Library of human understanding
Gravity shapes the universe at every scale, from falling apples to colliding black holes.
Every scale, live: from a wavefunction to a pendulum →
Gravity shapes the universe: from falling apples to black holes, the same fundamental law governs how mass bends spacetime.
This simulator shows a Schwarzschild black hole with mass ~10 M☉. The black disk is the event horizon; within it, not even light escapes. The orange circle shows the photon sphere where light orbits unstably. The red-yellow accretion disk glows as matter spirals inward, converting gravitational potential energy to heat. Notice how starlight bends around the black hole—this is gravitational lensing. The green text shows time dilation: near the photon sphere, time runs much slower than far away.
Superposition is a core quantum principle: a quantum system can exist in multiple states simultaneously until measured. A qubit on the Bloch sphere occupies a point in superposition; measurement collapses it to |0⟩ or |1⟩. The double-slit experiment shows wave-particle duality: without detection, particles behave like waves and interfere with themselves; with detection determining which slit they pass through, they behave like particles with no interference. Superposition isn’t the system “trying all answers”—it’s a coherent state where all possibilities coexist as a single quantum state until observation.
A quantum system exists in multiple states simultaneously until measured: α|0⟩ + β|1⟩ as a single coherent entity. Measurement collapses the superposition, forcing a choice.
The Schrödinger equation (iℏ∂ψ/∂t = Hψ) governs quantum dynamics. In a potential well, the wavefunction ψ(x,t) forms standing waves whose wavelengths fit precisely into the box. Only certain energies are allowed—this is quantization. The probability of finding a particle at position x is |ψ(x)|². Bound states oscillate in time but maintain constant probability density: they are stationary states (energy eigenstates).
The Schrödinger equation (iℏ∂ψ/∂t = Hψ) governs how the wavefunction evolves. In confined systems, only certain energies are allowed—energy quantization that explains discrete atomic levels. Stationary states remain stable indefinitely, their phase oscillating but probability unchanged.
The Dirac equation unified quantum mechanics and special relativity, predicting that electrons have spin-½ and that negative-energy solutions represent antiparticles. The positron, the electron's antimatter counterpart, was predicted by Dirac's mathematics before being experimentally discovered. This visualization shows how the energy spectrum has both positive (electron) and negative (positron) branches, separated by 2mc² where m is the electron mass.
Dirac unified quantum mechanics and relativity. The equation predicts both electron and positron solutions—opposite charges, same mass. Antiparticles were confirmed experimentally in 1932, launching quantum field theory.
The Pauli matrices σ_x, σ_y, σ_z form the generators of SU(2) rotations and represent spin operators. Each matrix acts on a two-component spinor, rotating or measuring spin in different directions. Together with the identity, they form a basis for all 2×2 Hermitian matrices and are fundamental to quantum computing and quantum mechanics.
Pauli matrices measure spin: eigenvalues of ±1 yield spin-up or spin-down. No two fermions can occupy the same quantum state—this antisymmetry explains atomic structure, chemistry, and why matter is solid.
String theory proposes that all fundamental particles are tiny vibrating strings. Different vibrational modes produce different particles: the lowest mode gives massless photons, higher modes give massive W and Z bosons, and even higher modes could produce gravity and other forces. The theory requires 10 or 11 total dimensions—the 4 we observe (3 space + 1 time) plus 6 or 7 others, compactified at the Planck scale and invisible to us.
All particles are vibrating strings; different modes produce electrons, photons, bosons. The theory requires 10 or 11 dimensions, with extra ones compactified at Planck scale. Unconfirmed experimentally.
The principle of least action states that physical systems evolve along paths that make the action S = ∫L dt stationary (usually minimum). The Lagrangian L = T − V (kinetic minus potential energy) encodes the dynamics. From this single principle emerge all equations of motion, including Newton's laws, Maxwell's equations, and Einstein's gravity. This unification is one of physics' deepest insights: nature optimizes action.
Physical systems evolve along paths that minimize the action S = ∫L dt, where L = T − V. This single principle yields Newton’s laws, Maxwell’s equations, and Einstein’s gravity. Noether’s theorem links symmetries to conservation laws.
Einstein's general relativity reveals that gravity is not a force but the geometry of spacetime itself. A massive object curves the spacetime around it—think of it as warping a rubber sheet. Other objects naturally follow geodesics (the straightest available paths) through this curved spacetime, which we perceive as gravitational attraction. The famous equation Gμν = 8πTμν (in natural units) relates the curvature of spacetime to the distribution of matter and energy. This unified view of gravity as geometry is one of the most elegant insights in physics.
Gravity is the geometry of spacetime. Mass curves spacetime, and objects follow geodesics through it. Einstein’s equations: matter tells spacetime how to curve, spacetime tells matter how to move.
Special relativity reveals that time and space are not absolute. As objects move faster, time runs slower (time dilation), and they become shorter in the direction of motion (length contraction). These are not illusions—they are real relativistic effects. The light clock thought experiment elegantly shows how maintaining a constant speed of light for all observers requires time to run at different rates in different reference frames. At everyday speeds these effects are imperceptible, but near light speed they become dramatic.
The Lorentz transformation lies at the heart of special relativity. Blue axes show the rest frame (S), while red axes show a frame moving with velocity v (S'). As velocity increases, the x' and t' axes tilt toward the light cone (the 45° diagonal). The spacetime interval, s² = −c²t² + x², is the same in all frames: this is spacetime's metric, its fundamental geometry. The transformation reveals why simultaneity is relative: two events at the same time in frame S occur at different times in frame S'.
Light speed is constant in all frames. Consequence: time dilates by γ = 1/√(1 − v²/c²), lengths contract, simultaneity is relative. At 99.9% light speed, moving clocks run slow and objects shrink to slivers.
Black holes: Schwarzschild radius r_s = 2GM/c² is the event horizon beyond which nothing escapes. The photon sphere at r = 3M creates the bright ring around black hole silhouettes. Gravitational waves from merging black holes ripple spacetime itself—detected by LIGO in 2015.
Henri Poincaré (1854–1912) was perhaps the most consequential mathematician-physicist of the 20th century. He developed special relativity symmetries (the Poincaré group), pioneered chaos theory through his work on the three-body problem, and founded modern topology. His insight that tiny differences in initial conditions lead to vastly different outcomes—sensitive dependence on initial conditions—revealed that deterministic systems can be unpredictable. This geometric view transformed how we understand dynamics.
The Poincaré group unifies all spacetime symmetries: Lorentz transformations and translations. Energy, momentum, angular momentum follow from symmetries via Noether’s theorem. Particles are irreducible representations—mass and spin are quantum numbers.
Poincaré discovered chaos and topology: small initial changes cause exponentially divergent outcomes (butterfly effect). The three-body problem has no closed form, yet chaotic orbits fill fractal structures. His vision transformed how physics understands determinism, complexity, and emergence.
Computed via Kauffman bracket: J(t) = t⊃−3W;[K], where W is the writhe
The simplest true knot with 3 crossings. Found in Celtic knots and molecular structures.
The [K] bracket is a Laurent polynomial invariant that can distinguish many knots. It satisfies:
From [K], the Jones polynomial J(t) is obtained by substituting A = t⊃−1/4; and scaling by the writhe.
The Jones polynomial appears in Witten’s Chern-Simons interpretation of knot invariants. In Chern-Simons gauge theory on 3D spacetime:
This perspective unified knot theory and quantum field theory, winning the 1990 Fields Medal for Edward Witten.
At first glance, knot theory—the study of how curves can tangle in 3D space—seems purely mathematical, disconnected from physics. But in 1989, Edward Witten discovered a profound connection: the Jones polynomial of a knot encodes the quantum expectation value of a Wilson loop in Chern-Simons gauge theory on three-dimensional spacetime.
This discovery unified two disparate fields. A knot—a topological object—appears as an observable in quantum field theory. When you compute the quantum expectation value of a loop in the gauge field using path integrals, the result is exactly the Jones polynomial of that loop’s knot type. This is not a coincidence; it reveals that gauge theory is fundamentally topological. The topology of space encodes the quantum behavior of fields.
The Kauffman bracket, shown above, is a polynomial invariant defined recursively through the smoothing relations at crossings. From it, we obtain the Jones polynomial by substituting A = t−1/4 and scaling by the writhe (the signed count of crossings). The remarkable fact is that this purely combinatorial invariant coincides exactly with the vacuum expectation value of the Wilson loop in Chern-Simons theory at a particular gauge-theoretic coupling.
Chern-Simons gauge theory is a topological quantum field theory in 3D—an exotic cousin of Yang-Mills theory that exists only in odd dimensions. The Chern-Simons action S = ∫ Tr(A ∧ dA + ⅔A ∧ A ∧ A) is insensitive to local deformations of spacetime metric; it depends only on the gauge connection A and global topology. Quantum fluctuations in this theory yield observables—Wilson loops—whose values are topological invariants of the loops themselves.
For a knot K, the Wilson loop W(K) is a path-ordered exponential of the gauge connection around the knot. The quantum average <W(K)> in Chern-Simons theory, computed at level k (a quantization parameter), yields a Laurent polynomial in q = e(2πi/k+2). In the limit k → ∞, this polynomial becomes precisely the Jones polynomial J(t), with t = q.
This connection revealed that knots have a deep quantum-theoretic significance. Topological invariants—properties unchanged by continuous deformation— emerge naturally in gauge theory as quantum expectation values. This perspective revolutionized knot theory, providing new tools and interpretations, and it earned Edward Witten the 1990 Fields Medal.
Einstein’s most famous equation reveals that mass and energy are interchangeable. A small amount of mass contains vast amounts of energy because light speed is so enormous. This explains why nuclear fission and fusion release such devastating energy from a small amount of matter. The Sun converts about 600 million tons of hydrogen to helium every second, converting just 4 million tons of matter into pure energy via E=mc². This is why the Sun has shone for 4.6 billion years and will continue for another 5 billion more.
Neutrinos are elementary particles with no electric charge and almost zero mass, though not all are massless. They interact through the weak force and gravity, not electromagnetism or the strong force. Roughly 100 trillion solar neutrinos pass through your body each second, almost all without touching an atom. A low-energy solar neutrino could cross a light-year of lead without interacting; the chance of a collision depends on its energy.
There are three flavours: electron, muon and tau, named for their charged-particle partners. A neutrino produced in one flavour can be detected as another after travelling. This flavour oscillation happens because each flavour is a quantum mixture of states with different masses. The discoveries at Super-Kamiokande and the Sudbury Neutrino Observatory earned Takaaki Kajita and Arthur B. McDonald the 2015 Nobel Prize in Physics and showed that neutrinos have mass, overturning the original massless-neutrino picture.
Nuclear fusion in the Sun's core makes electron neutrinos. Nuclear reactors and radioactive beta decay produce neutrinos or their antiparticles, antineutrinos. A collapsing star releases an enormous burst of neutrinos during a supernova. High-energy neutrinos can also come from particle collisions in the hot gas and jets around feeding black holes, outside the event horizon; they are not signals escaping from inside a black hole.
In 1930 Wolfgang Pauli proposed a light, neutral particle in a letter to explain the energy apparently missing from beta decay: an unseen particle carried it away, preserving energy conservation. In 1956 Frederick Reines and Clyde Cowan detected electron antineutrinos from the Savannah River nuclear reactor, confirming the particle's existence a quarter of a century later. Reines received the 1995 Nobel Prize in Physics for the detection; Cowan had died in 1974.
IceCube instruments a cubic kilometre of Antarctic ice because neutrino–atom collisions are so rare. When a neutrino interacts with an atomic nucleus, it can produce charged particles that travel faster than light travels in ice, though never faster than light in a vacuum. Those particles emit faint blue Cherenkov light. Buried light sensors record the flashes, allowing scientists to infer the neutrino's energy and direction rather than see the neutrino itself.
A simplified two-flavour model describes oscillation in a vacuum. The first sine-squared factor sets how strongly the flavours mix; the second makes the probability rise and fall as the travel distance grows relative to the neutrino's energy. Real neutrinos have three flavours, and matter can change their oscillation pattern.
P is the flavour-change probability, α and β are two different flavours, θ is the mixing angle, Δm² is the difference between squared neutrino masses in electronvolt squared (using c = 1), L is distance in kilometres, and E is energy in gigaelectronvolts; 1.27 converts these units.
1900 – 1958 · Physics
1918 – 1998 · Physics