GEOMETRY / MASS / AREA

A boundary.
A deeper geometry.

Turn a black hole’s mass into a shape you can explore. Follow the curved spatial slice to its horizon, then see the spherical area behind the Penrose mass bound.

INTERACTIVE 3D

FLAMM’S PARABOLOID

Space, in a single slice.

2D SPATIAL SLICE → 3D EMBEDDING

Exterior geometryHorizon at r = 2m

Drag to orbit · scroll / pinch to zoom

A constant-time equatorial spatial slice embedded in Euclidean 3D. The vertical direction illustrates distance geometry; it is not a literal funnel in space.

FOLLOW THE SUBSTITUTION

One example,
step by step.

Example check · not theorem verification

1. Convert physical mass to geometric length.

m = GM/c²

2. Compute the spherical horizon area.

A = 16πm²

3. Substitute that area into the lower bound.

√(A/16π) = √(m²) = m, since m > 0

4. Convert the bound back to solar masses.

M ≥ (c²/G)m = M

The analytic substitution proves equality for this established Schwarzschild example. The displayed numbers check that example with rounded readouts. The broader proposed theorem requires its manuscript assumptions and is not verified by this visualization. See exactly what is formalized ↗

Assumptions & coverage. The linked Lean scope concerns Cha–Khuri–Sakovich hyperboloidal initial data, end replacement preserving completeness and the dominant energy condition, and strictly future-timelike initial-data charge. It includes Schwarzschild equality examples. The general Bondi–Penrose inequality for possibly disconnected weakly trapped boundaries remains unformalized; a supporting connected-boundary inequality is conditional on an exterior Penrose inequality. This explorer checks a Schwarzschild equality example only. Consult the manuscript for its full hypotheses.

READ THE GEOMETRY

What the picture
actually means.

01 / THE SLICE

Curvature is about distances.

On a constant-time equatorial slice, radial distances are stretched relative to Euclidean space. The mesh represents that two-dimensional metric as a surface in three dimensions.

ds² = dr² / (1 - 2m/r) + r² dφ² z = +\sqrt[8m(r - 2m)], r > 2m

The rim at r = 2m is the limiting boundary. The extra embedding direction is not an additional physical dimension. Only one exterior sheet is shown.

02 / THE HORIZON

The area wraps a sphere.

The horizon has areal radius 2m. Its surface area is 4π(2m)² = 16πm². A disk drawn across the mesh’s rim would have only π(2m)² — one quarter of the spherical area.

A = 16πm² Double M \to double r s \to four times A

The sphere view is a separate representation of the horizon’s intrinsic area, not a sphere sitting inside the embedding surface.

03 / THE BOUND

Area sets a mass floor.

Under the relevant manuscript assumptions, the proposed Penrose inequality gives m ≥ √(A_min/16π), where A_min denotes the appropriate least enclosing area. In the Schwarzschild equality example, A_min = 16πm².

m \geq \sqrt(A min / 16π) M \geq (c²/G) \sqrt(A min / 16π)

Dividing all lengths by m makes every positive-mass Schwarzschild shape coincide. Changing mass changes its physical size.

An illustration, with a precise scope.

This is a classic Schwarzschild equality illustration associated with OpenAI’s proposed Penrose results. It does not verify the broader theorem, introduce a new gravity theory, or run a numerical-relativity simulation. The established Schwarzschild formulas are distinct from the proposed general result. The repository’s Lean scope note includes canonical Schwarzschild equality examples and states that the general Bondi–Penrose inequality remains unformalized.

A boundary.A deeper geometry.