Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature
- 95 Downloads
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.
Unable to display preview. Download preview PDF.
- 1.Marolf, D. (1999). Gen. Rel. Grav. 31, 919.Google Scholar
- 2.Epstein, L. C. (1994). Relativity Visualized, (Insight Press, San Fransisco), ch. 10,11,12.Google Scholar
- 3.Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (John Wiley & Sons, U.S.A.), p. 77.Google Scholar
- 4.Kristiansson, S., Sonego, S., and Abramowicz, M. A. (1998). Gen. Rel. Grav. 30, 275.Google Scholar
- 5.D'Inverno, R. (1998). Introducing Einsteins Relativity, (Oxford University Press, Oxford), p. 99–101.Google Scholar