The Proceedings of the Eighth International Conference on Creationism (2018)

( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 sin sin ds dt a t d d ds dt a t d d χ χ χ χ = − + + Ω = − + + Ω The “angular” radial coordinates χ are measured from the two poles of the closed cosmology. The angles 2 α and 2 α correspond to linear coordinates 1 r and 2 r , respectively. The parametric solutions for R k and t k in terms of the “cycloidal” parameter η are: ( ) ( ) ( ) (0) sin ( , ) 1 cos 2 (0) sin 2 k k k k k k a R a t χ η χ η η η η = + = + (0) k a for k=1,2 is the maximum radius of the regions 1 and 2 at maximum expansion at 0 η = . The parameter η ranges over [ ] , π π − . The big bang occurs at η π = − and the big crunch at η π = . The above solution is obtained from the general L-T solutions: ( ) ( ) ( ) ( ) 3/2 ( ) ( , ) 1 cos 2 ( ) ( ) , sin 2 ( ) k k k k k k k k k k k k M R E M t E χ η χ η χ χ η χ η η χ = + − = + − The function M and E are given by: 3 ( ) (0)sin k k k k M a χ χ = 2 1 ( ) sin 2 k k k E χ χ = − The joining of the three regions is performed by requiring continuity of the functions M and E. Clearly the function M is continuous if there are no delta-function surface layers, which is the case for our model. Aproperty of the closed universe is that the total gravitational mass is zero. Thus ( ) max 3 0 4 0, 0 3 r total d M r R dr dr π ρ   = =   ∫ In the Schwarzschild region 0 ρ = ; therefore the only contributions to total M are in the FLRW regions ( ) ( ) ( ) ( ) 1 max 2 3 3 1 2 0 3 3 1 1 2 2 4 4 0 0 0 3 3 4 0 (0, ) 0 (0, ) 3 r r r d d R dr R dr dr dr R r R r π π ρ ρ π ρ ρ     = +       = −   ∫ ∫ Therefore, 1 2 ( ) ( ) / 2 g M M r α α = = where 2 g r M = is the “gravitational” radius in the Schwarzschild region. Thus 3 3 1 1 1 2 2 2 ( ) (0)sin ( ) (0)sin / 2 g M a M a r α α α α = = = = (28) To relate the angular variable χ to the linear coordinate r, we define the linear radial coordinate of points in the FLRW regions by their projection onto the “ r” axis of the embedding space at maximal expansion. At the boundary points 1 r and 2 r we then have the following relation between χ and r ( ) 1 1 1 (0) 1 cos r a α = − 2 3 2 2 (0) cos r r a α = + In region 3, we proceed similarly and write the solution in Novikov coordinates: ( ) ( ) 2 2 2 2 2 2 1 2 R ds dt dr R d E r ′ = − + + Ω + Where the parametric solutions are: ( ) 2 0 1 (0, ) 4 g g R r r r r r = + − We can nowsolve for the “gravitational” center of the Schwarzschild region: ( ) 2 1 1 0 1 1 1 (0, ) (0) sin 4 g g R r r r r a r α = + − = Therefore ( ) 2 2 3 1 0 1 1 1 1 sin (0) sin 4 g g g r r r a r r α α   + − = =       And solving for r 0 ( ) 2 2 2 1 0 1 1 1 sin cos 4 g g r r r r α α − = 0 1 1 2 cot g r r r α = − (29) The matching at the other boundary yields: 0 2 2 2 cot g r r r α = − Using these relations, we obtain the embedding diagram shown in Figure 5. Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 35