Lorentzian

Lorentzian is an adjective derived from the Dutch physicist Hendrik Antoon Lorentz and is used across mathematics, physics, and statistics to describe several related concepts. It is most often associated with transformations, geometric structures, and probability distributions connected to Lorentz’s work on relativity and electromagnetism.

In special relativity, Lorentz transformations are linear changes of coordinates between inertial frames that preserve the spacetime interval in Minkowski space. They include boosts along spatial directions and rotations, and together with translations form the Lorentz–Poincaré group. A boost with velocity v involves the Lorentz factor γ = 1/√(1−v²/c²) and reduces to the Galilean transformation at low speeds.

In geometry, a Lorentzian metric is a nondegenerate symmetric bilinear form on a smooth manifold with signature (−,+,+,+) (or its opposite). A manifold equipped with such a metric is called a Lorentzian manifold and serves as the mathematical setting of general relativity, allowing classification of intervals as timelike, spacelike, or null, and enabling the study of curvature via the Einstein field equations.

In probability and signal analysis, the Lorentzian distribution (also called the Cauchy distribution) has probability density f(x; x0, γ) = (1/π) γ / ((x−x0)² + γ²). It is symmetric and heavy-tailed; moments do not exist in the usual sense. It arises in spectroscopy to model natural linewidths and in impulse response contexts, where it differs from the Gaussian.

The term also appears in other contexts as Lorentzian line shapes in optics and physics, reflecting the influence of Lorentz’s work on the behavior of resonances and spectral broadening.