Research
Recently, Smoller's main focus has been concerned with problems involving gravity mainly as described by (A) Einstein's equations of GR, but also (B) in the Newtonian framework as described by the classical Euler-Poisson equations in 3-spatial dimensions for compressible fluids. In the GR part, his work can be divided into two broad areas: first the study of shock and rarefaction waves and their various astronomical and cosmological applications, and second, the coupling of gravity to various "fundamental" forces mainly involving various stability questions for black holes (BH).
In Part (A), Smoller and his collaborators have studied the (linearized) stability for rotating and non-rotating BH's in the so-called Teukolsky equation (TE), framework. This equation is a second order wave like PDE depending on 3 real parameters, the mass M, the angular velocity a, and the spin s, and solutions of which describe various perturbations of a Kerr BH, under the natural hypothesis a<M. The Kerr BH reduces to the well-known spherically symmetric, stationary Schwarzschild BH, if a=0. For various values of s, the TE governs the linearized stability of the Kerr BH under various perturbations. Namely, the cases s=0,1/2, 1, and 2 correspond, respectively, to scalar fields, Dirac particles, electromagnetism, and gravitational waves. The mathematical problem is to show that for smooth compactly supported initial data, the solution of the TE decays in time. Smoller and his collaborators have solved this problem in the cases s=0, and s=1/2. For the case s=1/2, we proved that the solution decays generically at a rate t^(-5/6), and we have shown that Dirac particles must either enter the BH or escape to infinity. In addition, we have obtained probability estimates for these two cases. For the scalar wave case s=0, we obtained an integral representation for the propagator. We used this to prove decay in the case of a finite number of azimuthal angular momentum modes. We also gave a rigorous proof of the Penrose process, the energy extraction from a rotating BH. An important technical point is to prove that the outgoing wave has finite energy. We accomplished this together with F. Finster. With F. Finster we have also proved decay of solutions for the Teukolsky equation in a Schwarzschild background geometry, for all spin, s=0,1/2,1,2.
In a different direction, together with B. Temple, we have constructed exact shock wave solutions of the Einstein equations for perfect fluid which we interpret as the extension of the Oppenheimer-Snyder (OS) model to the case of non-zero pressure. To accomplish this, we derive equations that describe the general matching of a critically expanding, (k = 0), Friedmann-Robertson-Walker (FRW) metric to a new metric, the Tolman-Oppenheimer-Volkoff (TOV) metric, across a shock wave interface. In the cosmological interpretation of the FRW metric, the shock wave must lie at least one Hubble length from the center of the FRW spacetime. In this case we (rigorously) demonstrate the existence of a new class of global solutions in which the expanding FRW universe emerges behind a subluminal blast wave that explodes outward from the FRW center at the instant of the Big Bang. The shock wave then continues to expand, satisfying the entropy condition for shocks, all the way out until it weakens to the point where it continues naturally to an OS interface that emerges from the White Hole event horizon of an ambiant Schwarzschild metric. These shock wave solutions differ qualitatively from the OS solution, and 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. In this model the equation of state p=[(c^2)/3](ρ), correct for early Big Bang physics, is distinguished by the differential equations: for this equation of state alone does the shock wave emerge from the Big Bang at finite non-zero speed, the speed of light. We have recently also incorporated "inflation" into our shock-wave cosmological model.
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 phase, 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.
With Tao Luo we have recently studied the compressible Euler-Poisson equations in 3 spatial variables. We prove existence and nonlinear stability theorems for rotating star solutions which are axi-symmetric steady-state solutions of the equations. We apply our results to rotating White Dwarf stars, and rotating supermassive (extreme relativistic)stars, stars which are in convective equilibrium and have uniform chemical composition.
With F. Finster, we have developed a method for obtaining rigorous error estimates for approximate solutions of the Riccoti equation with real or complex potentials. This is established by deriving invariant region estimates for complex solutions of the Riccoti equation. These techniques can be applied to the WKB-Airy approximations of the corresponding Schrödinger equation.
With David Hoff (Indiana University), we proved 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.
With Ronguha Pan (GA Tech) we have proved that solutions of the compressible relativistic Euler equations, in (3+1) dimensions, develop singularities in finite time, for both finite and infinite initial energy. In the finite energy case, we show that any smooth solution with compactly supported initial data develops a singularity in finite time. For the infinite initial energy case, the smooth solution becomes singular provided that the initial data is a smooth compact perturbation of a constant state, and the radial component of the initial "momentum" is sufficiently large.
Tao Luo (Georgetown University) and Smoller have studied the compressible Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with a general equation of state. When the domain is a ball, and the angular velocity is constant, we obtain both existence and non-existence results, depending on the adiabatic gas constant. We also obtain several interesting properties of the solutions, some of them quite surprising.
Some of Smoller's most recent publications can be downloaded by going to lanl.arxiv.org.