Visualizing curved spacetime (Licentiate thesis)

In this thesis I show how one can circumvent the problem of negative distances, and visualize curved spacetimes after all. I do this using two completely different methods. I also re-derive an already existing method. Using the images of spacetime, one can explain how acceleration of particles here on Earth is caused by a curvature of spacetime rather than by a force. One can also, for instance, explain the gravitational slowing-down of clocks as a pure geometrical effect. Contents: 1 Introduction 1.1 This thesis 2 A geometrical introduction to gravitation 2.1 Spacetime, what is that? 2.2 Gravity and curved spacetime 2.3 Forces and the acceleration paradox 2.4 The full spacetime 2.5 Comments and conclusions 3 The absolute metric 3.1 The absolute line element 3.1.1 Black hole embedding 3.1.2 Comment regarding geodesics 3.2 Generalization to arbitrary spacetimes 3.2.1 A covariant approach 3.3 Freely falling observers as generators 3.4 On geodesics 3.4.1 Proof regarding geodesic generators 3.5 Covariant approach to photon geodesics 3.5.1 Photons in 1+1 dimensions 3.6 Photon geodesics in static spacetime 3.6.1 The reference freefaller coordinates 3.6.2 The absolute metric 3.6.3 Equations of motion in a general 1+1, time independent metric 3.6.4 Outmoving photons 3.6.5 Embeddings 3.7 Flat embeddings 3.7.1 Comments 3.8 Spacelike generators? 3.9 A mathematical remark 3.10 Comments 4 Metrics, geodesics and a!ne connections 4.1 Finding the metric from the geodesics 4.1.1 Coordinate curvature 4.1.2 The geodesic equation using a coordinate a"ne parameter 4.1.3 Equivalent a"ne connections 4.2 On the construction of the dual metric 4.2.1 Point-dual metrics 4.3 On the dual metric in freely falling coordinates 4.3.1 Finding the coordinate transformation to the freely falling coor- dinates 4.3.2 The dual metric in the freely falling coordinates 5 On whether the interior dual metric is a sphere 5.1 Conditions for spheres 5.2 The dual interior metric 5.3 Approximative internal sphere 5.4 Spheres in the Newtonian limit 6 The Epstein-Berg way 6.1 The main philosophy 6.1.1 Particle trajectories and mappings 6.1.2 Intuition about geodesics 6.1.3 Mathematics about geodesics 6.2 Embeddings 6.2.1 The Berg dynamical view 6.3 The Epstein internal space 6.3.1 Verification using the dual scheme 6.3.2 Comments 6.4 Comments Paper I Embedding spacetime via a geodesically equivalent metric of Euclidean signature