As the original thread had derailed with discussions about sign of integrals
and the usual written violence against who dares to try something against
Einstein, I have drop the integral issue and devote this OP to the most
important fraud, which is how he PLANTED a parameter to get Gerber's formula.
I want to be VERY PRECISE: This OP is not against GR which, IF I do, will
be in a separate post, in the future (part by part).
This story is to SHOW YOU ALL how I proved that Einstein committed
FRAUD in order to deliver his Nov.18, 1815 to the Prussian Academy of
Science.
Please, don't come to me with things like: "using Schwarzschild's metric it
has been proven right 1 million times". This is not the POINT that I'm
addressing here. The point it that HE CHEATED GREATLY with this paper
and, almost immediately, went out to shout loud and clear that the
achieved the first TRIUMPH of his GR, and THIS IS NOT TRUE.
Follow demonstration OF THE FRAUDULENT ACTION FUDGING THE PAPER:
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Einstein (11): (dx/dΦ)² = 2A/B² + α/B² x - x² + α x³
Φ = ∫ dx/√(2A/B² + α/B² x - x² + α x³) , between α₁ = 1/Aphelion and α₂ = 1/Perihelion
Quoting Einstein’s paper assertion: “Hereby we can with reasonable accuracy replace it with”
Φ = [1 + α/2 (α₁ + α₂) ] ∫ dx/√[- (x - α₁) (x - α₂) (1 - αx)] ≈ [1 + α/2 (α₁ + α₂)] ∫ (1 + α/2 x) dx/√[- (x - α₁) (x - α₂)]
Note 1) 1/α = 1/2954.13 m-1 ; α₁ = 1.43232E-11 m-1 ; α₂ = 2.17382E-11 m-1
Note 2) The replacement 1/√(1 - αx) ≈ 1 + α/2 x is correct between α₁ and α₂, with an error lower than 2.0E-15
Note 3) P(x) = √[- (x - α₁) (x - α₂)] = √[- x² + (α₁ + α₂) x - α₁ α₂] ; K = [1 + α/2 (α₁ + α₂)] ≈ (1 + 5.3265E-08)
Note 4) The parameter K is planted out of nowhere, with the fallacy 1 – K = 0, which is false at the end of the paper, where the difference α/2 (α₁ + α₂) is used. This fudge is essential, and deceiving in a naive follow up, but it contributes with 28.67” of arc (or 2/3) to the final target of 43”. This FRAUD is eliminated by using K = 1, exactly.
Making the rest of the development, following Einstein without any other correction, it is that:
Φ ≈ Φ₁ + Φ₂ = ∫ dx/P(x) + 1/2 α ∫ x/P(x) dx , between limits α₁ and α₂
Three analytic solutions are available for each integral (using ln, arcsin or arcsinh). The first one is used.
Making, in general, P(x) = √[a x² + b x + c]
Φ1(x) = ∫ dx/P(x) = 1/√a ln [(2ax + b)/2a + P(x)] = π rad/half orbit, by calculating Φ₁(α₂) - Φ₁(α₁)
Φ₂(x) = 1/2 α ∫ x/P(x) dx = 1/2 α { P(x)/a - b/(2a √a) ln [(2ax + b)/2a + P(x)] } = π 1/4 α (α₁ + α₂) rad/half orbit.
Note 5) In both cases, ln(-1) is solved though Euler’s identity: eiπ = -1 ln (-1) = iπ
Φ ≈ Φ₁ + Φ₂ = π + π 1/4 α (α₁ + α₂) = π [1 + 1/4 α (α₁ + α₂)] rad/half orbit
Note 6) Einstein’s equation previous to Eq. 12 is Φ = π [1 + 3/4 α (α₁ + α₂)] rad/half orbit.
This equation is false, and was achieved by fudging (cooking) inventing and
planting K = [1 + α/2 (α₁ + α₂)]. It can be seen the importance of the FRAUD,
because the result without it only gives 14.33” of arc/century (only 1/3 of the total). The difference with the real value and its origin is quite clear in
this explanation.
In one more step for a full orbit, Einstein multiplied Φ by 2 and subtracted 2π
in Equation 12. Being tricky to hide the result in geometrical units, he used
the equation of an ellipse to present the advance ε.
Note 7) α₁ + α₂ = 1/AP + 1/PE = (AP + PE)/
AP.PE = 2a/[a² (1 – e²)] = 2/[a (1 – e²)]. Replacing this in Φ gives
Φ = π [1 + 1/4 α (α₁ + α₂)] = π [1 + 1/2 α/[a (1 – e²)]] , which lead to the modified value of Equation 12:
(New Eq. 12) Φ = π [1 + 1/2 α/[a (1 – e²)]] , and making ε = 2Φ - 2π
(New Eq. 13) ε = α/[a (1 – e²)] , which is 1/3 of Gerber’s formula, and gives only 14.33” of arc/century.
Due to this fatal result, Einstein fudged the development by planting the
parameter K ≈ 1. But this apparently innocent planting, which differs from 1
in 1: 18.8E+06 (about 19 ppm), adds 28.67” to the final value of 43”.
Besides showing more details about the process of reverse engineering to
match Gerber's formula, I can provide more details on calculations performed in this OP.