Was Einstein's Original Formula for Transverse Mass Incorrect?

[pmb]

Never mind. I goofed on that transformation. I must be tired. :-)

Pete

 

[quartodeciman]

I've pretty much talked this to death already. Just one more comment or two and I'll quit.

I originally read paragraph 2 of §6 (Transformations of the Maxwell-Hertz equations...) incorrectly.

"If we apply to these equations the transformation developed in §3, by referring the electromagnetic processes developed there introduced, moving with the velocity v, we obtain the equations"

[§6.2]-[§6.7] {I won't write them}

"where β="... {the usual}
"."

I thought it meant Einstein claimed to get it all from the KINEMATICAL PART of his paper. But that couldn't be true for the electric/magnetic components, so he must have presaged what they would need to be from somewhere, hence my cookery with the Lorentz force equation.

I realized after one or two of our exchanges that I had missed the import of the word "obtain". He did what I call a physics-teacher move: knowing already what the component transformations need to be, he worked out [§6.2]-[§6.7] deliberately to make the terms stand out, all for the purpose of identifying them with components in the Maxwell-Hertz equations for the moving system.

A more interesting criticism of my thoughts would be to ask about a charge moving in the rest system K at a velocity u parallel to v, but not equal in magnitude or orientation. The velocity addition/subtraction theorems would be needed, I reckon.

Regards,
quart

 

[quartodeciman]

An answer to the question whether Einstein knew the F = q[E + v X B] relation ->

Pais {SITL,OUP(1982)} states on p. 124 that Lorentz in his 1895 paper included a set of corresponding-state equations:

x' = x - vt
t' = t - vx/c² --- {that's a scalar product, I guess}
E' = E + v X H/c --- {'X' is vector product operator}
H' = H - v X E/c
P' = P --- {'P' for electric polarization vector}
.

On p. 125 Pais states that Lorentz included K = e(E + v X H/c) for a moving ion and called it "electrische Kraft".

On p. 133 Pais states that Einstein knew the 1895 paper of Lorentz before 1905.

 

[pmb]

Originally posted by quartodeciman
An answer to the question whether Einstein knew the F = q[E + v X B] relation ->

Pais {SITL,OUP(1982)} states on p. 124 that Lorentz in his 1895 paper included a set of corresponding-state equations:

x' = x - vt
t' = t - vx/c² --- {that's a scalar product, I guess}
E' = E + v X H/c --- {'X' is vector product operator}
H' = H - v X E/c
P' = P --- {'P' for electric polarization vector}
.

On p. 125 Pais states that Lorentz included K = e(E + v X H/c) for a moving ion and called it "electrische Kraft".

On p. 133 Pais states that Einstein knew the 1895 paper of Lorentz before 1905.

Thanks - Great info. I know an historian of relativity. I'll be meeting with him in the future. I'll get back with his comments.

Pete

 

[quartodeciman]

In 1899 Lorentz wrote a paper including the true Lorentz transformations, a name that Poincaré invented.

x' = εγ(x - vt)
y' = εy, z' = εz
t' = εγ(t - vx/c²)

{"ε" is a scale factor}
.

{all this is included in ibid., ch. 6}

In a 1904 paper (included in The Principle Of Relativity anthology) Lorentz applied the Lorentz factor (β this time!) to the electric/magnetic component transformations, but showed a queer form of space-time transformations:

x' = βlx, y' = ly, z' = lz
t' = (l/β)t - βlv/c²

{"l" a scale factor}
.

Of course, these transformations were just a mathematical formalism to Lorentz, with no "real" space/time significance. Einstein evidently didn't know these papers when he wrote his first relativity paper. He must have divined the necessity of a multiplier by himself, or from what he knew of Poincaré's reviews of the subject.

It strikes me as odd that the co-producer of the Fitzgerald-Lorentz contraction idea didn't use the factor in his earlier (e.g.1895) papers.

 

[quartodeciman]

Something suddenly sticks out. Einstein DID know the Lorentz 1904 paper after all. The reason is notation.

Up until at least 1900, Lorentz used the letter "γ" to represent the transformation factor, following Voldemar Voigt (1887), the first researcher to consider transformations involving functions of v/c. During the early 1900s, Lorentz was publishing work about electron theory and, specifically,calculating the electron mass. In this research, he was running against the theory of Max Abraham, and the letter "β" was used to represent v/c there. I doubt Lorentz would have reused "β" to also represent the transformation factor in his electrodynamics at that time. It is (apparently) not until the Lorentz 1904 paper that "β" suddenly started getting used (don't know why) for the transformation factor, rather than "γ". Einstein used "β" the same way in his first relativity paper (1905). That is a big coincidence, if Einstein didn't know Lorentz's 1904 paper.

Miller {AESTOR,Addison-Wesley(1981)} states (section 1.15.1) that Einstein "maybe" knew this paper of Lorentz.