(John) Blake Temple

Shock waves are steep fronts that propagate when the convective motion of a fluid dominates the diffusion. A sonic boom off the wing of a plane, the propagation of grid-lock in traffic, waves breaking on the shoreline, flame fronts in combustion, the leading edge of a nuclear explosion, tidal waves, the bore of water let loose when a dam breaks, saturation waves in an oil reservoir, adsorption lines in chromatography, the water-hammer problem in pipelines, all are examples of the same phenomenon as understood by the modern mathematical theory of shock-waves. The mathematical theory of shock-waves has provided a unified conceptual picture capable of describing all of the above applications in terms of a single theory. Most interestingly, the modern theory has explained how {\em entropy} and {\em time-irreversibility}, (concepts that originally were understood only in the context of ideal gases), could be given meaning in an arbitrary conservative, first order system of nonlinear partial differential equations, much more general than gas dynamics. The conclusion: Shock-waves introduce increase of entropy into the dynamics of solutions. Shock waves are also one of the celebrated applications of the theory of distributions to nonlinear equations.\footnotemark[1]

Professor Temple has spent two decades contributing to the mathematical theory of shock waves. His early work was on geometrical aspects of shock wave interactions, c.f. [1],[2]. In 1965, James Glimm introduced the idea that the study of weak solutions with shock waves could be reduced to a study of wave interactions. Glimm's paper revolutionized the subject, and his ideas and methods of analysis remain the foundation of the mathematical theory to this day. Temple's recent focus has been on the theory of shock-wave propagation in general relativity.

In Einstein's theory of general relativity, gravitational {\em forces} turn out to be just anomalies of spacetime {\em curvature}, and the propagation of curvature through spacetime is governed by the {\it Einstein equations} \footnotemark[2]. In [3], Temple and his co-worker Smoller were the first to construct an exact shock-wave solution of the Einstein equations that models a blast wave. In this model, the shock-wave discontinuity is in the curvature of spacetime itself. Temple and Smoller have gone on to construct several examples of shock waves in general relativity, including a model for the expanding universe in which a shock wave is present at the leading edge of the expansion of the galaxies that we measure by the Hubble constant, [4]. If such a shock wave were present in the actual universe, then increase of entropy due to the shock would imply that the present universe cannot be time reversed back to micro-seconds after the Big Bang. In the standard model of the Big Bang, the expansion of the universe is assumed to be {\em infinite} at every time after the Big Bang, and time-reversible all the way back to the beginning, as well, and so the Smoller-Temple model is quite controversial. Temple's recent work with Jeff Groah on the initial value problem, [5], has led to his interest in the time-asymptotics of solutions after the formation of a black hole, and interest in the question of whether the Big Bang itself can be accounted for within the framework of general relativity, or whether its explanation lies beyond the theory. As Temple says: "The theory of shock waves opens up new doors for exploring the dynamics of solutions of these complicated equations\footnotemark[3]."

\footnotetext[1]{Shock waves are fronts across which solutions are discontinuous. Since the solutions are discontinuous, they do not satisfy the underlying partial differential equations in the classical sense. The theory of distributions, originally invented for linear equations, can be used to calculate the correct jumps and propagation speeds for shock waves, and it also gives meaning to the notion of a general weak solution of the equations.}

\footnotetext[2]{General relativity began in 1915, when Albert Einstein introduced his famous equations that describe the time evolution of the gravitational field. In Einstein's theory the gravitational field is an indefinite metric tensor $g$ defined on spacetime, and it evolves according to the Einstein equations, which can be written in the compact form $G=8\pi T.$ Here $G$ is the Einstein curvature tensor, (a second order operator on $g$ which Einstein borrowed from Riemann's 1864 theory of curvature), and the components of $T,$ the so called {\it stress energy tensor}, are the energy-momemtum densities and their fluxes, the sources of the gravitational field in Einstein's theory.}

\footnotetext[3]{Einstein's equations $G=8\pi T$ are expressed in terms of conceptually simple ideas, but the underlying equations that $G=8\pi T$ determine are fiercly complicated. A calculation shows that if every term in the definition of $G$ and $T$ were written down explicitly, without use of summation signs, it would take some 250 pages of standard text to simply write down the full set of partial differential equations that $G=8\pi T$ encodes.}

Selected publications

[2] The large time stability of sound waves (with R. Young), Comm. Math. Phys 179, 417-466 (1996).

[3] An astrophysical shock-wave solution of the Einstein equations (with J. Smoller), Phys. Rev. D, 51, 2733-2743 (1996).

[4] Shock-wave cosmology inside a black hole}, with J. Smoller, (to appear) Proceedings of the National Academy of Sciences, published online before print September 12, 2003, 10.1073/pnas.1833875100.

[5] Shock-wave solutions of the Einstein equations: Existence and consistency by a locally inertial Glimm scheme}, with J. Groah, (to appear) Memoirs of the AMS.

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