Embedding spacetime via a geodesically equivalent metric of Euclidean signature

Details Embedding spacetime via a geodesically equivalent metric of Euclidean signature Embedding spacetime via a geodesically equivalent metric of Euclidean signature

2007-08-17

arxiv.org/abs/0708.2381v1

Gen.Rel.Grav.33:1207,2001

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

28 pages, 17 figures. As compared to the published version there are corrections to Eqs. 46-49 (no impact on the discussion) a

Scientific paper

10.1023/A:1012037418513

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.

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