SR’s and AD’s Mechanics 2-1 Force in SR[1] Using the relativistic treatment of a force we have (2.1) In the F’ system (2.1) The problem consists of finding d^2x’ / dt’^2 Taking the derivative of the sum velocity equation (1.13) (2.3) Setting dx/dt = vx we obtain (2.4) To obtain the well known relativistic formula (2.5) it is necessary to set vx = v. This simplification makes no physical sense. It has no justification. There is no physical relationship between these two velocities. One of them, vx, is the velocity of point P in F, and the other one is the relative velocity v of a frame with respect to the other frame. In section 1-6 we discussed what happens regarding this arbitrary simplification in SR to yield the equation for acceleration and mass variation. If this last equation (2.5) is replaced in equation (2.2) the expression (2.6) yields the SR equation for mass variation. This mass was originally called “longitudinal mass” by Einstein[2] to distinguish it from what he called by himself “transversal mass” (2.7) This is also called “orthogonal mass” even though no one knows the origin of this term. As it is well known that the equation generally used in mechanics (And the only one that agree well with experimental results) has the exponential 1/2, that is, the square root. (To see a step by step derivation of equation (2.3) goes to Appendix A1.) 2-1-1 “Mass increase” and “Physical reality” When a particle is accelerated inside an accelerator, its kinetic energy increases, and since its mass increases with an equivalent amount of energy, there is no energy conservation. To explain this contradiction, SR talks about how "mass increase in the SR equation has no physical reality." Simultaneously, "mass increase" is the most significant factor in the momentum equation. If "mass increase" does not exist, momentum is equal to that of Classic Mechanics and, consequently, all the values calculated with SR are wrong. Particle Physics is today founded on momentum conservation as calculated with the SR momentum equation, using increasing mass. This means that the fundamental basis of modern Particle Physics is wrong. (See A8). In conclusion, to explain the non-conservation of energy using the argument "mass increase has no physical reality," SR arrives at a larger contradiction and incorrect results. 2-2 Force in AD. With the classical definition of force (2.8) The derivation of v with respect to dt’ is, using equation (1.15) (2.9) Replacing in equation (2.8) (2.10) If we call the expression m = mo (1 - b^2)^1/2 (2.11) the Autodynamics expression for mass variation, the equation (2.10) for force is written (2.12) Equation (2.11) tells us that when the particle velocity increases the mass decreases. This is equivalent to the decay process. Contrarily to what happen in SR (See 2-2-1), “decreasing mass” in AD has real physical meaning (See Appendix A8). This will be analyzed later after the derivation of AD kinetic energy. 2-3 AD Kinetic Energy. From the theoretical point of view and for practical application, we shall examine two different cases, accepting the general concept of mass-energy equivalence. 2-3-1 First application All available energy in AD comes from the particle rest mass energy with the total energy being: Et = mo c^2 (2.13) If a particle starts to move (decay), it will take energy from the rest mass energy and Et = m c^2 + Ec (2.14) where m < mo , remembering (2.11) and replacing Et get from (2.13) mo c^2 = mo (1 - b^2)^1/2 c^2 + Ec (2.15) Finally we have Ec = mo c^2 [ 1 - (1 - b^2)^1/2 ] (2.16) Equation (2.11) and equation (2.16) of AD kinetic energy form a coherent system. The AD kinetic energy equation increases to a maximum that represent the particle total rest-mass energy. Equation (2.16) yield Newton’s equation at law velocities.[3] 2-3-2 Rest mass plus KE. If a particle receives energy from an external medium, the particle energy will be the sum of its rest mass energy plus the kinetic energy added to it from the outside, yielding the expression: mo c^2 + Ec (2.17) This extra energy added to the rest mass energy is the cause why Pauli needed to postulate the Neutrino to carry off excess of energy in the SR KE equation when it was applied to decay cases. Replacing this equation in the AD kinetic energy equation (2.16) we have: Ec = (mo c^2 + Ec) - (mo c^2 + Ec) (1 - b^2)^1/2 (2.18) Simplifying, it yields Ec = mo c^2 [(1 - b^2)^-1/2 - 1] (2.19) The SR equation. Expressed in words, the equation with mo c^2 (rest mass energy), equation (2.16), represents the kinetic energy when a particle decays, that is to say, when there is auto-transformation of mass into kinetic energy. In equation (2.17) mo c^2 + Ec represents the particle kinetic energy when it receives energy from an external medium, equal to the very well known SR equation (2.19). It is possible to say that AD’s kinetic energy is a new general equation that represents two cases: decay (AD with moc^2), or energy from the external medium (SR’s kinetic energy equation). Thus we can say that SR is a sub-set of AD.(See R XIX). 2-4 AD’s Momentum. In keeping with the classical definition of momentum we have in Autodynamics p = m v = mo (1- b^2)^1/2 v (2.20) The value of v increases up to the finite limit of c and the value of mass decreases to zero as v tends to c. Momentum increases initially only to decreases later on, which clearly indicates a maximum values. This maximum Autodynamic momentum is found for v = 0.7 c. 2-4-1 A Remark on Momentum. It is necessary to keep in mind that the Autodynamics equation applies when there is decay. When the kinetic energy and mass of a particle is known, we must use equation (2.16), but the momentum equation will be p = mv v = mv b c (2.21) because mv now represents the real measured moving mass. 2-5 AD Energy-Momentum relation. The equation relating energy and momentum becomes: [(4moc^2E-2E^2-mo^2c^4)^2+4p^2moc^6]^1/2 = mo^2 c^4 (2.22) (See Appendix A2 for a step-by-step derivation of this equation and a remark about its units.) 2-6 AD sum velocity. Even though in Autodynamics there are no frames in relative motion, different but successive phenomena could be seen via kinetic energy (KE) conservation that yield the “sum velocity equation.” If a particle decays in two steps, (step 1 and step 2), energy and momentum must be conserved if the same particle is analyzed and supposing that it reaches a velocity given by the “sum velocity equation.” This does not happen with the SR equations, which apply its KE, and momentum equations to a phenomenon that simultaneously is calculated with the sum velocity equation. (See A7 and R-VIII) For successive decays the particle will reach the velocity given by: b3 = {1-[(1-b1^2) * (1- b2^2) ] }^1/2 (2.24) It is very easy to demonstrate that the equation is general, that is, it could be used for any number of successive phenomena: bn = {1-[(1-b1^2) *........ *(1-bn-1^2) ] }^1/2 (2.25) Unlike the SR equation, the AD equation applies equally kinetically as well as dynamically. (See Appendix A3 for a step-by-step derivation of this equation [1] .- This was the classical derivation of force. Given that this derivation drives SR to the incorrect solution, the force was thereafter defined by the derivation of momentum: F= dp/dt. With this derivation, a question arises: where does mass variation come from that is used in this equation? [2] .- ”The Principle of Relativity,” by H. A. Lorentz, A. Einstein, H Minkowski and H. Wey, with Notes by A. Sommerfeld, translated by W. Perrett and G. B Jeffery, first published in 1923 in the USA by Dover Publications, Inc., page 63. [3] .- Even though the AD’s author knows this, was also pointed out by Zeng Xinchuan, at Sichuan University, in a 1990 private communication.