Wormhole Spacetime Lab (2D)

What is simulated. A static, spherically symmetric, traversable wormhole of the Morris-Thorne kind with zero tidal redshift (no horizon, no gravitational time dilation). In the proper-distance coordinate l, measured from the throat along a radial line, the space-time interval is

b is the throat radius (smallest sphere area radius), and n sets how sharply the neck flares: n = 2 is the Ellis drainhole r = √(b² + l²) (Ellis 1973, Bronnikov 1973, Morris and Thorne 1988); a large n keeps the neck almost cylindrical for longer. The condition for a throat that opens up again on both sides is the flare-out condition r'(0) = 0 and r'' > 0 there.

The embedding diagram. The equatorial plane of that geometry has the 2D metric dl² + r²dφ². It can be drawn as a surface of revolution in flat 3D space with radius ρ = r(l) and height z(l) when dz² + dρ² = dl²:

The page integrates this with Simpson's rule for every n. The picture shows only space (a slice at fixed time) and only one plane of it, so it is a map, not a tunnel through anything: the upper and lower sheets are two asymptotically flat regions, either two universes or two distant places in one, and the "extra dimension" in the drawing is not physical.

The shortcut. The mouth is the circle where r = R = K b (K is the "mouth radius / throat radius" slider); beyond it space is almost flat. The proper distance from throat to mouth is l_m = (Rⁿ - bⁿ)^1/n, so the way through is 2 l_m. The two mouths are also separated by a distance D through ordinary space (a free parameter: nothing in the metric fixes it), and the shortcut factor is D / (2 l_m). A probe at speed v = βc needs t = path / (βc) in the frame of the mouths and τ = t / γ on its own clock, γ = 1/√(1 - β²).

Tides. For a radial plunge the lateral tidal acceleration per metre of a probe at speed βc is

The page reports the largest value along the whole path for a 2 m body in units of g. Morris and Thorne asked for less than 1 g; the green line in the tide graph is that limit. A small throat or a fast probe crosses it quickly because the tide grows as (β / b)².

The matter it needs. Einstein's equations G = 8πG T/c⁴ say what matter such a geometry requires. For this ultrastatic space G_tt = R⁽³⁾/2 with R⁽³⁾ = -4r''/r + 2(1 - r'²)/r², so

The energy density is negative at the throat (Ellis) and ε + p_r is negative wherever r'' > 0: the null energy condition is violated, so the matter is "exotic". Integrated between the mouths, the Ellis geometry needs a mass-energy of -(c² b / G) atan(l_m/b), which tends to -πc²b/(2G), about -1 × 10⁴ solar masses for b = 10,000 km. No such matter is known to exist.

Honest simplifications. (1) The geometry is static and perfectly symmetric; real perturbations would make a wormhole unstable (Ellis is linearly unstable). (2) D is a free parameter, and the picture of a probe racing an "ordinary route" is a thought experiment, not a prediction. (3) The probe is a point on a radial geodesic of the equatorial slice; light bending, redshift of a moving observer and the energy needed to move the mouths are not simulated. (4) Throat radii of 10³ to 10⁷ km are chosen so the tide of a human-sized object can be small; the mathematics itself has no preferred scale.

Checks run in the page. (1) The numerical embedding height at l = 3b equals b asinh(3). (2) The arc length of the drawn embedding curve equals the proper length l_m for the current settings, which must hold because dz² + dρ² = dl². (3) The numerically integrated exotic mass-energy of the Ellis geometry equals -(c² b/G) atan(l/b). (4) The flare-out condition holds and the analytic r'' agrees with a finite difference of r'. The 2D and 3D pages use the identical model.js, so the same settings give the same numbers on both.