Explore a simplified breakdown of the Einstein field equations

Explore Albert Einstein's Zürich notebook and review his notes on relativity

Within the pages are calculations, sketches and his first attempts at deriving what would be the field equations for gravity. It also documents some dead ends and the difficulties Einstein encountered as he describes part of the mathematics of what would become general relativity, describing it as "too involved."

Mathematician David Hilbert developed general relativity equations before Einstein

Although working independently of one another, the two exchanged correspondence in the lead up to November 1915, when Hilbert submitted their gravitational field equations five days before Einstein presented their work. An error in Hilbert's formulation and its nonpublication until March 1916 would leave credit for general relativity to Einstein.

View a breakdown of the mathematics behind the Standard Model

The Lagrangian version of the theory—related to the system's energy—presents a vast mathematical equation, with different components representing various force interactions and types of particles, including ghost particles to cancel out mathematical redundancies.

Einstein and the fourth dimension

How Einstein's greatest theoretical achievement contains a simple but mind-bending proposition—the dimension of time is physically connected to the three observable spatial dimensions.

Special relativity shows how motion at high speeds reshapes time, space, and energy

Einstein’s 1905 special theory of relativity revealed that the speed of light is absolute in the universe, forcing time and space to vary with one's speed. This idea led to the famous equation E=mc2, which shows that mass and energy are interchangeable.

Explore an interactive map of the unified theories of everything

This interactive concisely describes many highly complex theories ranging from quantum gravity and black holes to dark matter and energy. Supplementary resources, key questions, and possible solutions accompany each description. As of 2026, no developed theory provides an accepted explanation of all observed phenomena in the universe at all scales.

Time disappears when encompassing all quantum configurations across the universe

The Wheeler-DeWitt equation, a modified version of the Schödinger equation—the formula that describes the evolution of the quantum configuration of a system—describes the quantum state of space-time across the universe. The transformation causes the variable of time to disappear, suggesting it is not a fundamental, physical parameter of the cosmos.

One of Einstein's papers challenged quantum theory but helped confirm entanglement

In 1935, Einstein suggested that quantum mechanics was incomplete due to its strange implications about entangled particles. Their challenge led to decades of experiments that eventually proved distant particles could affect each other instantly.

Einstein created a mathematical term for anti-gravity to keep the universe static

General relativity predicted that the gravity of all the matter in the universe would cause it to collapse, contradicting the belief at the time that the universe was constant and unchanging. Albert Einstein added the cosmological constant to his equations, which acted like anti-gravity, to "fix" his math.

Explore a visualization of curved spacetime under general relativity

While depictions involving bends in elastic fabrics are common to illustrate gravity, they are circular representations that rely on gravity in the real world to depict gravity in the model. The river model instead shows how bends in spacetime convert speed through time into speed through space, producing acceleration.

How neutron stars serve as laboratories for testing Einstein's theory of gravity

Data from a 16-year study of an orbiting pair of spinning neutron stars showed that measurements of gravitational wave energy and the bending of light around the system were consistent with predictions made by general relativity to an accuracy of at least 99.99% across seven tests. (Some readers may experience a paywall.)

Albert Einstein's equations predicted black holes decades before their discovery

For nearly 50 years, scientists dismissed black holes as implausible oddities, despite solutions to the equations indicating their existence. But the 1970s discovery of intense X-rays from Cygnus X-1 and its massive hidden companion flipped the consensus.

View a gallery of Einstein rings

As light from a distant galaxy spreads in all directions, rays that pass around a massive object, such as another galaxy, may be bent around it. If the massive object is positioned between the distant galaxy and our telescopes, we may observe this deflected light forming a ring around it.

The Navier-Stokes equations include the fluid version of Newton's second law

The first equation is the fluid version of the conservation of mass, which prevents incompressible fluids from gaining or losing matter. The second equation replaces acceleration and inertia, which is resistance to acceleration, with fluid pressure and viscosity.

The interaction of quantum fields with black hole curvatures makes Hawking radiation

The classic explanation of Hawking radiation depicts virtual photons emitted from a black hole's event horizon. A more accurate explanation involves null geodesics, Bogoliubov transformations, and scattered quantum field nodes to explain the emission of thermal radiation.

The weakening of dark energy suggests a dynamic universe with evolving forces

The discovery, which came from analyzing extensive data from a major cosmological survey, challenges the assumed constancy of dark energy in modern physics. Understanding dark energy's behavior is crucial for predicting the universe's long-term evolution.

The one-electron theory describes the universe as having only one electron

The thought experiment explains why every electron has the same charge and mass as every other electron, and describes every positron—the antimatter version of the electron—as that same electron traveling backwards in time. Among other foundational challenges, the model suggests that the universe contains equal numbers of electrons and positrons, yet normal matter is much more common than antimatter.

Gravity, electromagnetism, and two nuclear forces make up nature’s fundamental forces

While gravity is the most familiar force, it is also the weakest and the only one without a particle to mediate the force. The weak nuclear force is responsible for radioactive decay and transmutation of elements, while the strong nuclear force holds quarks together.

A naked singularity is an infinitely dense point without a black hole's event horizon

Their existence would reveal where Einstein’s theory breaks down and quantum gravity takes over. The prevailing idea—cosmic censorship—suggests these singularities are always hidden behind black hole horizons, shielding the rest of the universe from their unpredictability.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

Earth's total mass is equivalent to 13 quadrillion Egyptian pyramids of Khafre

Newton's law of universal gravitation was used to calculate the value after Henry Cavendish torsion balance experiments determined the missing value of the universal gravitational constant from Newton's equation.

13 unanswered questions about black holes

Here's a quick rundown on black hole basics, the cosmic enigmas digesting massive swaths of the universe and destabilizing our fundamental ideas of space, time, and matter. Learn how they got their name, what happens in the unfortunate instance you fall into one, and more with this list of 13 questions about black holes.

According to the physics of black holes, our 4D reality may be a hologram—a projection of information coded on a 2D surface on the edge of the universe.

Accurate climate models incorporate calculus to account for complex feedback loops

The simplest models assume uniformity across the Earth, with incoming energy from the sun being offset by the planet's albedo, or reflectiveness, which prevents the radiation from being fully absorbed. Incorporating elements such as atmospheric layers, latitude bands, geographic variability, and emissivity yields coupled differential equations quantifying Earth's interconnected systems.

The selfie camera exists thanks to Einstein's discovery of the photoelectric effect

Smartphone cameras utilize sensors such as CMOS or CCD to capture light by converting it into electrical signals through the photoelectric effect. Einstein explained this effect by demonstrating that light behaves as both particles and waves simultaneously, a discovery that earned him the Nobel Prize in physics.

Explore our favorite visualization of the immense scale of space, from an astronaut to the size of the observable universe

Against his wishes, Albert Einstein's brain was removed before his cremation, and the pathologist responsible stored it for decades in mason jars within a beer cooler.

How fusion—which releases energy when two lighter atoms combine to form a heavier atom—differs from fission

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

While the sun's gravity keeps the planets in orbit, their gravity also pulls on the sun, causing it and everything else in the solar system to orbit around a mutual balance point called the barycenter—the true center of the solar system.

Stars appear to twinkle because turbulence in the motion of molecules in Earth's atmosphere causes light to bend and wiggle rapidly on its way to the surface—a phenomenon called scintillation.

Visualize the path of every solar eclipse from 1900 to 2099 and see what each looks like when observed from different locations on Earth.

The boiling temperature of water decreases with increasing elevation because the molecules need less energy to overcome the lower atmospheric pressure associated with thinner air.

Use this tool to calculate the dollar value of your parental mental load, from meal planning to coordinating after-school care, then see resources that help generate solutions.

While the first exchange-traded fund (a 'basket' of different assets that can be bought and sold on a stock exchange) in the US only launched in 1993, the global ETF market has grown to more than $10T in the short time since.

Georgia O'Keeffe's flower paintings are often assumed to be sexual, though the artist claimed she was simply interested in highlighting the beauty of flowers that most people neglect

The price of gas is a composite made up of several different factors, with the federal gas tax making up less than 20% of what drivers pay at the pump.

Three Ebola species are responsible for most large outbreaks of the disease, yet the only approved Ebola vaccine is known to work against just one of them.

How Amazon operates a 'flywheel' business model, a self-reinforcing cycle that drives continuous growth

Warren Buffett's former professor Benjamin Graham is known as the 'father of value investing,' a strategy in which investors seek to buy stocks trading below their intrinsic worth.

Meet the 56 signers of the Declaration of Independence—most of whom didn't put pen to paper until August 2, 1776—and see their original signatures.

Waves off the coast of Nazaré, Portugal, can reach over 100 feet and are caused by the area's deep underwater canyon geography

Discover the catastrophic consequences of a failure of the US electrical grid—something that is more likely than many Americans think.

The Brothers Grimm were lawyers before they were storytellers, but became interested in fairy tales as they began investigating how rules and customs had been passed down through cultures.

English mathematician Ada Lovelace wrote the first computer algorithm a century before computers existed and hypothesized that computing devices would one day be able to compose music—foreseeing the generative AI systems of today.

A 2019 Facebook event called on attendees to "Storm Area 51, They Can't Stop All of Us," resulting in countless memes, the creation of a music festival, and eventually the arrests of those who did try to approach the area.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.