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

unacceptable. Rather a solution based on Christian presuppositions requires us to take an initial surface occurring at a finite time in the past. For example, we should take the solution manifold to be the set characterized by: { } 0 ( , , , ) : M T X T T θ ϕ = > for some finite value 0 T (in general a function of , , X θ ϕ ) specifying the time of creation. This is one example of excising an open subset of a solution of EFE. Such an initial condition being an open subset of the KS solution is mathematically consistent with GR. As it turns out GR provides no single answer to the question of simultaneity and when “in time” the singularities “occur.” In fact, GR allows the singularities to “occur” at any causally consistent spacelike surface. This can be illustrated for the case of an external world line that remains forever within the R-region labeled I . Referring to Figure 3, cf., for example, Misner et al. (1973, p.528), we display several spacelike surfaces through the Schwarzschild space-time. Any of these could be a surface of simultaneity. The surfaces are temporally ordered: AA’ is earlier than BB’ etc. Each of these surfaces could be taken as an actual surface representing the present “now.” And, according to presentism and Christian theism, one such spatial hypersurface must be selected as “now” and, also, there must be an initial hypersurface corresponding to the first moment of creation, since the extension of region I to t = −∞ is inconsistent with Christian theism. As time progresses from each successive “now” ( A to B to C to D) the proper time on each world line intersecting those surfaces does, of course, progress at different rates, according to the proper time integral. For example, the time registered between space-time events “ A ” and “ B ” by any clock (inertial or not) along a world line ( ) a γ λ is given by the integral: ( ) ( ) B A d d g d d d α β αβ γ λ γ λ τ λ λ λ = ∫ (2) Thus, we emphasize, presentism does not deny that local clocks (in particular, non-inertial ones) tick at different rates. Presentism preserves all the differential structure of SR and GR and thus is mathematically consistent with SR and GR. 5. L-T Chronological Enigmas Before analyzing chronology enigmas in general, we now examine the general class of time dependent spherically symmetric solutions of the EFE. These are referred to as the Lemaître-Tolman (L-T) solutions. These solutions provide the foundation for the analysis of the chronological enigmas. Also, the FLRW cosmologies are a special case of the L-T solutions and can thus be analyzed in terms of the parameters of the L-T class. 6. A Survey of Spherically Symmetric Inhomogeneous models (L-T Models) The general solution for the EFE for an inhomogeneous spherically symmetric space time was developed in detail many years ago (Tolman 1934; Bondi 1947). Frolov and Novikov (1998) give a succinct summary of the process of solving those equations which we follow here with minor changes in notation. Plebański and Krasiński (2006) is also a highly recommended reference with detailed analyses of inhomogeneous cosmological solutions. The L-T models are based on the time evolution of a spherically symmetric (but otherwise inhomogeneous) dust cloud (no pressure) in comoving coordinates. These are solutions that result from a stress energy tensor that depends only on the mass density and is a function of t and r only: ( , ) T t r u u µν µ ν ρ = u µ is the four-velocity vector field of the dust. It can be shown that the metric interval for the general case of an inhomogeneous spherically symmetric space-time in comoving coordinates of freely falling particles is given by the form: ( ) 2 2 2 2 2 , ( , ) rr ds dt g t r dr R t r d = − + + Ω (3) The coefficient of 2 dt is 1 − since all clocks are radially free-falling at constant comoving coordinate r and thus register the “cosmic” time 2 2 dt ds = − . Note that ( ) , R t r is no longer a radial coordinate but a function of the comoving coordinate r and the proper time t . However, the area of a sphere at time t and radius r is still 2 4 ( , ) R t r π . The EFE with cosmological constant 0 Λ = and a pressureless “dust” then reduce to the following set of independent equations: 2 1 ( ) ( ) 2 M r R E r R − =  (4) ( ) ( ) ( ) 2 , 1 2 rr R g t r E r ′ = + (5) ( ) 2 4 ( , ) M r t r R R πρ ′ = ′ (6) In these equations: ( ) M r is gravitational mass within a radius r from the center of symmetry. It is not to be confused with the total invariant rest mass that appears in stress-energy tensor by way of the invariant density ρ . ( ) M r measures the rest mass energy plus Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 20 Figure 3. Spacelike hypersurfaces within the maximal extension of the Schwarzschild geometry.