A visual introduction to curved geometry for physicists

Abstract

This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build intuition for all types of Riemannian and Lorentzian manifolds of constant curvature. A visual derivation of the Thomas precession is given, showcasing the utility of differential geometry while also pointing a spotlight at certain intricacies of Minkowski space crucial from a pedagogical perspective. In addition, a straightforward method to generate some Carter-Penrose diagrams -- suitable for students with no differential geometry knowledge -- is presented, and a new method of indicating distortion on spacetime diagrams is shown.

Figures (19)

• Figure 1: When pulled taut, a length of string will necessarily pick the shortest distance between two points on a sphere. This is a geodesic.
• Figure 2: The construction of a geodesic (dashed line) between two points as the intersection of the plane spanning the origin and the two points with the sphere's surface. The thick line is the closest distance in the $E^3$ space, which also lies on this plane.
• Figure 3: A particularly simple geodesic path for parallel transport. We see that despite going back to the same place, our vector has rotated $\theta$ degrees, which is exactly the amount of meridians we crossed on the equator.
• Figure 4: Top: A strip of orange peel in the shape of a spherical triangle can be flattened on a surface, yielding three straight lines at right angles. Parallel transport around it becomes trivial, and is represented here by toothpicks inserted into the peel. Bottom: When overlaid back onto the surface, we see in spherical geometry our lines make a loop, and the transported vector appears to have rotated.
• Figure 5: When placed on a rotating Earth at a certain latitude, the Earth's rotation makes the laboratory perform an apparent movement in the inertial frame. The path is clearly non-geodesic, therefore we should expect a phase shift.