Relativity ★★★ Advanced

🕳 Schwarzschild Geodesics

Orbits around a non-rotating black hole in Schwarzschild spacetime. Watch GR perihelion precession, find the photon sphere (r = 1.5rs), the innermost stable circular orbit ISCO (r = 3rs), and launch plunging geodesics that cross the event horizon.

r/rs = — v/c = — Precession = —°/orbit Orbit type: —
(dr/dφ)² = r⁴/L̃² [Ẽ² − (1−rs/r)(1 + L̃²/r²)]  |  rphot = 1.5rs  |  rISCO = 3rs

Schwarzschild Spacetime & Geodesics

In flat space (Newton) orbits are perfect ellipses that never precess. In Schwarzschild spacetime the effective potential (per unit mass) is Veff(r) = (1−rs/r)(1 + L̃²/r²), adding a GR term −rsL̃²/r³ that makes close orbits unstable.

Key radii (rs = Schwarzschild radius = 2GM/c²):

  • r = rs — event horizon; nothing can escape
  • r = 1.5 rs — photon sphere (unstable circular light orbit)
  • r = 3 rs — ISCO: innermost stable circular orbit
  • r > 3 rs — stable circular orbits exist

Mercury's famous perihelion precession of 43 arcsec/century is this same GR correction applied to the Sun's weak gravity field.

About this simulation

In flat Newtonian space, orbits are perfect stationary ellipses. Around a Schwarzschild black hole, the geodesic equation (dr/dφ)² = r⁴/L̃²·[Ẽ² − (1−rs/r)(1+L̃²/r²)] adds a general-relativistic correction term that makes close orbits precess and can even fling infalling matter past the point of no return. This simulation numerically integrates that equation with RK4 stepping in r(φ) for a given conserved energy Ẽ and angular momentum L̃/rs, letting you launch a particle from any starting radius r0/rs and watch whether it settles into a stable rosette, a razor-thin photon orbit, or a one-way plunge through the event horizon.

🔬 What it shows

A curve traces a test particle's trajectory in polar coordinates (r, φ) around a non-rotating black hole of radius rs. Dashed reference circles mark the event horizon (r=rs), the photon sphere (r=1.5rs) where light itself can circle the hole, and the ISCO (r=3rs), the innermost radius where a stable circular orbit is possible.

🎮 How to use

Drag the Energy Ẽ, Angular Momentum L̃/rs, and r0/rs sliders, or jump straight to one of five presets — Circular ISCO, Precessing Ellipse, Photon Orbit, Plunging, Escape — then hit Launch. The stats bar reports the current radius, precession per orbit, and orbit classification; Clear resets the canvas.

💡 Did you know?

Mercury's real perihelion precesses by 43 arcseconds per century purely because of this same GR correction term applied to the Sun's much weaker gravity — the calculation that convinced physicists general relativity was right.

Frequently asked questions

What is the Schwarzschild radius (rs)?

It's the radius rs = 2GM/c² at which the escape velocity from a non-rotating mass equals the speed of light, defining the event horizon of a Schwarzschild black hole. Nothing, not even light, can escape from inside r = rs once it crosses.

What is the photon sphere?

The photon sphere sits at r = 1.5 rs, where light itself can, in principle, orbit the black hole in an unstable circular path. Any perturbation sends a photon either spiraling in or escaping outward, which is why it appears as a sharp dashed ring in the simulation rather than a place you can safely park an orbit.

What does ISCO mean and why does it matter?

ISCO stands for innermost stable circular orbit, located at r = 3 rs for a Schwarzschild black hole. Inside this radius, circular orbits become dynamically unstable and any inward perturbation causes the orbiting body to spiral into the horizon — which is why accretion disks around real black holes have an inner edge near this radius.

Why do the orbits precess instead of forming closed ellipses?

In Newtonian gravity a bound orbit is a fixed ellipse that never rotates. General relativity adds an extra 1/r³ term to the effective potential, so each orbit's perihelion advances by a small angle every revolution — the "Precessing Ellipse" preset makes this rosette pattern easy to see.

What happens in the "Plunging" and "Escape" presets?

The Plunging preset starts a particle with too little angular momentum to resist gravity at its radius, so it spirals inward and crosses the event horizon at r = rs. The Escape preset gives the particle enough energy (Ẽ ≥ 1) to overcome the black hole's pull entirely and fly off to large r, never to return.