Visualizing curved spacetime

In this thesis I derive two fundamentally different methods of visualizing curved space-times, the dual and the absolute. The images are nice pedagogical tools for teaching General Relativity. I also review the concept of space-eigentimes. In the dual scheme, I start from the equations of motion in a 1+1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element. I then find an-other metric, with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent dual metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles take the shortest paths. Thus, one can visualize how acceleration in a gravitational field is explained by particles moving on straight lines in a curved spacetime. In the absolute scheme, I start from an arbitrary Lorentzian spacetime with a given field of timelike four-velocities uµ. I then do a coordinate transformation to the local Minkowski system comoving with the given four-velocity at every point. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new absolute metric can be covariantly related to the original one. This method is particularly well suited for visualizing gravitational time dilation and the horizon of Black Holes.