# 3: Schwarzschild Orbits


• 3.1: The Geodesic Equation
The previous chapter dealt with the rules of geometry in Schwarzschild spacetime. If we want to look at motion, we need to look beyond the metric to something called the geodesic equation.
• 3.2: Constants of Motion
By applying the geodesic equation to the Schwarzschild metric, we can see what an inertial worldline looks like.
• 3.3: Effective Potential
An effective potential energy is not technically a potential energy but can still be used to obtain qualitative information about motion.
• 3.4: Local Inertial Reference Frames
The constants of motion E and L from a previous section are expressed in global coordinates, which means that they don't correspond to anything directly measurable. To turn them into something measurable, we have to use a global-to-local coordinate transformation.
• 3.5: Inside the Black Hole
To see what happens inside of a black hole, we need to use a metric other than the Schwarzschild metric since the Schwarzschild metric has problems at r=2M . There is actually another spherically symmetric solution to the Einstein Field Equations called the Global Rain metric. The word "rain" is a reference to the fact that the T-coordinate is measured by steadily infalling clocks.

Thumbnail: The overall geometry of the universe is determined by whether the Omega cosmological parameter is less than, equal to or greater than 1. Shown from top to bottom are a closed universe with positive curvature, a hyperbolic universe with negative curvature and a flat universe with zero curvature. (Public Domain; NASA).

This page titled 3: Schwarzschild Orbits is shared under a not declared license and was authored, remixed, and/or curated by Evan Halstead.