Herglotz worked in the fields of
seismology,
number theory,
celestial mechanics, theory of
electrons,
special relativity,
general relativity,
hydrodynamics,
refraction theory. • 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 (during a conversation reported by
Arnold Sommerfeld) 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 (an inverse problem). There, Herglotz solved a special integral equation of Abelian type. • The
Herglotz–Noether theorem stated by Herglotz (1909) and independently by
Fritz Noether (1909), 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 R_3, by which he classified the one-parameter Lorentz transformations into loxodromic, parabolic, elliptic, and hyperbolic groups (see Möbius transformation#Lorentz transformation). • In 1911, he formulated the
Herglotz representation theorem which concerns
holomorphic functions
f on the
unit disk D, with Re
f ≥ 0 and
f(0) = 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 :: \forall z \in D \ \ f (z) \ = \ \int_{\partial D} \frac{\lambda + z}{\lambda - z}\ d\mu(\lambda). : The theorem also asserts that the probability measure is unique to
f. • In 1911, he formulated a relativistic
theory of elasticity. In the course of that work, he obtained the
vector Lorentz transformation for arbitrary velocities (see History of Lorentz transformations#Herglotz (1911)). he also contributed to
general relativity. Independently of previous work by
Hendrik Lorentz (1916), he showed as to how the contracted
Riemann tensor and the
curvature invariant can be geometrically interpreted. ==Selected works==