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

Thus obtaining 1 2 2 2 2 2 2 2 2 2 1 1 M dz ds dr r d dr r d r dr ϕ ϕ −       = − + = + +               With solution: ( ) 2 0 2 8 z z r M M − = + We have introduced the constant z 0 which specifies the center of symmetry for later convenience. In anticipation of later analysis, we change the variables as follows: r R z r → → so that the embedding is specified as: ( ) 2 0 2 8 r r R M M − = + Now r is no longer the Schwarzschild radial coordinate (“curvature coordinate”) but is the Novikov comoving radial coordinate. The function R, which corresponds to the Schwarzschild radius , is now a function of r and is the radius of the spherical shell at Novikov radius r . We now discuss the Novikov coordinates for the Schwarzschild solution. 2. Novikov Coordinates We will also find it useful to utilize a version of Novikov coordinates for the maximal Schwarzschild manifold (Misner et al. 1973, pp. 826-7). Our path to a YEC cosmology will be via stitching together subsets of various space-times. One of these space-times will be the FLRW space-times which consists of freely falling matter (hence inertial) and which utilize “comoving” coordinates. Comoving coordinates have a great interpretational advantage since the coordinates are based directly on physical principles. Also, since Novikov coordinates are an example of comoving coordinates for the Schwarzschild geometry, they will be useful for facilitating the joining conditions between the solutions. When we do this, all coordinates are comoving and the time coordinate becomes the proper time of all freely falling observers in the cosmological solution. The joining conditions are easier because the L-T class of spherically symmetric inhomogeneous solutions all use a global comoving coordinate system. In a comoving coordinate system each comoving observer is freely falling and assigns his elapsed proper time as the time coordinate for each event along his world line. For these coordinate systems the tt g component of the metric tensor is necessarily -1. The spatial coordinates are constant for each observer and consist of the spherical coordinates ( ) , θ ϕ and (a function of) the radial coordinate R which labels the observer’s initial position. As the observer free-falls in the gravitational field he follows a geodesic. That geodesic is uniquely determined by the initial point and the four- velocity. Each event along the geodesic is thus determined by the “arc-length” (i.e. proper time t ) along the geodesic and the initial position R , θ and ϕ . The set of observers for Novikov coordinates consists of observers falling from rest at distinct “distances” R from an arbitrarily chosen instant of time. A convenient choice is to assume the clocks are synchronized at the instant of maximal radius. Novikov originally used a dimensionless coordinate related to the Schwarzschild R (recall our change of coordinate labels above) coordinate by: 1 2 R r M = − For our analysis we will use a dimensional Novikov radial coordinate defined as: 0 4 1 2 R r r M M = + − In this case our metric, in the notation of the L-T solutions (cf. equation (5)), becomes: ( ) ( ) 2 2 2 2 2 2 1 2 R ds dt dr R d E r ′ = − + + Ω + (where we now omit the asterisk and, as mentioned previously, set the radius function as R(t,r) rather than r). The function E(r), which is a measure of the free-falling particle’s energy, will be discussed below when we examine time-dependent spherically symmetric space-times of the Lemaître, Tolman, Bondi (L-T) family. In each of the FLRW regions the solution is: Dennis ◀ Young earth relativistic cosmology ▶ 2018 ICC 34 Figure 15. Profile of embedding diagram of “bar bell” cosmology. Construction of global radial comoving coordinate “ r ” r 1 r 2 r R 1 1 χ α = 2 2 χ α = ( ) 1 0 a ( ) 2 0 a ( ) 1 0 a 3 r