Physic Labs

Theory of relativity

The Schwarzschild solution and black holes

Explore the Schwarzschild geometry outside a spherical mass: read the event horizon at rs=2GM/c2r_s=2GM/c^2 on the embedding funnel and observe how spacetime curvature and gravitational time dilation grow near the horizon.

Research

Equipment

  • 3D embedding-funnel canvas of the Schwarzschild spatial slice
  • Slider “Schwarzschild radius” (rₛ scale)
  • Slider “Tốc độ thời gian” (animation time speed)
  • “Tạm dừng” (pause) button; orbit drawn for illustration

Procedure

  1. Locate the horizon

    Rotate the funnel by dragging the canvas and find the throat of the surface: its rim marks r=rs=2GM/c2r=r_s=2GM/c^2, the event horizon. The embedding height is only a visual aid — physical space has four dimensions and cannot be drawn fully on a flat sheet.

  2. Change the mass scale

    Move the Schwarzschild-radius slider and watch the funnel throat widen or narrow. Since rsr_s is proportional to M, doubling the mass doubles the horizon size; note how the innermost stable circular orbit, at r=3rsr=3r_s, shifts outward with it.

  3. Estimate real horizons

    Pause the animation and compute by hand: for the Sun (M≈2×1030M\approx2\times10^{30} kg) rs≈3r_s\approx3 km, and for Earth only about 9 mm. Explain why the ordinary surfaces of these bodies lie far outside their rsr_s, so the Schwarzschild vacuum solution applies only outside them.

Simulation

Experiment history

Einstein published the field equations of general relativity in November 1915. Within weeks, Karl Schwarzschild — serving on the Russian front — found the first exact solution, the geometry outside any spherical nonrotating mass, and sent it to Einstein in early 1916. Schwarzschild died later that year, before the solution's deepest implication was understood. The radius rsr_s seemed a mere singularity puzzle until the 1960s, when the work of Oppenheimer and Snyder (1939) on gravitational collapse, the Kerr solution (1963), and John Wheeler's advocacy turned “black holes” — Wheeler's coinage from 1967 — into physical objects. Direct evidence arrived with gravitational waves from merging black holes detected by LIGO in 2015 and the Event Horizon Telescope images of 2019 and 2022.

Related physicists

Albert Einstein

Related library topics

The Schwarzschild solution and black holes