criticisms of the einstein field equation - Alpha Institute for Advanced ...

3.4. DERIVATION OF SCHWARZSCHILD SPACETIME
Nonetheless, R(r) is still an a priori unknown function, and so it cannot be
arbitrarily asserted that R(0) = 0; contrary to the assertions of the astrophysical
scientists. It is now quite plain that the ‘transformations’ used by the Standard
Analysis in going from Eq. (3.11) to Eq. (3.12) are rather pointless, since all
the relations are contained in Eq. (3.11) already, and by its pointless procedure
the Standard Analysis has confused matters and thereby introduced a major
error concerning the range on the quantity r in its expression (12) and hence
in its expression (1). One can of course, solve Eq. (3.11d), subject to R µν = 0,
in terms of R(r), without determining the admissible form of R(r). However,
the range of R(r) must be ascertained by means of boundary conditions fixed
by the very form of the line-element in which it appears. And if it is required
that the parameter r appear explicitly in the solution, by means of a mapping
between the manifolds described by Eqs. (3.10) and (3.11), then the admissible
form of R(r) must also be ascertained, in which case r in Minkowski space is a
parameter, and Minkowski space a parametric space, for the related quantities
in Schwarzschild space. To highlight further, rewrite Eq. (3.11) as,
ds 2 = A (Rc) dt 2 − B (Rc) dR 2 c − R 2 2 2 2
c dθ + sin θdϕ , (3.13)
where A (R c) , B (R c) , R c (r) > 0. The solution for R µν = 0 then takes the form,
ds 2 =
1 + κ
dt
Rc 2
− 1 + κ
−1 dR
Rc 2 c − R 2 2 2 2
c dθ + sin θdϕ ,
R c = R c(r),
where κ is a constant. There are two cases to consider; κ > 0 and κ < 0. In
conformity with the astrophysical scientists take κ < 0, and so set κ = −α,
α > 0. Then the solution takes the form,
ds 2 =
1 − α
R c
dt 2 −
1 − α
R c
−1 dR 2 c − R 2 2 2 2
c dθ + sin θdϕ , (3.14)
R c = R c(r),
where α > 0 is a constant. It remains to determine the admissible form of Rc (r),
which, from Section II, is the inverse square root of the Gaussian curvature of
a spherically symmetric geodesic surface in the spatial section of the manifold
associated with Eq. (3.14), owing to the metrical form of Eq. (3.14). From
Section II herein the proper radius associated with metric (14) is,
Rp =
dR c
1 − α
R c
= R c (R c − α) + α ln
Rc +
Rc − α + k, (3.15)
where k is a constant. Now for some ro, Rp (ro) = 0. Then by (15) it is required
that Rc (ro) = α and k = − α ln √ α, so
Rp (r) =
Rc +
Rc (Rc − α) + α ln
Rc − α
√ , (3.16)
α
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