2001/07/01 by Rickard Jonsson · 1 citation
Mathematics · Physics and Astronomy · #Embedding #Euclidean geometry #Experimental and Theoretical Physics Studies #Geodesic #Gravitational field #Mathematics and Applications #Metric (unit) #Relativity and Gravitational Theory #Schwarzschild geodesics #Schwarzschild metric #Schwarzschild radius #Spacetime #World line #gr-qc
paper · pdf · doi:10.1023/a:1012037418513
published as Gen.Rel.Grav.33:1207,2001 · 28 pages, 17 figures. As compared to the published version there are corrections to Eqs. 46-49 (no impact on the discussion) and minor cosmetical updates in the figures. An addendum is also included, 5 pages, 2 figures
openalex publication_date 2001/07/01 · arxiv created 2007/08/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path. Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime. Freedom in the dual metric allows us to display, with substantial curvature, even the weak gravity of our Earth. This may provide a nice pedagogical tool for elementary lectures on general relativity. I also study extensions of the dual metric scheme to higher dimensions. In an addendum I extend the analysis concerning the shape of an embedding of the dual spacetime of a line through a planet of constant proper density.