One equation for all of gravity
In 1915 Einstein replaced "gravity is a force" with "gravity is geometry". This single tensor equation contains falling apples, GPS corrections, black holes, gravitational waves and the expansion of the universe. Hover each piece.
"Spacetime tells matter how to move; matter tells spacetime how to curve."— John Archibald Wheeler
Bend the grid
Drag the slider: the grid is squeezed toward the mass — gently far away, dramatically close in. The shape here is artistic shorthand; the true equatorial geometry of Schwarzschild is the funnel under ②.
Solving the equation
"Solving" Einstein's equation means finding the metric gμν — the function that tells you what every clock and ruler does. Exact solutions are rare and precious. This explorer shows you the four most famous ones. ↓
Schwarzschild — one number bends everything
Outside any non-spinning spherical mass, the geometry is fixed by mass alone. The same formula covers the Earth (barely curved), the Sun (light grazing it bends by 1.75 arcseconds — the 1919 measurement that made Einstein famous) and black holes.
Launch a planet
Effective potential (the physicist's x-ray)
The orbit "rolls" in this landscape. Newton's valley extends forever; Einstein's has a cliff at small r — the extra −1/r³ term. Fall off the cliff and no force in physics brings you back.
Make it real — choose a mass
Kerr — the black hole that drags spacetime around
Real stars spin, so real black holes spin. Spin doesn't just rotate the hole — it rotates spacetime itself. Near a fast Kerr hole you physically cannot stand still, and matter falling in can release up to 42% of its rest-mass energy. That is why quasars outshine galaxies.
Spin it up
a/M = Jc/GM² — in plain words: how close to the maximum possible spin. At a/M = 1 the two horizons merge into a single extremal one; push beyond and it dissolves entirely, baring the ring singularity. Nature seems to stop just short (the Thorne limit, ≈0.998, set by the disk photons the hole swallows).
Mass anchor (for real numbers)
Reissner–Nordström — give the hole an electric charge
A thought experiment more than an astrophysical object (real plasma neutralises charge almost instantly) — but a profound one. Charge adds a repulsive term to the geometry, splits the horizon in two, and pushed too far it threatens to expose the singularity itself.
Charge it
Why two horizons?
Gravity pulls like −2M/r, charge pushes back like +Q²/r². Near the centre the repulsion wins (1/r² beats 1/r), so after the point of no return there's a second surface where infall stops being compulsory. Between them: the shaded no-return zone.
FLRW — the universe as a solution
Assume only that the cosmos is, on huge scales, the same everywhere and in every direction. Einstein's equation then collapses into one line — the Friedmann equation — whose solution a(t) is the size-history of the universe. Galaxies don't fly through space; space itself stretches between them.
Build a universe
Radiation Ωr = 8.5×10⁻⁵ included automatically (it ran the show for the first ~50,000 years). Curvature Ωk = 1 − ΣΩ.
Choptuik — the critical edge of collapse
Not a closed-form metric, but a discovery (Matthew Choptuik, 1993) and the reason this whole tool exists. Throw a pulse of energy inward: too weak and it disperses; too strong and it forms a black hole. Tune it to the knife-edge between the two and a stunning universal structure appears — the same numbers for every kind of pulse. It is a phase transition, for gravity itself.
Throw a pulse inward
The two universal numbers
γ is the reciprocal of the single growing mode of the critical solution (1/γ ≈ 2.67) — that's why it forgets which pulse you threw in. Universality: a critical phenomenon, mathematically kin to a magnet at its Curie point.
The shape of space — in 3D
"Adding a dimension": take a real 2D slice of a curved spacetime and bend it into 3D so that distances along the surface equal the true distances a ruler would measure. Drag to rotate. These embedding diagrams are exact — but read the honest caveat: they show curved space, not curved time, and most of what we feel as gravity lives in the time part.
Controls · drag the surface to rotate
Ripples in spacetime — curvature as a field in time
The full 4D picture you asked for: at every point of space, a vector shows which way curvature is pushing two nearby free-floating masses — and as time passes, the vectors change. This is a gravitational wave sweeping through a 3D lattice of test masses (exactly what LIGO detected in 2015). The arrows are the genuine geodesic-deviation field a = ½ ḧ·ξ — a piece of the Riemann curvature tensor, animated.
Polarization & frequency
Reality check honest
Real strains are ~10⁻²¹. LIGO's 4-km arms move about 10⁻¹⁸ m — a thousandth of a proton's width. A 1-metre ring would shift ~10⁻²² m, a millionth of a proton. The pattern, timing and direction of the arrows are exact; only the size is blown up ~10²⁰× so you can see anything at all.