#### pmb

In his book "Albert Einstein's Special Theory of Relativity," Arthur Miller, pp. 328-331, the author explains the transverse mass error. This error was also noted in the now well known paper "Does mass really depend on velocity, dad?" Carl G. Adler, Am. J. Phys., 55(8), Aug 1987 page 742

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It should be noted that Einstein's original formula for transverse

mass was incorrect. It was corrected by Planck in 1906. Planck was the

first to introduce the formula m_o*v/sqrt[1-(v/c)^2].

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But it seems there is some misinformation being passed around the web on this subject (notably by someone who was banned from here but is still posting misinformation elsewhere). Clarity is in order.

In the 1905 paper Einstein examines the motion of a charged particle which is instantaneously at rest in the primed system in which there is a pure electric field. At t' = 0 the charge is at the origin of the primed system S'. In the unprimed frame S the particle is moving in the +x direction and at t = 0 the charge is at the origin of S. Einstein writes for S'

Einstein uses (X,Y,Z) for (E_x, E_y, E_z) and (L,M,N) for (B_x, B_y, B_z). I use the regular E_x, etc. m = rest mass

First Einstein considers the particle to be at rest in S

m*d^2x/dt^2 = qE_x

m*d^2y/dt^2 = qE_y

m*d^2z/dt^2 = qE_z

That's the first equation. It holds since m*d^x/dt^2 is the force since the particle is instantaneously at rest and therefore gamma = 1. Then he addresss the moving partricle which is at rest in S' and therefore (I use x', y', z' while Einstien uses Greek letters). Remember in S' there is only an electric field

m*d^2x'/dt^2 = qE'_x

m*d^2y'/dt^2 = qE'_y

m*d^2z'/dt^2 = qE'_z

Now Einstein transforms to S

d^2x/dt^2 = (q/m*gamma^3)E_x

d^2y/dt^2 = (q/m*gamma)(E_y - vB_z)

d^2z/dt^2 = (q/m*gamma)(E_z + vB_y)

rewritten

gamma*md^2x/dt^2 = qE_x = x-component of Lorentz Force

gamma*m*d^2y/dt^2 =q(E_y - vB_z) = y- component of Lorentz Force

gamma*m*d^2z/dt^2 = q(E_z + vB_y) = z- component of Lorentz Force

These are the correct equations of motion.

Einstein makes the error of equationg m*gamma^2 d^2/dt^2 = q*gamma(E_y - vB_z) with qE'_y and thus he gets the wrong value for the transverse mass (same with the other components). In effect what he does is fail to transform the force. He writes an expression like

MA = F' and then calls M transverse mass. This of course is incorrect since to call the M 'mass' one has to have all components in the same system. So if F = MA then one can call M a 'mass.'

For details see -

http://www.geocities.com/physics_world/sr/ae_1905_error.htm

Pete