The Maths of General Relativity (7/8) - The Einstein equation

Physicists employ the energy-momentum tensor as the single mathematical tool describing all matter, energy, and their motion within the universe, from galaxies to elementary particles.

Albert Einstein formulated this equation as the centerpiece of general relativity, relating spacetime geometry to matter-energy content. Physicists use it to predict gravitational phenomena from planetary orbits to black hole formation.

Mathematicians Gregorio Ricci-Curbastro and his student Tullio Levi-Civita developed these mathematical tools decades before Einstein recognized their physical significance for describing gravitational curvature.

Physicists and mathematicians struggle to find exact solutions to Einstein’s equation except in highly symmetric cases. Computational physicists develop sophisticated numerical methods and supercomputer simulations to approximate solutions for realistic scenarios.

Physicists and cosmologists confront this fundamental relationship when solving gravitational problems, recognizing that geometry and matter cannot be determined independently but must be solved together self-consistently.

Theoretical physicists discovered a mathematically equivalent formulation of Einstein’s equation by exploiting a symmetry property specific to four-dimensional spacetime, providing alternative problem-solving approaches.

Karl Schwarzschild found this exact solution to Einstein’s equation in 1916, remarkably while serving on the Russian front during World War I. Astrophysicists apply it to model spacetime around stars, planets, and nonrotating black holes.

Karl Schwarzschild’s solution predicted this critical radius, though its full significance for black hole physics wasn’t understood until decades later. Astrophysicists now use it to identify black hole event horizons throughout the universe.