Classroom Aid - Non-Euclidean Geometry
Channel: David Butler
Date: 2023-02-10

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In this segment of the “How Fast Is It” video book, we cover the geometry of general relativity. We start with the Elevator Thought Experiment, and show how it represents a gravitational field and how it predicts the bending of light. This sets the stage for the Equivalence Principle. This leads to the reconciliation of Newton’s two definitions for mass. Which, in turn, leads to the idea that the existence of a mass bends space. To understand the bending of space, we cover the basics of Euclidian and non-Euclidian Riemann geometry. We include spherical and hyperbolic geometries along with the nature of their respective geodesics. We actually measure geodesic deviation above the Earth. For a fuller understanding, we cover the definition of metrics and curvature in terms of tensors. With the general Riemannian Curvature Tensor in hand, we find the subsets that reflect the behavior of space within a volume. We then cover how Einstein mapped this geometry to space-time to produce the Einstein Curvature Tensor. And finally, we describe the Energy-Momentum tensor that identifies the nature of a volume of matter-energy, which is the source of the space-time curvature. Setting these equal to each other with an appropriate conversion factor gives us Einstein’s general relativity field equations.