🕳 Schwarzschild Spacetime — Embedding Diagram & Time Dilation

A 2D cross-section of curved spacetime around a non-rotating black hole. The funnel is Flamm's paraboloid — a faithful embedding of the Schwarzschild spatial metric — and the curve below plots the real gravitational time-dilation factor √(1 − r_s/r). Drag the mass and observer-distance sliders to see the event horizon, photon sphere, and time dilation respond to the actual Schwarzschild formulas.

Schwarzschild radius r_s—

Photon sphere r_ph=1.5r_s—

Observer r—

Time dilation √(1−r_s/r)—

1 s here = —

Event horizon r=r_s Photon sphere r=1.5r_s Observer

Mass M = 10 M_☉ Observer distance r/r_s = 6.0

r_s = 2GM/c² | dτ/dt = √(1 − r_s/r) | z(r) = 2√(r_s(r − r_s)) | r_photon = 1.5 r_s

What you're looking at

The funnel in the top half of the canvas is Flamm's paraboloid: if you take a flat "slice" through Schwarzschild spacetime at one instant and embed its curved geometry into ordinary 3D space so distances look right, you get exactly this shape, z(r) = 2√(r_s(r − r_s)). It is not decoration — the funnel's steepness at a given r is the same spatial curvature that appears in the full metric.

The lower graph plots the gravitational time-dilation factor √(1 − r_s/r): a clock sitting at radius r ticks at this fraction of the rate of a clock infinitely far away. At r = r_s the factor hits zero — time stops relative to a distant observer, which is why the event horizon is a one-way door. Move the mass slider across ordinary stellar masses up to a supermassive black hole like Sagittarius A*, and drag the observer marker toward the horizon to watch both curves respond to the same real physics.