

The Numerical Value Relation Of $\Sigma$ And $\Sigma_a$'s Measurement Data of $\Sigma$ And $\Sigma_a$'s Simultaneously Taking Measure of the Same Focus-Length of the Wave Front Surface of Light Emitted from $\Sigma$'s Origin and $\Sigma_a$'s Origin on $\Sigma_a$'s Position: Considering on $\Sigma_a$'s position, besides simultaneously measuring the same photon coming from the source stationary at $\Sigma_a$'s origin we also simultaneously measuring the another same photon coming from the source stationary at $\Sigma$'s origin: As the third postulate, if there is not measurement, both the wave front surface of light emitted from the $\Sigma_a$'s origin and from the $\Sigma$'s origin must be radius $r_a=t_aC$ sphere (because not been destroyed by measurement, so, both are radius $r_a=t_aC$ sphere — otherwise the measurement data of the light source's being "in motion" must be different from "in stationary") merely the centre of the $r_a=t_aC$ sphere from the $\Sigma_a$'s origin is at the $\Sigma_a$'s origin while the centre of the $r_a=t_aC$ sphere from the $\Sigma_a$'s origin is at the $\Sigma_a$'s origin being "in motion" with $\Sigma$ on $\Sigma_a$'s position. When there are simultaneous measurement of $\Sigma$ and $\Sigma_a$, the wave front surface of light emitted from the $\Sigma_a$'s origin being changed by the simultaneous measurement of $\Sigma$ and $\Sigma_a$, the wave front surface radius $r_a=t_aC$ sphere is changed into ellipsoid (27) (the centre of the sphere from the $\Sigma_a$'s origin is rightward divided another focus of the ellipsoid please note the distance between the two focuses of $\Sigma_a$'s measurement data is $2c_a$ of the (27) i.e. $2c_a=t_a(C_ax-C_ai)$); analogously the wave front surface of light emitted from the $\Sigma_a$'s origin (the wave front surface radius $r_a=t_aC$ sphere when there is not measurement on $\Sigma_a$'s position) is changed into ellipsoid (24) [under(25)] (the centre of the sphere from the $\Sigma_a$'s origin is leftward divided another focus of the ellipsoid (24)] [under(25)] please note the distance between the two focuses of $\Sigma_a$'s measurement data is $2c_a$ of the (24)] i.e. $2c_a=t_a(C_ax-C_ai)$). On the other hand, on $\Sigma_a$'s position the mass center of the two referenced weights $M_{00}$ and $M^+ = M_0/(a_{44}+va_{11})$ is on the length of $t_av$ (the $\Sigma_a$'s measurement data of the distance between the two origins in $\Sigma$ and $\Sigma_a$'s simultaneous measurement) and incises the length of $t_av$ to $l$ and $(t_av-l)$, where $l$ is the distance between the $M_{00}$ (at $\Sigma_a$'s origin) and the mass center of the $M_{00}$ and $M^+$. The $\Sigma_a$ seeing $\Sigma_a$ sphere centre being leftward divided another focus and $\Sigma_a$ self's sphere centre being rightward divided another focus, are only because the simultaneous measurements of $\Sigma$ and $\Sigma_a$ (nobody nearby and both $M_{00}$ and $M_{00}$ close to zero). If you ask that on $\Sigma_a$'s position how much the interaction impact of the simultaneous measurements of $\Sigma$ and $\Sigma_a$ is? We can only answer that we don't know, but we can confirm: From the new added postulate in 1.3 we can confirm that $\Sigma_a$'s measurement data of the two focuses' $2c_a$ and $2c_a^+$ show the magnitude of the interaction impact of simultaneous measurement of $\Sigma$ and $\Sigma_a$ on $\Sigma_a$'s position! Although in 3.2 we have known "we can consequently get" $s(a)$, $b)$, $c)$, $d)$ and the reversed case of $a)$, of $b)$, of $c)$, of $d)$, however, whether it is "$2c_a$ is in inverse ratio of the $\Sigma_a$'s self' referenced weight $M_{00}$ and in direct ratio of the $\Sigma_a$'s referenced weight $M^+$ in direct ratio of the $\Sigma_a$'s referenced weight $M_{00}$ and in inverse ratio of the $\Sigma_a$'s referenced weight $M^+$ or "$2c_a$ is in inverse square ratio of the $\Sigma_a$'s self' referenced weight $M_{00}$ and in direct square ratio of the $\Sigma_a$'s referenced weight $M^+$", $2c_a$ is in direct square ratio of the $\Sigma_a$'s referenced weight $M_{00}$ and in inverse square ratio of the $\Sigma_a$'s referenced weight $M^+$".

we do not know. From the new added postulate in 1.3, and from in 3.2 we have known "we can consequently get"'s a), b), c), d) and the reversed case of a), of b), of c), of d) on $\Sigma_a$'s position we can only confirm "$M_a0$ and $M^-$ who is bigger, the mass center of the $M_a0$ and $M^-$ will drift off from the middle of the $t_a$ to whom, whom's measurement data will be disturbed less, whom's ellipsoid two focus length will be less" i.e. on $\Sigma_a$'s position the simultaneous measurements of $\Sigma$ and $\Sigma_a$ disturb shown in $2c_a$ and $2c_a^-$ should be in accord with $2c_a/2c_a^- = l/(t_a\nu-l)$. While on $\Sigma_a$'s position the mass center of $M_a0$ and $M^-$ must be in accord with $l(M_a0+M^-) = (t_a\nu)/M^-$ i.e. $l/(t_a\nu-l) = M^-/M_a0$, then we get $2c_a/2c_a^- = l/(t_a\nu-l) = M^-/M_a0$. When $2c_a = t_a(C_ax-C_ax), 2c_a^- = t_a[(C_ax^-+v)-(C_ax^--v)]$, (12), (14) and $M^- = M_0/(a_{44}+v_{44})$ being placed in $2c_a/2c_a^- = M^-/M_a0$, under (1)d we get (please see appendix X)

$$\frac{(C_ax-C_ax^-)(a_{11}-C_x-C_x)}{(C_x-C_x)} = \frac{M_0}{M_a0}$$

(30)'

Determine The Element of (1)B Coefficient Matrix when we Re-Re-Reduce The Case

However, taking (30)' as an added equation to $f_3=0$, (19), (26) as four simultaneous equations to determine $C$. $C_ax, C_ax^-, C_x$ is not a good idea, for (30)' contains unknown quantity $a_{11}$ (please note: adding $\varphi_3=a_{44}/a_{11}$ will add more unknown quantities $a_{11}$ and $a_{44}$). Although having gone through reduce the case since 3, only when $\Sigma=\Sigma_a$ can we get (1)c. A little generally it may be $M_0\neq M_a0$ we cannot get (1)c, even re-reduce the case since 4.3, we still cannot find out the element of (1)b coefficient matrix, we can only confirm: 1)The light source is just on one focus of the wave front ellipsoid surface of light. 2)If the speed (or kinetic energy) is great enough, a small mass object can go beyond the light which coming from the light source being stationary in a big referenced weight reference system. 3)In this case it must be $a_{33}=a_{22}=a_{11}$ and $a_{41}=0$ in (1)b i.e. (1)b becomes (1)d and then we know: If the space length contract it must contract in all directions (instead of Einstein special relativity's only contract in the direction of motion and if dilate it will dilate in all directions as quasars apparent superluminal expansion observed in astrophysics (Blandford RD [3], Cohen MH [4]).

It must be pointed out that the Lorentz transformation of the special relativity is merely an approximate formula of the (1)c ignore the new added postulate to assume $C_a=C_a0$. Please note although $t=(t_a+x_a\rho)(1+v\rho)^{-1/2}$ in (1)c while the factor $\rho(1+v\rho)^{-1/2}$ like $-v/C^2$ is infinitely small when $v<<C$. So, the Lorentz transformation of the special relativity is not contrary to (1)c, not contrary to the $a_{41}=0$ of (1)d.

It also must be pointed out that in general the referenced weight mass may not be a particle the reference system's origin is on the center of the referenced weight mass, there are other objects and other reference systems joining simultaneously to measure with $\Sigma$ and $\Sigma_a$, the speed of a photon from a stationary light source is associated with all of the mass' space distribution and all of the reference systems joining to measure with $\Sigma$ and $\Sigma_a$, leading the $a_{ij}$(i,j) = 1,2,3,4) are the function of not only $M_a0$, $M_0$, $m_0$, $m_{10}$, $m_{20}$,……, the corresponding speeds 0, $v$, $u_a$, $u_{a1}$, $u_{a2}$,……in $\Sigma_a$, $\omega_x\xi\psi$,……variable representing the simultaneous measurements' disturbance, but also $x_a, y_a, z_a, t_a$ and $x, y, z, t$, there is not "it approximately is $a_{ij}=\beta_{ij}=y_{ij}=a_{ij}$", the $a$ and $a_{11}$ in 4.3.1 #the $a$ and $a_{11}$ in 4.3.2, we cannot "bring $a=a_{11}$ in 4.3.1 into $a_{41}=(a_{44}/v) [(a_{a11}-1)]$ in 4.3.2 leading $a_{41}=0$", (please see the explanation after (3) in 3.1).

Therefore the relation about $(x,y,z,t)$ and $(x_a,y_a,z_a,t_a)$ generally is (1)a, and the $a_{41}$ of (1)a may be analogous the $a_{11}=\rho(1+v\rho)^{-1/2}$ in (1)c, the $t$ in (1)a may be analogous the $t=(t_a+x_a\rho)(1+v\rrho)^{-1/2}$ in (1)c might go to time-reversal in some case, as the observation of time-reversal non-invariance in the neutral-kaon system published by CERN in 1998 [38, 39]. But we still believe that (1)a will still not disobey the conclusions "if the speed (or kinetic energy) is great enough, a small mass object can go beyond the light which comes from the light source rest in a big referenced weight reference system" and "if the space length contract it must contract in all directions instead of Einstein special relativity's only contract in the direction of motion" etc educated from 4.3.1 and 4.3.2, though determining the $a_{ij}(i,j)=1,2,3,4$ of (1)a is very difficult or impossible.

Please note in 4.1 because two reference system's origins are in a short way off then (16) and (17) are in action, there $a_{41}=0$ (i.e. $f_3=a_{41}/a_{11}=0$), the $v$ is not limited i.e. it may be $v\rightarrow0$ or $C$ or $>C$. While usually it may be $v<<C$ or $v\rightarrow0$ but $v\neq0$ leading $f_1\neq0$ and $f_2\neq0$, only $f_3$ can allow $f_3=a_{41}/a_{11}=0$. Therefore, in the ending of 4.2 abnegating $a_{44}|f_1=f_2$, adopt $a_{44}|f_2=f_3$ to get (1)c is right. The physical meanings of $f_3=a_{41}/a_{11}=0$ i.e. $C_ax,C_x-C_ax,C_x+C_ax$ i.e. $C_ax/C_x-C_ax/C_x-v2/C$ is clear. However, with $f_3=0$, (19) and (26) three simultaneous equations we still cannot determine $C_ax,C_ax,C_x,C_x$ four unknown quantities (please note: adding $\varphi_3=a_{44}/a_{11}$ will add more unknown quantities $a_{11}$ and $a_{44}$). Now, the $C_ax,C_ax,C_x,C_x$ and $a_{11}a_{44}$ of the (1)d still are unknown quantities. —If

we re-re-reduce the case: adding v＜＜C, can we find out the C-ax Cax C-x Cx and _a_11 _a_44 of the (1)d?

Taking note of that only both the time length in motion dilate and the space length in motion contract are in action, can we be able to completely explain the Michelson- Morley experiment [30] and almost all of these experiments are taken under v＜＜C. So, under re-re- reduce the case (i.e. adding v＜＜C), we can from (2) and (4) get $$ = \frac {\alpha}{a _ {4 4} + v a _ {4}} $$ + (31)

44 41 a

11 (or from (3) and (5) it must be (a_44+_va_41)_a_11/_a_44=α/_a_44 i.e. (_a_44+_va_41)_a_11 =α). Here α is a constant waiting to be determined. From (31) φ3=_a_44/_a_11= 2 44 _a /α we get a_44=(αφ3)1/2, _a_11=_a_44/φ3=(α/φ3)1/2. Because when _v=0 it must be a_11=_a_22=_a_33=1 (please see (1)a|_v=_0 in _2.3) and C- ax=Cax=C-x=Cx=C, then φ_3=_a_44/_a_11=[_C-axCax(C-x+Cx)]/[C-xCx(C- $$ a _ {x} + C _ {a x}) ] = 1, \mathrm {s o}, \mathrm {i t} \mathrm {m u s t} \mathrm {b e} \alpha \equiv 1. \mathrm {B r i n g} \alpha \equiv 1, $$

a11=(α/φ3)1/2=(φ3)–1/2, a44=(αφ3)1/2=(φ3)1/2 into (1)d, (1)d

goes to

a

a (1)e

a t t φ

0 0 0 a

3 In (1)e the element of the coefficient matrix of (1)e is completely determined by φ_3. Bring _a_11=(φ_3)–1/2 into (30)′, then the (30)′ goes to

( ) −

0 ax ax (where φ_3 is shown in (18)). Please note that with _a_41=0 i.e. _f_3=0 the φ3 can be reduced (please see appendix XI). Now, in mathematics, with four simultaneous equations _f_3=0, (19), (26), (30), we can determine four unknown quantities _C-ax , Cax , C-x , Cx , and then from (18) get φ_3 (actually it is with _f_3=0 , (19), (26), (30), φ3=_a_44/_a_11, and (31) i.e. (_a_44+_va_41)_a_11=α≡1 six simultaneous equations we can determine six unknown quantities _C-ax , Cax , C-x , Cx , _a_11 , _a_44 ).

However, when we stand C-ax=γ with f_3=0 , (19), (26) three simultaneous equations, get _Cax=γC/(2_γ –_C), C- x=γ_2_C/[2_γ2 –(2γ–_C)(v+γ)], Cx=γ_2_C/[(2_γ–_C)(v+γ)] bring into (30), the equation about γ (=C-ax) is higher than 5 degree (so do we stand Cax or C-x or Cx ). While as we known, any polynomial equation of degree ≥5 with real or complex coefficients is not solvable by radicals [45]. Therefore, we have no more better choice but to determine the approximate value of C-ax, Cax, C-x, Cx and φ_3 when _v＜＜C. When we simultaneous f_3=0 , (19), (26), (30) we can get $$ \left. C _ {- a x} ^ {\prime} \right| _ {v = 0}, \left. C _ {a x} ^ {\prime} \right| _ {v = 0} $$ $$ \left. C _ {a x} ^ {\prime} \right| _ {v = 0}, \left. C _ {- x} ^ {\prime} \right| _ {v = 0}, \left. C _ {x} ^ {\prime} \right| _ {v = 0}, \left. \varphi_ {3} ^ {\prime} \right| _ {v = 0}, \left. \varphi_ {3} ^ {\prime \prime} \right| _ {v = 0} \mathrm {a n d} $$ $$ \left[ \rho_ {3} ^ {m} \right| _ {v = 0} $$ _v φ etc (please see appendix XI). Then, when v＜＜C, approximate formula (neglected more higher order infinitely small) of Taylor series expansion are (where ς =M_0/_Ma_0) ( ) (1 ) _ax C C v σ $$ \approx C + \frac {(- c)}{(1 + c)} $$ ( ) (1 ) ax C C v σ $$ \approx C + \frac {(- c)}{(1 +} $$ + , (1 ) ax C C v σ $$ \approx C + \frac {\sigma}{(1 + \sigma)} $$ σ − σ − + + $$ \frac {1}{(1 + \sigma)} v, C _ {x} \approx C + \frac {(- 1)}{(1 + \sigma)} v, $$ $$ , C _ {- x} \approx C + \frac {1}{(1 + \sigma)} v, $$ + , $$ \varphi_ {3} \approx \left. \phi_ {3} \right| _ {v = 0} + \left(\left. \phi_ {3} ^ {\prime} \right| _ {v = 0}\right) v + \frac {1}{2 !} \left(\left. \phi_ {3} ^ {\prime \prime} \right| _ {v = 0}\right) v ^ {2} + \frac {1}{3 !} \left(\left. \phi_ {3} ^ {\prime \prime} \right| _ {v = 0}\right) v ^ {3} = $$ v C σ σ − + + + +

2 ( 1) 1 0 ( ) 0 (1 )

1

1 $$ =, a _ {4 4} = \sqrt {\varphi_ {3}} \approx $$

a_11=_a_22=_a_33= ≈ φ ( 1) 1 ( ) (1 ) _v C σ

σ − + +

(32) + Of course all v/C_2 above should be replaced by –(_Cax–C-

ax–v)/CaxC-ax because (1)c is more precise than the Lorentz transformation. After all, as it approximately is α_ij= βij= γij= _a_ij, and then we can bring _a=a_11 in _4.3.1 into a_41=(_a_44/_v)[(a/a_11)–1] in _4.3.2, then leading a_41=0. Precisely it is αij≈βij≈γij≈_a_ij, i.e. precisely the _a_11 in _4.3.1 ≠ the a_11 in _4.3.2, we cannot bring the a=a_11 in _4.3.1 into the a_41=(_a_44/_v)[(a/a_11)–1] in _4.3.2 get a_41=0 i.e. _precisely the (1)e’s a_41 =(_a_44/_v)[(a/a_11)–1] should be infinitely small instead of 0. So, for some reason (please see later) we should approximately take _a ≈1 and then a_41≈ (_a_44/_v)[(1/_a_11)–1] then (1)e goes to Physical Science & Biophysics Journal

$$\begin{bmatrix} x \\ y \\ z \\ t \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{\phi_3}} & 0 & 0 & \frac{(-\nu)}{\sqrt{\phi_3}} \\ 0 & \frac{1}{\sqrt{\phi_3}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{\phi_3}} & 0 \\ \frac{\sqrt{\phi_3}}{(-\nu)} (1 - \sqrt{\phi_3}) & 0 & 0 & \sqrt{\phi_3} \end{bmatrix} \begin{bmatrix} x_a \\ y_a \\ z_a \\ t_a \end{bmatrix}$$

(where $\varphi_3$ is shown in (18) and its’ approximate formula (neglected more higher order infinitely small) of Taylor series expansion is shown in (32), more precise please see the ending of appendix XI). Because only so, then can we get $a_{41} = (a_{44}/v)[(a/a_{11})-1] \approx (a_{44}/v)[(1/a_{11})-1] = (a_{44}/v)(a_{44}-1)$ in the (1)f, and then in our (on earth) taking measure of a micro-particle ($\sigma = M_0/M_{a0} \rightarrow 0$) will it be $a_{41} = (a_{44}/v)(a_{44}-1) \approx (a_{44}/v)/v \approx -v/(C^2)$ being in accord with the $a_{41}$ of the coefficient matrix of the Lorentz transformation. It is obvious that it is (1)f’s $a_{41} = (a_{44}/v)(a_{44}-1)$ being $\rightarrow 0$ but $\neq 0$ leading the time might go to time-reversal in some case, as the observation of time-reversal non-invariance in the neutral-kaon system published by CERN in 1998 [38, 39].