Limiting Behavior of Solutions to the Einstein-Yang/Mills Equations

The ADM masses of particle-like solutions to the Einstein-Yang/Mills Equations tend to 2 as the number of nodes of the solutions increases. The same result is true for black hole solutions with event horizon less than 1. For event horizon $\rho > 1$ the ADM masses converge to $\rho + \rho^{-1} .$ These statements extend and correct ``An Investigation at the Limiting Behavior of Particle-Like Solutions to the Einstein-Yang/Mills Equations and a New black Hole Solutions'', by J. A. Smoller and A. G. Wasserman, in Comm. Math. Phys., 161, 365-389, (1994).Comment: 2 pages, um594

Cosmology with a shock wave

We construct the simplest solution of the Einstein equations that incorporates a shock-wave into a standard Friedmann-Robertson-Walker metric whose equation of state accounts for the Hubble constant and the microwave background radiation temperature. This produces a new solution of the Einstein equations from which we are able to derive estimates for the shock position at present time. We show that the distance from the shock-wave to the center of the explosion at present time is comparable to the Hubble distance. We are motivated by the idea that the expansion of the universe as measured by the Hubble constant might be accounted for by an event more similar to a classical explosion than by the well-accepted scenario of the Big Bang

Cosmology, Black Holes and Shock Waves Beyond the Hubble Length

We construct exact, entropy satisfying shock wave solutions of the Einstein equations for a perfect fluid which extend the Oppeheimer-Snyder (OS) model to the case of non-zero pressure, {\it inside the Black Hole}. These solutions put forth a new Cosmological Model in which the expanding Friedmann-Robertson-Walker (FRW) universe emerges from the Big Bang with a shock wave at the leading edge of the expansion, analogous to a classical shock wave explosion. This explosion is large enough to account for the enormous scale on which the galaxies and the background radiation appear uniform. In these models, the shock wave must lie beyond one Hubble length from the FRW center, this threshhold being the boundary across which the bounded mass lies inside its own Schwarzshild radius, $2M/r>1,$ and thus the shock wave solution evolves inside a Black Hole. The entropy condition, which breaks the time symmetry, implies that the shock wave must weaken until it eventually settles down to a zero pressure OS interface, bounding a {\em finite} total mass, that emerges from the White Hole event horizon of an ambient Schwarzschild spacetime. However, unlike shock matching outside a Black Hole, the equation of state $p=\frac{c^2}{3}\rho,$ the equation of state at the earliest stage of Big Bang physics, is {\em distinguished} at the instant of the Big Bang--for this equation of state alone, the shock wave emerges from the Big Bang at a finite nonzero speed, the speed of light, decelerating to a subluminous wave from that time onward. These shock wave solutions indicate a new cosmological model in which the Big Bang arises from a localized explosion occurring inside the Black Hole of an asymptotically flat Schwarzschild spacetime

Decay of Solutions of the Teukolsky Equation for Higher Spin in the Schwarzschild Geometry

We prove that the Schwarzschild black hole is linearly stable under electromagnetic and gravitational perturbations. Our method is to show that for spin $s=1$ or $s=2$, solutions of the Teukolsky equation with smooth, compactly supported initial data outside the event horizon, decay in $L^\infty_{loc}$.Comment: 32 pages, LaTeX, 2 figures, error in expression for energy density of gravitational waves correcte

Absence of Zeros and Asymptotic Error Estimates for Airy and Parabolic Cylinder Functions

We derive WKB approximations for a class of Airy and parabolic cylinder functions in the complex plane, including quantitative error bounds. We prove that all zeros of the Airy function lie on a ray in the complex plane, and that the parabolic cylinder functions have no zeros. We also analyze the Airy and Airy-WKB limit of the parabolic cylinder functions.Comment: 25 pages, LaTeX, 7 figures (published version

Shock-Wave Cosmology Inside a Black Hole

We construct a class of global exact solutions of the Einstein equations that extend the Oppeheimer-Snyder (OS) model to the case of non-zero pressure, {\em 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 behind 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 OS interface, (bounding a {\em 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 new cosmological model in which the Big Bang arises from a localized explosion occurring inside the Black Hole of a Schwarzschild spacetime.Comment: Small corrections that significantly improve the result

Rotating Fluids with Self-Gravitation in Bounded Domains

In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state $P=e^S\rho^{\gamma}$. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant $\gamma$. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density . This is physically striking and in sharp contrast to the case of the nonrotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.Comment: 37page