Time

Time is the quantity that orders events in succession and measures how long they last. It is one of the most basic quantities in physics, and it is also one of the oldest subjects of philosophy. In everyday life, time is the basis of clocks, calendars, and schedules, and in science it appears as the independent variable in almost every equation of motion. Physicists measure it in SI seconds with extreme precision, yet no one has settled what time fundamentally is, whether it flows, or why it seems to run in only one direction. These questions link the subject to classical mechanics, special relativity, general relativity, thermodynamics, and quantum mechanics.

This entry first covers the basic concept, how time is measured, and how ancient cultures understood it. It then follows the main scientific revisions: Newton's absolute time, the relativistic treatment of time as part of spacetime, and the thermodynamic entropy-based account of the arrow of time. Next it turns to philosophical debates about the reality of passage and the status of the present, to psychological and biological timing, and to the role of time in mathematics and computing. It ends with the cosmological origin of time and the unresolved problem of time in attempts to unify quantum theory with gravity.

Basic concept and defining features

Time is, at its most basic, the framework that places events in order and allows the stretches between them to be measured. Philosophers and physicists usually separate two roles. The first is ordering: for any two events, one can ask which came earlier or whether they were simultaneous. The second is metric: a number assigned to a process that states how long it lasted. One analysis of physical theories treats these as two separate functions, and argues that the ordering role is the more fundamental of the two. In Newtonian physics the metric role is derived from the ordering: the length of a process is the difference between the time labels of its final and initial events.

Instants and intervals

An instant is a point on the time axis, a moment at which something happens. An interval is the stretch between two such points. If the instants are $t_1$ and $t_2$, the interval is $\Delta t = t_2 - t_1$. Both concepts are needed because they answer different questions. A clock reading names when an event occurs, while an interval says how long something took, which is what the laws of motion usually require. Some accounts hold that a duration is an ordered collection of instants rather than their sum, since a duration can be divided into shorter durations but not into instants. This distinction matters for the continuum: it implies that a stretch of time is not built up from indivisible moments in any simple additive way.

Time as a parameter and time as an observable

In the equations of classical mechanics, time is an independent variable. Positions and velocities are specified as functions of $t$, and the equations predict how these quantities evolve from one value of $t$ to the next. Time is not itself calculated from the equations; it is supplied from outside. Newton's picture reflects this: he treated time as a uniform background against which motion takes place, and for him clocks were localized physical objects, such as a pendulum clock, that approximated that background. Leibniz, by contrast, held that the relations between events were the only temporal reality, and that time was not a separate entity against which things change.

Time is also something that is directly observed, in the sense that it is read from instruments. Physicists have sometimes summed this up by defining time as what a clock reads. Special relativity keeps the parameter role but makes it frame-dependent: within a single reference frame, coordinate time fixes when each event occurs, how long processes last, and which events are simultaneous with one another. When relativity introduces proper time along a worldline, however, the duration of a process is no longer determined by its endpoints alone, as the twin paradox illustrates.

Time and change

Time is recognized through change. Ordinary experience registers that time has passed because things move, grow, decay, or repeat, and some accounts treat time as empirically tied to a sequence of occurrences. Aristotle took this relation to be constitutive: time is a measure of motion with respect to before and after. Newton reversed the dependence, imagining time as a container that exists whether or not anything happens within it, a view that the early modern theologian and mathematician Isaac Barrow had already argued for against Aristotle. Leibniz sided with the relational side of the debate and defined time as the order of successive events.

The practical method of measuring time rests on the same principle. Clocks create their units from a repetitive process, and the larger units of day, month, and year come from the rotation of the Earth and the orbit of the Moon around the Earth. This makes the question of what time fundamentally is harder to settle: a clock must already presuppose regular change in order to register the passage of time, and regular change is what we use to recognize time in the first place.

Measurement and units

Early timekeeping relied on natural motions that could be read against a scale. The Sun's shadow on a sundial was the first of these tools. The earliest verified sundial is an Egyptian shadow clock from about 1500 BCE, and sundials work only while the sun shines. Their readings also depend on the season and latitude, so their accuracy was limited even in daylight. Water clocks (clepsydrae) extended timekeeping into the night. Water dripping through a small hole lowered a level marked with hour lines. Early models could drift by roughly 15 minutes a day, and later designs with constant-head reservoirs improved on them, but they still lacked precision. Hourglasses were valued less for accuracy than for dividing time into reliable, repeatable intervals. They were especially useful at sea, where ship motion disrupted water clocks.

Mechanical clocks replaced these devices from the late medieval period onward, but their regulation was the key problem. Christiaan Huygens introduced the pendulum clock in 1656, and its regular swing greatly improved daily accuracy on land. Pendulum clocks still performed poorly at sea, where temperature, humidity, and motion disturbed them. John Harrison's H4 marine chronometer of 1761 kept time to within about five seconds over an 81-day voyage, which showed that a portable mechanical timekeeper could solve the longitude problem. Huygens's earlier spiral balance spring had already brought portable watches to within about a minute a day. In the twentieth century, quartz crystal oscillators, whose vibrations are very stable under electrical excitation, replaced mechanical regulators in many clocks. They drifted far less than pendulum clocks, but they remained dependent on the material's properties and on temperature.

From the Earth's rotation to atomic standards

For most of history the second was defined by astronomy. Before 1960 it was defined as 1/86,400 of the mean solar day, but Earth's irregular rotation made this unsatisfactory. The International Astronomical Union then tied the second to the tropical year 1900, and in 1967 the General Conference on Weights and Measures (CGPM) adopted an atomic definition. Atomic clocks work by locking an oscillator to the resonant frequency of atoms, and the first commercial atomic clock, the Atomichron, was sold in the 1950s.

Since 1967 the second has been based on a hyperfine transition in the ground state of caesium-133. In the current SI wording, the caesium frequency $\Delta\nu_{\mathrm{Cs}}$ is fixed at exactly 9 192 631 770 Hz, so one second equals the duration of 9 192 631 770 cycles of that radiation. The CGPM added a note in 1997 specifying an atom at rest at 0 K, and the 2019 SI revision changed the wording of the definition without changing its content, turning the frequency into an exact defining constant. Because the definition refers to an atom at rest and free of external fields, it describes proper time under idealized conditions. Clock corrections for gravity and motion are applied when the second is used in time scales such as International Atomic Time (TAI) and Coordinated Universal Time (UTC).

UTC and leap seconds

UTC is the civil time standard used worldwide, and it is maintained from atomic clocks. Because Earth's rotation is slowing and fluctuating, UTC is periodically adjusted so that it stays within a second of UT1, a time scale tied to the actual rotation of the planet. A leap second is inserted at irregular intervals for this purpose. Recently the trend has been different: since about 2020, Earth has sometimes completed a rotation slightly faster than 86,400 seconds, so a negative leap second has become conceivable.

In 2022 the 27th CGPM passed Resolution 4, which calls for the maximum allowed difference between UT1 and UTC to be increased by 2035. The CGPM proposed that UT1 and UTC could drift apart by about a minute without a leap second being added for at least a century. Russia voted against the proposal and asked for a later date, and the 2026 CGPM was tasked with settling the details. Critics of abolition point to disruption for computing and navigation systems that rely on UTC, while supporters argue that the link to the rotation of the Earth would remain in a looser form.

Ancient conceptions and early philosophy

Timekeeping in Mesopotamia and Egypt

Systematic time reckoning grew out of observing the sky in the river valleys of Mesopotamia and ancient Egypt. The Sumerians developed the base-60 system, and the Babylonians inherited it. Evidence from proto-cuneiform tablets from Uruk and Jemdet Nasr, dated to about 3200–3100 BCE, shows that time was already being recorded in this way. The base suited astronomy because 60 has many divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, 30), which made fractions and division simpler. Babylonian astronomers divided the ecliptic into 30° arcs, roughly one per month, and used these arcs to track the Sun through the zodiac. That practice is the origin of the 360-degree circle and of the division of the hour into 60 minutes and the minute into 60 seconds. Babylonian months began with the lunar crescent, and intercalary months were added so that the lunar calendar stayed in step with the seasons.

Egyptian practice combined a civil calendar with star observation. The civil calendar had 365 days: twelve months of thirty days plus five extra days. It had no correction for the extra quarter day in each year, so over centuries it drifted against the seasons. Daytime hours were tracked with shadow clocks, and night hours were tracked with the decans, groups of stars that rose in sequence at roughly ten-day intervals. Water clocks (clepsydrae) were also used. All of these instruments had clear limits. Sundials gave hours that varied with the season, because the Egyptians and Babylonians divided the daylight into equal parts whose length changed through the year. Water clocks depended on flow rates that could change with temperature and with the level of water in the vessel. Neither could give reliable precision at the level of minutes.

Aristotle: time as the number of motion

In Book IV of his Physics, Aristotle defined time as the "number of movement in respect of the before and after." His argument began from perception. We notice time only when we notice change: if the mind passes through a state in which nothing distinguishable happens, we do not judge that any time has elapsed. From this he concluded that time is neither motion itself nor something wholly independent of motion. Because motion is continuous, time is continuous too. The "now" serves as a boundary between the earlier and the later, and before and after are ordered by their distance from the present. Aristotle also faced the objection that counting seems to require a mind. He answered that what is counted, the motion, can exist without a soul, and that time is the numerable aspect of that motion. Modern commentators disagree about whether Aristotle meant time as a measure or as an order of changes.

Plato: the moving image of eternity

In the Timaeus, Plato describes a divine craftsman who, wishing to make the world as like its eternal model as possible, made "a moving image of eternity." This image is time. Eternity (Greek aiōn) rests in unity, while time moves in accordance with number, and it came into being together with the heavens. The Sun, Moon, and planets were fashioned partly to define and measure time. Scholars debate whether Plato's eternity is timeless or everlasting. Later Platonists, such as Plotinus, were the ones who gave the formula its enduring phrasing. Plato himself offered the cosmogony as a "likely story," not as certain doctrine.

Beginnings, cycles, and the eternal cosmos

Ancient thinkers disagreed about whether time had a beginning. Aristotle held that motion, and therefore time, had no beginning, because a first motion would itself need a prior cause and time before it. For him the cosmos was eternal. The Stoics, by contrast, taught that the cosmos was periodically consumed in fire and reborn in an identical cycle, so history itself repeated. Traditions such as the Indian yuga cycles and Greek ideas of a "Great Year" also imagined cosmic time returning to its starting point. Plato's account of the creation of time, by contrast, gave the world a beginning in myth. Later Christian thinkers such as Philoponus in the sixth century and Augustine argued that time was created together with the world, in opposition to the Aristotelian idea of an eternal cosmos. Their arguments shaped medieval debate on this issue.

Calendars and astronomical cycles

Calendars are human systems that impose regular units on three natural rhythms that do not divide evenly into one another. The day is set by the rotation of the earth relative to the sun. The month follows the phases of the moon, and the year is set by the seasons, which track the Earth's orbit. A tropical year is approximately 365.2422 days long and a synodic month is approximately 29.5306 days long, so a tropical year is approximately 12.36826 months long. Every calendar design is a compromise over that mismatch.

Lunar, solar and lunisolar designs

Early lunar calendars began each month when the new crescent was first seen. The Babylonians originally had a lunar calendar with months that started with the first visible appearance of the New Moon crescent. This automatically synchronised the lunar calendar with the Moon. Keeping the year in step with the seasons required an extra month when the year drifted too far from the spring equinox. Later, Babylonian astronomers adopted a fixed scheme. From around the fourth century before the Christian Era the Babylonians switched to a more rigid system with 7 intercalary lunar months in 19 solar years, a pattern now called the Metonic cycle.

Lunisolar calendars combine these two tracks. Twelve lunations amount to about 354 days, so a year of twelve months falls well short of the solar year, and every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month. Different cultures chose different rules for when to insert that month. Some use regular cycles such as the 19-year Metonic cycle, as in the Hebrew calendar and in the determination of Easter, while others calculate lunar phases, as in the Hindu lunisolar and Chinese calendars. Purely lunar calendars, by contrast, do not intercalate, since they do not seek to synchronise with the seasons. Solar calendars avoid the problem altogether. The Egyptian civil calendar, for example, used a fixed year of 365 days, although it showed traces of its origin in the earlier lunar calendar, which continued to be used alongside it for religious and agricultural purposes.

The Julian and Gregorian reforms

In 46 BCE Julius Caesar reformed the Roman calendar. His "Julian" calendar was no longer dependent on the observation of the new moon, but followed an algorithm of introducing a leap day every four years. This dissociated the calendar month from lunation. The Julian year was slightly too long, and the error accumulated. The difference was about 10 days at the beginning of the 16th century. The drift mattered because the date of Easter depended on the spring equinox. By the 1570s the equinox had moved toward March 11, far from the March 21 date set at the Council of Nicaea in 325 CE.

Pope Gregory XIII responded with the bull Inter gravissimas, issued on February 24, 1582. The reform had two parts. Friday, October 15, 1582 was decreed to directly follow Thursday, October 4. It also changed the leap-year rule. Every year divisible by four is a leap year, except for years that are divisible by 100 but not also divisible by 400. This gives an average year of 365.2425 days, and the calendar is still in use as the international civil standard. Adoption was uneven. It was quickly adopted by much of Catholic, but not Protestant, Europe. Sweden made a partial change in 1712 by adding a February 30, while Great Britain and its colonies resisted Gregorian reforms for two centuries. Britain adopted the reformed calendar in 1752, and historians still have to reconcile "Old Style" and "New Style" dates from that period.

Time zones and standard clock time

Calendars organise days, but the question of what hour it is had a separate history. Before the nineteenth century, time of day was a local matter, and most cities and towns used some form of local solar time, maintained by a well-known clock. Railways made this unworkable. In Britain, London time was adopted as the standard time, and the Great Western Railway was the first to adopt it, in November 1840. By 1855 nearly all public authorities, such as churches and town halls, set their clocks to 'railway time', displayed on station clocks by station masters who adjusted them according to the signals from Greenwich.

In North America, the railroads adopted a zone system on November 18, 1883. The Canadian engineer Sandford Fleming had proposed dividing the day into twenty-four zones of one hour each. In October 1884 the International Meridian Conference in Washington, D.C., agreed to divide the world into 24 hourly time zones based on the Greenwich meridian. The conference had no power to enforce its recommendations. It took several decades for the gradual worldwide adoption of a twenty-four-hour time-zone-based system centered on Greenwich Mean Time (GMT). In the United States, zones were not legally required until the Standard Time Act of 1918, despite the earlier railway practice.

Newtonian absolute time

Newton's definitions in the Principia

In the Scholium that follows the definitions of Isaac Newton's 1687 Principia, time is split into two kinds. The first is absolute, true, and mathematical time, which Newton described as a flow that belongs to time itself and has no reference to anything outside it. He also gave it the alternative name duration Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. The second kind is relative time, meaning any perceptible, external gauge of duration that rests on motion. Hours, days, months and years belong here, and people ordinarily use them in place of true time Relative, apparent, and common time is any sensible and external measure of duration by means of motion; such a measure—for example, an hour, a day, a month, a year—is commonly used instead of true time.

Newton held that the flow of absolute time cannot be altered, even though motions can be sped up or slowed down All motions can be accelerated and retarded, but the flow of absolute time cannot be changed. Because absolute time is defined without reference to events, it would still exist if nothing changed. Commentators read this as a sign that Newton treated the flow as continuous rather than made of discrete ticks That Newton takes absolute time to "flow equably" is a good indication that he thought that the flow of time was continuous and not discrete.

Newton's practical motivation came from astronomy. Ordinary solar days vary in length, so astronomers apply corrections to obtain a more accurate time scale for celestial motion. Newton used this to argue that the time we use in daily life is only an approximation of a truer underlying time For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time. Absolute space was defined in parallel with absolute time, and the two were treated as the stage on which bodies move and forces act For this reason he spoke of absolute space and absolute time, so as to distinguish these entities from the various ways by which we measure them.

Universal simultaneity

The Scholium does not spell out simultaneity explicitly, but the framework implies it. If one uniform flow governs every part of the universe, then any two events that carry the same reading of absolute time are simultaneous for every observer, whatever their state of motion. Two clocks moving relative to each other would therefore agree about how time passes. Newton's relativity of uniform motion works in the same spirit: the motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line. Under this scheme, a change of uniform velocity does not change the time coordinate. This is the Galilean structure that later textbooks use.

This assumption was overturned in the twentieth century. In 1905 Albert Einstein argued that time passes at different rates for different observers, which rejects the universal flow altogether in 1905, Albert Einstein changed altogether our notion of time, through the formulation of the special theory of relativity and stating, in particular, that time flowed at different rates for different observers. The consequences are developed in special relativity.

Leibniz's relational critique

Gottfried Wilhelm Leibniz attacked the Newtonian picture in his correspondence with Samuel Clarke, who acted as Newton's spokesman, during 1715 and 1716. The exchange ended when Leibniz died in 1716 The exchange, which consists of ten letters, began in 1715 and ended in 1716 with Leibniz's death; Leibniz and Clarke each produced five letters. Leibniz's core claim was that time has no existence apart from events. He described it as an order of successions, parallel to space as an order of coexistences I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions.

His most cited argument concerns the principle of sufficient reason. If time were a container that existed on its own, then God's choice to create the world at one moment rather than an earlier one would have no reason behind it, since the two alternatives would look the same. Leibniz concluded that times considered apart from events are nothing at all times, considered without the things or events, are nothing at all, and that they consist only in the successive order of things and events. In the Leibnizian view, when no events occur there are no relations to define and therefore no time He believes that when there are no events, there are no relations to define, leaving no time.

Clarke's reply drew a sharp distinction between the order of events and the amount of time between them. Order and interval are different things, so the same sequence can be preserved while the durations separating its members differ the distance, interval, or quantity of time or space, wherein one thing follows another, is entirely a distinct thing from the situation or order. This answer, which keeps duration as a quantity separate from the ordering of events, is the foundation of the Newtonian position. Later relational approaches, which try to derive duration from event relations alone, inherit the same problem.

Role in mechanics and prediction

Absolute time gives classical mechanics a single parameter $t$ that applies to every body. In Newton's laws of motion, force and acceleration are linked through derivatives with respect to this parameter: $$\mathbf{F} = m\,\frac{d^2\mathbf{x}}{dt^2}$$ Because each body obeys the same $t$, the equations of motion can be integrated along one timeline, and the state of a system at a later instant follows from its state at an earlier one. This is the basis for the deterministic, Laplacian picture of classical prediction, in which knowing the positions and velocities of all bodies at one moment fixes the motion at all other moments. The same assumption of an invariant time parameter remained untouched until relativity replaced the universal flow with frame-dependent time coordinates.

Special relativity and the relativity of time

In the late nineteenth century, most physicists took light to be a wave in a medium called the luminiferous ether, which was assumed to be at rest in an absolute frame. If that were true, light's speed measured on Earth should depend on the direction of Earth's motion through the ether. In 1887 Albert Michelson and Edward Morley tested this with an interferometer that compared light travelling along and across Earth's orbital motion. Michelson reasoned that, if the speed of light were constant with respect to the proposed ether through which Earth was moving, that motion could be detected by comparing the speed of light in the direction of Earth's motion and the speed of light at right angles to Earth's motion. No difference was found. The null result was later explained by the Lorentz–FitzGerald contraction of the arms, but it ultimately led to the proposal by Albert Einstein in 1905 that the speed of light is a universal constant. Einstein's special theory rested on two postulates: the laws of physics take the same form in all inertial frames, and light has the same speed in every such frame. Once that was accepted, time itself had to be treated differently.

The most important consequence is the relativity of simultaneity. Two events that are simultaneous for one observer are generally not simultaneous for an observer moving relative to the first. Einstein illustrated this with a train and lightning strikes at both ends of a platform: an observer at the midpoint of a moving train sees the flashes at different times, because she is moving toward one light and away from the other. Consequently, the duration between two events, and even the spatial separation between them, depends on the frame. The transformations that connect these measurements are the Lorentz transformations. For a clock moving at speed $v$, the Lorentz factor is $\gamma = 1/\sqrt{1-v^2/c^2}$, and a time interval $\Delta t_0$ measured on the moving clock appears in the rest frame as $\gamma\,\Delta t_0$. This is time dilation; the same factor shortens lengths along the direction of motion, which is length contraction. Time dilation is symmetric in the sense that each observer sees the other's clock running slow, which is why the frame-dependence is a property of the relations between observers rather than of any one clock.

The time measured along a clock's own worldline is called proper time. Proper time is the same for all observers, and it is the quantity that a clock actually records. The combination of space and time into spacetime is what makes these frame-dependent intervals consistent with a single invariant quantity, and it is the geometrical framework that general relativity later extended to include gravity.

Experiments have confirmed these predictions many times. Cosmic-ray muons provide a classic example. Muons are created high in the atmosphere and have a mean lifetime of about 2.2 microseconds in their rest frame. At 0.99c, a naive classical estimate gives a range of roughly 660 metres, which is far too short to reach sea level, yet many muons are detected there. This apparent paradox was resolved by special relativity's prediction of time dilation. In 1941 Rossi and Hall compared muon counts at altitude and at sea level, and the 1963 Frisch and Smith measurements on Mount Washington refined the test. Particle accelerators have since verified the effect with much higher precision, showing that the lifetime of fast-moving muons in storage rings is extended by the expected Lorentz factor. In 1971 Hafele and Keating flew caesium atomic clocks around the world on commercial airliners; the measured differences in elapsed time matched the predictions of special relativity together with the gravitational corrections from general relativity.

The twin paradox is the best-known thought experiment built on these effects. One twin stays on Earth while the other travels at high speed to a distant point, turns around, and returns. The traveller is younger on arrival. Each twin can initially regard the other as moving, and so each expects the other to be younger, which seems contradictory. The usual resolution is that the situations are not symmetrical: the travelling twin changes inertial frames at the turnaround, while the stay-at-home twin remains in a single inertial frame. Calculating each twin's proper time over the whole journey, using a consistent frame choice, gives a definite answer: the traveller's path through spacetime is shorter in proper time. Some authors have argued that the resolution must involve general relativity through the equivalence principle, but others maintain that acceleration plays no role in the analysis, other than to provide the asymmetry whereby the traveller must occupy more than one inertial frame, and the relativity of simultaneity is the key, not gravity. Within special relativity alone, the paradox therefore reduces to a careful accounting of which clock was in which frame at each moment.

General relativity and time in curved spacetime

In general relativity, time stops being a single background shared by all events and becomes one of the four coordinates of spacetime, whose geometry is shaped by mass and energy. What a clock actually records is the proper time along its own path. For a static, non-rotating mass, the Schwarzschild solution of 1916 gives the relation between proper time and the coordinate time $t$ used by a distant observer:

Here $r_s$ is the Schwarzschild radius. A clock held at fixed radius $r$ therefore ticks at the rate $d\tau/dt=\sqrt{1-r_s/r}$, which is always less than one. The coordinate time is a bookkeeping label; only the proper time corresponds to what a physical clock reads.

Gravitational time dilation

The equivalence-principle implies that clocks in a stronger gravitational field run more slowly than clocks higher up. Early predictions of this effect were extremely small, and the first decisive laboratory test came in 1959, when Robert Pound and Glen Rebka at Harvard measured the frequency shift of gamma rays from iron-57 over a vertical height of 22.5 metres. They used the Mössbauer effect to obtain a narrow enough resonance. Their result agreed with the predicted fractional shift $\Delta\nu/\nu = gh/c^2$, which is about $2.6\times10^{-15}$ for their setup. Later experiments, including the 1971 Hafele–Keating flights of caesium clocks around the Earth and the 1976 Gravity Probe A rocket test with a hydrogen maser, confirmed the effect with increasing precision.

Satellite navigation

The Global Positioning System is the most familiar everyday case. A GPS satellite experiences two competing effects. Its orbital speed makes its clock run slow by about 7 microseconds per day, as predicted by special relativity, while its weaker gravitational potential makes it run fast by about 45 microseconds per day. The net gain is about 38 microseconds per day. Because a light-travel time error of that size corresponds to kilometres of range error, uncorrected GPS positions would drift by roughly 10 km per day. Engineers handle this by setting the satellites' atomic oscillators to a slightly lower frequency before launch, so that after the orbital effects act, the clocks tick at the ground-referenced rate. Ground receivers then apply further corrections. Published figures differ slightly in rounding (sources cite values from about 38 to 39 microseconds per day), but the magnitude and necessity of the correction are not in dispute.

Black holes and the limits of coordinate time

Near a black hole, the factor $\sqrt{1-r_s/r}$ approaches zero as $r$ approaches $r_s$. For a distant observer, an object falling toward the horizon appears to slow down, its light increasingly redshifted, and it seems never to cross. For the infalling observer, however, the crossing takes a finite proper time, and nothing special is felt at the horizon itself. The divergence of the Schwarzschild coordinate time at $r=r_s$ is therefore a property of the coordinate chart rather than of the physics. Coordinates such as Eddington–Finkelstein or Kruskal–Szekeres cover the horizon smoothly, showing that the boundary is a coordinate singularity while the horizon is a regular region of spacetime.

Inside the horizon, $r$ becomes a time-like coordinate, so the radial decrease toward $r=0$ is unavoidable for any infalling path. There the curvature becomes infinite and proper time along such a path comes to an end at a singularity. Because the classical description breaks down there, the singularity marks the region where a theory of quantum-gravity is expected to be needed. More generally, in curved spacetime a single global coordinate time that every clock agrees on usually exists only within limited regions or special symmetric cases, so the notion of a universal clock reading from Newtonian physics must be replaced by path-dependent proper time.

Time in quantum mechanics

Time as an external parameter

In non-relativistic quantum mechanics, a system's state is described by a wave function that evolves according to the Schrödinger equation:

Here $t$ is a parameter that specifies the moment at which the state is evaluated. It is not an operator acting on the state space. Position, by contrast, is an observable: each coordinate is represented by an operator whose eigenvalues are possible measurement outcomes, and the wave function assigns probabilities to them. Standard theory therefore treats time as a background variable that is external to the dynamics of any particular system. This asymmetry is the source of most of the difficulties discussed below.

The time–energy relation

The Heisenberg-style expression $\Delta E\,\Delta t \gtrsim \hbar/2$ cannot be read like the position–momentum relation, because $\Delta t$ is not the spread of any observable. The rigorous form was given by Mandelstam and Tamm in 1945. For an observable $\hat A$ with no explicit time dependence, they showed that $\Delta A\,\Delta H \ge \tfrac{\hbar}{2}\,|d\langle A\rangle/dt|$. Dividing through defines a characteristic time $\tau_A = \Delta A / |d\langle A\rangle/dt|$, the time needed for the mean of $A$ to shift by about its own spread. This quantity satisfies $\tau_A\,\Delta H \ge \hbar/2$. The relation is therefore a bound on how fast a state can change, and it is used to set quantum speed limits and to estimate the lifetimes of decaying states. Its interpretation remains disputed: some analyses argue that the derivation and the inequality cannot be regarded as universally valid.

Measuring time and the missing operator

The deeper issue is whether time can be an observable at all. Pauli argued in 1926 and again in 1933 that a self-adjoint operator $\hat T$ canonically conjugate to $\hat H$, with $[\hat T,\hat H]=i\hbar$, would force the Hamiltonian to have a spectrum unbounded in both directions. Since energy for most systems is bounded below, he concluded that a time operator must be abandoned. Later work has questioned how general this conclusion is, and some authors argue that Pauli's implicit assumptions are inconsistent in a single Hilbert space. Alternative programs treat time as a positive operator-valued measure (POVM) describing the statistical distribution of event times, rather than as a conjugate operator. In these approaches the expectation value of such a measure gives an average event time. Other proposals use quantum clocks, in which time emerges from entanglement between a clock subsystem and the rest of the universe, or they invert the roles of time and energy. None of these is universally accepted, and in practice time is still read from an external clock treated as a classical reference.

Tunneling and time-dependent experiments

Tunneling raises the question of how long a particle spends under a classically forbidden barrier, and the question has no agreed definition. A key difficulty is that the start of the tunneling process is not sharply defined; one study notes that there is no obvious reason to assume the electron enters the barrier at the instant of maximum field. The main experimental tool is the attoclock (attosecond angular streaking). Its first demonstrations in 2008, by Eckle and colleagues, used an elliptically polarized laser pulse to ionize atoms and read the emission angle of the electron as a clock. Those early results were interpreted as consistent with zero delay. Later measurements at lower laser intensities, including a 2014 experiment and a 2017 study in noble gases, reported a non-zero tunneling time. A 2020s experiment on the hydrogen atom did not detect a measurable delay. A 2022 streaking method reported a tunneling time close to zero within an accuracy of a few attoseconds, and a 2025 phase-resolved attoclock tied the measurement to the carrier-envelope phase of the laser. A 2020 review concluded that the decade of attoclock experiments had ended inconclusively, leaving room for a finite tunneling time. The results are therefore sensitive to the choice of observables and models, which is why the attoclock is best understood as a test of the theory of strong-field ionization as much as a direct clock for quantum processes.

Thermodynamic arrow of time

The most widely discussed physical account of time's direction comes from thermodynamics. The line of work that led to entropy grew out of attempts to understand the efficiency of heat engines and the Carnot cycle. Clausius developed the idea of entropy while studying the efficiency of heat engines and the Carnot cycle. In 1865 Clausius summarized his findings in a famous claim, quoted as "The entropy of the universe tends to a maximum." In modern form, this is the statement that the entropy of an isolated system cannot fall during a spontaneous process, so that $dS \ge 0$ for such processes.

Boltzmann's statistical reading

Clausius treated entropy as a macroscopic quantity and had no molecular picture of it. Ludwig Boltzmann supplied one. His H-theorem of 1872 tried to derive the growth of entropy from molecular collisions in a gas, but it required the Stosszahlansatz (molecular chaos hypothesis)—itself a time-asymmetric assumption. Boltzmann's later and more durable idea was to count microscopic configurations. In this picture entropy is proportional to the logarithm of the number $W$ of microstates compatible with a given macrostate, $S = k_B \ln W$, where $k_B$ is the Boltzmann constant and the statistical description is set out in statistical mechanics. A system drifts toward states with more microstates simply because there are overwhelmingly more of them. Boltzmann insisted that the Second Law is only statistical, entropy increases most of the time, not all the time. The decrease in entropy is not an impossibility but is only highly improbable.

The name "arrow of time"

The explicit link between entropy and the direction of time is usually credited to Arthur Stanley Eddington. The phrase itself appeared in 1927, in his popular book <i>The Nature of the Physical World</i>, where Eddington coined the term "Arrow of Time" ... He connected "Time's Arrow" to the one-way direction of increasing entropy required by the second law of thermodynamics. Eddington also distinguished a separate cosmological direction tied to the expansion of the universe, which is treated in its own section. The historical association is contested. Arieh Ben-Naim has argued that there is no relationship between either entropy or the second law and the so-called arrow of time. He regards this false association between the Second Law and time as having first been suggested by Arthur Eddington. Other authors respond that the connection survives in a refined, statistical form.

Time-symmetric laws and the reversibility problem

The microscopic laws do not single out a direction. Newton's equations, Maxwell's equations, the Schrödinger equation, and the Einstein field equations are all invariant under time reversal (or, more precisely, under CPT symmetry). A time-reversed film of a collision looks as lawful as the forward one. This raised an objection in the 1870s and 1880s: if every velocity were reversed, entropy should fall. Boltzmann conceded that a reversed motion could lower entropy, though only for a limited time, and he argued that a gas could not return to a highly ordered initial state in practice. In the second half of the XIX century, Maxwell and Boltzmann became aware of the tension between the linear time evolution and the reversibility of the fundamental equations of motion, and later Poincaré showed in 1890 the eternal recurrence of motion in phase space. Recurrence means that a bounded system eventually returns close to its starting state, which again shows that entropy growth is a matter of probability over long times rather than a law of the equations.

The standard resolution places the asymmetry in the initial conditions rather than in the dynamics. The prevailing account holds that the universe began in an extraordinarily ordered, low-entropy state, and that the second law records its departure from that state. Roger Penrose proposed a gravitational version in which the initial singularity has vanishing Weyl curvature, while the final singularity is Weyl-dominated. David Albert later formalized the requirement as a past hypothesis that the early universe had very low entropy.

Limits and open questions

The statistical picture has concrete consequences. With the advent of single molecule techniques, research has shown that entropy can spontaneously decrease for small systems consisting of a single particle or a few particles. These transient reversals are quantified by fluctuation theorems, which predict that their probability falls off rapidly with the size of the system and the time window considered. Some authors have also questioned the foundations of the entropy argument itself. Statements like "entropy always increases" are meaningless; entropy, in itself does not have a numerical value. Therefore, one cannot claim that it increases or decreases. Entropy changes are meaningful only for well-defined thermodynamic processes in systems for which the entropy is defined.

Whether entropy alone accounts for all temporal asymmetries remains disputed. The thermodynamic arrow is often linked to the psychological arrow of memory and to the causal arrow of cause and effect, but whether these follow from entropy growth or from something more basic is an open question. The status of the initial low-entropy condition, which is itself unexplained, is one of the central problems discussed in later sections.

Other arrows and asymmetries of time

The second law of thermodynamics is only one of several ways in which the world distinguishes past from future. Other asymmetries appear in memory, in the way causes and effects are ordered, in the large-scale structure of the universe, and in some corners of particle physics. Physicists and philosophers disagree about how many of these are independent, and whether they can all be traced back to one origin.

The psychological arrow

We remember the past but not the future. Stephen Hawking gave this asymmetry the name "psychological arrow of time." It is a record-keeping asymmetry: traces such as memories, fossils, and written records accumulate in the direction we call past. Hawking also argued that this arrow is tied to the thermodynamic one. In a hypothetical contracting universe, where entropy would fall, he suggested the psychological arrow would reverse as well. Some physicists have described it as not well defined in physical terms, and usually treat it as subsumed under the thermodynamic arrow, which in turn is sustained by cosmic expansion.

Whether entropy actually explains the asymmetry of traces is contested. Mlodinow and Brun (2014) followed Boltzmann's approach and expected the psychological and thermodynamic arrows to line up. Critics have argued that such a link must be physically necessary, not a coincidence of coextension, before it counts as an explanation. Other researchers have sought a time-symmetric formulation of quantum theory in which the psychological arrow emerges from boundary conditions rather than from entropy directly.

The causal arrow

Causes generally precede their effects, and we can influence the future but not the past. Philosophers and physicists have tried to ground this in the structure of causal relations rather than in the thermodynamic arrow. In electromagnetism, the observed radiation is retarded: waves spread outward from their sources, while the time-reversed solutions, in which waves converge onto a source, are not seen in practice. Both possibilities are allowed by the field equations, so the asymmetry comes from the choice of boundary conditions.

The causal asymmetry is also central to scientific method. Techniques such as causal modeling and time-series analysis depend on the fact that the direction of time matters for inferring which variable drives which, even though the basic equations of classical mechanics make no such distinction. Some recent work treats the causal arrow as more fundamental than the thermodynamic one, building toy models in which causal structure is imposed first and entropy-like behavior emerges afterward.

The cosmological arrow

The universe is expanding, and this gives one of the clearest large-scale asymmetries. Observations of redshifts of distant galaxies, as described by Hubble's law, show that the scale of the cosmos grows with time. Expansion alone does not produce an arrow, but it provides the setting in which entropy can increase.

The more demanding question is why the early universe was so smooth. Roger Penrose's Weyl curvature hypothesis proposes that the Weyl tensor, which measures the part of spacetime curvature that is not fixed by local matter and energy, vanishes at the initial singularity. This would explain why the early universe had very low gravitational entropy. Gravity then drives matter to clump into stars, galaxies, and black holes, and because black hole entropy is so large, total entropy keeps rising. The hypothesis remains a conjecture, and some researchers regard it as an extra assumption added to standard cosmology rather than a derived result. The cosmic microwave background supports the smoothness of the early universe, but does not on its own settle the question.

Does entropy account for everything?

A common view is that the other asymmetries are consequences of the thermodynamic arrow combined with a low-entropy initial condition. Time-symmetric dynamical laws, solved with asymmetric boundary conditions, could in principle produce the radiation, expansion, and inhomogeneity arrows. Some theorists propose that the CP asymmetry in weak interactions, which is connected to the approximate symmetry under combined charge conjugation and parity, might instead arise from a spontaneously broken symmetry of the Hamiltonian. That would make it independent of entropy in its origin.

Critics of the entropy-first view argue that low-entropy beginnings only restate the problem, since the question "why was the early universe so ordered?" remains open. Some propose that a past-boundary condition, sometimes called the Past Hypothesis, should be stated as an independent law. Others argue that the causal and psychological arrows must be treated as separate facts. The question is still unresolved, and the answer depends on whether one regards time asymmetry as a property of the laws, of the boundary conditions, or of the observers who record events.

Cosmic time and the origin of time

Cosmic time and the age of the universe

In standard cosmology, cosmic time is the time coordinate that tracks the history of an expanding universe for a typical observer, and the age of the universe is the interval since the scale factor of the universe would have shrunk to zero. Within the Big Bang framework this age is a computed quantity rather than a directly observed one. It follows from the expansion rate and the matter content, combined in the integral

where $z$ is the redshift and $H(z)$ the expansion rate at that epoch. The best-fitting value is about 13.8 billion years. The 2018 analysis of the cosmic microwave background by the Planck satellite, interpreted within the standard $\Lambda$CDM model, gave 13.787 ± 0.020 billion years. Earlier NASA WMAP results gave a value of 13.77 billion years with a larger uncertainty, and Atacama Cosmology Telescope data later produced a figure consistent with Planck.

The age was once a serious problem. Some stellar ages, derived from stellar evolution models applied to the oldest stars, seemed to exceed the cosmological age. Most of these stellar estimates now carry large error bars and are consistent with the cosmological value. The matter is not fully settled, however. Measurements of the present expansion rate, parameterized by the Hubble constant, give values that differ from the Planck-based inference, and some groups have proposed ages of roughly 12.5 to 13 billion years. Whether this discrepancy reflects measurement error or a need to revise the standard model remains open.

Observational foundations

Three lines of evidence underpin the picture. The first is the expansion of space. In the late 1920s, redshift measurements of distant galaxies showed that their light was stretched toward longer wavelengths, and the amount of stretching grew with distance. Hubble's linear relation between distance and recession velocity implied that the universe is expanding. The inverse of the Hubble constant, roughly 14 billion years for the currently measured values, provides a rough estimate of the age, which is why the expansion rate and the age are tightly linked.

The second line of evidence is the cosmic microwave background, discovered accidentally in 1965. It is radiation released when the universe had cooled enough for atoms to form, roughly 370,000 years after the start of the hot phase. Its nearly uniform temperature, with tiny fluctuations of about one part in 100,000, records the conditions of that early epoch. Because general relativity connects these fluctuations to the rate of expansion, cosmologists can run the model backward and infer the elapsed time.

The third line is the agreement between the predicted and observed abundances of light elements such as hydrogen, helium and lithium, produced in the first few minutes. Together these results make the hot, dense early phase of the universe well supported, even though the description of its very earliest moments is much less secure.

Inflation and the question of a first moment

The hot Big Bang and the first instant of time are not the same thing. Early descriptions often equated the two, with the expansion emerging from a singularity of infinite density. Since the early 1980s, the theory of cosmic inflation, developed by Alan Guth, Alexei Starobinsky, Andrei Linde and others, has proposed an earlier phase in which the universe expanded exponentially, doubling in size at least 80 times within a tiny fraction of a second. Inflation ends in a process called reheating, which converts the energy driving the expansion into hot particles and begins the familiar hot Big Bang.

Inflation explains features that the standard hot model leaves unexplained, including the large-scale uniformity of the universe and the origin of the density fluctuations seen in the microwave background. These fluctuations are traced to quantum fluctuations stretched to cosmic scales during inflation. The same exponential stretching, however, dilutes almost any trace of the conditions that existed before it. Physicists therefore generally regard the pre-inflationary state as unknown and, in practice, possibly unrecoverable from observations. Models of eternal inflation go further, suggesting that inflation produces an effectively unbounded set of separate regions, though these ideas are speculative.

Singularities, beginnings, and beginningless models

Extrapolating the classical equations of general relativity backward suggests a singularity, a point where density and curvature become infinite and the equations break down. Hawking and Penrose showed that singularities are generic under broad assumptions, so the classical theory points toward a beginning. Yet the theory cannot be trusted at the extreme densities involved, and most physicists expect that quantum gravity would modify the picture near the earliest times. The earliest interval that can be meaningfully discussed, the Planck epoch, is a regime where present-day physics is not known to hold.

Several alternatives avoid a first instant. In the Hartle–Hawking proposal, the universe has no sharp initial boundary in imaginary time, so asking what happened before it becomes meaningless, and time in the usual sense does not extend back to a first moment. Bouncing models, in which a previous contraction turns into the present expansion, and quantum models in which the singularity is replaced by a smooth transition, are also studied. None of these has direct empirical support. Future measurements, such as searches for primordial gravitational waves in the polarization of the microwave background, could constrain inflationary scenarios, but they are not expected to reveal what lay before inflation itself. The status of questions about time before the Big Bang therefore depends on theory, not on established observation.

Philosophy of time: passage, presentism, and eternalism

McTaggart's A-series and B-series

In 1908 the British philosopher J. M. E. McTaggart distinguished two ways of arranging moments in time. The B-series rests on the relations of earlier-than and later-than, and first, events might be ordered by the relation of earlier than, which gives a series McTaggart calls the B-series. The A-series adds a second ordering: designating some moment within the B-series as the present moment gives a series that McTaggart calls the A-series. On this picture, an event moves from future to present to past, so its A-properties change while its B-relations stay fixed.

McTaggart's argument has two steps. He first held that change requires the A-series, because the B series without the A series does not involve genuine change, since B-series positions are forever "fixed" while A-series positions are constantly changing. He then argued that the A-series contradicts itself. The properties of being future, present and past are mutually exclusive, yet each event must eventually possess all of them. McTaggart concluded that, since neither the A-series nor the B-series can order the time series, time is unreal. Most later philosophers rejected this conclusion, but the vocabulary he introduced organized the whole later debate about passage.

Presentism and relativity

Presentism holds that only the present is real. What has happened has lapsed, and what will happen has not yet come into being. In its usual formulation, presentism is the belief that everyone shares a unique, 'real' present state while their common past states no longer exist and their common future states are yet to exist.

The chief pressure on presentism comes from special relativity. Observers in different states of motion assign different times to distant events, so there is no frame-independent "now" that spans space. Eternalism follows from special relativity because relativistic reference frames exist whose observers disagree about the temporal ordering of spacelike separated events. Hilary Putnam's version of an argument traced to Rietdijk states the conclusion directly: presentism plus the relativity of simultaneity implies eternalism, i.e. the block universe. Philosopher of science Dean Rickles reports that the consensus among philosophers seems to be that special and general relativity are incompatible with presentism.

Presentists have tried several responses. Some posit a privileged reference frame whose simultaneity judgments are the true ones, even though no experiment could distinguish it; Dean Zimmerman is among those who defend this option. Christian Wüthrich argues that absolute simultaneity is available to presentists only if they reject either empiricism or relativity itself. General relativity complicates the picture further, since some reformulations of gravity do posit a preferred global simultaneity, which keeps the debate open at the level of physics.

Eternalism and the block universe

Eternalism holds that all points in time are equally real. The past, the present, and the future are all real. Within four-dimensional spacetime, this view is often described as the block universe, a picture in which all of the points in space-time are fixed and unchanging. On this view, "past" and "future" are relational labels. Eternalism does not necessarily do away with the concepts of past and future, but considers them more like directions, so that whether something is in the past or future depends entirely on your frame of reference.

Eternalism has its own costs. A static block seems to leave no room for the felt movement of the present, and it is hard to say what "happening" means within it. Intermediate positions such as the growing block, in which past and present are real but the future is not, try to preserve a privileged present, but they face difficulties of their own in reconciling with time dilation and the relativity of simultaneity.

Illusion, consciousness, or physical fact?

The apparent flow of time can be read in three main ways. The first treats it as an illusion. B-theorists accept that time does not really pass; the entire timeline simply is. Critics press the explanatory problem: if all times exist equally, the question is why we experience a flow at all, and the B-theory has no obvious answer.

The second locates passage in consciousness. Here the experienced present is indexical, anchored to the subject who experiences it, and the apparent movement of "now" reflects the structure of awareness rather than of the world. This view explains the felt difference between past and future without granting physical reality to becoming, though it faces the objection that a theory of experience alone cannot account for the asymmetry found in physical processes.

The third treats passage as a physical fact. Most A-theorists hold that time is real and has both B-series and A-series characteristics. Some have tried to reconcile this with relativity by treating becoming as ontologically private, a proposal that remains controversial. Proposals of this kind often look to the second law of thermodynamics and the entropy gradient as a candidate physical marker of direction, but whether that asymmetry can ground passage itself is disputed, and it is treated in a separate section.

Time in perception and biology

Perceiving duration and timing

Elapsed time has no dedicated sense organ. The brain reconstructs duration from many cues, and psychologists separate timing, the accuracy with which an interval is estimated or reproduced, from time experience, how long an interval feels. The two can diverge. Attention, emotional arousal, novelty and memory change how long a stretch of time seems while leaving the accuracy of timing judgements largely intact. The classic framework is the internal-clock model, which posits a single central timekeeper that governs estimates across scales from milliseconds to several seconds. Rival accounts argue that the timing process itself differs with the duration being judged, so no single pacemaker is needed (Interval timing; psychology).

Neural mechanisms from milliseconds to seconds

Neuroscientists commonly divide temporal processing into four ranges: microseconds, milliseconds, seconds, and circadian rhythms. Sub-second timing is needed for speech, music, dance and motor control, and it depends heavily on the cerebellum, which becomes especially important when brief intervals must be tracked relative to precise events. Lesion evidence points the same way: in one study, patients with cerebellar damage were impaired at both 400 ms and 4 s, whereas prefrontal damage produced deficits mainly at the longer interval. Intervals of several seconds and more involve circuits linking the striatum and substantia nigra to the prefrontal cortex, which are also used when timing is combined with other cognitive demands. Conscious representation of intervals appears to rely on prefrontal and parietal cortex. An early behavioural test by Rammsayer and Lima in 1991 found that intervals of 50 to 100 ms were unaffected by a secondary task, while multi-second intervals were clearly disrupted, one of the first pieces of evidence for separate mechanisms. Brain imaging remains hard to interpret, because timing tasks also recruit attention, working memory and motor systems, and activation may reflect those processes rather than timing itself (brain; neuroscience).

Circadian rhythms and internal clocks

Circadian rhythms are endogenous cycles of roughly 24 hours. In mammals the master pacemaker is the suprachiasmatic nucleus (SCN) of the hypothalamus. At the molecular level, a transcription–translation feedback loop drives the cycle: the heterodimer BMAL1/CLOCK activates the period and cryptochrome genes, and their protein products eventually suppress their own transcription, producing an oscillation close to one day. Because the intrinsic period is not exactly 24 hours, the clock must be reset every day. Light is the dominant cue. It reaches the SCN through the retinohypothalamic tract from intrinsically photosensitive retinal ganglion cells, which release glutamate and PACAP. The response depends on phase: in dark-adapted rodents, light early in the subjective night delays the rhythm, light late in the subjective night advances it, and light in the middle of the subjective day has essentially no effect. Within the SCN, VIP-expressing neurons are important for light entrainment, while AVP-expressing neurons help set the period. The SCN in turn coordinates peripheral clocks in other organs. Clocks of this kind are also found outside the SCN: in mammals the retina has light-driven local clocks, and in birds, reptiles and fish the retina and pineal gland can act as oscillators (Circadian rhythm; earth; physiology).

Developmental timing and the cell cycle

Development proceeds through an ordered series of stages, and the rate of that progression is itself regulated. Biologists distinguish clocks, which rely on oscillations, from timers, which behave more like hourglasses by tracking the accumulation or concentration of a factor. The best-studied developmental clock is the segmentation clock of vertebrate embryos. Oscillating gene expression spreads through the presomitic mesoderm as a wave that travels toward the anterior end, and a new somite forms when the wave reaches the front. Experiments on cultured single cells show that the slowing of oscillations as cells mature is driven largely from within each cell, while extrinsic signals such as fibroblast growth factor can lengthen the duration of the intrinsic program. Within a single cell, the cell cycle is commonly described by a "clock plus checkpoint" model: a cyclical sequence of states, with checkpoints in G1 and G2 that must be passed before DNA synthesis and mitosis respectively. In multicellular organisms, timing is coordinated across tissues. In a nematode, a timer built from the transcription factor MYRF-1 and the PERIOD-like repressor LIN-42 produces once-per-stage pulses that are phase-locked across somatic tissues, and it licenses a checkpoint needed for growth and molting. Because individual timing mechanisms must be kept in step to preserve the overall developmental sequence, evolutionary changes in developmental timing are a major source of differences between species (cell-cycle; biology).

Time in mathematics, statistics, and computation

In mathematics, time is usually the independent variable of a model rather than a quantity the model predicts. A differential equation relates a quantity to its rate of change with respect to time, so the state at one instant determines how the system evolves. The state variables form a phase space, and an autonomous system, written $\dot{x} = f(x)$, has a right-hand side that does not depend on time explicitly. Models in astronomy, chemistry, biology, economics and engineering often take this form. Time can also be discrete: difference equations relate the value at step $t+1$ to the value at step $t$, and iterated maps such as the logistic map generate complex behaviour from very simple rules. A non-autonomous system can be rewritten as an autonomous one by adding time as an extra state variable, which shows that the role of time in these equations is formal as much as physical.

Statistics treats time in two main ways. A time series is a sequence of observations ordered by time, analysed for trend, seasonality and autocorrelation. Its order carries information that a random sample lacks, and forecasting must rely only on past values, so the direction of time is built into the method. A stochastic process is a family of random variables indexed by time. Its most important continuous example is Brownian motion, first described physically in the context of particles suspended in fluids and mathematically developed in the early twentieth century by Bachelier, Einstein and Wiener. The standard version starts at zero, has continuous paths, and has independent Gaussian increments, so that $W_t - W_s$ is normally distributed with variance $t-s$ for $t > s$. Its variance thus grows linearly with elapsed time. The stochastic differential equation $dX_t = \mu\,dt + \sigma\,dW_t$ adds a steady drift and a noise term, a model used in finance and in physics. Brownian motion can be obtained as the limit of rescaled random walks.

A Markov chain is a process in which the distribution of the next state depends only on the present state, not on the full history. In a time-homogeneous chain the transition probabilities do not change from step to step, and they can be collected into a transition matrix. Markov chains underlie simulation methods such as Markov chain Monte Carlo, and Brownian motion is a continuous-time Markov process. Both are used to model phenomena where memory of the past is absent or can be summarised by the current state.

In computer science, "time" usually means the number of elementary operations an algorithm performs, not clock seconds. The running time is expressed as a function $T(n)$ of input size $n$, and time complexity is described with asymptotic notation: $O(\cdot)$ gives an upper bound, $\Omega(\cdot)$ a lower bound, and $\Theta(\cdot)$ a tight bound. Scanning an unsorted list takes $O(n)$ steps, binary search on a sorted array takes $O(\log n)$, and nested loops over the input often give $O(n^2)$. Any comparison-based sorting algorithm needs $\Omega(n \log n)$ comparisons in the worst case. Constant factors and hardware speed are deliberately ignored, which makes the measure independent of any particular machine. Classes such as P and NP, discussed under computational-complexity, are defined in the same way.

Real computers also need to agree on the time. Crystal oscillators drift, so machines must be synchronised. The Network Time Protocol, developed by David Mills in the 1980s, arranges servers in strata: a stratum-1 server is linked directly to a reference clock such as a GPS receiver or atomic clock, and each further level is one step further from that source. The client estimates its clock offset from the round-trip time of request and reply messages. Accuracy ranges from about a millisecond over ordinary internet paths to microseconds with hardware timestamping, as in the Precision Time Protocol (IEEE 1588). Distributed systems often avoid physical time altogether: logical clocks of the kind introduced by Lamport in 1978 order events by causal relations. Timestamps are commonly stored as counts of seconds since 1 January 1970 UTC, and a 32-bit signed counter will overflow in January 2038. Operating systems also rely on time-sharing, introduced at MIT's CTSS in the early 1960s, in which the processor is divided into short slices so that many users can work on one machine.

Cultural and social conceptions of time

Societies have pictured time in different shapes. Many religious and philosophical traditions describe it as cyclical, with ages or cosmic periods that repeat, while others, especially those shaped by Christianity and Judaism, describe history as a single sequence with a creation and a final end. Hindu yugas and Buddhist kalpas are often cited as cyclical frameworks, while the Abrahamic traditions treat history as directed toward a decisive end. The contrast is partly a matter of emphasis, since many cultures combine recurring rituals with a sense of unique historical events. Mircea Eliade's influential account of "eternal return" argued that archaic societies used ritual to re-enter a sacred origin time, a claim that later scholars have debated. The distinction between cyclical and linear time also shapes how people understand religion, morality, and the meaning of the past.

The most influential modern account of how timekeeping shapes social life is E. P. Thompson's 1967 essay "Time, Work-Discipline and Industrial Capitalism" in Past & Present. Thompson argued that pre-industrial labour was often organized by task and natural rhythm, and that the spread of clocks and watches in the eighteenth and nineteenth centuries helped create a new discipline in which time was measured, sold, and policed. He described a shift from "time passed" to "time spent" and identified a growing concern with "time-thrift," the sense that time not used productively was wasted. Factory owners, employers, and later schools used fines, timesheets, and supervision to enforce this regime, which accompanied the rise of the Industrial Revolution. Later historians have qualified Thompson's thesis. Glennie and Thrift (1996) argued that his account understated the plurality of temporal experience, and Birth (2022) showed that seasonally variable hours and itinerant labour in pre-industrial England complicate the picture of a simple transition.

Standardized clock time also had a spatial dimension. Before the railways, most towns kept local mean solar time, so that clocks differed from one place to the next. In North America, railway companies adopted standard railway time on 18 November 1883, a change that became known as the "Day of Two Noons." The International Meridian Conference in Washington in October 1884 then selected the meridian through Greenwich as the prime meridian and adopted Greenwich-based reference time, although the vote was not unanimous and some countries, notably France, adopted the system only gradually. The result was the modern system of time zones, which linked clock time to telegraph networks, rail timetables, and international commerce. Standard time thus illustrates how a technical convention can reshape everyday habits and social coordination.

Calendars and timekeeping remain socially constructed even when they rest on astronomical cycles. Lunar, solar, and lunisolar systems reflect different community priorities, such as religious observance, agriculture, or political authority, and calendar reforms have often been contested for religious or national reasons. Historical narrative also shapes how people experience the past, present, and future. Historians and storytellers organize events into periods, origins, and turning points, and these structures influence collective memory and identity. Paul Ricoeur's Time and Narrative (1983–1985) argued that narrative gives human temporal experience its coherence, linking the chronological sequence of events to the lived sense of an unfolding life.

Today, global digital networks, shift work, and synchronized computing have extended the logic of standard clock time into almost every sphere of activity. Yet cultures continue to hold different senses of rhythm, punctuality, and the relation between generations, so the modern standardized clock coexists with older and more varied ways of living in time.

Unresolved problems and research frontiers

The most sharply defined open problem concerns quantum gravity. When general relativity is quantized in canonical form, the constraint equations produce the Wheeler–DeWitt equation, and that equation contains no time variable, unlike the Schrödinger equation of ordinary quantum mechanics. The Hamiltonian turns out to be a combination of constraints multiplied by Lagrange multipliers, so it vanishes whenever the constraints hold, which makes a time-dependent Schrödinger equation impossible to write down. Researchers trace the cause to general relativity itself. Classical physics and standard quantum mechanics treat time as an external parameter, like a clock outside the system, whereas in general relativity time is entangled with the geometry of the system, so a closed quantum-gravitational system seems to lose its conventional time. Proposed remedies typically promote some variable to the role of a clock, such as a massless scalar field, but these methods raise gauge-dependence problems, and they have mostly been developed for cosmological models.

A related question is whether time is fundamental or emergent. Some approaches attempt to derive it from more basic relational structure. In shape dynamics, the variables are naturally separated into dimensionless true degrees of freedom and a single variable that serves the role of time. This picture echoes Leibniz's insistence that time is nothing over and above the order of events. If something like it were correct, the smooth four-dimensional spacetime of general relativity would be an effective description valid at large scales, and the central task would become showing how it arises from a more primitive structure. Progress is slow because the regime where such emergence would matter lies near the Planck scale, where direct tests are not available.

The origin of the thermodynamic arrow of time is another frontier. The second law gives entropy a direction, but the microscopic laws are essentially time-symmetric, and statistical mechanics explains the one-way trend only if the universe began in a very special, low-entropy state. The second law describes entropy growth but assumes these special initial conditions without explaining them. Roger Penrose argued that the Big Bang is special because entropy was very small back then, compared to what it could have been. He estimated the relevant phase-space ratio at roughly 10^10^123, and he proposed the Weyl curvature hypothesis, which requires the initial Big Bang state to have very small Weyl curvature. The most widely favored response is cosmic inflation. Most authors accept inflationary models as the mechanism for producing this early low gravitational entropy, although Penrose rejects this route. Critics note that inflation itself appears to require a special starting state, and some argue it makes the puzzle worse. The alternative of low-entropy fluctuations from an equilibrium background avoids a special initial state, but these models often assume time-symmetric boundary conditions, which is not obviously more compelling. Whether the low-entropy initial condition is a brute fact, a consequence of a deeper law, or something else remains unsettled.

Finally, general relativity admits solutions with closed timelike curves, which allow worldlines to return to their own past. Physicists have long known that some solutions to the theory of general relativity contain closed timelike curves—for example the Gödel metric. Igor Novikov proposed that physics on such spacetimes is globally self-consistent, so that the principle does not allow a time traveller to change the past in any way at all, but it does allow them to affect past events in a way that produces no inconsistencies. Making this rigorous is difficult, because it is generally impossible to freely specify initial conditions on a spacelike hypersurface in such spacetimes, which undermines the predictability that the Cauchy problem normally provides. Hawking's chronology protection conjecture instead proposes that the laws of physics prevent such loops from forming. It suggests that the laws of physics might rule out time machines, but a full quantum treatment of the relevant regimes is still lacking. Whether these constraints are imposed by general relativity, by quantum gravity, or by neither remains an open question.

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