Einstein cheated with his fraudulent derivation of Lorentz transforms

This can be a long and complex post, because calculus and deep reasoning
are neccesary. The analysis is based on everything he wrote in the
points §2 and §3 of his famous 1905 paper.

The most important part, where he cheated by inserting wrong/false
assumptions, is when he derived the equation

∂τ/∂x′ + v/(c² – v²) ∂τ/∂t = 0

which solution allowed him to derive the relationship τ = Θ(v) β (t -
vx/c²), being β the currently known γ factor and Θ(v) assumed to be Θ(v)
= 1, as Lorentz did in his 1904 paper.

I've attached a drawing which represent the most relevant aspects of his
point §3.

FOR THOSE WHO ARE NOT INTERESTED IN THIS SUBJECT, I SUGGEST TO STOP
READING NOW.

This post is strictly based on two things:

1) How he FORCED a relationship between t, τ and x' = x - vt by mixing
results obtained from the PERCEPTION of time at the rest frame
K(x,y,z,t) and the assumed relationship between this variables in the
moving frame k(ζ, η, ζ,, τ).

2) How he WRONGFULLY applied Taylor's series solutions when a reference
x' on k is made infinitesimally small. The use of x' in k is a TRICK to
replace the rigid rod r_AB used in his point §2, as it can visualize in
the attached graphic. Now, using x' allowed him to replace the rigid rod
r_AB with a length variable x' simulation (from ζ=0 to  ζ=x'), being
that (as in §2) mirrors and stop clocks are placed at both ends IN THE
MOVING FRAME (ζ=0 and  ζ=x').

To start with, this is the first order Taylor expansion for a
multivariable function, which as to be STRICTLY OBSERVED in the paper
(it didn't happen):

f(x+dx, y+dy, z+dz, t+dt) = f(x, y, z, t) + dx ∂f/∂x + dy ∂f/∂y + dz
∂f/∂z + dt ∂f/∂t

I'll try to explain systematically the steps that he did in the first
part of §3:

1) RIGHT:  He used the equation 1/2 (τ₀ + τ₂) = τ₁ for times registered
in the moving frame k. No problem here, as long τ is the time MEASURED
in the moving frame.

2) WRONG: Immediately after, he WRONGFULLY mixed perceptions of time (as
derived in §2 for the STATIONARY FRAME K, FORCING a relationship between
t, τ and x' = x - vt, which is A FALLACIOUS ASSERTION. The he came with
a modified equation from 1) as follows:

1/2 [τ(0,0,0,t) + τ(0,0,0,t + x'/(c - v) + x'/(c + v)] = τ[(x',0,0,t +
x'/(c-v)]
Simplyfing it by eliminating y=0 and z=0, this equation changes (with no
errors) to

1/2 [τ(0,t) + τ(0,t + x'/(c - v) + x'/(c + v)] = τ[(x',t + x'/(c-v)]

Using the following relationships (developed in §2 for PERCEPTION in
frame K)

Δτ₁ = x´/(c – v)
Δτ₂ = x´/(c – v)+ x´/(c + v)

the above equation can be WRONGFULLY re-written as

1/2 [τ(0,t) + τ(0,t + Δτ₂)] = τ(x',t + Δτ₁) ; this is the equation where
Taylor is applied.

Observe that on the left side, ζ=0, because the ray of light departs and
come back, after reflection on x', to the origin of k(ζ, η, ζ,, τ). On
the right side ζ=x', where the reflecting mirror is placed.

Observe, also, that on the right side x' CAN'T BE ZERO (only
significantly small). Otherwise, the equation 1/2 (τ₀ + τ₂) = τ₁ lose
completely its meaning (no reflection).

Also observe that, on the right side, there is not a differential
increment Δx' accompanying x'. So Taylor expansion has to take this
SERIOUSLY (but here Einstein cheated). I repeat Taylor expansion
formula, to the first order only:

f(x+dx, y+dy, z+dz, t+dt) = f(x, y, z, t) + dx ∂f/∂x + dy ∂f/∂y + dz
∂f/∂z + dt ∂f/∂t

A) Being the equation  1/2 [τ(0,t) + τ(0,t + Δτ₂)] = τ(x',t + Δτ₁),
applying Taylor to right side gives

τ(x',t + Δτ₁) ≈ τ(x',t) + Δτ₁  ∂τ/∂t = τ(x',t) + x´/(c – v)  ∂τ/∂t

But Einstein wrote

τ(x',t + Δτ₁) ≈ τ(0,t) + x´ ∂τ/∂x' + x´/(c – v)  ∂τ/∂t

Which are the fallacies and falsitudes here?

1) Using τ(0,t) INSTEAD OF τ(x',t), with the excuse of x' being
infinitesimally small.

2) Adding A FALSE x´ ∂τ/∂x', which is an fraudulent addition to Taylor
expansion around x'.

Without these FALSE MODIFICATIONS, Einstein would never had reached his
final equations. So, with the excuse of x' infinitesimally small, he
MODIFIED the thought experiment by making believe that the reflection at
time τ₁ still occurs at x'=0, being that this means that the points of
emission, reflection and reception of the ray of light occurs EVEN WHEN
all the three points HAVE NO DEPTH, because they are located in the same
coordinate ζ = 0, which is IMPOSSIBLE even for a thought experiment.

Since this point onwards, ALL THE RESULTS ARE INVALID, because he NEVER
EVER can reach the equation

∂τ/∂x′ + v/(c²-v²) ∂τ/∂t = 0

B) Being the equation  1/2 [τ(0,t) + τ(0,t + Δτ₂)] = τ(x',t + Δτ₁),
applying Taylor to left side gives

1/2 [τ(0,t) + τ(0,t + Δτ₂)] ≈ 1/2 [τ(0,t) + τ(0,t) + Δτ₂ ∂τ/∂t]

1/2 [τ(0,t) + τ(0,t) + Δτ₂ ∂τ/∂t] = 1/2 [τ(0,t) + τ(0,t) + [x'/(c - v) +
x'/(c + v)] ∂τ/∂t]

This equation IS CORRECT.

C) Both Taylor results are displayed in place in this CORRECT EQUATION:

1/2 [τ(0,t) + τ(0,t) + [x'/(c - v) + x'/(c + v)] ∂τ/∂t] = τ(x',t) +
x´/(c – v)  ∂τ/∂t

D) Both Taylor results are displayed in place in EINSTEIN'S EQUATION:
1/2 [τ(0,t) + τ(0,t) + [x'/(c - v) + x'/(c + v)] ∂τ/∂t] = τ(0,t) + x´
∂τ/∂x' + x´/(c – v)  ∂τ/∂t

E) After a little of algebra, results from C) are:

2 τ(0,t) + x'/(c - v) ∂τ/∂t + x'/(c + v)] ∂τ/∂t = 2 τ(x',t) + 2 x´/(c –
v)  ∂τ/∂t

2 τ(0,t) + x'/(c + v)] ∂τ/∂t = 2 τ(x',t) + x´/(c – v)  ∂τ/∂t

As it can be seen, even when x' is infinitesimally small, YOU CAN'T
CANCEL 2 τ(0,t) with 2 τ(x',t). And EVEN IF YOU DO (wrongfully using
x'=0), the equation is left as

x'/(c + v) ∂τ/∂t = x´/(c – v)  ∂τ/∂t  ; then you cancell x' from both
sides

1/(c + v) ∂τ/∂t = 1/(c – v)  ∂τ/∂t

[1/(c + v) - 1/(c – v)] ∂τ/∂t = 0

v/(c² – v²) ∂τ/∂t = 0

This equation is QUITE DIFFERENT from the one published by Einstein:

∂τ/∂x′ + v/(c² – v²) ∂τ/∂t = 0

So, no Lorentz transformation (Poincaré, Lorentz) can be achieved.

This fatal flaw on the §3 PREVENT any further result in this point of
the paper.
F) Einstein could have kept following the development in the rest of
points, by just PLANTING the Lorentz-Poincaré transforms (as Lorentz
did), but the PROUD CHEATER couldn't do such move. He preferred to make
a fraudulent development, taking MOST of the people as gullible idiots.

There are many more FRAUDULENT assumptions in the rest of the paper,
like when he played with "slowly accelerated electrons". Not to mention
his BLATANT PLAGIARISM of the developments of Lorentz and Poincaré.