Einstein’s idea on space-time was to take the “physical object” as the premise. He said: “Space-time is not nec- essarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended” [5] . Einstein’s words remind people: the space-time is actually one kind of attributes of physical objects. In order to understand more deeply the space-time, it is necessary to more sufficiently face the object. It has long been known, “the actual objects of physical reality” are in hierarchical structure. In general, between different structural levels the objects could have fundamentally different concept and attributes. On the space-time attributes of the object, if accept the viewpoint of Einstein at his later years, namely the actual space-time must depend on the actual object, we will not be able to rule out such a possibility that the current understanding space-time will only be maintained to certain one very low-level particles. Starting from this assumption, we can further deduce as follows.

• When measuring such a particle, because the uncertain amount of the momentum and the energy can not be infinite, according to the Landau interpretation of the uncertainty principle, then the space uncertain amount and the time interval between twice measuring are impossible to zero. If abandoning the unreasonable infinity of the uncertain amount of momentum and energy, and taking a limited maximum value to replace them respectively, then the uncertain amount of space and the time interval will have the minimum value that are greater than zero. This indicates that in the actual observable sense the particles on this level should have a certain size.

• So-called the particle has “a certain size”, it means that the particle not only has a 3-dimensional space size, also has an intrinsic time size. Thus, according to the space-time unity of the special relativity, such a particle that has the nonzero and finite intrinsic scales of space-time should have an intrinsic 4-dimensional geometric shape.

• Taking into account when the particle is in rest, because there are the 3-dimensional space spherical symmetry and time translation symmetry, then the geometrical shape of particle should be a 4-dimensional cylinder that the timeline is taken as the rotational symmetry axis, and its axial height is the maximum value of the intrinsic time interval of particle.

• According to the above, the space and time of the 4-dimentional cylinder of particles must be the real physical space-time, rather than any formal mathematical space-time, such as the Minkowski space-time etc., so only Euclidean geometry is suitable for it.

• What particles are the final form that maintain the understandable time and space? According to the existing knowledge let us assume that they are particles of the quarks-leptons level. Of course, this is a mandatory assumption, of which correct or not need by testing its derived results later to judge.

Above these can be attributed to a basic postulate:

Having a nonzero and finite intrinsic time height is the most important feature of the particles 4-dimensional space-time cylinder model. The time height concept comes from the Landau interpretation of the energy-time uncertainty relation [6] , Landau believes that the particle energy E at one moment can be arbitrarily accurately

measure, in the uncertainty relation

is the difference of two accurate measurement values of energy, is the time interval between twice measurements. Therefore, in according to the meaning of particle’s intrinsic time height concept defined by the [Postulate], there is obviously:

The [Corollary 2] points out that the particle’s shape that is randomly observed at one moment is a 3-dimen- sional space cross-section of the particle’s 4-dimensional space-time cylinder in the coordinates-system of observer. Referring to Figure 1, according to the space rotational symmetry with the velocity u direction as axis, it can be seen that the 3-dimensional space cross-section is the rotational ellipsoid or its section, that is, an ellipsoid crown, an ellipsoid zone as well as the sphere or a cylinder in extreme situations. What specific shape is depends on two factors: 1) the angle, which is associated with the speed u according to (1); 2) the randomness that a moment of measurement occurs in the time height T. Referring to Figure 1, all types of 3-dimen- sional space cross-section are listed in Table 1:

In Table 1, denotes the angle between the short side and the diagonal in the rectangle that the particle’s 4-dimensional cylinder is projected on the plane of observer’s coordinate system, the speed corresponding to is

When i.e. (see Figure 1(a)), for any moment t in time intervals and we can define a dimensionless variable:

Similarly, when i.e. (see Figure 1(b)), for any moment t in time intervals and we can also define the dimensionless variable:

is called the random fluctuation variable, apparently there is

According to Figure 1 and (1) (3) (4) (9) (9’) (10), using the elliptic equation to make definite integrals the

Table 1Table 1

. The 3-dimensional space cross-sections of the 4-dimensional space-time cylinder of particles

	,	,	,	,
	Sphere with diameter D*
		Complete ellipsoid with major axis Λ1 and minor axis D*	Ellipsoid zones with same height (=Λ2), different end-plane diameters

volume V of 3-dimensional space cross-sections can be gotten:

where

is the volume of 3-dimensional space cross-section of rest particle relative to observer, that is the volume of sphere with diameter as.

According to Figure 1 further getting other features of the 3-dimensional space cross-sections as follows:

1) If and, it is a complete ellipsoid whose minor axis is and long axis is (especially when, , it becomes a sphere),

2) If but, it is an ellipsoid crown having different height and bottom area S,

where is the area of ellipsoid projection in observer coordinate plane.

3) If and, it is an ellipsoid crown having different height,

4) If and, it is an ellipsoid zone having different end-plane diameters and a same height,

5) If, it is a 3-dimensional space cylinder having the same height and different diameter D,

6) If, the particle is no observed at this moment.

To sum up, the differences as well as advantages of the PSTC model compared to other non-point particle models as follows:

1) The basis of establishment of PSTC model is realistic and reliable, it is based upon the spherical symmetry of 3-dimensional space, the translational symmetry of time, the space-time unity of relativity, the Heisenberg uncertainty principle, and its applying objects are quarks and leptons, which all are empirical facts and recognized principle, there are no additional imaginary content.

2) The PSTC model is simple, there is only 4-dimensional space-time, with no additional dimension.

3) For space-time, the PSTC model requires real physical space-time, i.e. Euclidean space-time, rather than any mathematical space-time, such as the Minkowski space-time etc. We know that the Minkowski geometry is one of expression forms of special relativity but has not the substantive contribution to the special relativity.

4) The PSTC model points out that the particles not only has a size of the space, in the time also has a non- zero and finite intrinsic “height”, the ratio of the time height T^* to the diameter D^*, , is a large number, which shows the difference between the particle’s intrinsic time scale and space scale.

5) The PSTC model can be used to develop the basic theory of physics, such as the development and improvement of special relativity.

In accordance with [Postulate], “the amount of mass” included inside the particle of quarks and leptons etc. must uniformly be distributed in the intrinsic 4-dimensional space-time cylinder, and further in accordance with [Corollary 2], the particle’s shape observed at one moment is a 3-dimensional space cross-section of the 4-dimen- sional cylinder, so the mass m of particle is proportional to the volume V of 3-dimensional cross-section, that is

where, m₀ denotes the rest mass of particle. Then (11) into (18) we obtain the more universal mass equation:

where on the speed please see (8), on the speed substitution variable please see (12).

To this equation interpretations and in-depth discussions as follows:

[I] The mass m is not a single variable function of the speed u, it depend on both of the speed u and the random variable. The variable that appears in the derivation process of equation makes the randomness concept automatically enter the relativistic equations.

[II] When, (19) has the maximum value form

It can be seen that m_R depends singly on u. When (20-1) happens to be the Lorentz-Einstein mass formula (In this case m_R and are called Lorentz mass and Lorentz energy). However when, the maximum form is another (20-2), in particular, the mass is not the infinity but a finite maximum value.

Because the new equation eliminates the infinity, so the theoretical suppression of the physical particles moving at light speed will be eliminated. When the electron’s maximum mass

[III] Because generally m is a double variable function on u and, even when the particle’s speed is constant, its mass m and its energy still at and change with. When, and , which shows in particles movement there is a strange “high speed but low mass (energy)” effect or HSLM effect for short.

[IV] With reference to Figure 1(a), it can be proved that when the probability that the particle appears in the HSLM state of (i.e.) is calculated by the following formula

Since Lorentz energy is singly corresponding to the speed, so the probability can be calculate by replacing u via (20-1)

The formula shows that the higher the particle’s speed u (i.e. the higher the Lorentz energy E_R), the greater the possibility appearing in HSLM state. Because is huge and there is always in the current observation, so the probability of particle in HSLM state is very small. Such as the electron’s, the probability of different energy electron appearing in HSLM state is shown in the following table (Table 2):

The probability of particle appearing in a neighborhood of

However the probability of particle in “normal state” of i.e. is

Because there is always in the current observation range, so i.e. and state is most probable, what described by the Lorentz-Einstein mass formula is just the mass-speed relationship in this state.

[V] It can be seen in Figure 1(b), when, the probability of particle appearing in each state is all equal, so the particle is in the fluctuation state of the mass and energy. Especially when, the following relation is deduced from (19-2b):

when, the projection of the particle’s 4-dimensional spacetime cylinder on the plane and plane of coordinate system of observer is a circle, in which there is the relationship of an arbitrary circumferential angle and:

(27) into (26) and using (21), to obtain

which shows that when a particle moving at the light speed, its energy changes proportional to the square of the sinusoidal of phase angle.

[VI] For the mass, according to the state probability to weighted average, we get the mean mass

The (29) shows:

1) The mean mass has included overall effects of the variable, it is only associated with the speed u and is applicable in the all speed range.

2) In the “lower energy” state of, (29) and the Lorentz-Einstein formula difference is very small.

3) For composite particles and macroscopic objects, as long as the rest mass have define, then (29) are generally applicable for physical objects of all levels.

4) The mass/energy of satisfying the conservation law are neither the nor the, but the.

5) (29) multiplied (3) to obtain

That is, the product of the particle’s apparent time height T multiplied by the mean mass is a Lorentz invariant.

6) When has the maximum

when

has the minimum value

It should be noted, according to the PSTC model, only to the particles of level of quarks/leptons the speed is meaningful, but for composite particles such a speed can not be defined. (32), (33) show that the particle in the lowest energy state is not in absolute motionless but in motion at the tiny speed.

7) Putting the total mean energy, the mean momentum and the rest energy into Equation (29), the new energy-momentum relation formula is obtained as follows

where

Compared with the old energy-momentum relationship

(34) than (36) multi-a, which indicates that the existence of so-called Lorentz invariance violation (LIV) is an inevitable result of the new special relativity, and the violation coefficient can be directly exactly calculate by (35). Equation (35) shows that the absolute value of LIV coefficient is dependent on the speed u of particle, that is the higher the u the smaller (not the larger!) the, when attains the minimum value. Table 3 lists LIV coefficients corresponding to different speeds. Of particular note is that if the particle speed is low to (m/s) = 0.289 (Å/s) then, (34) becomes the strange form

(37) suggests that people should re-understanding of the “stationary” concept, because such an extreme tiny speed ~0.289 (Å/s) has never been actually observed. About (29) and (34) we will do for a more in-depth discussion in Section 4.

By the state probability to weighted average, the maximum average length of the 3-dimensional space cross-sectional in the particle movement direction is got (see Figure 1, Table 1, (13) (14) (14’) (16)). According to the [Postulate], is the final standard ruler in the length measurement in the particle movement direction. Using, the improved Lorentz transformations can be derived:

and inverse transforms

where is the movement velocity of an inertial relative to another inertial along the x-axis, and. According to the reasoning mode of the classical theory of special relativity, from the above coordinate transformations it can be further derived that

the time dilation (40)

the “length contraction” (41)

the velocity addition law (42)

and

Using above those results we can design two ideal experiments respectively to evaluate the particle’s intrinsic time-space ratio and the particle’s intrinsic time height.

About this issue, the new special relativity gives three levels theoretical results which consistent with observations of HiRes and Auger: First, pointed up the existence of Lorentz invariance violation is inevitable and the LIV coefficient can also be strictly calculated by (35); Second, although existed the LIV effect, for above 4 × 10¹⁹ eV UHECR protons the calculated, which indicates that LIV effect too weak to change the GZK cutoff; Third, although existed the GZK suppression not rule out possible appearing a very few random events beyond GZK energy. Now, detailed as follows.

The people have known that the higher than 10¹⁸ eV UHECRs, in propagating from the extragalactic sources to the Earth, interact with CMB photons to result in a decline of their energy and flux and finally cause the spectrum suppression at (3 - 6) × 10¹⁹ eV, that is the GZK cutoff predicted by Greisen, Zatsepin and Kuzmin in 1960s [7] [8] . However by the 1990s, the AGASA [9] experiment results show an extension of the spectrum beyond the GZK cutoff, so the GZK mechanism based on the special relativity is challenged. For this reason, S. R. Coleman and S. L. Glashow [10] introduced a theory of Lorentz invariance violation (LIV), a tiny correction term is added into the energy-momentum relationship, namely

Thus, if the coefficient absolute value is just as small as ~10⁻²³ enough to eliminate the GZK cutoff. However, in 2008, the HiRes Collaboration announced to confirm the GZK cutoff with a standard deviation [1] and then the Auger collaboration gives the result consistent with HiRes and rejects a single power law spectrum above 4 × 10¹⁹ eV with a 6σ standard deviations [2] . Therefore the data of HiRes and Auger can set constraints on the LIV. The current common research way on this problem is from different values to calculate the energy spectrum, then comparison with the data of Auger and HiRes thereby find the most fitting value as the constraint on LIV, obviously this phenomenological study cannot reveal the nature of problem.

Here let us use the new Equation (34) of the special relativity to solve this problem. Although (34) and the Coleman-Glashow formula (57) both have the same form (I even deliberately use the same symbol), their origins and meanings are completely different. A detailed account of (34) characteristics and significance is as follows:

1) (34) is derived out from a basic Equation (29) of the new special relativity, so the occurring of is inevitable, which does not require reluctantly to add. In fact, the existence of so-called Lorentz invariance violation effect is one of inherent features of the new special relativity.

2) The LIV coefficient is a function of the speed and has the strict expression i.e. (35), so is a theoretical value independently calculated before the experimental observation, not by fitting observation data to adjust.

3) (29) points out, for all kinds of particle the physical quantity with common upper limit is just the speed i.e. the light speed c (not other physical quantities, e.g., energy). When, the particle energy is not the infinity but a limited maximum (see (31)), so for different particles according to their different rest energy E₀ there are different maximal attainable energies (and not different maximal attainable speeds).

4) (35) shows, the higher the speed u, the smaller (and not the larger!) the coefficient absolute, when it attains the minimum value.

5) When Lorentz energy and speed is one to one, so via (20-1) can replace u then (35) is rewritten as

In the energy region, (58) can be simplified to

For the UHECR spectrum ankle region protons and ⁵⁶Fe, we use (58’) to specifically calculate the LIV coefficient values and list to Table 4.

To above 4 × 10¹⁹ eV protons, the calculated. By comparison with some fitting calculations [11] [12] , it can be seen, although (34) has expected that there is the Lorentz invariance violation effect, the violation coefficient is too small to change the GZK suppression of UHECR spectrum, which is consistent with experimental results that the HiRes collaboration and the Auger collaboration confirm the GZK suppression with a 5σ standard deviation [1] [2] .

On the other hand, we know that events beyond GZK energy were also really recorded in cosmic ray observation, in order to explain this phenomenon, it is necessary to use the basic Equation (19-1). As described above, (34) is derived from (29), and (29) is derived from (19). The mass of composite particles is usually embodied as the mean mass described by (29) that has contained the all influence of the random variables, however, for the proton as most simple composite particle, when its energy is very high the random variable still has some influence, and therefore the high-speed low-energy (HSLM) effect has still a certain degree of performance.

Now let us imagine that a group of beyond the GZK energy protons from an extragalactic source which is more distant than the mean free path fly to the Earth. In this process, a very few of protons could appear in certain HSLM state in some time. Because at the HSLM state the energy and the relative cross-section are both smaller than the normal values, therefore the possibility that the HSLM protons interact with the CMB photons will be smaller than of the normal protons, thereby they can propagate farther than the normal mean free path. If arriving Earth before restored to the normal state, then the Earth will receive the over GZK energy event. Of course, such event is very rare. To summarize, although the tiny LIV coefficients will not change the GZK cutoff, yet not rule out the possibility that a very few of beyond GZK energy protons reach the Earth, moreover, occurring such small probability events beyond GZK energy is random and do not constitute a single power-law spectrum, which are all consistent with Auger observation [2] .

2002, J. Magueijo and L. Smolin modified Einstein’s special relativity and proposed the “doubly special relativity (DSR)”, of which one of four basic principles is the observer independence of the Planck energy [3] . Below we prove that this “basic principle” can be derived by the new special relativity.

Considering a particle at a velocity u moves along the x axis of an inertial frame, its energy and momentum are and relative to the observer in inertial frame. In another inertial frame at speed moving along the x axis relative the, the particle’s energy and momentum are transformed by (44) (45) respectively. When the particle speed relative to the observer of the inertial frame, its mass attains the maximum (see (31)), thus its energy and momentum attain also the maximum and. Putting and into (44) and (45), the calculations are and respectively, so whether in the or this particle’s energy all attain the maximum value of the same, however, the relative velocity v between two inertial systems has no any impact. The above calculation has proved an important result, that is, when moving at the strict light speed c, each particle can attain its own maximal mean mass (or energy), which is the invariant independent of the observers.

As a special case, we use (31) and (55) to calculate the maximum value of the mean mass of neutron

where, denotes the rest mass of neutron, is the Planck mass. (59) shows, the Planck mass (or energy) and the maximum value of the neutron average mass (or energy) both are in the same order of magnitude, their distinguish is only a constant factor. Because the maximum of neutron mean mass (or energy) is the invariant independent of the observers, so the Planck mass (or energy) also is necessarily the invariant independent of the observers, that is, there is the observer independence of the Planck energy.

In addition, we prove that an important formula of DSR can be derived by (29). According to (20-1), letting the Lorentz mass and the rest mass replace the speed ratio, then (29) is written in the following form

When, it can be simplified to

At the same speed the mass ratio of any kind of particles can be replaced with the neutron’s mass ratio, so

According to (59), the Planck mass and the neutron’s mean mass maximum are in the same order of magnitude, their difference is just on a constant factor, so can be substituted by to write (29) as

It is interesting, the DSR also has such a similar formula [3] , which even is as the highlight [13] .

Now it has been seen that the DSR’s one basic principle and a similar to (63) result all can be proved by the new special relativity. However, this does not mean that my new special relativity and the DSR are the same thing, in essence they are different in not only the theoretical fundamentals but also some typical results. For example, the new special relativity has proved that the Planck energy cannot serve as the common upper limit of the energy of all kinds of particles, and indicate that of importance is not the Planck energy but the parameter describing the particle’s intrinsic space-time feature. Another example, as described in Section 4.1, the formula (35) calculating LIV coefficient point out, the higher the speed, the weaker the LIV effect, thus to prophesy that the GZK cutoff of UHECR will not change. However, in theories of the DSR as well as superstrings, loop quantum gravity and so on there are no such contents.

According to (19.1), at a same speed, the mass m of HSLM electron in state is smaller than the mass of normal electrons in state, i.e.. In the electron storage ring, so when the beam passes through the accelerating electric field, HSLM electrons in the beam will be accelerated more quickly than normal electrons. If these abnormal electrons return to the most probable state before reaching the deflection magnetic field, their energy will certainly be higher than the normal electrons energy, thereby an abnormal emission of electrons much higher than the beam energy will appear in the RF cavity downstream direction (see Figure 3).

1) Strength: Using (19-1) (23’) (24) (15) it can be proved that numbers of super-high energy electrons may appear per hour

where, (MeV) denotes the energy of super-high electron; E_R (MeV) denotes the energy of beam; U (MV) denotes the accelerating voltage; I (mA) denotes the average current intensity; E₀ (MeV) is the rest energy of electron; e (C) is the electron charge. So in the numbers of super-high energy electrons may appear per hour

In existing electron storage rings, calculated N₂₁ values are all very small, such as, if E_R = 2000 MeV, U = 1.5 MV, I = 500 mA, about average 1 hour appearing 1 event of super-high energy electrons of 34,000 MeV.

2) Energy-positional distribution: Shown in Figure 3, the extension line of beam in straight section (i.e. the tangent line of beam in deflection section) as the y axis and the bending magnet starting point as the origin point to establish a coordinate system. For different electron storage rings, according to the beam energy, the magnetic field strength and the machine structure parameters, a relation between the energy of super-high energy electron and the injection point coordinate x at the absorber can be deduced. For example, the electron storage ring in Figure 3, if, there is

where r is the normal bending radius of beam in the bending magnetic field; a is the length of first bending magnetic field; b is the distance between the exit end of first magnetic field and the absorber.

The super-high electrons occurring in electron storage rings are very rare, but it is still feasible to detect such small probability events by existing experimental techniques, which because:

1) Electron storage rings have higher current and can run continuously for a long time, so to facilitate the long-term search and cumulative record. And, to many electron storage rings, in the RF cavity downstream direction occurring super-high energy electrons, the beam pipeline of synchrotron radiation are not installed and the vacuum chamber is only closed by the blind end, which facilitate the placement of detector.

2) The sampling electromagnetic calorimeter can be used as a detector, and such detectors are mature technologies. Because the copper blind-end is very thick, the super-high energy electron injects in which will occur electromagnetic cascade shower, so the detector will in fact accept such the secondary particles shower. To the sampling electromagnetic calorimeter, the copper blind-end is equivalent to its front-side absorber. Of course, it is best to place the detector to inside of the vacuum chamber of storage ring. In addition, taking into the residual gas bremsstrahlung along the straight section axis is a main interference, so in order to reduce its impact, the detector should be appropriately off-axis placed.