the Einstein field equations
/ INE-styne /
Why do planets orbit the Sun, why does time run slightly slower at sea level than on a mountain, and how do two black holes spiralling together send ripples across the universe? General relativity answers all three with one idea: mass and energy bend the geometry of spacetime itself, and things move along the straightest paths in that curved geometry. The Einstein field equations are the law connecting the two — they say exactly how matter curves spacetime. This is the deep end of PDEs.
Written compactly, the equations are G = (8 pi G / c^4) T, where G on the left is the Einstein tensor — a specific combination of curvatures (Ricci curvature and the metric) built from second derivatives of the unknown, the metric tensor g that encodes distances and times — and T on the right is the stress-energy tensor describing matter and energy. The unknown you solve for is the metric g itself: ten coupled functions of spacetime. Unpacked, this is a system of ten nonlinear second-order PDEs. They are not linear (curvature depends quadratically on the metric's derivatives), and in the right gauge they form a nonlinear hyperbolic system — wave-like, with a finite propagation speed: that finite speed is exactly why gravity travels at c and why gravitational waves exist and were detected.
These equations are spectacularly hard: the nonlinearity couples everything, and only a handful of exact solutions are known (Schwarzschild's black hole, the cosmological models, plane waves). Real predictions — colliding black holes, the early universe — require numerical relativity, solving the hyperbolic system on supercomputers. They mark the honest far frontier of this field: PDEs whose well-posedness was itself a hard theorem (Choquet-Bruhat), whose long-time behaviour and singularity formation are active research, and whose union with quantum mechanics remains unknown.
The 2015 LIGO detection of gravitational waves observed the spacetime ripple from two merging black holes — a direct confirmation of the wave-like (hyperbolic) character of the Einstein equations. Matching the observed waveform required years of numerical-relativity solutions of the full nonlinear system, since no closed-form merger solution exists.
Ten coupled nonlinear PDEs; hyperbolic, so gravity propagates at c and waves exist.
The compact 'G = 8 pi T' hides enormous complexity: it is ten equations, coordinate (gauge) freedom must be fixed before it is even a clean hyperbolic system, and most of its solutions are known only numerically. The elegance of the notation is not the simplicity of the mathematics.