Schwarzschild Innermost Orbit Stability
Circular orbits can exist yet be unstable, and the last stable orbit around a nonrotating black hole lies outside its horizon.
Free theory articleCircular orbits can exist yet be unstable, and the last stable orbit around a nonrotating black hole lies outside its horizon.
Concept & interpretation
In general relativity, gravity is modeled as the curvature of spacetime. For a non-rotating, uncharged black hole, the spacetime geometry is described by the Schwarzschild metric. Test particles follow geodesics in this curved spacetime. One intriguing consequence is that circular orbits can exist but might be unstable if they are too close to the event horizon, even outside it.
Interactive Experiment
Use the interactive simulation to explore circular geodesics and the effective potential in the Schwarzschild metric.
- Select the Circular radius to set the orbit of the test particle. Notice how the shape of the effective potential curve and the position of the stationary point change.
- Adjust the Potential probe radius to evaluate the effective potential V(x) at different distances for the current orbit's angular momentum.
- Observe the calculated values for Specific energy and Coordinate radial frequency squared. Verify that when the radius is below 6, the frequency squared becomes negative, indicating an unstable orbit.
Try it out in the module: Schwarzschild Innermost Orbit Stability.