General Relativity I – Geometry

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 Euclidean and non-Euclidean 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.

2 comments on “General Relativity I – Geometry
  1. James C. Johnson says:

    I am enjoying you video series, but have a question about how to interpret the constant of proportionality in the gravitational field equation (8*PI*G)/c^4. This has units of seconds squared (s^2) per mass-kilogram (mkg). What does this mean?

    • David Butler says:

      A constant of proportionality is a measured number that takes the variables in the equation that are proportional to each other and converts the proportionality to an equation. A more familiar example is Newton’s gravitational constant. We start with the fact that the force on an object in a gravitational field is proportional the product of the two mass and inversely proportional to the distance between them squared. To turn this into an equation we introduce the constant. In this case G has the units of 6.674×10−11 with units (N⋅m^2/kg^2) that fit the equation (i.e. newtons on one side and mass squared over distance squared on the other.)

      Does this help any?

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