ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

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April, 2007 PROGRESS IN PHYSICS Volume 2 1 Introduction On Line-Elements and Radii: A Correction Stephen J. Crothers Queensland, Australia E-mail: thenarmis@yahoo.com Using a manifold with boundary various line-elements have been proposed as solutions to Einstein’s gravitational field. It is from such line-elements that black holes, expansion of the Universe, and big bang cosmology have been alleged. However, it has been proved that black holes, expansion of the Universe, and big bang cosmology are not consistent with General Relativity. In a previous paper disproving the black hole theory, the writer made an error which, although minor and having no effect on the conclusion that black holes are inconsistent with General Relativity, is corrected herein for the record. In a previous paper [1] (see page 8 therein) the writer made the following claim: “the ratio χ > 2π for all finite Rp” Rp where Rp is the proper radius and χ is the circumference of a great circle. This is not correct. In fact, the ratio χ is Rp greater than 2π for some values of Rp and is less than 2π for other values of Rp. Furthermore, there is a value of χ for which χ = 2π, thereby making Rp = R R c, where Rc is p the radius of curvature. Thus, if the transitional value of the circumference of a great circle is χe, then 2 Correction — details χ < χe ⇒ χ Rp χ = χe ⇒ χ Rp χ > χe ⇒ χ Rp > 2π, = 2π, < 2π. Consider the general static vacuum line-element ds 2 = A(r)dt 2 − B(r)dr 2 − C(r) � dθ 2 + sin 2 θdϕ 2� , (1) A(r), B(r), C(r) > 0. It has been shown in [1] that the solution to (1) is ds 2 = � 1 − α � C(r) � dt 2 − 1 1 − α d √ C(r) � C(r) 2 − − C(r) � dθ 2 + sin 2 θ dϕ 2� , α < � C(r) < ∞, (2) where, using c = G = 1, Rc = Rc(r) = � C(r) = ��r � − r0�n + α n�1 n , Rp = Rp(r) = � Rc(r) (Rc(r) − α) + � � �Rc(r) + + α � � � � Rc(r) − α � � √ � α � , r ∈ ℜ, n ∈ ℜ + , r �= r 0, and where r0 and n are entirely arbitrary constants, and α is a function of the mass M of the source of the gravitational field: α = α(M). Clearly, lim ± r→r Rp(r) = 0 0 + and also lim ± r→r Rc(r) = α 0 + irrespective of the values of r0 and n. Usually α = 2m ≡ 2GM/c2 by means of a comparision with the Newtonian potential, but this identification is rather dubious. Setting Rp = Rc, one finds that this occurs only when Then Rc ≈ 1.467α. χe ≈ 2.934π α. Thus, at χe the Euclidean relation Rp = Rc holds. This means that when χ = χe the line-element takes on a Euclidean character. An analogous consideration applies for the case of a point-mass endowed with charge or with angular momentum or with both. In those cases α is replaced with the corresponding constant, as developed in [2]. 3 Summary The circumference of a great circle in Einstein’s gravitational field is given by χ = 2πRc , 2πα < χ < ∞. S. J. Crothers. On Line-Elements and Radii — A Correction 25 (3)
Volume 2 PROGRESS IN PHYSICS April, 2007 In the case of the static vacuum field, the great circle with circumference χ = χe ≈ 2.934 πα takes on a Euclidean character in that Rp = Rc ≈ 1.467α there, and so χe marks a transition from spacetime where χ < 2π to spacetime Rp where χ > 2π. Thus, Rp lim r→∞ ± χ = 2π, Rp(r) lim r→r ± χ = ∞ , Rp(r) 0 χ = 2π. Rp(r) lim χ→χ ± e Similar considerations must be applied for a point-mass endowed with charge, angular momentum, or both, but with α replaced by the corresponding constant β in the expression for Rp [2], β = α 2 + � α2 4 − (q2 + a2 cos2 θ) , q 2 + a 2 < α2 4 , a = 2L α , where q is the charge and L is the angular momentum, and so ��r � Rc = Rc(r) = − r � 0 n 1 + β n� n , r ∈ ℜ, n ∈ ℜ + , r �= r 0, where both r 0 and n are entirely arbitrary constants. References Submitted on December 26, 2006 Accepted on January 06, 2007 1. Crothers S. J. On the geometry of the general solution for the vacuum field of the point-mass. Progress in Physics, 2005, v. 2, 3–14. 2. Crothers S. J. On the ramifications of the Schwarzschild spacetime metric. Progress in Physics, 2005, v. 1, 74–80. 26 S. J. Crothers. On Line-Elements and Radii — A Correction
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