## In Memoriam: Abraham Haskel Taub

**September 10, 2001**

Abraham H. Taub, 1911-1999

Taub worked with Robertson on his doctoral thesis in mathematical cosmology and received his PhD in mathematics in 1935. He then moved to the newly established Institute for Advanced Study in Princeton, working as a postdoctoral fellow on differential geometry with Veblen. Taub's first published paper, with Veblen in 1934, was on the projective differentiation of spinors. Projective relativity theory had been developed by Veblen, Pauli, and others and was being intensively investigated at the time. The projective theories are essentially equivalent to the five-dimensional Kaluza-Klein theories. The work of Veblen and Taub was followed six months later by a paper of Taub, Veblen, and von Neumann on the Dirac equation in projective relativity.

In 1933, using a novel approach based on group theory, Robertson had summarized his work on relativistic cosmology in the *Reviews of Modern Physics*. Robertson's work and independent studies by A.G. Walker led to the spatially homogeneous and isotropic Robertson-Walker metric of standard cosmology. At about the same time, the general relativistic formulation of Dirac's equation had been developed by Weyl, Fock, Schrödinger, and Pauli, among others. In lectures delivered at Princeton University in 1934, Schrödinger discussed the Dirac equation in the Milne universe, and in 1935 Dirac published an equation for the electron in the de Sitter universe based on the embedding of de Sitter space in a flat five-dimensional space. Taub's third paper, published in 1937, discussed the Dirac equation in the spatially homogeneous cosmological spaces of Robertson. In this significant work [1], Taub showed that his results for de Sitter space were different from Dirac's.

By this time, Taub had joined the faculty at the University of Washington in Seattle, where he would be a professor of mathematics until 1948; there, he worked on various mathematical and physical aspects of spinors. His interest in general spinor fields, an area in which he made basic contributions, persisted throughout his academic career, and he returned to active research in this area after his retirement [2].

Taub's time in Seattle was interrupted by World War II: Called back to Princeton University to serve as a theoretical physicist from 1942 to 1945, he did research on shock waves in connection with national defense; in fact, he was the theoretician in an experimental group led by Walker Bleakney of Princeton. They worked on shock tubes, which provided a relatively simple means for studying blast waves. Taub developed the theory of the shock tube [3]. The reflection and refraction of shock waves, unlike the familiar linear case of light waves, give rise to the intrinsically nonlinear phenomena of Mach reflection and irregular refraction. The oblique reflection of shocks was discussed by von Neumann in 1943. Taub worked on the interaction of shock waves, the oblique refraction of plane shock waves, and the theory of Mach reflection of a plane shock from a rigid wall, especially the phenomena associated with the Mach stem. In Mach reflection---in contradistinction to regular reflection---the reflected shock meets the incident shock at a triple point (or line) that is some distance from the wall and is joined to it by a third shock wave (usually curved), called the Mach shock, or Mach stem. The nonlinear theory of this phenomenon is extremely complicated but crucial to an understanding of blast waves.

In 1946, Abe Taub received the Presidential Certificate of Merit for his defense-related work on shock waves. The main scientific results of this work were published in a series of papers in 1946-51. The theory was confronted with experiment in two papers, one with Bleakney (who originated the Mach-Zehnder interferometric method of studying shock diffraction) and the other with Fletcher and Bleakney, in the *Reviews of Modern Physics*. The excellent agreement of theory with experiment in most cases must have been a source of immense satisfaction for Taub, while the few cases of disagreement---such as the Mach reflection of weak shocks---pointed to the vast complexity of nonlinear dynamics.

In 1947, Taub returned to the Institute for Advanced Study as a Guggenheim postservice fellow. There, in addition to his research on shock waves, he worked on differential geometry and the groups of motions in Riemannian spaces. Among other accomplishments, he proved a theorem concerning the characterization of conformally flat spaces. Subsequently, he studied empty space-times that admit a three-parameter group of motions. Searching for a consistent formulation of Mach's principle in general relativity, he investigated, for the case of spatially homogeneous Ricci-flat spacetimes, the general solutions of Killing's equation for each of the nine types of transitive three-parameter continuous groups discussed by Bianchi. The three-dimensional Lie groups that are simply transitive on homogeneous 3-spaces had been classified by Bianchi in 1897. Taub recognized the significance of Bianchi's work for the construction of cosmological models. He presented this major work at the International Congress of Mathematicians in 1950, and it was published in the *Annals of Mathematics* in 1951 [4].

That paper has become a classic. It contains, among other things, an interesting Ricci-flat solution known as the Taub universe. At about the same time, Kurt Gödel independently constructed the first explicit spatially homogeneous expanding and rotating cosmological models with matter. The discoveries of Gödel and Taub have exerted a profound influence on the subsequent development of general relativity.

After the war, Taub made significant contributions to the relativistic theory of continua. Among his achievements were the first development of Hamilton's principle for a perfect fluid and other variational principles in general relativistic hydrodynamics, the circulation theorem, the relativistic Rankine-Hugoniot equations, and the stability of fluid motions in general relativity [5]. As an applied mathematician, he was the leading authority in relativistic hydrodynamics, and his work is indispensable in relativistic astrophysics.

In his investigation of the nonlinear phenomena of shock waves and relativistic hydrodynamics, Taub recognized the significance of numerical analysis. Inspired by von Neumann, he became a pioneer in computational hydrodynamics and computer science. He admired von Neumann's scientific genius and was the general editor of his collected works, published in six volumes in 1961-63.

In 1948, Abe Taub went to the University of Illinois as the chief mathematician associated with a project to build a computer based on von Neumann's plans. The computer, called ORDVAC, was completed in 1952 and delivered to the Aberdeen Proving Grounds. A subsequently built computer, ILLIAC, remained at Illinois and was the prototype for several other computers. Taub was head of the Digital Computer Laboratory at Illinois from 1961 until 1964, when he moved to the University of California, Berkeley, as director of the Computer Center (1964-68).

Taub was a professor of mathematics at Berkeley from 1964 until his retirement, in 1978. In 1980, on the occasion of his retirement, a collection of essays was published in his honor [6]. As professor emeritus of mathematics, he remained active in research until a few years before his death. He died on August 9, 1999, after a long illness.

During his distinguished career, Abe Taub had many postdoctoral associates and research students who have made significant contributions to computer science, applied mathematics, and general relativity theory. The Berkeley relativity seminars, which he organized, provided a lively environment for discussions of mathematical relativity and Lorentzian geometry. Taub was a member of a number of scientific societies and served on various advisory panels for applied mathematics. He edited the book *Studies in Applied Mathematics* (1971) and, with S. Fernbach, co-edited another, *Computers and Their Role in the Physical Sciences* (1970).

Abraham Taub is survived by his wife of 66 years, Cecilia Vaslow Taub, and their two daughters and one son, as well as one grandchild.

**Acknowledgments**The author is grateful to Robert Jantzen and Haskell Taub for much help and advice in the preparation of this brief biography.

**References**[1] A.H. Taub,

*Quantum equations in cosmological spaces*, Phys. Rev., 51 (1937), 512-525.

[2] A.H. Taub,

*Space-times admitting a covariantly constant spinor field*, Ann. Inst. Henri Poincaré, 41 (1984), 227-236.

[3] A.H. Taub,

*Determination of flows behind stationary and pseudo-stationary shocks*, Ann. Math., 62 (1955), 300-325.

[4] A.H. Taub,

*Empty space-times admitting a three parameter group of motions*, Ann. Math., 53 (1951), 472-490.

[5] A.H. Taub,

*Stability of general relativistic gaseous masses and variational principles*, Commun. Math. Phys., 15 (1969), 235-254.

[6]

*Essays in General Relativity*, F. J. Tipler, ed., Academic Press, New York, 1980.

*Bahram Mashhoon, Department of Physics and Astronomy, University of Missouri-Columbia. *