Here we present the derivation of the new set of equations termed， Lorentz transformations， and all the subsequent LORENTZ TRANSFORMATIONSWe consider two coordinate systems (frames of reference) one stationary S and one moving at some velocity v relative to S， then according to the two postulates of Relativity， stated in the main text， the displacement in both frames is of the same Therefore， we have (A-1) (A-2)We should note here that in the old Galilean transformations these equations would be (A-3)which is in direct contradiction to Postulate 2， a firm experimental Equations (A-1) and (A-2) can be written as(A-4) (A-5)That is， (A-6)We are interested in finding and in terms of x and That is， = (x， t) (A-7) = (x， t) (A-8)This is accomplished via the formation of two linear simultaneous equations as follows:(A-9) (A-10)where a11， a12， a21， and a22 are constants to be It is required that the transformations are linear in order for one event in one system to be interpreted as one event in the other system; quadratic transformations imply more than one event in the other Solution of problems involving motion begins with an assumption of their initial conditions; ， where does the problem begin?The classical assumption is to set = 0 at = Therefore， according to S， the system appears to be moving with a velocity v， so that x = We can obtain this from E (A-9) by writing it in the form = a11(x - vt) so that， when = 0， x = Therefore， we conclude that a12 = - We can write Equations (A-9) and (A-10) as (A-11) (A-12)Substituting and into Equation (A-6) and rearranging， we get (A-13)Since this equation is equal to zero， all the coefficients must That is，(A-14) (A-15) (A-16)Solving these equations we obtain(A-17) (A-18)where β = v/c and Thus， substituting these values in E (A-11) and (A-12) we obtain the famous Lorentz coordinate transformation equations connecting the fixed coordinate system S to the moving coordinate system :(A-19) (A-20)We may also obtain the inverse transformations (from system to S) by replacing v by –v and simply interchanging primed and unprimed This gives，(A-21) (A-22)VELOCITY TRANSFORMATIONSAs a direct consequence to these new transformations， all the other mathematical operations and physical variables follow For example， the velocity equations (though still the derivatives of the displacement) assume a new form， so the Lorentz form of the velocities is:From E (A-19) and (A-20) we have: (A-23) (A-24)Therefore:(A-25)ENERGY CONSIDERATIONSConsider a particle of rest mass m0 being acted by a force F through a distance x in time t and that it attains a final velocity The kinetic energy attained by the particle is defined as the work done by the force F The applicable equations are，(A-26)We note thatand thatSubstituting d(γv) in E (45) and integrating， we obtain (A-27)That is， (A-28)This says that K = (m – m0)c2 and finally one sees that the total energy is equal to the sum of the kinetic energy K and the rest energy E0 = ， E = K + Eo = γm0c2 = γE0， (A-29) where E0 = m0c2 and E = 给分吧
