Wormhole Embedding Diagram and Ray Deflection Lab: Ellis Throat, Geodesics, Traveller Clock and Tidal Forces (2D)

Presets

What is happening

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Live readouts

Graphs

Total sweep of the ray position angle against impact parameter (RK4 against elliptic integrals)

Throat profile r(l) for the chosen shape against the Ellis drainhole

Embedding profile: height z against radius rho for both universes

Magnitude of the (negative) energy density the geometry requires

Lateral tidal acceleration over 2 m along the ship path (green: 1 g)

Sky mapping: viewing angle in the cockpit against the sky angle it samples

Numerical checks

running...

How to use

Press nothing: the ship flies a trip from our universe (A) through the throat into the other one (B). The left view is the embedding diagram, the right view is what the pilot sees.
Drag in the cockpit to look around (dragging right moves the sky right); double-click to centre. Drag the diagram to orbit it, wheel to zoom. Buttons switch between the layouts.
Untick "Fly the trip automatically" and drag "Position along the trip" to stop anywhere. Near the throat the other universe fills a round window ringed by a lensed copy of our own sky. Turn on the sky grid to see the bending.
Raise the impact parameter p above 1: the ship no longer fits through, its geodesic turns back (white path) and the cockpit never shows the other side for long.
Change the throat shape n: longer neck, same radius. Watch the throat profile, the embedding curve and the energy density graphs change.
Lower the throat radius b or raise the speed: the lateral tidal acceleration over 2 m in the readout and in its graph exceeds 1 g (green line), and the trip would not be survivable.
Compare the sweep-angle graph: orange dots are the exact elliptic-integral values for the Ellis wormhole, the blue curve is the RK4 ray.

The science

What is simulated. A static, spherically symmetric, traversable wormhole of the Morris-Thorne kind with zero redshift (no horizon, no gravitational time dilation, so a ship's clock runs like a distant clock apart from its speed). In proper radial distance l (negative on the other side) the geometry is

ds² = -c²dt² + dl² + r(l)² (dθ² + sin²θ dφ²), r(l) = (bⁿ + |l|ⁿ)^1/n

The throat radius is b (slider, 10³ to 10¹⁰ m). For n = 2 this is the Ellis drainhole r = √(b² + l²) (Ellis 1973; Bronnikov 1973; Morris and Thorne 1988). Larger n keeps r close to b for longer, giving a longer, more cylindrical neck. The exponent n is a free modelling choice here, not something nature picks.

Embedding diagram. The equatorial plane (θ = π/2) 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, i.e.

dz/dl = √(1 - r'(l)²) Ellis: z = b asinh(l/b)

The surface shows only space (a slice at fixed time) and only one plane of it. It is a picture, not a tunnel through anything: the "upper" and "lower" sheets are two different universes (or two distant regions of one) joined at the throat, and no fourth spatial dimension is involved.

Light. Spherical symmetry makes every ray planar. With a conserved impact parameter L = r sinα the null geodesic obeys

dl/dλ = p_l, dφ/dλ = L / r², dp_l/dλ = L² r'(l) / r³, p_l² + L²/r² = 1

These are integrated with fourth-order Runge-Kutta in the page. A ray with L < b has no turning point and goes through the throat; a ray with L > b turns round at r = L and returns. So the throat looks like a round window of the other sky with a bright lensed copy of our own sky around it. For the Ellis case the total sweep of the position angle is known in closed form with the complete elliptic integral K: 2(L/b)K(L/b) for rays that cross and 2K(b/L) for rays that return (flat space would give π). The page checks the RK4 result against these.

Cockpit view. Because the problem is spherically symmetric, the sky direction a ray ends up on depends only on the angle α between the line of sight and the radial direction at the ship. The page tabulates that mapping once per ship position (a lookup table of thousands of RK4 rays; the 3D page does the lookup per pixel in a shader and the 2D page per pixel in JavaScript) and samples a sky from our universe or from the other one, according to where the ray ended. The two skies are made up (a blue-white sun and a Milky Way band in A, a red giant and orange dust in B); the lensing is real. The optional relativistic aberration uses the velocity transformation of a light ray, k = (k' + β_^[(γ-1)(k'·β_^) + γβ]) / (γ(1 + k'·β)); Doppler shifts and brightness changes are not drawn.

The traveller. A massive geodesic with Lorentz factor γ and angular momentum J = γ v p b (p is the impact parameter in throat radii): dp_l/dτ = J²r'/r³, dφ/dτ = J/r², p_l² + J²/r² = γ²v². Since g_tt = -1 the energy per unit mass dt/dτ = γ is constant, so the ship's clock ticks at 1/γ of the distant clock all the way (a straight dive from l = D reaches the throat after τ = D/(γ v) in the ship's time). With p > 1 the ship cannot cross: it swings round and comes back out on the same side.

Tidal forces. In this geometry the radial tidal stretch is exactly zero and the lateral tidal acceleration per metre for a ship at speed v is

a_lat = γ² v² (c²/b²) · r''(l)/r (radial dive; for oblique motion the mix of r''/r and (1 - r'²)/r² shown in the page)

For Ellis at the throat this is γ²v²c²/b². Morris and Thorne asked for less than 1 g over a 2 m body; for v = 0.05 c that needs b of about 10⁷ m, for v = 0.01 c about 2 × 10⁶ m. The readout shows the value for the chosen b and v along the whole path.

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

ε = c⁴ R⁽³⁾ / (16 π G b²) Ellis: ε = -c⁴ b² / (8 π G r⁴) < 0

and ε + p_r = -c⁴r''/(4πG r) < 0 wherever r'' > 0, a violation of the null energy condition. Integrated over the space the Ellis metric needs a total mass-energy of -(π/2) c² b / G, which is about -1 solar mass for b = 1 km (the page integrates it numerically and prints the value for your b). No known classical matter has negative energy density of this kind. Quantum effects (the Casimir effect) give small negative energy densities, but nothing close to what is required, and whether a stable wormhole can exist is open. Gravitational-wave and black-hole observations show no wormholes. This page is an exploration of a mathematical solution, not a claim that wormholes exist or can be built.

Honest simplifications. (1) Static geometry, perfectly symmetric; real perturbations would make a wormhole unstable (the Ellis solution is linearly unstable). (2) Only the equatorial plane is drawn in the 3D diagram, but the cockpit view is the full 3D sky. (3) The two skies are artificial, there is no accretion matter or dust inside the wormhole, and no Doppler or intensity shift. (4) The ship is a test body: its own mass does not change the geometry. (5) The Blender ship and station are scale-free decorations placed on the surface, drawn much larger than the real scale of the throat so that you can see them. (6) Rays are followed to a large distance (at least 60 b) and then continued as straight lines, which leaves an angular error of order b²/l²; the lookup table has a finite number of entries (the 3D page refines it when you stop moving).

Checks run in the page. Embedding height against b asinh(l/b); RK4 sweep angles against the elliptic-integral formulas for two crossing and two returning rays; conservation of the null condition; the critical impact parameter found by bisection equals b; finite-difference curvature r'' against the analytic formula; the tidal law against a second difference of the separation of two radial geodesics; the proper time of a straight dive; the energy constraint of the massive geodesic; and the total negative mass-energy against -(π/2)c²b/G.