We introduce the locally inertial Godunov method with dynamical time dilation, and use it to give a definitive numerical simulation of a point of shock wave interaction in general relativity starting from a new initial dataset. Prior work of Groah and Temple justifies meeting the Einstein constraint equations for the initial data only at the weak level of Lipshitz continuity in the metric. The forward time simulations, presented here, resolve the secondary wave in the Smoller-Temple shock wave model for an explosion into a static, singular, isothermal sphere. The backward time solutions indicate black hole formation from a smooth solution via collapse associated with an incoming rarefaction wave. A new feature is that space-time is approximated as locally flat in each grid cell so that Riemann problems and the Godunov method can be implemented. Clocks are then dynamically dilated to simulate effects of space-time curvature. Such points of shock wave interaction are more singular than points on single shock surfaces because the coordinate systems that make space-time locally flat on single shock surfaces (Gaussian normal coordinates), break down at points of shock wave interaction.
In this note, which is a first step in authors' longer calculation, we present a system of equations for computing the p = 0 evolution of corrections to the Standard Model of Cosmology consistent with corrections produced by spherical self-similar perturbations of the Standard Model known to exist during the radiation phase of the Big Bang. We record here that the asymptotic equations close at leading order, creating a subsystem which contains a stable rest point which does not correspond to the unperturbed critical zero pressure Friedmann spacetime. Corrections to the Standard Model lying within the domain of attraction of this stable rest point would produce, time-asymptotically, quadratic corrections to redshift vs luminosity, i.e., quadratic corrections to the Hubble constant going out from the center, that would be independent of initial conditions, and different from the predictions of the Standard Model. In future investigations authors will characterize the parameter values associated with self-similar waves from the radiation era that would give rise to corrections that lie within this domain of attraction. The results in this paper arose as a step in the authors' program to compute the corrections to redshift vs luminosity produced by self-similar waves from the radiation epoch, interpreting this as a baseline mechanism for the creation of anomalous accelerations that does not require extraneous fields like Dark Energy. Details will appear in their forthcoming paper.
We identify the condition for smoothness at the centre of spherically symmetric solutions of Einstein's original equations without the cosmological constant or dark energy. We use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the centre of symmetry when the pressure =0. In this phase portrait, the critical=0 Friedmann space-time appears as a saddle rest point which is unstable to spherical perturbations. This raises the question as to whether the Friedmann space-time is observable by redshift versus luminosity measurements looking outwards from any point. The unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of local uniformly expanding space-times whose accelerations closely mimic the effects of dark energy. A unique simple wave perturbation from the radiation epoch is shown to trigger the instability, match the accelerations of dark energy up to second order and distinguish the theory from dark energy at third order. In this sense, anomalous accelerations are not only consistent with Einstein's original theory of general relativity, but are a prediction of it without the cosmological constant or dark energy.
Abstract.It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term regularity singularity was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous (C 0,1 ), but no smoother, in any coordinate system of the C 1,1 atlas. An existence theory for shock wave solutions in C 0,1 admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but C 1,1 is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger C 1,1 atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of C 0,1 shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of smooth coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and 'real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of C 1,1 transformations. If essentially non-removable, it would argue strongly for a 'real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities.
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