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

geometric properties (for example, curvature, geodesics between points, length of geodesic within the polar region, etc.). This is also the case for the full 4-dimensional case of space-time solutions, the only difference being dimensionality and signature of the metric. The topological method uses the local (tensorial) feature of solutions of the EFE that imply that if ( ) , M g is a solution then “cutting” and removing any closed subset X from M is also a solution on the manifold ( ) , A A M g with A M M X = − and A A M g g = . Also, if ( ) , A A M g and ( ) , B B M g are two “cut out” solutions of the EFE in disjoint regions then the “stitching” together of ( ) , B B M g and ( ) , B B M g , with continuous boundary conditions is also a solution. We will use this method to show conceptually how an approximate “crude”model with a young earth neighborhood and an older remote universe can be constructed. To construct this model, we will join two regions consisting of different homogeneous densities. Each of these regions is thus a subset of the FLRW cosmology. The two regions will be connected by a vacuum region (“Einstein Rosen bridge”) or a “void” consisting of a subset of the Schwarzschild solution. This model is an example of a spherically symmetric inhomogeneous space-time. We will return to the FLRW space- time below. First, we look at the “chronology enigma” of the vacuum Schwarzschild space-time. 4. The Schwarzschild Chronology Enigma The best known spherically symmetric inhomogeneous solution is the vacuum Schwarzschild metric. Figure 2 shows the maximally extended solution in Kruskal-Szekeres (KS) coordinates (cf. Misner et al., pp. 827-35). (In the figure the coordinates X, T correspond to u, v in Misner et al.). Plotted in the figure are the contours of the Schwarzschild coordinates (r,t) in relation to the KS coordinates (X,T). This solution is sometimes referred to as the “eternal” black hole solution. It is actually a “white hole” at r=0 in the past and a “black hole” at r=0 in the future. It should be noted that the surfaces r=0 are spacelike; so, they are not a place but a time . They do not correspond to the world-line(s) of any physical particle(s) with mass. In this sense, the maximal vacuum Schwarzschild solution is completely devoid of mass, and is an example of pure curvature producing gravitational effects without matter. The entire manifold consists of four regions labeled in the figure I, II, III, IV. These four regions can be characterized as T and R regions according to the criterion whether the gradient of the coordinate of r is timelike (T- region) or spacelike ( R- region), cf. Novikov (2001), Frolov and Novikov (1998, pp. 24-5). A region is said to a T-region if the gradient of r is timelike: 0 r r µ µ ∂ ∂ < , i.e. the normal to the r=constant surface is timelike. In the R-regions the gradient of r is spacelike: 0 r r µ µ ∂ ∂ > , i.e. the normal to the r=constant surface is spacelike. The regions II and IV are T- regions since the gradient of r is timelike there, and thus r is not a time coordinate there. Regions I and III are R -regions and there, r is a spatial coordinate. {As an aside, a failure to recognize T and R regions has historically been the source of many misinterpretations of GR solutions.} The holes are said to be “eternal” since any observer who maintains a constant radial distance greater than r=2M (a world line solely in the R -region I , for example) have world lines that extend from proper time τ = −∞ to τ = ∞ . Such a world line would represent an observer who did not emerge from the past singularity and does not cross the future event horizon (and subsequently falling into the future singularity). On the other hand, any observer freely falling from r=0 in region IV to r=0 in region II has a finite temporal history. Such a world line would represent an observer who emerged from the past singularity (“white hole”) and crosses the future event horizon falling into the future singularity. This is the source of the temporal enigma. For, if time is real, then the white and black hole must be of finite temporal duration, yet the external region I is of infinite temporal duration. This presents the enigmatic question: “ When , relative to the time in the external R- region I did the singularities occur?” The time interval between a point on the “white hole” boundary to any point on the “black hole” boundary occurs in finite time. As such those temporally finite world-lines must ultimately be finished relative to the infinite temporal region I . This temporal enigma is akin to the Kantian antinomy (Kant, 1787, A426/B454) that there cannot be an actual infinite past. The argument, in a nut shell, is that the future is a potential infinity which can never be exhausted, yet this KS solution requires not a potential past infinity, but an actualized temporal past infinity consisting of events that have occurred . Time symmetry then implies a contradiction. For Christian theism, we reject out of hand an actualized infinite past since this would entail a “creation” co-eternal with God. Thus, this idealized empty “eternalist” Schwarzschild solution is rejected as theologically and physically Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 19 Figure 2. The maximal extension of the Schwarzschild space-time in Kruskal-Szekeres coordinates (X,T) . The Schwarzschild curves of constant r are displayed with the broken lines. The r=0 singularities are shown. For T<0 the singularity is a white hole; while for T>0 the singularity is a black hole. The orthogonal solid lines are constant Schwarzschild time coordinates ( t).