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."

Visualizing Einstein's special relativity

Before Einstein developed his theory of general relativity, he worked out what is known as special relativity—or why time is experienced different for different observers. The breakthrough rests on his theory that light travels at a constant...

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’s theory of special relativity

Special relativity, formulated by Einstein, describes how space and time are linked for objects moving at constant speeds. It states that the speed of light is constant for all observers, leading to time dilation (moving clocks run slower), length contraction (moving objects shrink), and mass-energy equivalence (E = mc²). This video breaks down how motion and time are fundamentally related.

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.

Albert Einstein's theory of gravity arose from what he described as 'the happiest thought of his life': realizing that a person who fell off the roof of their house would not feel their own weight as they fell.

Experimenting with space-time

Einstein's theory of general relativity—that mass literally bends space and time—can be difficult for the human brain to imagine. This interactive resource lets you create celestial objects, imagine their effect on space-time, and see how they interact with each other (best viewed on desktop).

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.

To develop general relativity, Einstein relied on his college friend, Marcel Grossman

Recognizing the mathematical complexities of generalizing special relativity with a new picture of gravity and acceleration, Albert Einstein had Grossman introduce him to tensor calculus. Years before developing the precursor to GR together, Einstein would use Grossman's lecture notes to prepare for his exams.

Quantum mechanics and general relativity—two of the most successful explanations of the physics seen across the universe—are logically incompatible with one another because they use contradictory models for space and time.

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.

Gravitational waves—ripples in spacetime that result from the motion of objects—were detected a century after Albert Einstein predicted their existence upon developing his theory of gravity.

Discoveries in electromagnetism, which suggested that light traveled at a single speed, became the catalyst for overturning Isaac Newton's view that all speeds are relative and could appear to change based on one's perspective.

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.

Within Einstein's picture of the universe, gravity is technically not a force because forces cause accelerations that deviate objects from their natural path, and moving solely due to gravity is natural.

The Standard Model reveals an underlying organization to the universe

Using the framework of quantum field theory, it structures all known particles into 12 matter particles, alongside five particles that carry three of the four fundamental forces. Although remarkably precise and predictive, it cannot yet explain gravity, dark matter, or dark energy.

While special relativity describes motion along a flat underlying fabric of the universe called spacetime, general relativity extends this theory to explain motion through curved spacetime.

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.)

Quantum field theory, which has made predictions accurate to 12 decimal places, explained

Instead of being made of indivisible particles, all visible matter in the universe is built from underlying quantum fields—smooth, invisible entities that behave like waves and extend throughout space and time. The theory suggests these fields are fundamental, and particles are disturbances that move through them.

Because some models of the universe do not incorporate time, it may be an emergent property born from our limited perception of the universe, rather than a fundamental property of reality.

The underlying fabric of the universe—spacetime—changes shape in the presence of mass and energy, which affects measurements of space and time.

Nearly 50 years before the first observational evidence, a soldier fighting on the Eastern Front in World War I found a mathematical solution to Einstein's equations that suggested the existence of black holes.

The mathematics of general relativity does not rule out the existence of wormholes

Such objects, also known as Einstein-Rosen bridges, would resemble funnel-like tunnels connecting two regions of spacetime, allowing for instant travel across the universe. However, such structures would be incredibly unstable and require undiscovered materials with negative energy to keep them open.

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.

Understanding time dilation

The faster you go, the slower time ticks for you. And the stronger the gravitational field acting on you, the slower time ticks for you. These phenomena – collectively called time dilation – were first explained by Albert Einstein’s theories of general and special relativity. In this 11-minute video, beloved astrophysicist Neil deGrasse Tyson explains them to comedian Chuck Nice.

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.

Scientists validated Albert Einstein's theory of gravity by measuring the apparent positions of stars via their starlight as it traveled near the sun on its way to Earth during a total solar eclipse.

Quantum field theory reveals that particles are ripples in universal fields

Instead of being made of indivisible particles, all visible matter in the universe is built from underlying quantum fields—smooth, invisible entities that behave like waves. The theory suggests these fields are fundamental, and particles are disturbances that move through them.

The graviton may be the last missing piece of the Standard Model

While the electromagnetic, strong nuclear, and weak nuclear forces can be explained through interactions with particles, gravity is instead best modeled through spacetime curvature. Discovering a quantum particle of gravity would unite the two pillars of modern physics—quantum mechanics and general relativity—toward developing a theory of everything.

The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

General relativity describes a singularity as the final moment in time

Using Penrose diagrams—mathematical structures that compress all of space and time—physicists can observe the path of light and which events can affect one another. The space-time maps help visualize the structure of black holes and describe what happens when objects fall in.

As of 2025, the Standard Model has at least five unresolved problems

Among the issues with the most successful theory of the universe at subatomic scales is its inability to explain the extreme imbalance observed between visible matter and antimatter, since both should be roughly equal. The Standard Model also omits gravity and cannot account for the observed rate of cosmic expansion.

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.

Photons are both particles of light and messengers of the electromagnetic force

Einstein's model of light as a wave and a massless particle of energy revolutionized physics, and scientists' application of the model led to groundbreaking technologies, including medical imaging and treatments, telecommunication signals, and probes of the quantum and cosmological realms.

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.

String theory replaces black hole singularities with fuzzballs of tangled strings

Black holes contain singularities where information about the material that fell in is lost, violating quantum mechanics. Instead, modeled as a bundle of strings, information is stored and slowly released as the black hole evaporates.

Under a model of quintessence, the strength of dark energy should change over time

The theory suggests that dark energy permeates space and may drive the universe to expand or contract throughout its history. The model's flexibility allows it to fit every possible observed behavior of the universe's expansion, rendering it unfalsifiable and untestable.

The universe's expansion is accelerating due to a mysterious force called dark energy

Astronomers discovered in the late 1990s that galaxies move apart faster over time, contradicting expectations that gravity should slow cosmic expansion. This prompted the revival of the "cosmological constant," a form of repulsive gravity first proposed by Albert Einstein.

Observations of colliding neutron stars by the Laser Interferometer Gravitational-Wave Observatory in 2017 confirmed that gravity travels at the speed of light.

Understanding how gravity works at the quantum scale is physics's greatest challenge

While the theory of general relativity explains how gravity works and quantum mechanics explains interactions at the subatomic scale, the two frameworks contradict one another. Developing a model of gravity mediated by particles can help complete the Standard Model.

Quantum field theory overestimated the value of dark energy by 120 orders of magnitude

Physicists in the 1960s used their most successful theory to determine whether the accelerated expansion of the universe arises from quantum fluctuations in empty space. However, the mismatch between vacuum energy density and dark energy was so severe that it has been referred to as "the worst theoretical prediction in the history of physics."

According to the equivalence principle, objects in a closed space cannot tell the difference between whether they are accelerating or experiencing a uniform gravitational field.

During his 'miraculous year,' Einstein published four groundbreaking papers

The 1905 "annus mirabilis" papers explained the photoelectric effect, Brownian motion, the absolutism of the speed of light, and mass-energy equivalence, revolutionizing scientists' understandings of space, time, mass, and energy. For the discovery of the photoelectric effect, Einstein won the 1921 Nobel Prize in Physics.

Black holes are formed from massive stars collapsing under their own gravity

This creates a region where spacetime curves infinitely, known as a singularity. Black holes' intense mass and energy allow them to trap light, distort time and space, and present challenges with reconciling Einstein's theory with quantum mechanics.

A breakdown of the experimental test results supporting the general theory of relativity

Besides accurately explaining the slight wobble observed in Mercury's orbit, general relativity predicted that light would bend around massive objects by an amount matching what was observed when background starlight bent around the sun during an eclipse. The theory also predicted the existence of gravitational waves and black holes decades before they were found.

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.