# Special relativity **Special relativity** is the theory of [[Spacetime|space and time]] that follows from two assumptions: that the laws of physics are the same in every unaccelerated frame, and that the [[Speed_of_light|speed of light]] in vacuum has the same value c in all of them. The second is incompatible with the everyday rule for adding velocities, and the price of keeping it is that simultaneity, duration and length become properties of events *as measured in a particular frame* rather than of the events themselves. What stays absolute is a combination of time and distance, the spacetime interval.[^idema-partii][^openstax-v3-ch5] In the microsim below the reader sets β = v/c over −0.95 to 0.95 and steers a [[Spacetime_diagram|Minkowski diagram]]. The moving frame's axes tilt toward the light line by atan(β) — 41.99° at β = 0.9 — and `gamma = 1/sqrt(1 − beta^2)` reads out.[^derived-sr] Three things happen at once: a light clock's ticks stretch as `dt = gamma·dtau`, a rod shrinks as `L = L0/gamma`, and two events simultaneous in one frame slide apart in the other. The sheared grid is the [[Lorentz_transformation|Lorentz transformation]], `x' = gamma·(x − v·t)` and `t' = gamma·(t − v·x/c^2)`.[^idema-partii][^spec-p40] A muon preset supplies the measurement: a particle that lives 2.19703 μs at rest, and should cover 659 m before decaying, reaches the ground from 15 km.[^schiller-muon][^derived-sr] On the [[Physics]] flagship this article is the *Special relativity* section of Part II — Core theories, and the root of the shared C25 sim family. Its siblings reuse this spec: [[Time_dilation|time dilation]] takes the light clock, [[Length_contraction|length contraction]] the rod, [[Relativity_of_simultaneity|relativity of simultaneity]] the sliding events, and [[Mass–energy_equivalence|mass–energy equivalence]] the same β slider, reading energy off it instead of geometry. ## Overview The theory is *special* because it treats a restricted case: frames in uniform relative motion, gravity absent; [[General_relativity|general relativity]], ten years later, removes both restrictions. What it supplies is a replacement for the rule connecting one observer's coordinates to another's. In Galilean physics that rule is x′ = x − vt with t′ = t, so velocities simply add: light moving at c in one frame would move at c ± v in another, and [[Maxwell's_equations|Maxwell's equations]] — which contain c explicitly, built from measured electric and magnetic constants — would change form from frame to frame.[^openstax-v3-ch5] Relativity keeps Maxwell's equations and changes the transformation instead. The replacement mixes time into space and back, reduces to the Galilean rule when v ≪ c, and leaves c invariant by construction.[^idema-partii] Everything usually listed as a "consequence" of relativity — [[Time_dilation|time dilation]], [[Length_contraction|length contraction]], the [[Relativity_of_simultaneity|relativity of simultaneity]], velocity addition, [[Mass–energy_equivalence|E = mc²]] — is algebra applied to that one substitution. ## History The theory resolved a nineteenth-century impasse rather than reporting a new observation. Electromagnetic theory gave a definite speed for light and appeared to need a medium to carry it; experiments looking for the Earth's motion through that medium found nothing. Hendrik Lorentz and others obtained the transformation now named after him as a device for explaining that null result while keeping the medium. Albert Einstein's 1905 paper discarded the medium and took the transformation as a statement about space and time themselves, derived from the two postulates rather than fitted to an experiment.[^einstein1905][^openstax-v3-ch5] Hermann Minkowski's 1908 reformulation supplied the geometry — a four-dimensional [[Spacetime|spacetime]] with an indefinite interval — in which the transformation is a rotation-like change of coordinates.[^minkowski1908] The experimental case has since become overwhelming: colliders such as the [[Large_Hadron_Collider|Large Hadron Collider]] and the whole of [[Elementary_particle|particle physics]] are designed around the theory.[^openstax-v3-ch5] ## Terminology Some vocabulary is worth fixing first. An *event* is a point of spacetime — a place and a time, with no duration or extent. A *frame of reference* is a system of rulers and synchronised clocks; an *inertial* frame is one in which a free body moves in a straight line at constant speed. *Proper time* τ is the time read by a clock carried along a worldline and *proper length* an object's length in its rest frame; both are frame-independent, which is why the formulas are written around them. β denotes v/c and γ the Lorentz factor 1/√(1 − β²): 1.25 at β = 0.6, 2.294 at β = 0.9, 22.37 at β = 0.999.[^derived-sr] And *observer* means a frame, not a person looking. ## Traditional "two postulates" approach to special relativity Einstein's own route takes two statements as given. The first is the principle of relativity: the laws of physics take the same form in all inertial frames, so no experiment inside a closed laboratory can reveal its uniform velocity. The second is that light propagates in vacuum at the same speed c in every inertial frame, whatever the motion of its source — a value now fixed by definition at 299,792,458 m/s.[^murphy-c][^openstax-v3-ch5] The two are enough. Requiring the transformation between frames to be linear, to form a group, and to leave c unchanged determines it up to sign conventions, and the result is the Lorentz transformation.[^idema-partii] That economy is why the approach became standard: nothing about the structure of matter, and nothing about rods contracting under forces between their atoms, needs to be assumed. ## Principle of relativity The first postulate is older than relativity: Galileo's argument that a ship's cabin below decks gives no clue to the ship's steady motion is the same statement, and [[Newton's_laws_of_motion|Newtonian mechanics]] satisfies it exactly, since [[Force|forces]] and accelerations survive the Galilean transformation unchanged. What changed in 1905 was the scope: Newton's laws are Galilean-invariant and Maxwell's are not, so one had to be rewritten. Einstein rewrote mechanics, and the test of that choice is that it reduces to Newton's to about one part in 10⁵ at β = 0.1 while leaving electromagnetism untouched.[^derived-sr][^idema-partii] The principle also says what relativity does *not* claim: no frame is preferred, so there is no sense in which the moving clock "really" runs slow. Each of two observers finds the other's clock slow, and both are right, because they compare along different surfaces of simultaneity. ## Lorentz transformation In standard configuration — axes parallel, motion along x, origins coincident at t = t′ = 0 — the transformation is `x' = gamma·(x − v·t)`, `t' = gamma·(t − v·x/c^2)`, with y and z unchanged; reversing the sign of v inverts it.[^idema-partii] The factor γ multiplies both equations, so it is not by itself the physical content; that lies in the mixing terms, −vt in the first and −vx/c² in the second. The second is easy to overlook and does most of the conceptual damage: the time coordinate of an event depends on *where* the event is. The sim draws the transformation rather than solving it. On a [[Spacetime_diagram|Minkowski diagram]] with ct upward, the primed time axis is the worldline of the moving origin and the primed space axis is the set of events the moving frame calls simultaneous, each tilted by atan(β); the two close on the 45° light line as β → 1 and never cross it.[^spec-p40] Moving the slider shears the grid rather than rotating it, preserving the interval instead of the distance. At β = 0.95 the tilt is 43.53°, only 1.5° short of the light line, while γ has reached just 3.20.[^derived-sr] ## Consequences derived from the Lorentz transformation Everything here is algebra applied to the two equations above, and the sim shows the three best-known consequences on one screen from one slider. Each is reciprocal: the primed frame says exactly the same about the unprimed one. ### Relativity of simultaneity Two events at the same time t but at different positions x₁ and x₂ in one frame have times in the other differing by t′₂ − t′₁ = −γ·v·(x₂ − x₁)/c², zero only if v = 0 or the events coincide. Simultaneity is a property of a pair of events *and a frame*. This is the root of nearly every apparent paradox in the subject, the [[Twin_paradox|twin paradox]] and the pole-in-the-barn included: the frames do not disagree about what happened, they slice spacetime along different surfaces. ### Time dilation A clock at rest in the primed frame has Δx′ = 0 between two ticks, and the transformation gives `dt = gamma·dtau`: the interval measured where the clock moves exceeds the proper time by γ.[^idema-partii] The sim's light clock shows the mechanism — a photon bouncing between two mirrors travels a longer, diagonal path when the clock moves, and since its speed is the same in both frames the tick takes longer. A muon's mean life at rest is 2.19703 μs, in which time light covers 659 m and the particle rather less.[^schiller-muon][^derived-sr] Muons made at 15 km should almost never reach the ground: at β = 0.999 the trip is 22.8 undilated lifetimes, a survival probability of 1.3 × 10⁻¹⁰. With dilation it is 1.02 proper lifetimes and survival is 0.36 — larger by 2.9 × 10⁹.[^derived-sr] ### Length contraction Measuring a moving rod means marking both ends at the same time in the measuring frame, and since simultaneity is frame-dependent so is the answer: `L = L0/gamma`, with L₀ the proper length. A rod measures 80 % of its rest length at β = 0.6 and 43.6 % at β = 0.9.[^derived-sr] Contraction and dilation are one fact seen from two frames, which the muon makes concrete: in its own frame nothing lives longer than 2.197 μs, and it reaches the ground because the 15 km of atmosphere is contracted to 671 m at β = 0.999.[^derived-sr] ## Optical effects What a camera records differs from what a frame measures, and the difference is optics rather than kinematics. Light from the far parts of a fast-moving object left earlier than light from the near parts, so a photograph of a passing cube shows it rotated rather than flattened. Aberration — the tilting of apparent directions with the observer's motion — follows from transforming a ray direction, and in the ultrarelativistic limit it packs almost the whole sky into a small forward cone. The [[Doppler_effect|Doppler]] shift gains a factor of γ from time dilation on top of the classical shift, and that factor survives even for motion exactly transverse to the line of sight, where the classical shift vanishes; for β = 0.6 directly approaching, the combined factor is exactly 2.[^derived-sr][^openstax-v3-ch5] Dragging effects — the partial carrying-along of light by a moving medium — follow from relativistic velocity addition to first order.[^openstax-v3-ch5] ## Dynamics Newtonian [[Momentum|momentum]] m·v is not conserved under the Lorentz transformation; the conserved quantity is `p = gamma·m·v`, with energy `E = gamma·m·c^2`.[^idema-partii][^openstax-v3-ch5] Since γ diverges as β → 1, the energy needed to reach c is infinite — the dynamical statement of the kinematic fact that the tilted axes never cross the light line. Force is no longer parallel to acceleration, and the second law is written most cleanly as a rate of change of four-momentum with respect to proper time. ### Equivalence of mass and energy Setting v = 0 in `E = gamma·m·c^2` leaves E = mc², a rest energy carried by a body simply for existing.[^idema-partii] Eliminating the velocity between the momentum and energy expressions gives the relation that holds for every particle, massless ones included: `E^2 = (p·c)^2 + (m·c^2)^2`, with m the [[Invariant_mass|invariant mass]] and the left side the time component of a [[Four-momentum|four-vector]].[^idema-partii] The kinetic part (γ − 1)mc² returns Newton's ½mv² at low speed and exceeds it by 7.3 % already at β = 0.3, where the two curves in the sibling sim part.[^derived-sr] The [[Mass–energy_equivalence|mass–energy equivalence]] page carries that ladder in full. ## Rapidity Velocities do not add in relativity, but rapidity does. Defining φ by β = tanh φ turns the Lorentz transformation into a hyperbolic rotation through φ, with γ = cosh φ and γβ = sinh φ, and successive boosts along one line simply add their rapidities. The impossibility of reaching c becomes the unremarkable statement that no finite sum of finite numbers is infinite: β = 0.9 is φ = 1.472 and β = 0.99 is φ = 2.647, so two boosts of 0.9 compose to 0.9945 rather than 1.8.[^derived-sr] It is also the natural variable for the diagram, since axis tilt and grid shear are both linear in it. ## Minkowski spacetime Minkowski's contribution was to see that the invariant of the theory is geometric. For two events separated by Δt and Δx the quantity `s^2 = (c·dt)^2 − dx^2` is the same in every inertial frame, and the Lorentz transformation is exactly the set of linear maps preserving it.[^idema-partii] The minus sign is the whole difference from Euclidean geometry, and it cuts the spacetime around any event into three regions: s² > 0 is timelike and can be causally connected, s² < 0 is spacelike and cannot, s² = 0 is the light cone. Since the sign of s² is invariant, so is the causal ordering of timelike-separated events — which is why effects cannot precede their causes. Quantities transforming like the coordinates — energy and momentum, charge density and current, the electromagnetic potentials — group into [[Four-momentum|four-vectors]], and laws written as relations between four-vectors hold in all frames automatically. ## Acceleration Special relativity handles accelerated motion perfectly well; what it cannot handle is gravity. An accelerating body is described at each instant by the inertial frame momentarily at rest with it, its proper time being the integral of dτ = dt/γ(t) along the worldline. Uniform proper acceleration — constant as felt on board — gives a hyperbolic worldline asymptotic to the light cone, so the speed approaches c without reaching it however long the engine burns. The asymmetry in the [[Twin_paradox|twin paradox]] enters here: the traveller's worldline is not straight, and between two events the straight worldline is the one of *greatest* proper time, the reverse of the Euclidean rule. ## Relativity and unifying electromagnetism Electricity and magnetism are one field seen from different frames. A line of static charge is purely electric in its rest frame; where the charges move it is a current with a magnetic field too, and the force on a test charge comes out the same once charge density, current and the fields are transformed.[^openstax-v3-ch5] This is the strongest historical argument for the theory: [[Maxwell's_equations|Maxwell's equations]] are not merely compatible with relativity but already relativistic, and the [[Lorentz_force|Lorentz force]] law is what a purely electric force looks like from a moving frame. ## Theories of relativity and quantum mechanics Combining special relativity with [[Quantum_mechanics|quantum mechanics]] was neither optional nor easy. A relativistic wave equation must treat time and space alike, which the [[Schrödinger_equation|Schrödinger equation]] does not, and writing one forces negative-energy solutions, [[Spin_(physics)|spin]] as a consequence rather than an addition, and antiparticles. The resolution is that such a theory cannot be about a fixed number of particles: where energy of order 2mc² is available, particles can be created, so it must be a field theory. The two fit together exactly in this way; it is [[General_relativity|general relativity]] and quantum mechanics that do not. ## Status Special relativity is as well tested as any theory in physics, and tested continuously rather than ceremonially, since every working collider and every muon-flux measurement is a test. Its status is a constraint rather than a model: any proposed law is expected to be Lorentz-invariant, and searches for violations are framed as bounds on small parameters.[^openstax-v3-ch5] Its domain is equally well known: where gravity matters, spacetime is curved and special relativity holds only locally, as the tangent-space approximation general relativity is built on. ## See also - [[Lorentz_transformation]] — the substitution from which everything here follows - [[Time_dilation]] — the light clock, as its own pair - [[Length_contraction]] — the rod - [[Relativity_of_simultaneity]] — the sliding events - [[Spacetime_diagram]] — the drawing the sim steers - [[Twin_paradox]] — the asymmetric path - [[Albert_Einstein]] - [[Mass–energy_equivalence]] — the sibling that reads energy off the same slider - [[General_relativity]] — what happens when gravity is put back ## Notes Explanatory notes are folded into the body of this page; the footnotes under References carry both the sources and the derivations. ## Primary sources - Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper." *Annalen der Physik* 17. Pages and DOI to pin. - Minkowski, Hermann (1908). "Raum und Zeit," address to the 80th Assembly of German Natural Scientists and Physicians, Cologne. Publication details to pin. ## References [^idema-partii]: Idema, Timon (2018). *Mechanics and Relativity*. Part II, "Special relativity" (pp. 120–169): the two postulates, the Lorentz transformation in standard configuration `x' = gamma*(x - v*t)`, `t' = gamma*(t - v*x/c^2)`, the invariant interval, time dilation `dt = gamma*dtau`, length contraction `L = L0/gamma`, relativistic momentum `p = gamma*m*v`, energy `E = gamma*m*c^2` and the energy–momentum relation `E^2 = (p*c)^2 + (m*c^2)^2` (page to pin). https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^openstax-v3-ch5]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 5, "Relativity" (pp. 183–240): the postulates, simultaneity, time dilation and the muon experiment, length contraction, the Lorentz transformation and velocity addition, the relativistic Doppler effect, relativistic momentum and energy, and the experimental status of the theory (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^murphy-c]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15, the speed of light c = 2.99792458 × 10⁸ m/s as used with `dE = dm*c^2` (p. 266). https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^schiller-muon]: Schiller, Christoph. *Motion Mountain, Vol. IV: The Quantum of Change*. Particle-lifetime table, muon mean life 2.19703 μs (pp. 128–129). https://open.umn.edu/opentextbooks/textbooks/the-adventure-of-physics-vol-iv-the-quantum-of-change [^derived-sr]: Computed for this article from the closed forms of [^idema-partii], with c = 299,792,458 m/s and the muon mean life of [^schiller-muon]. Lorentz factor: γ = 1.00504 (β = 0.1), 1.25 (0.6), 2.2942 (0.9), 3.2026 (0.95), 7.0888 (0.99), 22.366 (0.999). Axis tilt atan(β): 26.57° (0.5), 30.96° (0.6), 41.99° (0.9), 43.53° (0.95). Length contraction 1/γ: 0.800 at β = 0.6 and 0.436 at β = 0.9; 15 km contracts to 671 m at β = 0.999. Muon: c·τ = 658.65 m; crossing 15 km at β = 0.999 is 22.80 undilated mean lives (survival e⁻²²·⁸ = 1.26 × 10⁻¹⁰) against 1.019 proper mean lives with dilation (survival 0.361), a ratio of 2.87 × 10⁹; at β = 0.99 the figures are 23.00 and 3.245 lifetimes, survivals 1.02 × 10⁻¹⁰ and 0.0390. Kinetic energy (γ − 1)mc² against Newton's β²/2·mc²: 0.04828 against 0.04500 at β = 0.3, an excess of 7.3 %; at β = 0.1 the two differ by 8 parts in 10⁴ of the Newtonian value, i.e. about 1 part in 10⁵ of mc². Rapidity atanh β: 0.5493 (0.5), 1.4722 (0.9), 2.6467 (0.99); composing two boosts of β = 0.9 gives 0.99448. Approaching Doppler factor √((1+β)/(1−β)) = 2.000 at β = 0.6. [^spec-p40]: Matter & Energy Cluster contract, `_registry/plans/PHYSICS_SECTIONS.md` row P40: new root sim (`wt-rel.lorentz`), a Minkowski diagram the reader steers with β over −0.95 to 0.95. The moving frame's axes tilt by atan(β), γ reads out, a light clock's ticks stretch as `dt = gamma*dtau`, a rod shrinks as `L = L0/gamma`, and two events simultaneous in one frame slide apart in the other; the Lorentz transformation is drawn as the sheared grid, and a muon preset (2.197 μs at rest, reaching the ground from 15 km) supplies the measurement. The sim runs live because every quantity is closed form. Row P41, mass–energy equivalence, is a sibling built on this same spec family. [^einstein1905]: Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper." *Annalen der Physik* 17. Pages and DOI to pin. The paper that takes the two postulates as primitive and derives the transformation from them. [^minkowski1908]: Minkowski, Hermann (1908). "Raum und Zeit," address to the 80th Assembly of German Natural Scientists and Physicians, Cologne. Publication details to pin. The four-dimensional geometric reformulation used by this page's diagram. ## Further reading - Idema, Timon (2018). *Mechanics and Relativity*, Part II — a compact derivation of the transformation and its dynamics from the postulates. https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*, Chapter 5 — relativity at first-course level, with the muon experiment worked through. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 ## External links - The Wikipedia pair's *External links* section is the place to look for original works, popular expositions and visualisations; this page lists only the open texts above, which it has read. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Special_relativity.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Special relativity* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Special_relativity.html" data-title="Special relativity"></div> *Built from `MICROSIM_GUIDE/specs/sims/Special_relativity.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Special_relativity) : [Wikitube](https://en.wikitube.io/wiki/Special_relativity) · pinned revision [1374051308](https://en.wikipedia.org/w/index.php?oldid=1374051308) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P40 · sim pending (matter/Special_relativity).*