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

maximum expansion, the density is ( ) 0 ρ and the maximum radius is given by: ( ) 0 3 8 0 a πρ = (13) The lifetime of the universe is then evaluated as: ( ) ( ) ( ) 0 3 8 0 T t t a π π π π ρ = − − = = (14) 8. Relation of the homogeneous FLRW solution to the L-T solutions A special case of the spherically symmetric solutions is the FLRW homogeneous cosmologies which we described above. Here we show the relation of the FLRW solution to the functions M and E of the general L-T solution. For the FLRW cosmology the matter density is uniform and independent of space, and therefore a function of the comoving time only. For reference the parameterization of the metric for the FLRW solution is ( ) ( ) ( ) 2 2 2 2 2 K ds dt a t d f d χ χ = − + + Ω (15) The general L-T solution as discussed in the section above is given by: 2 2 2 2 2 1 2 R ds dt dr R d E  ′   = − + + Ω + (16) Comparison with the FLRW metric then yields: 2 2 2 2 ( ) 1 2 ( ) ( ) k R a t E R a t f χ  ′   = + = Therefore 1/2 2 1 ( ) ( ) ( ) 2 1 2 ( ) 1 4 k k k k R a t f f E f f χ χ χ − ′ ′ =  ′ = −   Specializing to the closed solution we have: 2 2 ( ) cos 2 ( ) cos 1 sin R a t E χ χ χ χ ′ = = − = − With the specialization of the density to a function of time, the functions E and M become (now using χ as the radial coordinate): ( ) ( ) ( ) ( ) ( ) 3 0 3 0 3 3 3 4 0 (0, ) 3 4 0 (0, ) 3 4 0 (0, ) 3 4 0 (0)sin 3 d M d R d d d R d R a χ χ π χ χρ χ χ π ρ χ χ χ π ρ χ π ρ χ   =     =   = = ∫ ∫ Using the relation given in equation (13) yields: ( ) 3 1 (0)sin 2 M a χ χ = We will use these relations to produce a globally inhomogeneous solution (but with piecewise locally homogeneous regions) which will exhibit the features for a YEC cosmology. 9. A Semi-closed inhomogeneous model A few years ago, I attended a presentation in which the speaker presented an inhomogeneous cosmology consisting of two separate regions which are subsets of the FLRWwith different total mass. The solution consists of a closed universe consisting of two spherical homogeneous FLRW regions of different uniform density connected by a cylindrical Schwarzschild section with no matter. I will refer to this class of solutions as the “bar bell” cosmologies. Bonnor (1956) also has considered such closed solutions in his investigations of nebulae formation. If we qualitatively diagram the time dependence of a radial cross section of the space-time we arrive at the notional Figure 4, which depicts the idea of such an inhomogeneous space-time. The left side and right side of the figure depicts regions of homogeneous density. The density of the left side is greater than the density of the region on the right side; hence, the lifetime of the region on the left is less than that of the region on the right by virtue of equation (14). The middle section is a spherically symmetric vacuum (zero matter density) and thus by Birkhoff’s theorem must be a section (subset) of the maximal Schwarzschild solution. In that figure, it is evident that one can slide the smaller region’s time of existence upward or downward. When one slides the smaller region on the left forward in time we have a cosmology in which the proper “time to the creation” is less than the proper “time to creation” of the larger region. In other words, the EFE do not specify when solutions occur globally. We will need to proceed from this qualitative notional solution by solving the EFE. Relying on the relativity principle, there is no preferred, i.e. detectable, spacelike surface of “now.” If T(x 0 , x 1 ,x 2 ,x 3 ) is a scalar function on the space-time such that T µ ∂ is a timelike (i.e. 0 g T T µν µ ν ∂ ∂ < ) covariant field then T=constant is a physically allowable “now.” This “bar bell” cosmology is an example of a solution obtained by stitching several solutions at the seams. We should note that the “stitching” method requires that the solutions join smoothly along the seams. This condition sometimes rules out many simplistic constructions. [Note: Exact solutions can be obtained provided the stitching condition at the junction hypersurface ( ) 0 f x = satisfies Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 22