Research | Summary of Recent Research
My recent past and current work is in these areas
General Relativistic Fluids
Einstein-Dirac-Yang/Mills Equations
Decay Results in the Kerr Geometry
The Penrose Process
Compressible Fluids
Error Estimates
In the paper [ 7 ] we construct the simplest solution of the Einstein equations that incorporates a shock wave into the standard Friedmann-Robertson-Walker (FRW) 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 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. The papers [ 8 , 9 , 12 ] are summaries of this and previous work.
In [ 13 ] we construct a class of global exact solutions of the Einstein equations that extend the Oppenheimer–Snyder model to the case of non-zero pressure, inside the black hole, yielding a new cosmological model by incorporating a shock wave at the leading edge of the expansion of the galaxies, arbitrarily far beyond the Hubble length in the FRW spacetime. Here the expanding FR universe emerges behind a subluminal blast wave that explodes outward from the FRW center at the instant of the big bang. The total mass behind the shock wave 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 a black hole of an asymptotically flat Schwarzschild spacetime. The complete details are given in [ 18 ]. In [ 17 ] we derive and analyze the equation, k≠0. By an asymptotic argument we show that the equation for state p=(c^2)(ρ)/3 plays the same distinguished role as it does when k=0. The papers [ 22 , 23 ] discuss the notion of an inflationary cosmology, and how inflation can be incorporated to our finite total mass spacetimes. The paper [ 26 ] gives a summary of our work, and the book [ 27 ] expounds the entire theory from first principles.
More recently, Temple and Smoller derived a new set of equations which describe a continuous one parameter family of expanding wave solutions of the Einstein equations such that the Standard Model of Cosmology is embedded as a single point in this family. All of the spacetimes in this family satisfy the equation of state for the pure radiation epoch, and represent a perturbation of the Standard Model. We derive a co-moving coordinate system and compare the perturbed spacetimes to the Standard Model. In this coordinate system we calculate the leading order correction to the redshift v. luminosity relation for an observer at the center of the expanding wave spacetime. This leading order correction contains an adjustable free parameter that induces an anomalous acceleration, (AA). Unlike the theory of Dark Energy based upon a cosmological constant, our theory provides a possible explanation for the AA of the galaxies, which is not ad-hoc and is derivable from physical first principles based upon the Einstein equations in a mathematically rigorous manner.
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Einstein-Dirac-Yang/Mills Equations
In [ 2 ] we show that there are no normalizeable black hole solutions of the Einstein-Dirac-Yang/Mills (EDYM) equations outside of the black hole; that is, the spinors must vanish. This paper generalizes the results in [ 1 ]. In [ 3 ] we give a summary of our work on the EDYM equations.
In [ 4 ] we consider a spherically symmetric, static system of a Dirac particle interacting with gravity and an SU(2) Yang-Mills field. The corresponding EDYM equations are derived. Using numerical methods (involving several parameters), we find different types of particle-like solutions of these equations and discuss their properties. In particular, we find stable particle-like solutions even for arbitrarily weak gravitational coupling. This result is surprising since physicists ignore gravity in these problems, because it is a far weaker force than the other forces. This leads them to renormalizable techniques in order to obtain bound states. Our results show that the inclusion of gravity (weak as it is), has a regularizing effect on the equations. This result is generalized in [ 11 ]. These works are summarized in [ 19 ], and [ 24 ].
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Decay Results in the Kerr Geometry
In [ 14 ] we prove that solutions of the Dirac equation in the Kerr geometry must decay. This means that a Dirac particle must either enter the black hole or escape to infinity. We obtain probability estimates for these choices. We also prove that the decay rate is t^(-5/6). This generalizes the results in [ 5 , 6 ].
In [ 20 , 25 ] we study the decay of the wave equation in the Kerr geometry. The problem is made especially difficult since the standard energy-density is not everywhere positive. In [ 20 ] we construct an integral representation for the propagator, and in [ 25 ] we use this to prove decay.
In [ 31 ] we consider Cauchy problem for the Teukolsky equation for any spin in the Schwarzschild geometry. We prove that solutions with smooth compactly supported initial data must decay.
All of the above works in this section are discussed, along with background details, in the survey paper [ 33 ].
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In [ 30 ] the Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting superradience, the wave analogue of the Penrose process, into a rigorous mathematical framework. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou’s result for the Penrose process. The main mathematical tools are our integral representation for the propagator derived in [ 20 ], together with a time-independent energy estimate for outgoing scalar waves in the Kerr geometry derived in [ 28 ].
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We prove in [ 10 ] that weak solutions of the Navier-Stokes equations for compressible fluid flow in one space dimension do not exhibit vacuum states, provided that no vacuum states are present initially. The solutions and external forces that we consider are quite general.
In [ 16 ] 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)(ρ^γ). 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 γ. We also obtain several physically interesting properties of the solutions concerning the relations of diameter, central density, angular velocity and total mass of rotating stars, some of them quite surprising 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 non-rotating 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.
In [ 21 ] we study singularity formation of smooth solutions of the relativistic Euler equations in (3+1)-dimensional spacetime, for both finite and infinite initial energy. For finite initial energy, we prove that any smooth solution with compactly supported initial data blows up in finite time. For the infinite initial energy case, we first prove the existence, uniqueness, and stability of a smooth solution if the initial data is subluminal and away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation of a non-vacuum constant state, we prove that the smooth solution blows up provided that the radial component of the initial “generalized” momentum is sufficiently large. Our methods are based upon ideas contained in T. Sideris’ well-known blowup result for non-relativistic fluids.
In [ 30 ] we prove the existence of rotating star solutions of the compressible isentropic Euler-Poisson (EP) equations on 3 spatial dimensions, with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty-Beals paper. We prove the nonlinear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star. We also prove local in time stability of W^(1, ∞) solutions where the perturbations are entropy-weak solutions of the EP equations. We also give a uniform (in time) a-priori estimate for entropy-weak solutions of the EP equations. In [ 29 ] these results are extended to rotating and non-rotating White Dwarf and rotating high density supermassive stars, stars which are in convective equilibrium and have uniform chemical composition.
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A method is presented in [ 35 ] for obtaining rigorous error estmates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invarialnt region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by glueing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations.