Relativity is a foundational theory in physics developed by Albert Einstein in the early 20th century, comprising two interrelated frameworks: special relativity (1905) and general relativity (1915). Special relativity revolutionized the understanding of space, time, and energy by establishing that the laws of physics are invariant in all inertial frames and that the speed of light in a vacuum is constant. It introduced concepts such as time dilation, length contraction, and the equivalence of mass and energy (E=mc²). General relativity extended these principles to include gravity, describing it as the curvature of spacetime caused by mass and energy, leading to predictions like gravitational time dilation, black holes, and gravitational waves.
1 Special relativity
1.1 Historical background
1.1.1 Maxwell's equations and the luminiferous ether
In the late 19th century, James Clerk Maxwell's equations unified electricity and magnetism, predicting that light is an electromagnetic wave propagating at a fixed speed. According to classical physics, waves require a medium; thus scientists postulated the existence of the luminiferous ether—a hypothetical, invisible substance filling all space through which light waves were thought to travel. The ether was assumed to provide an absolute reference frame against which the speed of light could be measured. However, Maxwell's equations themselves did not specify a preferred frame, leading to theoretical tensions.
1.1.2 Michelson–Morley experiment
The most famous attempt to detect the ether was the Michelson–Morley experiment (1887), performed by Albert A. Michelson and Edward W. Morley. Using an interferometer, they aimed to measure the Earth's motion through the ether by detecting a shift in interference fringes due to differences in the speed of light in perpendicular directions. To their surprise, the result was null: no fringe shift was observed, indicating that the speed of light is constant regardless of the Earth's motion. This negative result contradicted the ether hypothesis and posed a major challenge to classical physics, paving the way for Einstein's special relativity.
1.2 Fundamental postulates
1.2.1 Principle of relativity
The principle of relativity states that the laws of physics are the same in all inertial frames of reference (those moving at constant velocity relative to each other). This idea was not new—it had been recognized by Galileo and Newton—but Einstein extended it to include all physical laws, not just mechanics. In special relativity, no inertial frame is privileged; any experiment performed in one frame yields identical results when repeated in another.
1.2.2 Invariance of the speed of light
The second postulate asserts that the speed of light in a vacuum, denoted \( c \), is constant and independent of the motion of the source or the observer. This means that all inertial observers measure the same value for \( c \) (approximately \( 3 \times 10^8 \) m/s), regardless of their relative velocities. This postulate directly contradicts classical velocity addition and forces a revision of Newtonian concepts of absolute space and time.
1.3 Lorentz transformations
1.3.1 Derivation and implications
The Lorentz transformations are mathematical equations that relate the coordinates (space and time) of an event as measured in two inertial frames moving at constant relative velocity. They were first derived by Hendrik Lorentz in 1904 to explain the null result of the Michelson–Morley experiment, but Einstein reinterpreted them physically in 1905. For motion along the \( x \)-axis, the transformations are:
\[ t' = \gamma \left( t - \frac{vx}{c^2} \right), \quad x' = \gamma (x - vt), \quad y' = y, \quad z' = z, \]
where \( \gamma = 1/\sqrt{1 - v^2/c^2} \) is the Lorentz factor. These equations imply that time and space are not absolute; they mix together depending on the relative velocity. At low speeds (\( v \ll c \)), they reduce to the familiar Galilean transformations.
1.3.2 Relativistic velocity addition
The Lorentz transformations lead to a new rule for adding velocities. If an object moves with velocity \( u \) in one frame, its velocity \( u' \) in another frame moving at speed \( v \) along the same axis is given by:
\[ u' = \frac{u - v}{1 - \frac{uv}{c^2}}. \]
This formula ensures that the speed of light is invariant: if \( u = c \), then \( u' = c \), regardless of \( v \). It also shows that no combination of velocities can exceed \( c \), preserving the cosmic speed limit.
1.4 Consequences of special relativity
1.4.1 Time dilation
Time dilation is the phenomenon whereby a moving clock runs slower relative to a stationary observer. Specifically, if a clock moves at speed \( v \) relative to an inertial observer, the time interval between ticks measured by that observer is longer than the proper time (the interval measured by a clock at rest relative to itself). The relation is \( \Delta t = \gamma \Delta \tau \), where \( \Delta \tau \) is the proper time. This effect becomes significant only at speeds approaching \( c \).
1.4.1.1 Twin paradox
The twin paradox is a thought experiment illustrating time dilation. One twin travels at high speed on a round trip to a distant star, while the other remains on Earth. According to special relativity, the traveling twin ages less than the stay-at-home twin upon return. The paradox arises because from the traveling twin's perspective, the Earth twin appears to be moving, seemingly leading to symmetric aging. The resolution lies in the fact that the traveling twin undergoes acceleration when turning around, breaking the symmetry; only the twin who experiences acceleration returns younger. The effect has been confirmed experimentally with atomic clocks on airplanes.
1.4.2 Length contraction
Length contraction, also known as Lorentz contraction, states that an object moving relative to an observer is measured to be shorter along the direction of motion than its proper length. The contraction factor is \( 1/\gamma \). For example, a spaceship of proper length \( L_0 \) moving at speed \( v \) appears to a stationary observer to have length \( L = L_0 / \gamma \). Like time dilation, this effect is reciprocal and only noticeable at relativistic speeds.
1.4.3 Relativity of simultaneity
According to special relativity, events that are simultaneous in one inertial frame may not be simultaneous in another frame moving relative to it. This is a direct consequence of the finite speed of light and the relativity of time. For instance, if two lightning strikes hit the front and back of a moving train simultaneously in the train's frame, an observer on the ground will see the strike at the front first if the train moves forward. There is no absolute "now" across all frames; simultaneity is frame-dependent.
1.4.4 Mass–energy equivalence
Perhaps the most famous equation in physics, \( E = mc^2 \), expresses the equivalence of mass and energy. It states that an object's rest energy \( E \) is equal to its mass \( m \) times the speed of light squared. This implies that mass can be converted into energy and vice versa. The equation is derived from relativistic dynamics and has been verified in nuclear reactions, particle-antiparticle annihilation, and the operation of nuclear power plants and weapons. It also means that an object's total energy includes both its rest energy and its kinetic energy.
1.5 Spacetime
1.5.1 Minkowski metric
Hermann Minkowski, in 1908, reformulated special relativity by introducing a four-dimensional spacetime, where time is treated as a fourth dimension (often denoted as \( ct \)) alongside the three spatial dimensions. The geometry of flat spacetime is described by the Minkowski metric:
\[ ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2, \]
where \( ds \) is the spacetime interval. Unlike Euclidean geometry, the time coordinate has a negative sign, reflecting the distinct nature of time. This metric is invariant under Lorentz transformations, meaning all inertial observers agree on the value of \( ds^2 \).
1.5.2 Light cones
A light cone is a geometric representation of causality in spacetime. For a given event (point in spacetime), the light cone consists of all points that can be reached by light signals emanating from it (future light cone) or from which light signals can reach it (past light cone). Events inside the future light cone are causally influenced by the event; those outside cannot be reached without exceeding the speed of light. The light cone structure separates spacetime into regions: timelike (inside), lightlike (on the cone surface), and spacelike (outside). Events with spacelike separation are not causally connected.
1.6 Experimental confirmations
1.6.1 Muon decay
Muons are unstable subatomic particles created when cosmic rays strike the Earth's atmosphere. They have a mean lifetime of about 2.2 microseconds at rest. Due to their high speed (close to \( c \)), time dilation causes them to survive much longer in the Earth's frame, allowing them to travel many kilometers to the surface. The number of muons detected at sea level matches the relativistic prediction, providing direct evidence for time dilation.
1.6.2 Particle accelerator tests
Particle accelerators routinely confirm special relativity. For example, the lifetimes of fast-moving particles like pions and kaons match the time dilation formula. Additionally, the relativistic increase in mass (or energy) is observed when particles are accelerated to near-light speeds; the required energy far exceeds Newtonian predictions. The Large Hadron Collider and other accelerators rely on relativistic formulas for beam dynamics and collision energies.
2 General relativity
2.1 From special to general
2.1.1 Equivalence principle
The equivalence principle is the foundation of general relativity. It states that in a sufficiently small region of spacetime, the effects of a uniform gravitational field are indistinguishable from those of a constant acceleration. This is often illustrated by a thought experiment: an observer in an accelerating elevator feels a force indistinguishable from gravity. Conversely, a freely falling observer feels weightless. This principle implies that gravity is not a force in the traditional sense but a manifestation of spacetime curvature.
2.1.1.1 Weak and strong versions
The weak equivalence principle (WEP) states that the inertial mass and gravitational mass of any object are equal, leading to the universality of free fall. It has been tested to high precision, e.g., by lunar laser ranging experiments. The strong equivalence principle (SEP) extends this to all laws of physics, including gravitational effects themselves. It asserts that the outcome of any local experiment is independent of the external gravitational environment. General relativity satisfies the SEP, while some alternative theories do not.
2.1.2 Mach's principle
Mach's principle, named after Ernst Mach, is a philosophical idea that influenced Einstein. It roughly states that the inertia of an object arises from its interaction with the distribution of mass in the universe. In general relativity, this principle is partially realized: the local inertial frames are determined by the overall distribution of matter, though the extent to which Mach's principle is fully implemented remains debated. Einstein initially hoped general relativity would incorporate it completely, but modern interpretations vary.
2.2 Mathematical framework
2.2.1 Tensor calculus
General relativity uses tensor calculus, a mathematical language that describes geometric objects in curved spaces. Tensors are generalizations of scalars, vectors, and matrices that transform in a specific way under coordinate transformations. Key tensors include the metric tensor \( g_{\mu\nu} \), the Riemann curvature tensor \( R^\rho_{\sigma\mu\nu} \), and the stress–energy tensor \( T_{\mu\nu} \). Einstein's field equations are tensor equations, ensuring they are valid in all coordinate systems.
2.2.2 Metric tensor and geodesics
The metric tensor \( g_{\mu\nu} \) defines the distance (spacetime interval) between nearby points in curved spacetime. In general relativity, it describes the gravitational field. Geodesics are the curves that particles and light follow when moving freely under gravity—they are the straightest possible lines in curved spacetime. The geodesic equation generalizes Newton's first law: objects in free fall move along geodesics, which can be derived from the equivalence principle.
2.2.3 Einstein field equations
The Einstein field equations (EFE) are the core of general relativity:
\[ R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, \]
where \( R_{\mu\nu} \) is the Ricci curvature tensor, \( R \) is the scalar curvature, \( \Lambda \) is the cosmological constant, \( G \) is Newton's gravitational constant, and \( T_{\mu\nu} \) is the stress–energy tensor. The left side represents the curvature of spacetime, while the right side represents the distribution of matter and energy. The equations are highly nonlinear and describe how mass-energy curves spacetime, and how this curvature in turn governs the motion of matter.
2.3 Physical predictions
2.3.1 Gravitational time dilation
Gravitational time dilation predicts that time runs slower in stronger gravitational fields. A clock at a lower gravitational potential (closer to a massive body) ticks more slowly than one at a higher potential. For a static observer, the relation is \( \Delta \tau = \sqrt{1 - 2GM/(rc^2)} \, \Delta t \) for a non-rotating mass. This effect has been confirmed experimentally.
2.3.1.1 Pound–Rebka experiment
The Pound–Rebka experiment (1959) measured gravitational time dilation by using the Mössbauer effect at Harvard University. Gamma rays emitted from a source at the top of a tower (22.6 m height) were compared with a receiver at the bottom. The frequency shift due to gravity matched general relativity's prediction to within about 1%. This was one of the first terrestrial verifications of gravitational time dilation.
2.3.2 Bending of light
General relativity predicts that light follows curved paths in a gravitational field. The curvature is twice that predicted by Newtonian gravity (due to the effect of gravity on space as well as time). When a light ray passes near a massive object, its path is deflected. The deflection angle for a ray grazing the Sun is about 1.75 arcseconds.
2.3.2.1 Eddington expedition
The Eddington expedition of 1919, led by Arthur Eddington, photographed stars near the Sun's edge during a solar eclipse. By comparing the star positions to their known locations, they confirmed that the Sun's gravity bent the starlight by an amount consistent with Einstein's prediction. This made Einstein a global celebrity and provided dramatic evidence for general relativity.
2.3.3 Precession of Mercury's perihelion
The orbit of Mercury precesses (its perihelion advances) at a rate of about 574 arcseconds per century. Newtonian gravity, accounting for perturbations from other planets, could explain 531 arcseconds, leaving a residual of 43 arcseconds per century unexplained. General relativity precisely accounts for this anomalous precession through corrections to Newtonian gravity, a major early success of the theory.
2.3.4 Gravitational redshift
In a gravitational field, light emitted from a lower potential (e.g., near a massive body) is redshifted when observed at a higher potential. This is a consequence of gravitational time dilation: the frequency of light is effectively lowered as it climbs out of the gravitational well. Gravitational redshift has been measured in sunlight (solar lines), in the light from white dwarfs, and in terrestrial experiments like the Pound–Rebka experiment.
2.3.5 Black holes
Black holes are regions of spacetime where gravity is so strong that nothing, not even light, can escape. They form when massive stars collapse or through other high-density processes. General relativity predicts their existence as a logical consequence of the field equations.
2.3.5.1 Schwarzschild solution
The Schwarzschild solution, obtained by Karl Schwarzschild in 1916, is the first exact solution to the Einstein field equations. It describes the spacetime around a non-rotating, uncharged spherical mass. The solution includes a critical radius, the Schwarzschild radius \( r_s = 2GM/c^2 \), inside which an event horizon exists. For the Sun, \( r_s \approx 3 \) km; for Earth, about 9 mm.
2.3.5.2 Event horizon and singularity
The event horizon is the boundary of a black hole; once crossed, no information or matter can return to the outside. At the center of a Schwarzschild black hole lies a singularity—a point of infinite curvature where the known laws of physics break down. Rotating (Kerr) black holes have a ring singularity and an ergosphere. The existence of singularities suggests that general relativity is incomplete, requiring a quantum theory of gravity.
2.3.6 Gravitational waves
Gravitational waves are ripples in spacetime produced by accelerating masses, especially during cataclysmic events like black hole mergers. They propagate at the speed of light and are a prediction of general relativity (1916). For decades they were only indirectly observed through the orbital decay of binary pulsars.
2.3.6.1 LIGO detection
The Laser Interferometer Gravitational-Wave Observatory (LIGO) made the first direct detection of gravitational waves in September 2015, announced in February 2016. The signal, GW150914, came from the merger of two black holes about 1.3 billion light-years away. The observation matched general relativity's predictions with high precision and earned the 2017 Nobel Prize in Physics. Since then, multiple detections have been made, opening a new window on the universe.
2.4 Cosmological applications
2.4.1 Expanding universe
General relativity, when applied to the universe as a whole, predicts that spacetime can expand or contract. In the 1920s, Alexander Friedmann derived solutions to the field equations for a homogeneous, isotropic universe, showing that the universe must be either expanding or contracting. Edwin Hubble's 1929 observations of galactic redshifts confirmed the expansion, establishing the Big Bang model.
2.4.2 Big Bang theory
The Big Bang theory describes the universe's evolution from an extremely hot, dense state about 13.8 billion years ago. General relativity describes the expansion and the subsequent cooling, leading to structure formation. The theory is supported by cosmic microwave background radiation, light element abundances, and galactic redshits. However, the initial singularity remains a challenge for physics.
2.4.3 Dark energy and the cosmological constant
In 1998, observations of Type Ia supernovae revealed that the universe's expansion is accelerating. This is attributed to dark energy, a mysterious form of energy that counteracts gravity. The cosmological constant \( \Lambda \) in Einstein's field equations can represent dark energy—a constant energy density of empty space. The current standard model of cosmology, ΛCDM, includes dark energy (about 68% of the universe's energy density) and explains the accelerated expansion with general relativity.
2.5 Tests and current research
2.5.1 Solar system tests
General relativity has passed a variety of tests within the solar system, including the precession of Mercury's perihelion, bending of light by the Sun (verified by radio astronomy and spacecraft missions), gravitational time dilation (Pound–Rebka, GPS corrections), and the Shapiro time delay (delay of radar signals passing near the Sun). All results are consistent with Einstein's theory to high precision.
2.5.2 Strong-field tests (binary pulsars)
Binary pulsars, such as the Hulse–Taylor binary (PSR B1913+16), provide tests of general relativity in strong gravitational fields. The orbital decay of the pulsar system matches the energy loss predicted by gravitational wave emission to within 0.1%, providing indirect confirmation of gravitational waves. Other tests include the measurement of relativistic precession, time dilation, and frame dragging in strong-field regimes.
2.5.3 Challenges and extensions
Despite its successes, general relativity faces challenges. It does not incorporate quantum mechanics, leading to singularities (black hole and Big Bang) that are not fully understood. Dark matter—an unseen form of matter inferred from galactic rotations—is not explained by general relativity itself, though the theory is used to model its effects. Alternative theories, such as modified Newtonian dynamics (MOND) or scalar-tensor theories, attempt to address these issues, but none have supplanted general relativity as the standard theory of gravity. Current research includes testing the theory with gravitational wave detections, precision cosmology, and attempts to formulate a quantum theory of gravity.