2005/02/09 by Rickard Jonsson
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Coordinate system #Coordinate time #Cosmology and Gravitation Theories #General relativity #Geodesic #Geometry #Gravitational field #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Metric tensor #Minkowski space #Noncommutative and Quantum Gravity Theories #Physics #Quantum mechanics #Spacetime #Special relativity #Theoretical physics #Theory of relativity #World line #gr-qc
paper · pdf · doi:10.1119/1.1830500
published as Am.J.Phys.73:248,2005 · 15 pages, 20 figures
openalex publication_date 2005/02/09 · arxiv created 2007/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities uμ. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a two-dimensional original metric, the absolute metric may be embedded in three-dimensional Euclidean space as a curved surface.