General Relativity and Gravitation

, Volume 33, Issue 7, pp 1207–1235

Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature

  • Rickard Jonsson

DOI: 10.1023/A:1012037418513

Cite this article as:
Jonsson, R. General Relativity and Gravitation (2001) 33: 1207. doi: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.

Embedding spacetime dual metric geodesics signature change 

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Rickard Jonsson
    • 1
  1. 1.Department of Astronomy and AstrophysicsChalmers University of TechnologyGöteborgSweden

Personalised recommendations