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*ω* = 235696.8871 km/s, necessary for its transition to a massless state, was determined by two independent methods. Zeroing of the mass of matter also occurs at a temperature T =2.17 . 10^{36} m_{o} (m_{o} is the rest mass of the particle in grams). At higher temperatures or speeds of movement, the mass of bodies becomes negative. The resulting formulas made it possible to calculate the masses of X,Y-bosons equal to 4.606 . 10^{-9} g. The temperature of the Superunification of the four fundamental interactions, including gravity, is estimated to be 4.72 . 10^{31} K.

Possible properties and mechanics of the interaction between a particle with a negative mass and a particle with a positive mass were considered in studies [1-5], in which it was shown that no fundamental physical laws contradict the existence of substances with a negative mass. The theory of the emergence of a substance with a negative mass is based on the ideas of E.L. Feinberg [7] about relativistic changes in the length scales in the direction of motion of a body, time, mass, and acceleration of a moving body as a result of the action of certain forces is described in [6]. The idea of E.L. Feinberg's decomposition of the force acting on the body into components responsible for changing the dynamic characteristics of the body is hypothetical since there is no evidence of its validity in the literature.

The purpose of this work is to search for other independent evidence of the formation of a substance with a negative mass, which would confirm or refute the conclusions of [6], and at the same time assess the correctness of the approach of E.L. Feinberg.

The composition of the force F acting on the body, which is expended on changing the parameters of the motion of a physical object, according to [6,7] can be represented as

$F={F}_{v}+{F}_{m}+{F}_{l}+{F}_{\tau},\text{(1)}$

Where the cost of force F_{v} is directly to change the speed v of the object in the direction of Force F, F_m is the cost of force F to change the mass *m* of the object, Fl is the cost of force F to change the length *l* of the object in the direction of force F, F is the cost of force Fτ to change the rate time τ = ∆t. There is no change in the length of the object in directions perpendicular to the speed of the object in accordance with SRT. In the future, since all components of the acting force F, as well as the length, speed, and acceleration of the object is estimated only in the direction of the force F, then in the following formulas, for the convenience of recording and perception of information, their vector nature will not be fixed, and equality (1) can be written:

$F={F}_{v}+{F}_{m}+{F}_{l}+{F}_{\tau},\text{(2)}$

Where

${F}_{v}={m}_{0}\cdot \frac{dv}{dt}={m}_{0}\cdot a,\text{(3)}$

${F}_{m}=v\cdot \frac{dm}{dt},\text{(4)}$

${F}_{l}=\frac{{m}_{0}v}{{l}_{0}}\cdot \frac{dl}{dt},\text{(5)}$

${F}_{\tau}=-\frac{{m}_{0}v}{{\tau}_{0}}\cdot \frac{d\tau}{dt},\text{(6)}$

$m=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}},\text{(7)}$

$l={l}_{0}\sqrt{1-{v}^{2}/{c}^{2}},\text{(8)}$

$\tau =\frac{{\tau}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}},\text{(9)}$

And *m _{0}*,

Substituting formulas (7 - 9) into equalities (4 - 6) and differentiating, we get:

${F}_{m}={m}_{0}a\cdot \frac{{v}^{2}/{c}^{2}}{{\left(1-{v}^{2}/{c}^{2}\right)}^{1.5}},\text{(10)}$

${F}_{l}=-{m}_{0}a\cdot \frac{{v}^{2}}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}},\text{(11)}$

${F}_{\tau}=-{m}_{0}a\cdot \frac{{v}^{2}/{c}^{2}}{{\left(1-{v}^{2}/{c}^{2}\right)}^{1.5}}\text{(12)}$

Substituting expressions (3, 10 - 12) into equality (2), we obtain:

$F={m}_{0}a\left(1-\frac{{v}^{2}}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}}\right).\text{(13)}$

Formula (13) shows that the costs of the acting force for increasing the body mass and slowing down time are mutually compensated. This indirectly indicates the dependence of the time rate on the object on the body mass.

Since the mass depends on the speed of the body according to the formula (7), then, in order to note this circumstance, we write the formula (13) in the form:

$F=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}\cdot a\left(\sqrt{1-{v}^{2}/{c}^{2}}-{v}^{2}/{c}^{2}\right).\text{(14)}$

$F={m}_{0}{a}_{0}.\text{(15)}$

Where

${a}_{0}=a=\frac{F}{{m}_{0}}.\text{(16)}$

Substituting (16) into (14), we get:

$F=\frac{{m}_{0}}{+\sqrt{1-{v}^{2}/{c}^{2}}}\cdot {a}_{0}\left(+\sqrt{1-{v}^{2}/{c}^{2}}-{v}^{2}/{c}^{2}\right).\text{(17)}$

The formula for the dependence of the acceleration of the body *a _{v}* on its speed, at the current time, will look like this:

${a}_{v}={a}_{0}\left(\sqrt{1-{v}^{2}/{c}^{2}}-{v}^{2}/{c}^{2}\right).\text{(18)}$

Table 1 presents the characteristics of a body moving with acceleration, calculated by formulas (7, 18), as well as the magnitude of the mass defect

$\Delta {m}_{r}={m}_{r}-{m}_{0},\text{(19)}$

Acceleration *a _{r}*, and mass

From formula (18) and Table 1, it can be seen that the acceleration of the body will decrease with increasing body speed, and at some critical speed, *v =* *ω* will be equal to zero. To find *ω*, we solve the equation

$\sqrt{1-{v}^{2}/{c}^{2}}-{v}^{2}/{c}^{2}=0\text{(20)}$

Where

$v=\omega =c\sqrt{\frac{\sqrt{5}-1}{2}}\approx 0.7862c\approx 235696.8871\text{}km/s\text{(21)}$

When squaring equality (21)

${v}^{2}={\omega}^{2}=\frac{\sqrt{5}-1}{2}{c}^{2}\approx 0.61803{c}^{2}=\frac{1}{\Phi}{c}^{2}\text{(22)}$

We obtain that the square of the critical velocity *ω* is the golden ratio of the squares of velocities not exceeding the speed of light, i.e. he divides the range of squares of these speeds in the extreme and average ratio into two parts so that the boundary of this section is the geometric mean between these parts. However, in most literature sources, the reciprocal value Φ = 1.618034 is taken as the value of the golden section [9].

When the speed of the object is greater than the critical one, the acceleration of the body, defined by formula (18), formally becomes negative. If this were the case, then the bodies that have reached such a speed should turn back. In reality, we know that elementary particles, for example, in hadron colliders, reach speeds close to the speed of light, and no negative accelerations are observed. However, if we apply the principle of reinterpretation [8], and replace the negative value of the body acceleration with a negative value of the mass, then from such a transformation it follows that after the body reaches the critical speed *ω*, the internal structure of the body is rearranged in such a way that the body mass becomes negative, the acceleration of the body positive, and the body continues to move in the same direction.

The acceleration of a physical body is determined through the differentiation of its speed

$v=\frac{dx}{dt},\text{(23)}$

Where *dx* is the distance traveled by the body in time *dt*. Any movement of the body is due to the forced interactions of this body with the surrounding matter and its characteristics. Thus, the interval *dx* and the time interval *dt*, included with equality (23), are external, depending on the speed of the body, parameters by which, in this case, the speed is found. In the process of motion, the quantities *dx* and *dt* do not change the speed of the point, but, on the contrary, they themselves change depending on the change in speed, so that

$dx=df\left(v,\tau \right),\text{(24)}$

$dt=df\left(\tau \right),\text{(25)}$

Where $\tau $ is the proper time of the moving body.

Relations (24, 25) show that in formula (23) the differential *dt* of the argument *t* in this case does not depend on *t*. In this regard, the second differential of the function *f (
$\tau $
)* is equal to the differential of its first differential:

$d\left[df\left(\tau \right)\right]={d}^{2}f\left(\tau \right)\text{(26)}$

Using the well-known formula for the second derivative in terms of differentials [10]

$\frac{{d}^{2}y}{d{x}^{2}}=\frac{dx\cdot {d}^{2}y-dy\cdot {d}^{2}x}{d{x}^{3}},\text{(27)}$

Body acceleration can be defined as

$\frac{dv}{dt}=\frac{{d}^{2}x}{d{t}^{2}}-\frac{dx}{dt}\cdot \frac{{d}^{2}f\left(\tau \right)}{d{t}^{2}}.\text{(28)}$

For *df(
$\tau $
)* = *d
$\tau $
*, equality (28) will look like

$\frac{dv}{dt}=\frac{{d}^{2}x}{d{t}^{2}}-\frac{dx}{dt}\cdot \frac{{d}^{2}\tau}{d{t}^{2}}\text{(29)}$

or, taking into account (23), we obtain

$\frac{dv}{dt}=\frac{{d}^{2}x}{d{t}^{2}}-v\cdot \frac{{d}^{2}\tau}{d{t}^{2}}\text{(30)}$

On the right side of equation (30), both terms are, in essence, partial derivatives of the path and proper time, so equality (30) can be written as:

$a=\frac{dv}{dt}=\frac{{\partial}^{2}x}{\partial {t}^{2}}-v\frac{{\partial}^{2}\tau}{\partial {t}^{2}}.\text{(31)}$

Based on the dependence of the body length on its speed according to formula (8), the change in its length will be equal to

$dl=d{l}_{0}\sqrt{1-{v}^{2}/{c}^{2}}\text{(32)}$

To determine the reduction of the object's proper time (acceleration of the time rate), we consider that the signal moving at the speed of light will cover the distance *dl* in the time *d
$\tau $
*:

$\frac{dl}{c}=d\tau =\frac{d{l}_{0}\sqrt{1-{v}^{2}/{c}^{2}}}{c}=dt\sqrt{1-{v}^{2}/{c}^{2}},\text{(33)}$

Where

$\frac{\partial \tau}{\partial t}=\frac{d\tau}{dt}=\sqrt{1-{v}^{2}/{c}^{2}},\text{(34)}$

$\frac{{\partial}^{2}\tau}{\partial {t}^{2}}=\frac{\partial \sqrt{1-{v}^{2}/{c}^{2}}}{\partial t}=\frac{\partial \sqrt{1-{v}^{2}/{c}^{2}}}{\partial v}\cdot \frac{\partial v}{\partial t}\text{(35)}$

$\frac{{\partial}^{2}\tau}{\partial {t}^{2}}=-\frac{v}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}}\cdot \frac{\partial v}{\partial t}\text{(36)}$

The second derivative of the path with respect to time at the moment of starting is equal to

$\frac{{d}^{2}x}{d{t}^{2}}=\frac{\partial v}{\partial t}={a}_{0},\text{(37)}$

Where *a _{0}* is determined by the formula (16). Substituting formulas (36, 37) into (31), we obtain

${a}_{v}=\frac{dv}{dt}=\frac{\partial v}{\partial t}\left(1-\frac{{v}^{2}}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}}\right)={a}_{0}\left(1-\frac{{v}^{2}}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}}\right)\text{(38)}$

Newton's second law in accordance with (38) will have the form:

$F={m}_{0}a={m}_{0}{a}_{0}\left(1-\frac{{v}^{2}}{{c}^{2}\sqrt{1-{v}^{2}/{c}^{2}}}\right).\text{(39)}$

And since the mass of the body depends on its speed according to the formula (7), it is rational to represent the formula (39) in the form:

$F={m}_{v}{a}_{v}=\frac{{m}_{0}}{+\sqrt{1-{v}^{2}/{c}^{2}}}\cdot {a}_{0}\left(+\sqrt{1-{v}^{2}/{c}^{2}}-{v}^{2}/{c}^{2}\right)\text{(40)}$

Formula (40) coincides with formula (17), and the dependence of the acceleration *a _{v}* on the body velocity (38) is determined by the formula (18). Thus, in this section, we obtained the same results as in [6], but without using the idea of E.L. Feinberg on the admissibility of expanding the action of a force on a change in the dynamic characteristics of a body, which confirms his hypothesis.

From the data in Table 1 it follows that the body mass should increase with increasing body speed, but at a speed equal to *ω* ≈ 0.7862c ≈ 235696.8871 km/s due to the principle of reinterpretation at this speed, the internal structure of the body is restructured in such a way that the body mass becomes negative. With a further increase in the speed of the body in absolute value, the absolute value of the body mass will continue to increase. Such a dependence of the body mass on its speed suggests that the mass of the body, in itself, does not determine the amount of matter in the body, but the amount of matter can be determined by the modulus of the mass value. The existence of bodies with negative mass suggests that, in addition to the amount of matter contained in them, material bodies also have a gravitational charge, which, depending on their speed, and, possibly, for other reasons, can be both positive and negative. And negative. Curious is the fact that, as follows from Table 1, the acceleration of the body *a _{r}* with increasing body speed the value

Numerous experiments at Hadron colliders have not yet made it possible to fix the formation of substances with a negative mass. If the idea of changing the parameters of a moving body from its speed is correct, the fact that bodies with a negative mass were not detected would indicate that the transformation of bodies with a positive mass into bodies with a negative mass either does not take place, or the transformation does not affect the instruments for monitoring the course of the experiment. In an appreciable way, or to the fact that, externally, with respect to observation instruments, the behavior of a substance with a negative mass is the same as with a positive mass.

From the Equivalence Principle (EPE) of inert *m _{i}*, active

$F={m}_{i}a\text{(41)}$

$F={m}_{i}a=-G\frac{{m}_{p}{m}_{a}}{{r}^{2}},\text{(42)}$

Where the minus sign in formula (42) indicates the attraction of interacting bodies to each other [13]. Consider the interaction of two bodies of positive mass:

As follows from formula (42), if the masses of both bodies are positive, then the bodies are attracted (Figure 1). Formula (41) indicates that the accelerations of both bodies are positive, i.e. coincide with the direction of the acting forces. Therefore, the bodies move towards each other. Now consider the case when the masses of both bodies are negative (Figure 2):

Substituting the values of the masses into formula (42), we obtain that bodies with negative mass must also attract. Let us consider the case when two bodies have masses of different signs (Figure 3)

When the masses are substituted into formula (42), the result will be positive. Therefore, the bodies must repel. However, on closer examination, we will stumble upon "pitfalls". If the mass is negative, then the acceleration is also negative, which follows Newton's second law (41). Therefore, the body will move in the opposite direction to the acting force. Let's review the previous examples with this conclusion in mind. As can be seen from Figure 4, bodies with negative mass will repel, and bodies with masses of different signs will move in the same direction in the direction of motion of a body with positive mass, and with additional acceleration, since in addition to the external force acting on the bodies, there is also the force of their gravitational influence Each other.

In accordance with the above, if the EPE is correct, then the formed particles with negative mass should repel each other, which will lead to the expansion of the beam of moving particles.

In the direction of motion of the particle beam, there will be a stratification of particles into avant-garde particles repelled from behind the flying particles, and particles lagging behind. But since an external accelerating force *F* still acts on all particles of the beam, then in general, all particles of the beam will move with acceleration.

If the principle of reinterpretation is not applied, i.e. consider it erroneous, then, as can be seen from Table 1, the mass of the body would remain positive, and the acceleration of the body *a _{v}*, when the body speed

In [13], the New Equivalence Principle (NPE) is substantiated, taking into account the possibility of the existence of particles of negative mass:

$\left|{m}_{i}\right|=\left|{m}_{a}\right|=\left|{m}_{p}\right|,\text{(43)}$

From which it follows that the amount of matter in the body is determined by the modulus of its mass:

$\left|{m}_{i}\right|=\left|-{m}_{a}\right|=\left|-{m}_{p}\right|=+m\text{(44)}$

It is easy to see that, on the basis of NPE, two bodies with both positive and negative masses will be attracted to each other, and two bodies with different masses will repel each other. Obviously, a substance with a negative mass will move away with acceleration from a substance with a positive mass and form colonies. In hadron colliders, when particles reach a velocity above the critical velocity *v = **ω*, their mass (Table 1) becomes negative. But they will be attracted to each other in the same way as particles with positive mass.

Therefore, they cannot be distinguished from particles with a positive mass. However, at speeds slightly exceeding the critical speed, particles of both signs will be present in the beam of moving particles. They will repel each other, which will lead to some expansion of the beam. Thus, one of the signs of the formation of matter with a negative mass can be the expansion of a beam of moving particles in hadron colliders. Another sign of the existence of matter with a negative mass, if the NPE is correct, is the presence in our Universe of huge islands of matter with a negative mass remote from our part of the Universe. If EPE is correct, then matter with negative mass should be scattered throughout the Universe. In both cases, matter with negative mass will manifest itself as one of the types of "dark matter", the need for the existence of which has already been explained [14].

As is known [14], the average kinetic energy of the translational motion of particles is written as

$w=\frac{\stackrel{}{\overline{m{v}^{2}}}}{2}=\frac{3}{2}kT,\text{(45)}$

Where *k* = 1,3807·10^{-16} erg/*K* – Boltzmann's constant, *T* – absolute temperature К. Specifically, for the critical velocity *ω*, equality (45), taking into account (7), can be written

$\frac{{m}_{0}{\omega}^{2}}{\sqrt{1-{\omega}^{2}/{c}^{2}}}=3kT\text{(46)}$

Noticing that

$\sqrt{1-{\omega}^{2}/{c}^{2}}=\frac{1}{\Phi},\text{(47)}$

Find

${m}_{0}=\frac{3kT}{{\omega}^{2}\Phi}\text{(48)}$

$T=\frac{{m}_{0}{\omega}^{2}\Phi}{3k}\text{(49)}$

After substituting the numerical values of the constants k, *ω*, Φ into equality (49), we obtain a simple formula for calculating the transition temperature of a substance to a massless state:

$\u0422=2.17\cdot {10}^{36}{m}_{0}\text{K,(50)}$

where mo is the rest mass of the particle in grams.

Grand Unification theories predict that a system of particles heated to the temperatures of the Grand Unification becomes a system of massless particles and, upon subsequent cooling, will pass through special temperature points at which this system will experience phase transitions due to spontaneous symmetry breaking, and some of the particles in each of these transitions will gain mass. On the other hand, from Table 1 it can be seen that the dependence of the body mass on its velocity at the critical velocity of the body *ω* undergoes a transition from positive to negative values. This indicates that at this speed the value of the mass of the body is zero. It follows from this that formula (48) allows one to find the rest mass of a particle if the temperature at which the particle, in accordance with the Higgs theory [14,15], loses mass is known, and formula (49) makes it possible to determine the corresponding transition temperature to the massless state if the mass of the particle is known. Table 2 shows the values of *m _{o}* Mand

At *T* ≈ 10^{28}K, the Grand Unification of fundamental interactions ends [14,15], the strong interaction is separated from the electroweak one, and the carriers of the Grand Unification X- and Y-bosons acquire masses ≈10^{15}-10^{16} GeV/c^{2} [14] (according to [14] ≈10^{14} GeV /c^{2}). The rest of the particles remain massless. The mass of these bosons calculated by the formula [48] turned out to be 2.586·10^{15} GeV/c^{2} and is more accurate.

The carriers of the Superunification forces of all four fundamental interactions, including gravity [14], are Planck particles with a mass of 1.222·10^{19} GeV/c^{2}. According to formula (49), we find that the Planck particles should become massless at a temperature of 4.732·10^{31}K.

Black holes are formed as a result of gravitational contraction and accretion of matter. The speed of particles falling towards the center of the Black Hole increases as it approaches the center and eventually reaches the critical speed *ω*, after which the mass of the falling particles becomes negative. In accordance with the EPE, such particles should repel each other, and then the Black holes should fly apart, but this is not observed. On the contrary, NPE requires the attraction of negative particles to each other, their merging with the formation of a nucleus with a negative mass, which makes the Black Hole a stable formation and a candidate for the role of a type of “dark matter”.

It is proved that the force acting on the body can be decomposed into components responsible for relativistic effects. Formulas for calculating the mass and transition temperature of an elementary particle to a massless state are derived. It has been established that as the body speed increases to the critical speed *ω* ≈ 235696.8871 *km/*, the body mass increases to the value Φ≈1.618034*m _{0}*, and upon reaching the critical speed, the body undergoes an internal restructuring, as a result of which the body mass becomes negative. A substance with a negative mass can be obtained by increasing the body temperature, which depends, in accordance with formula (49), on the rest mass of the original substance. From what has been said, it follows that the minimum speed of the body movement necessary for the transition of the mass of the substance to a state with zero or negative mass for all bodies is the same and equal to

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