About Schwarzschild Geodesics
This simulation traces the paths of test particles and photons through the curved spacetime around a non-rotating, uncharged black hole described by the Schwarzschild metric. Each geodesic obeys the radial energy equation (dr/dτ)² = Ẽ² − V²eff(r), with the effective potential V²eff = (1 − Rs/r)(1 + L̃²/r²). The orbit is integrated step by step using a fourth-order Runge–Kutta scheme in geometric units (G = c = 1).
You choose a massive particle or a massless photon, then set the conserved angular momentum L̃, the energy Ẽ, and the black hole mass M with the sliders. Launch fires a new geodesic from its outer turning point, and a live inset plots V²eff(r) against the Ẽ² line. The dashed rings mark the photon sphere at 1.5Rs and the ISCO at 3Rs. The same physics explains Mercury's anomalous perihelion precession and the shadow imaged by the Event Horizon Telescope.
Frequently Asked Questions
What is a Schwarzschild geodesic?
A geodesic is the path that a freely falling object follows through curved spacetime. The Schwarzschild solution describes the spacetime around a spherical, non-rotating, uncharged mass, so a Schwarzschild geodesic is the orbit of a particle or light ray near such a black hole, determined entirely by spacetime curvature rather than any force.
What does the simulation actually compute?
It integrates the radial equation of motion derived from the Schwarzschild metric using the effective potential V²eff(r) = (1 − Rs/r)(1 + L̃²/r²). At each step it advances the radius r, the radial velocity dr/dτ, and the angle φ with a fourth-order Runge–Kutta method, then maps r and φ to screen coordinates to draw the orbit.
What do the sliders and buttons do?
The Massive and Photon buttons switch between a particle with rest mass and a massless light ray. The three sliders set the dimensionless angular momentum L̃, the energy Ẽ, and the black hole mass M. Launch starts a fresh orbit from its outer turning point, while Clear erases all stored trails.
What is the Schwarzschild radius Rs?
The Schwarzschild radius is Rs = 2GM/c², the location of the event horizon. In the simulation's geometric units Rs equals 2M, and the stats bar shows that for one solar mass it is about 2.95 km. Anything that crosses this radius cannot escape, not even light.
Why do the orbits precess instead of closing?
In Newtonian gravity a bound orbit is a perfect, closed ellipse. General relativity adds an extra term to the effective potential, so the angle swept between successive closest approaches is slightly more than 360 degrees. The result is a rosette: the orbit slowly rotates, which is exactly the perihelion precession Einstein first explained.
What are the ISCO and the photon sphere?
The ISCO, or innermost stable circular orbit, sits at r = 3Rs (6M); inside it no stable circular orbit for a massive particle exists. The photon sphere at r = 1.5Rs is the radius where light can orbit on an unstable circular path. Both are marked by dashed rings in the canvas.
What makes an orbit plunge into the black hole?
If the particle's energy Ẽ² exceeds the peak of the effective potential, there is no inner turning point to stop it, so the radius falls monotonically until it crosses the horizon. Lowering the angular momentum L̃ flattens that potential barrier, which is why low-L orbits plunge while high-L orbits stay bound.
What do L̃ and Ẽ mean physically?
Ẽ = E/mc² is the conserved energy per unit rest mass, and L̃ = L/mc is the conserved angular momentum per unit mass, both made dimensionless by the particle's mass and the speed of light. They are constants of the motion arising from the time and rotational symmetry of the Schwarzschild metric, which is why the whole orbit can be derived from just these two numbers plus M.
Is this simulation physically accurate?
The equations are the genuine Schwarzschild geodesic equations, integrated with a fourth-order Runge–Kutta scheme, so qualitative features such as precession, the ISCO, the photon sphere, and plunging trajectories are correct. It assumes motion in a single plane and uses geometric units; numerical step size limits precision very close to the horizon, so it is a teaching tool rather than a research-grade integrator.
How does this connect to real astrophysics?
The same effective potential governs Mercury's 43-arcsecond-per-century perihelion shift, the tight orbits of stars around the Milky Way's central black hole Sagittarius A*, and the bending of light that produces a black hole's shadow. The photon sphere directly shapes the bright ring seen in Event Horizon Telescope images of M87* and Sgr A*.
Why are rotation and charge ignored?
The Schwarzschild metric is the simplest exact black hole solution, valid only for a non-rotating, uncharged mass. Real black holes usually spin, which requires the more complex Kerr metric and introduces frame dragging and an ergosphere. This simulation deliberately uses Schwarzschild so the core ideas of geodesics, precession, and the ISCO stay clear.