Einstein–Rosen Bridge
Shape · (x, y, z) = f(u, v, t)
((a +(v − π)²/(4 · a)) · cos(u), v − π, (a +(v − π)²/(4 · a)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
The space around a black hole at one instant, drawn as a surface. The distance from the axis is a + (v − π)²/(4a), and the height is v − π, from −π to π. The throat radius a is on a slider at 1.
Solved for the height, that is ±2√(a(r − a)): a parabola on its side, turned about the vertical axis. At the throat the surface is vertical. It leans to 45° where r = 2a, and flattens slowly outward, reaching 3.47 from the axis at the edges.
What the shape means
Einstein's equations for a single mass give the Schwarzschild geometry. Its space at one moment is curved, and drawing it as a surface in an imagined extra dimension shows how. Only the distances measured along the surface are real; the height is not a direction in space.
The throat is at the Schwarzschild radius, a = 2GM/c²: 2.95 km for the Sun, 8.9 mm for the Earth. Walking out along the surface from the throat to the edge here covers 4.16, although the edge is only 2.47 further from the axis. That extra length is the curvature.
The top sheet is our universe and the bottom one a second, joined through the throat. Robert Fuller and John Wheeler showed in 1962 that the throat pinches shut faster than light can cross it, so nothing gets through.
History
Ludwig Flamm drew this surface in 1916, the year after general relativity, and it is often called Flamm's paraboloid. Albert Einstein and Nathan Rosen described the bridge between the two sheets in 1935. Michael Morris and Kip Thorne worked out in 1988 what a wormhole that can be crossed would need.
Try
- Replace a + (v − π)²/(4a) with a cosh((v − π)/a) in both x and z. This is the Ellis wormhole, one that can be crossed: its surface is a catenoid, the same as at the middle of Helicoid to Catenoid.
- Drag a to 2. The throat doubles, and the edges reach only 3.23 from the axis, since a wider throat flares more slowly.
- Set Span of v (×π) to 1. Only the lower sheet is drawn, ending at the throat.
- Press Front in the deck to see the parabola that the surface is turned from.