Specific Energy, Lorentz Factor & WIMP Annihilation

JAVIER VIAÑA

College of Engineering and Applied Science, University of Cincinnati, Cincinnati, Ohio, USA Email: [email protected]

Abstract. The time dilation formulas of both the Special Relativity and General Relativity could be studied using a factor dependent on specific energy. Should such factor be used to define the relativistic mass, the equation that arises is an approximation of the mass and energy relation. This mathematical definition of mass is finally compared to the equations that define Dark Matter Annihilation into charged states via loop-level processes.

Keywords. Specific energy — relativistic time — relativistic mass — dark matter

1. Introduction 푑퐸퐴 푟퐵 휀퐴 = (2) 푟퐵 푑푚 The way time and mass is understood has accurately 퐴푟퐵 predicted most of the research that has been carried out over Being 푑푚퐴 the mass, and 푑퐸퐴 the energy of the particle the last century. But there are still many uncertainties in the 푟퐵 푟퐵 universe which lack sufficient understanding of these two 퐴 that has 퐵 as a reference. variables, such as Dark Matter. For that reason, the present Let 푆 be considered a set of particles 퐴. paper, an alternative mathematical perspective of the ∫ 휀 푑푚 = ∫ 푑퐸 (3) Lorentz Factor, is proposed. 퐴푟퐵 퐴푟퐵 퐴푟퐵 The resulting expressions of time and mass are applied to The total energy of the set 푆 (퐸 ) will be the integral of three different cases to provide a brief comparison between 푆푟퐵 the current proven knowledge from Special and General all the energetic contributions. Relativity and the present theory.

Finally a fourth application is considered to benchmark the ∫ 휀퐴푟 푑푚퐴푟 = 퐸푆푟 (4) proposed formulation with the insights of photon production 퐵 퐵 퐵 from WIMP annihilation into charged states via loop-level The energy contribution made by each particle 퐴 of the set processes (휒휒 → 훾푋) (Bertone 2010; Coogan, Profumo, can be the same regardless of the particle (condition (5)), Shepherd 2015). 휀퐴 = 휀푆 (5) 푟퐵 푟퐵 2. Methodology If so, 휀 is constant throughout the mass of the set and 퐴푟퐵 therefore can be extracted from the integral. A particle of infinitesimal mass (푑푚) can be identified in 휀 ∫ 푑푚 = 퐸 (6) space-time with the three position coordinates (푥, 푦, 푧) and 푆푟퐵 퐴푟퐵 푇표푡푟퐵 its time (푡). This particle also contains a differential energy On the other hand, the mass of the set will be the sum of all (푑퐸), even though its mass is infinitely small. However, its the differential masses that compose it, specific energy (휀) is much higher, 푚푆 = ∫ 푑푚퐴 (7) 푑퐸 푟퐵 푟퐵 휀 = (1) 푑푚 Thus,

Energy, position, time and even mass need a reference. 퐸푆푟 휀 = 퐵 (8) Kinetic energy, for example, requires a zero-speed 푆푟퐵 푚푆 reference. Similarly, gravitational energy is associated with 푟퐵 its corresponding null potential. In fact, the same thing If it is necessary to apply the formulas described below for a happens to the mass; its value depends on the observer. non-differential mass set, condition (5) has to be verified. Thus, the specific energy of a particle of infinitesimal mass, Otherwise, portions of the subject matter where said 퐴, can be redefined by taking another particle, 퐵, as a condition is verified should be considered. reference, Let 퐴 and 퐵 be two particles of infinitesimal mass, 푑푚퐴 and 푑푚퐵 respectively, whose energy states are different. Consider also a light beam 퐷 moving with a speed 푐.

1 Specific Energy, Lorentz Factor & WIMP Annihilation

2 Said spatial separation is defined below as the product of 휀 = 푐 퐷푟 2 퐴 the equivalent velocity (푣푒푞) and the time of the particle (푡).

휀퐵 = 휀 푟퐴 Ω = 푡 푣푒푞 (15) The equivalent speed is understood as that which would be necessary for all the specific energy of the particle to be specific kinetic energy. 퐴 퐵 퐷 푐2 Knowing that the expression of the specific kinetic energy 휀퐴 = 휀 휀 퐷 = 푟퐵 푟퐵 2 is,

Figure 1. Specific energy differences of the particles 푣 2 휀 = 푒푞 (16) 푘푖푛 2 The specific energy of particle 퐴 taking 퐵 as a reference is, The equivalent speed of 퐴 (based on reference 퐵) will be, 휀 = 휀 (9) 퐴푟퐵 푣푒푞 = √ 2 휀퐴 (17) By reciprocity, the specific energy of particle 퐵 taking 퐴 as 퐴푟퐵 푟퐵 a reference is, Therefore, the equivalent separation between particles 퐴 and 휀퐵 = 휀 (10) 푟퐴 퐵 seen from 퐵 is, Particle 퐷 has only kinetic energy. This is defined as,

Ω퐴푟 = 푡퐵 √ 2 휀퐴푟 (18) 1 2 퐵 퐵 퐸푘푖푛 = 푚 푣 (11) 2 It should be noted that the time of the particles is not the 푣 and 푚 are the velocity and mass of the particle same. Since 푣푒푞퐴 is the equivalent velocity of the particle respectively. Using the expression (1) the specific kinetic 푟퐵 퐴 observed from 퐵, it is necessary to use the time of the energy can be obtained, observer, which in this case is 퐵. 푣2 This transformation is carried out for each specific energy 휀 = (12) 푘푖푛 2 of Fig. 1, obtaining Fig. 2. Ω퐷 = 푡퐴 푐 Particle 퐷 has no mass, but as seen in (12) its specific 푟퐴 energy does not depend on mass, Ω퐵 = 푡퐴 2 휀 In the system described according to Fig. 1, 퐷 travels at 푟퐴 speed 푐. The speed of light is independent of the reference frame, therefore, 2 푐 퐴 퐵 퐷 휀 = (13) 퐷푟퐴 2 Ω = 푡 2 휀 Ω퐷 = 푡퐵 푐 퐴푟퐵 퐵 푟퐵

푐2 Figure 2. Equivalent distances between the particles 휀퐷 = (14) 푟퐵 2 These equivalent distances (defined by 푑푖푠푡) are related to The particles of Fig. 1 can be infinitely close in three- each other. However, they differ according to the reference dimensional space. In fact, they both could be in the exact from which they are observed. same point of the universe. Since the derivative is ( ) ( ) considered over the mass, not over the volume, they would 푑푖푠푡 퐴, 퐷 푟퐴 ≠ 푑푖푠푡 퐴, 퐷 푟퐵 (19) still be different particles, even in such an extreme 푑푖푠푡(퐵, 퐷) ≠ 푑푖푠푡(퐵, 퐷) (20) condition. 푟퐵 푟퐴 However, despite their proximity, they are not the same To compare (19) and (20) the factor 푘 is used as seen in particles, their specific energy differentiates them. (21) and (22), Therefore, in order to distinguish the energy states of each 푑푖푠푡(퐴, 퐷) = 푘 ( 푑푖푠푡(퐴, 퐷) ) (21) particle, the 푥, 푦, 푧 position is not enough. In other words, 푟퐴 푟퐵 the universe characterized by 푥, 푦, 푧, 푡 is not adequate to ( ) ( ) 푑푖푠푡 퐵, 퐷 푟퐵 = 푘 ( 푑푖푠푡 퐵, 퐷 푟퐴 ) (22) make the comparison of the energies discussed in the present study. Then, Instead, an equivalent two-dimensional universe is used. Ω = 푘 ( Ω + Ω ) 퐷푟퐴 퐷푟퐵 퐴푟퐵 (23) This universe is defined by two variables Ω, 푡. Ω is the equivalent spatial separation of the particles due to their Ω = 푘 ( Ω − Ω ) 퐷푟퐵 퐷푟퐴 퐵푟퐴 (24) specific energy.

Specific Energy, Lorentz Factor & WIMP Annihilation 2 Substituting the values of the equivalent distances, 1 푇′ = 푇 푐 푡퐴 = 푘 ( 푡퐵 푐 + 푡퐵 √ 2 휀 ) (25) 휀 (33) 1 − 2 √ 푐 푐 푡퐵 = 푘 ( 푡퐴 푐 − 푡퐴 √ 2 휀 ) (26) 2 Considering the common factor, The specific energy of the particle 퐴 with respect to particle 퐵 is entirely kinetic. Thus, 푐 푡퐴 = 푘 푡퐵 ( 푐 + √ 2 휀 ) (27) 2 1 1 2 퐸퐴 푚퐴 ( 푣퐴 ) 푚퐴푣 푣2 푐 푡 = 푘 푡 ( 푐 − √ 2 휀 ) (28) 푟퐵 2 푟퐵 2 (34) 퐵 퐴 휀 = 휀퐴 = = = = 푟퐵 푚 푚 푚 2 Due to the symmetry of the problem, it is not possible to 퐴 퐴 퐴 If this value (34) of the specific energy is substituted in solve 푘 using the information from a single equation. It is necessary to incorporate the information of both to obtain (33), then it can be seen how the relation that arises is exactly the one defined by (32). the parameter 푘. Therefore (27) and (28) must be

multiplied, obtaining (29). 3.2. Gravitational effect in time 2 2 2 푐 푡퐴푡퐵 = 푘 푡퐴 푡퐵 ( 푐 − 2 휀 ) (29) Simplifying, (35) is the formula that defines time dilation due to 1 gravitational effect (Chou et al. 2010), 푘 = 1 휀 ′ 1 − (30) 푇 = 푇 √ 푐2 2퐺푀 (35) √1 − 2 푅 푐2 The factor obtained in (30) allows one to relate the variables The previous expression could also be rewritten as (33) of two reference systems whose specific energies differ. where the particle considered is only submitted to the effect The last part of the development is analogous to the one of gravity, and thus its specific energy is only gravitational, made to get the Lorentz Factor (31) (Einstein 1905; Einstein 퐸퐴 푚 푔 푅 퐺푀 퐺푀 1915; Cenko et al. 2015). Indeed, the mathematical form of 푟퐵 퐴 (36) 휀 = 휀퐴푟 = = = 푔 푅 = 2 푅 = both factors (30) and (31) is very similar, 퐵 푚퐴 푚퐴 푅 푅 1 It is curious that both equations (32) and (35) have (33) as a 훾 = 푣2 (31) common ancestor. √1 − 푐2 3.3. Mass and energy relation However, due to the initial transformations, the result is different. In the next section, (30) will be applied to In (37) is expressed the equation that relates the relativistic different cases to observe the distinction between the mass or total mass (푚푇표푡) with the rest mass (푚0) and the current theories and the one developed in this study. Lorentz Factor (Roche 2005),

푚0 3. Applications 푚푇표푡 = 푣2 (37) √1 − 푐2 3.1. Velocity effect in time Equation (38) relates 푚푇표푡 and 푚0 if the proposed factor As it can be seen in (31), the Lorentz Factor depends on (30) is considered. both the velocity (푣) and the speed of light (푐). This 푚0 푚푇표푡 = parameter (훾), defines the time dilation due to the velocity 휀 1 − (38) of a particle (Einstein 1916; Francis et al. 2013), √ 푐2 2 1 푇′ = 푇 Equation (38) is an equation that relates the rest mass, the 2 (32) 푣 relativistic mass, and the total specific energy of a certain √1 − 2 푐 particle. But the famous equation (39) already relates those The relativistic time (푇′) can be understood as the time of a variables (Rainville, Thompson, Myers et al. 2005). certain particle 퐴 whose velocity with respect to a reference 2 (39) 퐵 is 푣. 퐸 = 푚푒푥푡푟푎 푐 Should the parameter suggested in (30) be used, the Where the extra mass (푚푒푥푡푟푎) is the difference between previous expression becomes, the total mass (푚푇표푡) and the rest mass (푚0), that is,

푚푇표푡 = 푚0 + 푚푒푥푡푟푎 (40) Expressing (39) in a different form,

3 Specific Energy, Lorentz Factor & WIMP Annihilation

The resulting plot is of interest, since both functions are 퐸 = ( 푚 − 푚 ) 푐2 (41) 푇표푡 0 actually tangent at the point where 푦 is equal 1 (Fig 3.). It is To make (41) comparable to (38), both equations should thus observed that for those values close to 1, the error have the variable of specific energy 휀. For such purpose, incurred is very small, while when being far from 1, the (41) can be modified as follows, error can be very high. In other words, when 푚 and 푚 are similar (usual in 퐸 푚푇표푡 − 푚0 2 푇표푡 0 휀 = = 푐 (42) small particles, where 휀 is often small), both formulas are 푚푇표푡 푚푇표푡 applicable; (38) and (43). In fact, for the low-mass 푚0 experiments carried out on Earth (small values of 푚 ), 휀 = (1 − ) 푐2 (43) 푒푥푡푟푎 푚푇표푡 if (43) was correct, no significant difference between (38) and (43) would be appreciated in the measurements. Equations (43) and (39) are the same, but it would be truly Both formulas are very similar, and that is remarkable given remarkable if this last equation (43) is exactly equal to the fact that their origins are completely different. equation (38), which is proposed in this research. Each one understands the mass in its own way, and perhaps Let equations (38) and (43) be compared. To do so, this is indeed the most interesting point to think about. equation (38) will be rewritten as, 푚 1 Equation (39) says that mass is a linear property; that can be 푇표푡 = calculated adding up their parts simply with a sum. Indeed, 푚0 휀 1 − (44) (39) and (40) together form a system of two equations. √ 푐2 Formula (39) without (40) is meaningless. But equation (38) 2 is suggesting that mass depends on its energy, and that the Now (43) will be substituted in (44) and if both are the sum of the parts (푚 + 푚 ) is not the same as the same, the resulting combined equation should lead to an 0 푒푥푡푟푎 whole (푚푇표푡). identity, To see more in detail the relation between these two 푚 1 푇표푡 = versions of the theory, (38) and (43) will be compared using 푚 푚 (45) 0 √2 0 − 1 퐸 instead of 휀. To do so, (44) will be transformed as 푚푇표푡 follows, 푚0 Calling 푦 the quotient of , 푚푇표푡 1 푚푇표푡 = 푚 1 1 0 퐸 (49) = (46) 1 − 2 푦 √ 푐 √2 푦 − 1 푚푇표푡 2 Simplifications end in two functions, one on the left of the 퐸 푚 2 equality and one on the right, 1 − = ( 0 ) 푐2 푚 (50) 1 푚푇표푡 2 푇표푡 푓1 = (47) 푦 2 2 푐 푚0 퐸 = 푚푇표푡 (1 − ( ) ) (51) 1 2 푚푇표푡 푓2 = (48) √2 푦 − 1 2 2 푐 푚0 퐸 = ( 푚푇표푡 − ) (52) The functions defined by 푓1 and 푓2 are not the same. Thus, 2 푚푇표푡 equations (38) and (43) are not the identical. To see how big Equation (52) is exactly the same as (38), but allows the the difference is, in Fig. 3 both are plotted together having 푦 calculation of the bonding energy, if 푚 and 푚 are as the independent variable. 0 푇표푡 known. Fig. 4 and Fig 5. show the difference between (41) and (52) 2 in a three-dimensional space if 푚0 and 푚푇표푡 are the independent variables. 1.

1 푓2

. 푓1 푦 . . 1 1. 2 2. Figure 3. Tangency of functions 푓1 y 푓2

Specific Energy, Lorentz Factor & WIMP Annihilation 4 21 1 퐸 ( ) 푐2 푥2 퐹 = 푧 − (푦 − ) = (55) 1 2 푦 0.5 0 Being the point 푃 any point belonging to the line 푚0 = - 0.5 푚푇표푡, thus 푃(푚, 푚, ). Calculating the partial derivatives, -1 푐2 2 푥 -1.5 퐹 = − (− ) (56) -2 푥 2 푦 100 80 2 2 60 10 푐 푥 40 60 80 퐹푦 = − (1 + ) (57) 20 40 0 2 푚 ( 푘푔 ) 20 2 푦 푇표푡 0 0 푚0 ( 푘푔 )

퐹푥 = 1 (58) Figure 4. Differences between equations (41) and (52) Substituting 푃, 1 21 퐸 ( ) 2 푐 2 푚 2 퐹푥|푃 = − (− ) = 푐 (59) 1 2 푚 0.5 푐2 푚2 퐹 | = − (1 + ) = −푐2 (60) 0 푦 푃 2 푚2 - 0.5 퐹푧|푃 = 1 (61) -1 Finally, -1.5 2 2 푐 ( 푚0 − 푚) − 푐 (푚푇표푡 − 푚) + (퐸 − ) = (62) -2 0 40 20 0 60 20 40 80 2 2 60 80 100 퐸 = 푐 (푚 − 푚) − 푐 ( 푚 − 푚) (63) 100 푚 ( 푘푔 푇표푡 0 푚푇표푡 ( 푘푔 ) 0 ) 퐸 = (푚 − 푚 ) 푐2 = 푚 푐2 (64) Figure 5. Differences between equations (41) and (52) 푇표푡 0 푒푥푡푟푎 Thus, proving that the plane is indeed tangent to the curved These representations (Fig. 4 and Fig 5.) are of vital surface. importance. The plane represents equation (41) while the But still, there is some hidden relation between equations curved surface refers to equation (52). (41) and (52) that has not been covered. In order to see it, Both equations provide the same value of the energy when the variable 푚푇표푡 will be extracted from equation (52), obtaining, 푚0 and 푚푇표푡 are equal, and therefore 퐸 is zero. In the immediate vicinity of this line, (52) is a good approximation 2 2 퐸 2 of (41). But when leaving the adjoining margin, the errors 푚푇표푡 − 푚푇표푡 − 푚0 = (65) 푐2 become much more noticeable, even reaching infinitely different values. 2 퐸 4 퐸2 ± √ + 4푚 2 Interestingly, the plane defined by (41) is tangent to the 푐2 푐4 0 (66) 푚 = curved surface (52). In fact, it is tangent along the entire line 푇표푡 2 퐸 = . This is an uncommon feature between two three- Considering only the positive value of the mass, dimensional functions. 2 Both surfaces predict a negative energy in case 푚0 > 퐸 퐸 푚 = + √ + 푚 2 (67) 푚푇표푡, positive when 푚0 < 푚푇표푡, and null in case the 푇표푡 푐2 푐4 0 masses are equal. 퐸2 The following development shows how the plane is tangent, 2 Assuming that is negligible in comparison to 푚0 , then, across the straight line, to the curved surface. 푐4 The equation of the plane tangent to a given point 푃 of a 퐸 푚 = + 푚 (68) function 퐹 is given by, 푇표푡 푐2 0 Which in the end leads to, 퐹 | (푥 − 푥 ) + 퐹 | (푦 − 푦 ) + 퐹 | (푧 − 푧 ) = 푥 푃 푃 푦 푃 푃 푧 푃 푃 (53) 퐸 = (푚 − 푚 ) 푐2 (69) Being 퐹, 푇표푡 0 푐2 푚 2 In such a way that the equation (41) could be understood as 퐹 = 퐸 − ( 푚 − 0 ) = (54) 2 푇표푡 푚 퐸2 푇표푡 an approximation of (52), if is much smaller than 푚 2. 푐4 0 Renaming with 푥, 푦, 푧, On the other hand, it should be remembered that 퐸 =

5 Specific Energy, Lorentz Factor & WIMP Annihilation

푚 푐2 is actually a specific case in which the particle 푐2 푚 2 0 (74) considered has no velocity according to the reference. If it 퐸 = 푚푇표푡 (1 − 2) 2 푚푇표푡 has velocity, the expression becomes (Okun 2009), Equation (74) is exactly the same equation as the one 퐸2 = (푚 푐2) 2 + (푝푐)2 (70) predicted in this theory (52). Equation (52) has been obtained from a mathematical 푝 is the linear momentum of the study particle. However, development that understands mass as a dependent variable this generalization is not necessary with the proposed of the specific energy. But equation (71) is a well-known equation (52), since it already considers all the specific relation developed from energy conservation laws, used to energy according to the desired reference. seek explanations for the observed data of Dark Matter If the previous holds, it would imply that the energy cannot presence in nearby halos. The fact that they are both the be converted into matter nor vice versa, but rather the same it is not only an impressive result, but also a energy affects the weight of matter. clarification of what the real nature of Dark Matter could be.

6. Conclusions 5. WIMP annihilation via loop-level processes This development studies from a mathematical perspective Many are the challenges that yet have not been solved with the possible influence of the specific energy in the Lorentz the current understanding of mass. Dark Matter is indeed Factor and its implication in the definition of time and mass. one of those challenges (Evrard, Metzler, Navarro 1996; The major differences between the Theory of Relativity and Merritt 2006; Navarro, Frenk, White 1997; de Blok et al. the one proposed are shown in Table I. This ultimately 2001; Wang et al. 2016). There are indisputable differences summarizes that both formulas in the right column are the between predictions and observations related to certain same equation, but ordered in two different ways, while the gravitational effects in the universe, and in order to use two on the left are different formulas. Going back to Fig. 4 current theories of gravitation, it would be necessary to have and Fig. 5, it can be seen that there is a very important more mass than what is observed. Indeed, that is the origin difference between both solutions. Although in the small of Dark Matter. Dark Matter should exist to reconcile both world of the experiments carried out on Earth both theories observations and theories. But its nature is so complex that predict similar results, in the vastness of outer space, they its existence has never been proved. lead to completely different predictions. The so-called weakly-interacting massive particles, or WIMPs, are studied for being a potential candidate to Table I. Differences between theories explain the nature of Dark Matter (Sanders 1990; Borriello, Salucci 2001; Zaharijas, Hooper 2006; Gnedin et al. 2004). WIMP annihilation into charged states produces photons via Theory of Relativity Theory proposed loop-level processes. When two WIMP particles 휒휒 푚0 푚0 푚푇표푡 = annihilate each other at close to zero relative velocity into 푚푇표푡 = 휀 푣2 1 − 훾푋, an energy of 퐸훾 is released (Abdo et al. 2010; Goodman √ √ 2 1 − 2 푐 Ibe, et al. 2010), as it is given by (71). 푐 2 2 푚푋 2 2 퐸훾 = 푚휒 (1 − 2) (71) 푐 푚 4 푚휒 2 0 퐸 = ( 푚푇표푡푎푙 − 푚0 ) 푐 퐸 = ( 푚푇표푡 − ) 2 푚푇표푡 푚휒 refers to the mass of a single WIMP particle 휒 and 푚푋 refers to the remaining mass after the annihilation, both masses measured in energy units. Thus, the total mass of Dark Matter before and after the annihilation can be expressed as 푚푇표푡 and 푚0 respectively ((72) and (73)), where 푚푇표푡 is twice the 푚휒 (since there are two WIMP particles). Applying the transformation to get 푚푇표푡 and 푚0 in mass units instead of energy units, the resulting masses are, 2 푚 푚 = 휒 (72) 푇표푡 푐2 푚 푚 = 푋 (73) 0 푐2 Substituting 푚휒 and 푚푋 in (43),

Specific Energy, Lorentz Factor & WIMP Annihilation 6 References

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