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 푑퐸퐴푟 퐵 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.