Joel Smoller has done research in shock-wave theory, Navier-Stokes equations, systems of reaction-diffusion equations, dynamical systems (Conley Index Theory), and bifurcation theory (symmetry-breaking bifurcations). Recently, Smoller has been concerned with problems involving gravity, as described by Einstein's Theory of General Relativity (GR), on two different scales: (A) elementary particles, (which involves the coupling of gravity to other "fundamental" forces), and (B) astrophysical shock-waves. The first, a continuation of work with S.-T. Yau, (Harvard) and Felix Finster, (Max Plank Institute, Leipzig) studies the equations which describe the nonlinear coupling of gravity to both the Dirac equation, (describing quantum-mechanical particles), and to a Yang/Mills field, (describing nuclear forces), the EDYM equations. The second, an on-going project with Blake Temple, (University of California, Davis), is concerned with the constuction of shock-wave solutions of Einstein's equations, and the application of such solutions to problems in astrophysics.
In part (A), Smoller and his collaborators studied the interaction of Dirac particles (fermions) with non-abelian gauge fields and gravity. First, we showed that for arbitrarily weak gravitational interaction, we can find (linearly) stable static, spherically symmetric bound-state (particle-like) solutions. Second, we considered static, spherically symmetric black-hole solutions of the EDYM equations, and we proved that the spinors ( wave functions), cannot be normalized. (One interpretation of this latter result is that the Dirac particles either enter the black-hole, or go off to infinity.) Both of these results are quite surprising. Finally, we showed that time-periodic solutions of the Dirac equation, in a Reissner-Nordstrom black-hole background, canot exist. This result was extended, together with Niky Kamran (McGill University), to the case of "rotating", (axi-symmetric), black-holes in the presence of quantum mechanical particles, and we again showed that time-periodic solutions of the Dirac equation in an axi-symmetric black-hole geometry cannot exist. We have also recently proved that solutions of the Cauchy problem for the Dirac equation, with initial data outside of the event horizon, decay generically at a rate t-5/6. We have also obtained probability estimates for when the Dirac particle enters the black hole, or escapes to infinity. We have recently considered the scalar wave equaion in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation for the propagator which expresses the solution as a superposition of solutions or the radial and angular ODEs which arise in the separation of variables. In particular, we prove the completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry,
In part (B), Smoller, together with Blake Temple, have constructed shock-wave solutions of Einstein's equations for a perfect fluid, which can model explosions on any scale. We applied these results to propose an alternative scenario to the usual Big-Bang theory. We showed that our dynamical shock-wave model is consistant with both the observed values of the Hubble constant and the cosmic background microwave radiation temperature. We calculated the position of our shock-wave to be of Hubble length, [(10)^10 light -years], from us. Furthermore, as our model contains a true fluid-dynamical shock-wave, entropy increases across the shock as time proceeds, resulting in a consequent loss of information about the details of the early explosion.
We put forth in a rigorous mathematical setting, a new
Cosmological Model in which the expanding Friedmann-Robertson-Walker
(FRW) universe emerges from an event more similar to a classical
explosion---there is a shock wave at the leading edge of the
expansion---than the standard scenario of the Big
Bang. We believe that general relativity pretty much forces such a
solution on you as soon as you try to relax the assumption in the
standard model that the expansion of the galaxies is of infinite mass and extent
at each fixed time.
Most importantly, in these new models, the explosion is large enough
to account for the enormous scale on which the galaxies and the cosmic
background radiation appear uniform.
There are a number of remarkable twists that arise in these new GR
blast wave solutions. First of all, in order that the shock wave lie
beyond one Hubble length from the FRW center, (so as to agree with astronomical observations), it
follows that the universe begins inside a time reversed Black Hole, (a
White Hole). One Hubble length is the threshold radius beyond which
the total mass in the universe bounded by that radius lies inside its own
Schwarzschild radius---that is, 2M/r>1 beyond one Hubble length---and
thus when the shock wave lies beyond one Hubble length, the shock wave
solution evolves inside a Black Hole. We derive the relevant equations, and
discuss some surprising implications. One interesting consequence is that the entropy condition, which
introduces time-irreversability, chooses the explosion over the
implosion. Another consequence of the entropy condition is that it implies that the
shock must eventually weaken until it emerges from the Black Hole,
(through the White Hole event horizon), as a zero pressure
Oppenheimer-Snyder solution. Asymptotically, for large time, the
explosion settles down to something like a giant supernova of finite
mass and extent, but on an enormous scale---a localized mass expanding into
an asymptotically flat Schwarzschild spacetime, everywhere outside the
Black Hole. The main new technical point is the introduction of a new
solution of the Einstein equations, (which we call the TOV metric inside the
Black Hole), that cuts off and bounds the total mass of the FRW spacetime at
finite radii inside the Black Hole. But he biggest surprise to us is that unlike shock matching outside
the Black Hole, the equation of state, p=(1/3)(rho)---the equation of state
at the earliest stage of Big Bang physics---is mysteriously 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. (The shock wave then
decelerates to a sub-luminous wave at all times after the Big Bang.) These solutions describe, in exact formulas, the global dynamics of
strong gravitational field solutions of the Einstein equations, and the
setting, inside the Black Hole, is pretty much unexplored territory for
analysis.
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.
