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Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature
2001
General Relativity and Gravitation
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
doi:10.1023/a:1012037418513
fatcat:6m3iv4wxvfgxveyiqzuvxzbgqa