Geodesic Convergence: Gravity as Spacetime Curvature
Einstein revolutionized physics by recognizing that gravitational attraction results not from forces but from freely-falling objects following converging geodesics in curved spacetime, replacing Newton’s action-at-a-distance with geometric inevitability.
Parallel Transport Reveals Spacetime Curvature
Mathematicians and physicists studying differential geometry use parallel transport to distinguish curved surfaces from flat ones, revealing fundamental properties that govern gravitational effects in Einstein’s theory.
Christoffel Symbols: Derivatives of Basis Vectors
Named after Elwin Bruno Christoffel, these connection coefficients are calculated by physicists and mathematicians from the metric tensor to enable differentiation on curved manifolds where basis vectors themselves change from point to point.
Riemann Curvature Tensor: Complete Description of Geometry
Bernhard Riemann introduced this fundamental tensor in the 19th century, providing mathematicians and physicists with the definitive tool for characterizing curved spaces used throughout Einstein’s general relativity.
Ricci Tensor Measures Volume Changes in Curved Spaces
Gregorio Ricci-Curbastro developed this simplified curvature measure that Einstein later placed at the heart of his field equations, relating spacetime geometry directly to matter-energy distribution.
Ricci Scalar: Single Number Characterizing Average Curvature
Cosmologists and relativists use the Ricci scalar to characterize symmetric spaces and universe-scale geometries where curvature is uniform in all directions, simplifying the full tensor description to a single manageable number.
Minkowski Spacetime: Zero Curvature in Empty Space
Hermann Minkowski formulated this flat spacetime geometry underlying special relativity, providing the baseline against which general relativity measures gravitational curvature caused by matter and energy.
Spherical Geometry: Positive Curvature and Geodesic Convergence
Mathematicians studying spherical trigonometry and geographers mapping Earth’s surface first encountered positive curvature, later formalized through Riemann’s framework and applied to cosmological models of closed universes.