The 1905 SR Paper with Commentary and Corrections (was: physics formulary)
- From: Rock Brentwood <markwh04@xxxxxxxxx>
- Date: Fri, 22 Feb 2008 15:31:28 -0800 (PST)
On Feb 21, 5:40 pm, Rock Brentwood <markw...@xxxxxxxxx> wrote:
Why not use gifs as shown in this example:
http://www.fourmilab.ch/etexts/einstein/specrel/www/
I moved into PDF because of this.
... I redid the 1905 paper (a) into PDF and (b) updating some of the notation.
This is going out on the web shortly.
http://federation.g3z.com/Physics/index.htm#Einstein1905a
Derived from a translation of Einstein's 1905 original landmark paper
on Special Relativity, the notation has been updated and additional
commentary (as well as corrections) provided.
The version currently on-line is working off a compilation from 1999
that, in turn, came from a 1923 English translation. I'm set to go
back to the 1905 original and do the translation myself.
Some comments on the problematic features of the original paper. This
is speaking as an editor, had I been assigned the task of refereeing
the paper.
As I noted in the paper, the functional argument needed serious
reworking. First, at some point the assumption of linearity is simply
deposited in. There are plenty of ways to realise the light hypothesis
non-linearly, examples can be seen (and illustrated) under
http://federation.g3z.com/Physics/index.htm#Alexandroff
These are the conformal transformations. More to the point: one does
not need to assume the functions are even differentiable, let alone
linear. The conformal nature of the transformations follows by a
purely geometric argument. From this, in turn, one gets
differentiability and, finally by assuming the "plane at infinity" is
fixed, and the "affine space" preserved, one gets linearity. The
result are the Poincare' transformations. Assuming the origin is fixed
(as was pointed out in the paper), the Poincare' transformations drop
down to the Lorentz transformations.
At one point, he references the "time" when "t = 0 = t'" -- completely
missing the point of the paper, itself! This is not a time, it's a
plane at an instant. Namely, it's the YZ plane at the instant it meets
the Y'Z' plane.
The second major issue is the use of the Maxwell-Hertz variant of
electrodynamics (or, for that matter, even Maxwell-Lorentz theory).
The point would have been better made by going back to the so-called
"macroscopic" Maxwell equations, thereby also unclouding the issue. In
fact, Einstein consistently kept out the 4th equation of each set and
at one point (the t = 0 = t' "time" comment), was caught in the
process of thinking in the Newtonian mould.
The Maxwell equations written out in their full glory CLEARLY show the
regularity that Einstein was trying to capture, but which was obscured
in his paper:
0 + dD^x/dx + dD^y/dy + dD^z/dz = rho
-dD^x/dt + 0 + dH_z/dy - dH_y/dz = J^x
-dD^y/dt - dH_z/dx + 0 + dH_x/dz = J^y
-dD^z/dt + dH_y/dx - dH_x/dy + 0 = J^z
and
0 + dB^x/dx + dB^y/dy + dB^z/dz = 0
-dB^x/dt + 0 - dE_z/dy + dE_y/dz = 0
-dB^y/dt + dE_z/dx + 0 - dE_x/dz = 0
-dB^z/dt - dE_y/dx + dE_x/dy + 0 = 0
It gets even more obvious (in the process also paving the way to the
mass energy relations) when the force and power laws are joined onto
these relations,
F_x = e (E_x + 0 + v^y B^z - v^z B^y)
F_y = e (E_y - v^x B^z + 0 + v^z B^x)
F_z = e (E_z + v^x B^y - v^y B^x + 0)
-P = e (- v^x E_x - v^y E_y - v^z E_z)
And also with the field-potential relations
E_x = -dA_x/dt + d(-phi)/dx, B^x = dA_z/dy - dA_y/dz
E_y = -dA_y/dt + d(-phi)/dy, B^y = dA_x/dz - dA_z/dx
E_z = -dA_z/dt + d(-phi)/dz, B^z = dA_y/dx - dA_x/dy
In particular, when the potentials are substituted in the force and
power laws, writing the force and power, respectively, as derivatives
of the momentum and kinetic energy
F_x = dp_x/dt, F_y = dp_y/dt, F_z = dp_z/dt, P = dT/dt,
one gets something that screams out the regularity:
d(p_x + e A_x)/dt = d/dx (A_x v^x + A_y v^y + A_z v^z - phi)
d(p_y + e A_y)/dt = d/dy (A_x v^x + A_y v^y + A_z v^z - phi)
d(p_z + e A_z)/dt = d/dz (A_x v^x + A_y v^y + A_z v^z - phi)
d((-T) + e (-phi))/dt = d/dt (A_x v^x + A_y v^y + A_z v^z - phi)
where the partial derivatives on the right are with v^x, v^y, v^z kept
fixed.
A flaw in the derivation of the transformation law can be seen from
the field equations, when written correctly, as above. Einstein did
not write down the most general transformation law. In the absence of
sources (rho = 0, (J^x, J^y, J^z) = (0,0,0)), one has cross-
transformations between (D,H) and (E,B):
D' = D + theta B; H' = H - theta E.
E' = E - lambda H, B' = B + lambda D.
In fact, ONLY the latter can be ruled out in the presence of sources.
One STILL has the former set. This cannot be ruled out. It's a
surviving residual of complexity transformation that resides in the
little group of the field equations-with-source.
Had the analysis been done right, one would have posed the following
question, instead. What is the most general relation between (D,H) and
(B,E) that accords with the light hypothesis? In fact, the answer is
MORE general than Maxwell-Hertz theory or Maxwell-Lorentz theory and
is of the form
D = epsilon E + theta B, H = epsilon c^2 B - theta E.
This leads to a different symplectic structure (and different
Lagrangian and Hamiltonian dynamics) than usual, and (when the
constitutive coefficients are not constant) a non-trivial dynamics
with respect to theta.
There is nothing that theoretically mandates either of the
coefficients be constant. The only requirement that Lorentz invariance
makes -- assuming the electrodynamic theory comes out of a Lagrangian
form (another oversight of the 1905 paper, not to tie this or anything
else to a Lagrangian formulation, when even Maxwell wrote part of his
treatise in the language of Lagrangian dynamics) -- is that the
Lagrangian be functions of the Lorentz invariants,
L = L(I, J)
I = 1/2 (E^2 - B^2 c^2), J = E.B
with coefficients
epsilon = dL/dI, theta = dL/dJ
likewise functions of I and J. That's all that one can say from
Lorentz invariance.
But none of this could even be addressed, because it's filtered out by
the stipulation of Maxwell-Hertz dynamics, where the (equivalent of
the) Lorentz relations (D = epsilon_0 E, B = mu_0 H) are built into
the very notation, itself.
.
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