Gustav Herglotz

Biography

Work

• In 1904, Herglotz defined relations for the electrodynamic potential which are also valid in special relativity even before that theory was fully developed. Hermann Minkowski pointed out that the four-dimensional symmetry of electrodynamics is latently contained and mathematically applied in Herglotz' paper.
• In 1907, he became interested in the theory of earthquakes, and together with Emil Wiechert, he developed the Wiechert–Herglotz method for the determination of the velocity distribution of Earth's interior from the known propagation times of seismic waves. There, Herglotz solved a special integral equation of Abelian type.
• The Herglotz–Noether theorem stated by Herglotz and independently by Fritz Noether, was used by Herglotz to classify all possible forms of rotational motions satisfying Born rigidity. In the course of this work, Herglotz showed that the Lorentz transformations correspond to hyperbolic motions in, by which he classified the one-parameter Lorentz transformations into loxodromic, parabolic, elliptic, and hyperbolic groups.
• In 1911, he formulated the Herglotz representation theorem which concerns holomorphic functions f on the unit disk D, with Re f ≥ 0 and f = 1, represented as an integral over the boundary of D with respect to a probability measure μ. The theorem asserts that such a function exists if and only if there is a μ such that
• In 1911, he formulated a relativistic theory of elasticity. In the course of that work, he obtained the vector Lorentz transformation for arbitrary velocities.
• In 1916, he also contributed to general relativity. Independently of previous work by Hendrik Lorentz, he showed as to how the contracted Riemann tensor and the curvature invariant can be geometrically interpreted.

  Selected works

• Gesammelte Schriften / Gustav Herglotz, edited for d. Akad. d. Wiss. in Göttingen by Hans Schwerdtfeger. XL, 652 p., Vandenhoeck & Ruprecht, Göttingen 1979,.
• Vorlesungen über die Mechanik der Kontinua / G. Herglotz, prepared by R. B. Guenther and H. Schwerdtfeger, Teubner-Archiv zur Mathematik; vol. 3, 251 p.: 1 Ill., graph. Darst.; 22 cm, Teubner, Leipzig 1985.
• Über die analytische Fortsetzung des Potentials ins Innere der anziehenden Massen, Preisschriften der Fürstlichen Jablonowskischen Gesellschaft zu Leipzig, VII, 52 pages, with 18 Fig.; Teubner, Leipzig.
• Über das quadratische Reziprozitätsgesetz in imaginären quadratischen Zahlkörpern, Ber. über d. Verh. d. königl. sächs. Gesellsch. d. Wissensch. zu Leipzig, pp. 303–310.