This lab visualizes the exterior equatorial spatial geometry of a non-rotating black hole and places a clock on a stable circular orbit. It contrasts the coordinate radius with the proper radial distance from the Schwarzschild radius R_s = 2GM/c², and shows how an orbiting clock runs slower than a distant one.
• A Schwarzschild radius boundary, a Flamm spatial embedding bowl, a circular geodesic orbit, a proper radial distance path and an orbiting clock comparison. • Sliders for central mass (1 to 100 solar masses) and orbit radius in units of R_s (4 to 10), plus toggles for the spatial grid and explanatory markers. • Readouts: Schwarzschild radius in km, coordinate orbital period in ms, proper radial distance and coordinate radial span from R_s, orbiting dτ/dt and embedding height. • Experiments: at fixed r/R_s lengths and period scale with mass; a closer stable orbit accumulates less proper time per coordinate second.
The embedding height is w/R_s = 2√(r̄ − 1) with r̄ = r/R_s, and the proper radial distance is ℓ/R_s = √[r̄(r̄ − 1)] + ln(√r̄ + √(r̄ − 1)). The coordinate period is T = 2π√(r³/GM) and the orbiting clock rate is dτ/dt = √(1 − 3R_s/(2r)). Proper distance exceeds coordinate distance because space is curved.
The model covers a non-rotating, uncharged Schwarzschild exterior. The bowl is an embedding of space, not gravity acting in an extra physical direction, and it is not a full spacetime diagram. Only stable circular test-particle orbits are offered, with no inspiral or gravitational radiation. One orbit takes 8 animation seconds; the physical period is reported separately.
No. It is an embedding device that lets a curved two-dimensional space be drawn in three dimensions. It does not mean gravity acts in an extra physical dimension.
No. The orbit radius slider starts at 4 R_s so only stable circular orbits are shown. The innermost stable circular orbit of Schwarzschild is at 3 R_s, and unstable orbits are outside the model.
At a fixed r/R_s, physical lengths and the orbital period scale with the mass, while dimensionless ratios such as the clock rate stay the same.
It combines gravitational potential and orbital motion, giving dτ/dt = √(1 − 3R_s/(2r)). A closer orbit gives a smaller value, meaning fewer proper seconds per coordinate second.