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

(a negative) gravitational binding energy. For example, for closed solutions the total gravitational mass can be zero even though ρ (which is always non-negative) is not. ( ) E r is the energy and curvature at a given comoving radius r . ( ) E r is required to satisfy the inequality ( ) 1/ 2 E r > − . For ( ) 0 E r < we have a closed universe with positive curvature which expands from an initial “big bang” to a maximum radius then collapses to a final “big crunch.” For ( ) 0 E r = the universe is open and flat (zero curvature), while for ( ) 0 E r > the universe is open and hyperbolic (negative curvature). An important and interesting feature of these equations is that for fixed r the matter in that “shell” evolves independently from the rest of the matter in the universe that is at a radius > r . This is the same as the Newtonian effect that the matter outside a spherical shell does not affect the motion of matter interior to the shell. Each shell of constant r is in fact the equation of a geodesic. Further, each shell can be given its own initial conditions specified by the arbitrary functions M(r) and E(r) and the shell will evolve according to the standard Friedman cosmological model. This observation will play an important part in the YEC solution and its interpretation later. [It should be noted that the functions must satisfy some rather general conditions to avoid surface layers and shell crossings. The details of these conditions are not essential to the overall discussion here. The interested reader is referred to the papers by Hellaby and Lake (1985) or Hellaby (1987) for the details.] The last equation can be integrated to obtain ( ) M r ( ) ( ) 3 0 4 0, (0, ) 3 r d M r dr r R r dr π ρ   =   ∫ (7) ( ) M r gives the amount of gravitating mass at radius less than r. In the following we restrict our attention to closed solutions with 1/ 2 ( ) 0 E r − < < . For this case, a general solution to equations (4) - (6) can be found by introducing the cycloidal parameter π η π − ≤ ≤  . This choice for η corresponds to maximum expansion at 0 η = . ( ) ( ) ( ) 3/2 ( ) sin 2 ( ) B M r t t r E r η η − = + − (8) ( ) ( ) ( ) 2 ( ) ( ) ( , ) 1 cos cos 2 ( ) ( ) 2 M r M r R t r E r E r η η = + = − − (9) ( ) B t r is a constant of integration and an arbitrary function of r . In the literature, it is referred to as the local “time to the Big Bang.” Note that this solution is the general inhomogeneous solution for a pressureless dust universe. Another useful representation for this solution can be obtained to express ( ) , R t r implicitly in terms of t and r is obtained by solving for η in the equation for ( ) , R t r 1 ( ) ( , ) cos 2 ( ) E r R t r M r η −   − =     Substituting this into the equation for t then gives the implicit equation for R as a function of t and r : ( ) ( ) 2 1 3/2 1 2 ( ) ( ) 2 ( ) 2 ( ) cos 2 ( ) ( ) 2 ( ) B M r E r R t t r M r R E r R E r M r E r −   − − = + +   −   (10) The “big bang” surface is given by 0 R = when η π = − . When we set η π = − in equation (8) we get for the past singularity: ( ) ( ) 3/2 ( ) 2 ( ) B M r t t r E r π − = − − . Since we choose (arbitrarily) t=0 as the time of the big bang we obtain: ( ) ( ) 3/2 ( ) 2 ( ) B M r t r E r π = − Thus, explaining the term “time to the big bang.” A special case of the general spherically symmetric solution is the homogeneous FLRW solution. We now briefly discuss the homogeneous FLRW solution since it will play a central role in the conceptual development. 7. The homogeneous models. The FLRW cosmology The FLRW space-time is described by the metric: ( ) ( ) ( ) 2 2 2 2 2 K ds dt a t d f d χ χ = − + + Ω (11) ( ) 2 2 2 sin 1 0 sinh 1 K K f K K χ χ χ χ  = + = =   = −  The value of K determines whether the geometry is closed or open. K=+ 1 is the closed solution, K= 0 is the open conformally flat solution, and K=- 1 is an open hyperbolic space-time. The FLRW space-time is the solution for a pressureless and homogeneous “dust” cloud. Recall that this solution uses “comoving” coordinates. The particles are all free falling (only gravity is present), and for any “particle” in the space-time the coordinates θ and ϕ are constant. We will be using these “comoving” coordinates exclusively in our analysis. In these coordinate systems t is the proper time registered by the freely falling clocks. The scale factor ( ) a t can be written in terms of a parametric equation using the cycloidal variable η . ( ) ( ) ( ) ( ) 0 0 1 cos 2 sin 2 a a a t η η η η η = + = + (12) We have chosen the parameter η to be in the interval [ ] , π π − . Here a 0 is the radius of the universe at maximumexpansion corresponding to 0 η = . The universe expands from the singularity a=0 at η π = − to the maximum at 0 η = . It then collapses to the singularity a=0 at η π = . Note that with these choices all particles are synchronized so that they register proper time 0 t = at maximum expansion. The radius of maximum expansion and the lifetime of the solution is determined by the matter density ( ) t ρ in the universe. At Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 21