Embedding spacetime via a geodesically equivalent metric of Euclidean signature (Q5955542)

The author considers a two-dimensional Lorentzian metric of the form \(ds^2 = a(x) dr^2 + c(x) dt^2\). He constructs a Riemannian 2-dimensional metric (called dual metric) which has the same geodesic as \(ds^2\). Realizing this dual metric on some surface in the Euclidean 3-space, one can visualize the movement of free particles in the initial Lorentzian 2-space. The dual metric is not unique. The author considers several examples and shows that the construction of a dual metric gives a useful pedagogical tool for elementary lectures on general relativity. A problem of construction of a dual metric for 3-dimensional Lorentzian metric is discussed. In general there is no dual metric even for a diagonal time-independent Lorentzian 3-metric. The author proposes to consider a ``partially dual metric'' which gives a model for geodesics with a fixed energy.