How to smooth a crinkled map of space-time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equations.

We present the authors' new theory of the RT-equations ('regularity transformation' or 'Reintjes-Temple' equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Γ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

An instability of the standard model of cosmology creates the anomalous acceleration without dark energy.

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 p=0. In this phase portrait, the critical k=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.

Causal dissipation for the relativistic dynamics of ideal gases.

We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier-Stokes equations.

No regularity singularities exist at points of general relativistic shock wave interaction between shocks from different characteristic families.

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C0,1 to C1,1 in a neighbourhood of points of shock wave collision in general relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier Proc. R. Soc. A publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Thus, our result implies that regularity singularities (a type of mild singularity introduced in our Proc. R. Soc. A paper) do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original Proc. R. Soc. A publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein-Euler equations remains open.

Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation.

CURRENT THEORIES OF DISSIPATION IN THE RELATIVISTIC REGIME SUFFER FROM ONE OF TWO DEFICITS: either their dissipation is not causal or no profiles for strong shock waves exist. This paper proposes a relativistic Navier-Stokes-Fourier-type viscosity and heat conduction tensor such that the resulting second-order system of partial differential equations for the fluid dynamics of pure radiation is symmetric hyperbolic. This system has causal dissipation as well as the property that all shock waves of arbitrary strength have smooth profiles. Entropy production is positive both on gradients near those of solutions to the dissipation-free equations and on gradients of shock profiles. This shows that the new dissipation stress tensor complies to leading order with the principles of thermodynamics. Whether higher order modifications of the ansatz are required to obtain full compatibility with the second law far from the zero-dissipation equilibrium is left to further investigations. The system has exactly three a priori free parameters χ,η,ζ, corresponding physically to heat conductivity, shear viscosity and bulk viscosity. If the bulk viscosity is zero (as is stated in the literature) and the total stress-energy tensor is trace free, the entire viscosity and heat conduction tensor is determined to within a constant factor.

Expanding wave solutions of the Einstein equations that induce an anomalous acceleration into the Standard Model of Cosmology.

We derive a system of three coupled equations that implicitly defines a continuous one-parameter family of expanding wave solutions of the Einstein equations, such that the Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. By approximating solutions near the center to leading order in the Hubble length, the family reduces to an explicit one-parameter family of expanding spacetimes, given in closed form, that represents a perturbation of the Standard Model. By introducing a comoving coordinate system, we calculate the correction to the Hubble constant as well as the exact leading order quadratic correction to the redshift vs. luminosity relation for an observer at the center. The correction to redshift vs. luminosity entails an adjustable free parameter that introduces an anomalous acceleration. We conclude (by continuity) that corrections to the redshift vs. luminosity relation observed after the radiation phase of the Big Bang can be accounted for, at the leading order quadratic level, by adjustment of this free parameter. The next order correction is then a prediction. Since nonlinearities alone could actuate dissipation and decay in the conservation laws associated with the highly nonlinear radiation phase and since noninteracting expanding waves represent possible time-asymptotic wave patterns that could result, we propose to further investigate the possibility that these corrections to the Standard Model might be the source of the anomalous acceleration of the galaxies, an explanation not requiring the cosmological constant or dark energy.

Shock-wave cosmology inside a black hole.

We construct a class of global exact solutions of the Einstein equations that extend the Oppeheimer-Snyder model to the case of nonzero pressure, inside the black hole, by incorporating a shock wave at the leading edge of the expansion of the galaxies, arbitrarily far beyond the Hubble length in the Friedmann-Robertson-Walker (FRW) spacetime. Here the expanding FRW universe emerges be-hind a subluminous blast wave that explodes outward from the FRW center at the instant of the big bang. The total mass behind the shock decreases as the shock wave expands, and the entropy condition implies that the shock wave must weaken to the point where it settles down to an Oppenheimer-Snyder interface, (bounding a finite total mass), that eventually emerges from the white hole event horizon of an ambient Schwarzschild spacetime. The entropy condition breaks the time symmetry of the Einstein equations, selecting the explosion over the implosion. These shock-wave solutions indicate a cosmological model in which the big bang arises from a localized explosion occurring inside the black hole of an asymptotically flat Schwarzschild spacetime.