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Prerequisites 

Time in physics is defined by its measurement: time is what a clock reads.^{[1]} In classical, nonrelativistic physics it is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic energy and timedependent fields. Timekeeping is a complex of technological and scientific issues, and part of the foundation of recordkeeping.
Before there were clocks, time was measured by those physical processes^{[2]} which were understandable to each epoch of civilization:^{[3]}
Eventually,^{[9]}^{[10]} it became possible to characterize the passage of time with instrumentation, using operational definitions. Simultaneously, our conception of time has evolved, as shown below.^{[11]}
In the International System of Units (SI), the unit of time is the second (symbol: ). It is a SI base unit, and it is currently defined as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." ^{[12]} This definition is based on the operation of a caesium atomic clock.
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The UTC timestamp in use worldwide is an atomic time standard. The relative accuracy of such a time standard is currently on the order of 10^{−15}^{[13]} (corresponding to 1 second in approximately 30 million years). The smallest time step considered observable is called the Planck time, which is approximately 5.391×10^{−44} seconds  many orders of magnitude below the resolution of current time standards.
Both Galileo and Newton and most people up until the 20th century thought that time was the same for everyone everywhere. This is the basis for timelines, where time is a parameter. Our modern conception of time is based on Einstein's theory of relativity, in which rates of time run differently depending on relative motion, and space and time are merged into spacetime, where we live on a world line rather than a timeline. Thus time is part of a coordinate, in this view. Physicists believe the entire Universe and therefore time itself^{[15]}^{[dubious – discuss]} began about 13.8 billion years ago in the big bang. (See Time in Cosmology below) Whether it will ever come to an end is an open question. (See philosophy of physics.)
In order to measure time, one can record the number of occurrences (events) of some periodic phenomenon. The regular recurrences of the seasons, the motions of the sun, moon and stars were noted and tabulated for millennia, before the laws of physics were formulated. The sun was the arbiter of the flow of time, but time was known only to the hour for millennia, hence, the use of the gnomon was known across most of the world, especially Eurasia, and at least as far southward as the jungles of Southeast Asia.^{[16]}
In particular, the astronomical observatories maintained for religious purposes became accurate enough to ascertain the regular motions of the stars, and even some of the planets.
At first, timekeeping was done by hand by priests, and then for commerce, with watchmen to note time as part of their duties. The tabulation of the equinoxes, the sandglass, and the water clock became more and more accurate, and finally reliable. For ships at sea, boys were used to turn the sandglasses and to call the hours.
Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously built a mechanical clock as an astronomical orrery about 1330.^{[17]}^{[18]}
By the time of Richard of Wallingford, the use of ratchets and gears allowed the towns of Europe to create mechanisms to display the time on their respective town clocks; by the time of the scientific revolution, the clocks became miniaturized enough for families to share a personal clock, or perhaps a pocket watch. At first, only kings could afford them. Pendulum clocks were widely used in the 18th and 19th century. They have largely been replaced in general use by quartz and digital clocks. Atomic clocks can theoretically keep accurate time for millions of years. They are appropriate for standards and scientific use.
In 1583, Galileo Galilei (1564–1642) discovered that a pendulum's harmonic motion has a constant period, which he learned by timing the motion of a swaying lamp in harmonic motion at mass at the cathedral of Pisa, with his pulse.^{[19]}
In his Two New Sciences (1638), Galileo used a water clock to measure the time taken for a bronze ball to roll a known distance down an inclined plane; this clock was
Galileo's experimental setup to measure the literal flow of time, in order to describe the motion of a ball, preceded Isaac Newton's statement in his Principia:
The Galilean transformations assume that time is the same for all reference frames.
In or around 1665, when Isaac Newton (1643–1727) derived the motion of objects falling under gravity, the first clear formulation for mathematical physics of a treatment of time began: linear time, conceived as a universal clock.
The water clock mechanism described by Galileo was engineered to provide laminar flow of the water during the experiments, thus providing a constant flow of water for the durations of the experiments, and embodying what Newton called duration.
In this section, the relationships listed below treat time as a parameter which serves as an index to the behavior of the physical system under consideration. Because Newton's fluents treat a linear flow of time (what he called mathematical time), time could be considered to be a linearly varying parameter, an abstraction of the march of the hours on the face of a clock. Calendars and ship's logs could then be mapped to the march of the hours, days, months, years and centuries.
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Lagrange (1736–1813) would aid in the formulation of a simpler version^{[23]} of Newton's equations. He started with an energy term, L, named the Lagrangian in his honor, and formulated Lagrange's equations:
The dotted quantities, denote a function which corresponds to a Newtonian fluxion, whereas denote a function which corresponds to a Newtonian fluent. But linear time is the parameter for the relationship between the and the of the physical system under consideration. Some decades later, it was found that the second order equation of Lagrange or Newton can be more easily solved or visualized by suitable transformation to sets of first order differential equations.
Lagrange's equations can be transformed, under a Legendre transformation, to Hamilton's equations; the Hamiltonian formulation for the equations of motion of some conjugate variables p,q (for example, momentum p and position q) is:
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in the Poisson bracket notation and clearly shows the dependence of the time variation of conjugate variables p,q on an energy expression.
This relationship, it was to be found, also has corresponding forms in quantum mechanics as well as in the classical mechanics shown above. These relationships bespeak a conception of time which is reversible.
By 1798, Benjamin Thompson (1753–1814) had discovered that work could be transformed to heat without limit  a precursor of the conservation of energy or
In 1824 Sadi Carnot (1796–1832) scientifically analyzed the steam engines with his Carnot cycle, an abstract engine. Rudolf Clausius (1822–1888) noted a measure of disorder, or entropy, which affects the continually decreasing amount of free energy which is available to a Carnot engine in the:
Thus the continual march of a thermodynamic system, from lesser to greater entropy, at any given temperature, defines an arrow of time. In particular, Stephen Hawking identifies three arrows of time:^{[24]}
Entropy is maximum in an isolated thermodynamic system, and increases. In contrast, Erwin Schrödinger (1887–1961) pointed out that life depends on a "negative entropy flow".^{[25]} Ilya Prigogine (1917–2003) stated that other thermodynamic systems which, like life, are also far from equilibrium, can also exhibit stable spatiotemporal structures. Soon afterward, the BelousovZhabotinsky reactions^{[26]} were reported, which demonstrate oscillating colors in a chemical solution.^{[27]} These nonequilibrium thermodynamic branches reach a bifurcation point, which is unstable, and another thermodynamic branch becomes stable in its stead.^{[28]}
In 1864, James Clerk Maxwell (1831–1879) presented a combined theory of electricity and magnetism. He combined all the laws then known relating to those two phenomenon into four equations. These vector calculus equations which use the del operator () are known as Maxwell's equations for electromagnetism.
In free space (that is, space not containing electric charges), the equations take the form (using SI units):^{[29]}
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where
These equations allow for solutions in the form of electromagnetic waves. The wave is formed by an electric field and a magnetic field oscillating together, perpendicular to each other and to the direction of propagation. These waves always propagate at the speed of light c, regardless of the velocity of the electric charge that generated them.
The fact that light is predicted to always travel at speed c would be incompatible with Galilean relativity if Maxwell's equations were assumed to hold in any inertial frame (reference frame with constant velocity), because the Galilean transformations predict the speed to decrease (or increase) in the reference frame of an observer traveling parallel (or antiparallel) to the light.
It was expected that there was one absolute reference frame, that of the luminiferous aether, in which Maxwell's equations held unmodified in the known form.
The MichelsonMorley experiment failed to detect any difference in the relative speed of light due to the motion of the Earth relative to the luminiferous aether, suggesting that Maxwell's equations did, in fact, hold in all frames. In 1875, Hendrik Lorentz (1853–1928) discovered Lorentz transformations, which left Maxwell's equations unchanged, allowing Michelson and Morley's negative result to be explained. Henri Poincaré (1854–1912) noted the importance of Lorentz' transformation and popularized it. In particular, the railroad car description can be found in Science and Hypothesis,^{[30]} which was published before Einstein's articles of 1905.
The Lorentz transformation predicted space contraction and time dilation; until 1905, the former was interpreted as a physical contraction of objects moving with respect to the aether, due to the modification of the intermolecular forces (of electric nature), while the latter was thought to be just a mathematical stipulation.^{[citation needed]}
Albert Einstein's 1905 special relativity challenged the notion of absolute time, and could only formulate a definition of synchronization for clocks that mark a linear flow of time:
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B.
But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an "A time" and a "B time."
We have not defined a common "time" for A and B, for the latter cannot be defined at all unless we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A. Let a ray of light start at the "A time" t_{A} from A towards B, let it at the "B time" t_{B} be reflected at B in the direction of A, and arrive again at A at the “A time” t′_{A}.
In accordance with definition the two clocks synchronize if
We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—
 If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
 If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
—Albert Einstein, "On the Electrodynamics of Moving Bodies" ^{[31]}
Einstein showed that if the speed of light is not changing between reference frames, space and time must be so that the moving observer will measure the same speed of light as the stationary one because velocity is defined by space and time:
Indeed, the Lorentz transformation (for two reference frames in relative motion, whose x axis is directed in the direction of the relative velocity)
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can be said to "mix" space and time in a way similar to the way a Euclidean rotation around the z axis mixes x and y coordinates. Consequences of this include relativity of simultaneity.
More specifically, the Lorentz transformation is a hyperbolic rotation which is a change of coordinates in the fourdimensional Minkowski space, a dimension of which is ct. (In Euclidean space an ordinary rotation is the corresponding change of coordinates.) The speed of light c can be seen as just a conversion factor needed because we measure the dimensions of spacetime in different units; since the metre is currently defined in terms of the second, it has the exact value of 299 792 458 m/s. We would need a similar factor in Euclidean space if, for example, we measured width in nautical miles and depth in feet. In physics, sometimes units of measurement in which c = 1 are used to simplify equations.
Time in a "moving" reference frame is shown to run more slowly than in a "stationary" one by the following relation (which can be derived by the Lorentz transformation by putting ∆x′ = 0, ∆τ = ∆t′):
where:
Moving objects therefore are said to show a slower passage of time. This is known as time dilation.
These transformations are only valid for two frames at constant relative velocity. Naively applying them to other situations gives rise to such paradoxes as the twin paradox.
That paradox can be resolved using for instance Einstein's General theory of relativity, which uses Riemannian geometry, geometry in accelerated, noninertial reference frames. Employing the metric tensor which describes Minkowski space:
Einstein developed a geometric solution to Lorentz's transformation that preserves Maxwell's equations. His field equations give an exact relationship between the measurements of space and time in a given region of spacetime and the energy density of that region.
Einstein's equations predict that time should be altered by the presence of gravitational fields (see the Schwarzschild metric):
Where:
Or one could use the following simpler approximation:
Time runs slower the stronger the gravitational field, and hence acceleration, is. The predictions of time dilation are confirmed by particle acceleration experiments and cosmic ray evidence, where moving particles decay more slowly than their less energetic counterparts. Gravitational time dilation gives rise to the phenomenon of gravitational redshift and delays in signal travel time near massive objects such as the sun. The Global Positioning System must also adjust signals to account for this effect.
According to Einstein's general theory of relativity, a freely moving particle traces a history in spacetime that maximises its proper time. This phenomenon is also referred to as the principle of maximal aging, and was described by Taylor and Wheeler as:^{[32]}
Einstein's theory was motivated by the assumption that every point in the universe can be treated as a 'center', and that correspondingly, physics must act the same in all reference frames. His simple and elegant theory shows that time is relative to an inertial frame. In an inertial frame, Newton's first law holds; it has its own local geometry, and therefore its own measurements of space and time; there is no 'universal clock'. An act of synchronization must be performed between two systems, at the least.
There is a time parameter in the equations of quantum mechanics. The Schrödinger equation^{[33]} is
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One solution can be
where is called the time evolution operator, and H is the Hamiltonian.
But the Schrödinger picture shown above is equivalent to the Heisenberg picture, which enjoys a similarity to the Poisson brackets of classical mechanics. The Poisson brackets are superseded by a nonzero commutator, say [H,A] for observable A, and Hamiltonian H:
This equation denotes an uncertainty relation in quantum physics. For example, with time (the observable A), the energy E (from the Hamiltonian H) gives:
The more precisely one measures the duration of a sequence of events the less precisely one can measure the energy associated with that sequence and vice versa. This equation is different from the standard uncertainty principle because time is not an operator in quantum mechanics.
Corresponding commutator relations also hold for momentum p and position q, which are conjugate variables of each other, along with a corresponding uncertainty principle in momentum and position, similar to the energy and time relation above.
Quantum mechanics explains the properties of the periodic table of the elements. Starting with Otto Stern's and Walter Gerlach's experiment with molecular beams in a magnetic field, Isidor Rabi (1898–1988), was able to modulate the magnetic resonance of the beam. In 1945 Rabi then suggested that this technique be the basis of a clock^{[34]} using the resonant frequency of an atomic beam.
See dynamical systems and chaos theory, dissipative structures
One could say that time is a parameterization of a dynamical system that allows the geometry of the system to be manifested and operated on. It has been asserted that time is an implicit consequence of chaos (i.e. nonlinearity/irreversibility): the characteristic time, or rate of information entropy production, of a system. Mandelbrot introduces intrinsic time in his book Multifractals and 1/f noise.
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Signalling is one application of the electromagnetic waves described above. In general, a signal is part of communication between parties and places. One example might be a yellow ribbon tied to a tree, or the ringing of a church bell. A signal can be part of a conversation, which involves a protocol. Another signal might be the position of the hour hand on a town clock or a railway station. An interested party might wish to view that clock, to learn the time. See: Time ball, an early form of Time signal.
We as observers can still signal different parties and places as long as we live within their past light cone. But we cannot receive signals from those parties and places outside our past light cone.
Along with the formulation of the equations for the electromagnetic wave, the field of telecommunication could be founded. In 19th century telegraphy, electrical circuits, some spanning continents and oceans, could transmit codes  simple dots, dashes and spaces. From this, a series of technical issues have emerged; see Category:Synchronization. But it is safe to say that our signalling systems can be only approximately synchronized, a plesiochronous condition, from which jitter need be eliminated.
That said, systems can be synchronized (at an engineering approximation), using technologies like GPS. The GPS satellites must account for the effects of gravitation and other relativistic factors in their circuitry. See: Selfclocking signal.
The primary time standard in the U.S. is currently NISTF1, a lasercooled Cs fountain,^{[35]} the latest in a series of time and frequency standards, from the ammoniabased atomic clock (1949) to the caesiumbased NBS1 (1952) to NIST7 (1993). The respective clock uncertainty declined from 10,000 nanoseconds per day to 0.5 nanoseconds per day in 5 decades.^{[36]} In 2001 the clock uncertainty for NISTF1 was 0.1 nanoseconds/day. Development of increasingly accurate frequency standards is underway.
In this time and frequency standard, a population of caesium atoms is lasercooled to temperatures of one microkelvin. The atoms collect in a ball shaped by six lasers, two for each spatial dimension, vertical (up/down), horizontal (left/right), and back/forth. The vertical lasers push the caesium ball through a microwave cavity. As the ball is cooled, the caesium population cools to its ground state and emits light at its natural frequency, stated in the definition of second above. Eleven physical effects are accounted for in the emissions from the caesium population, which are then controlled for in the NISTF1 clock. These results are reported to BIPM.
Additionally, a reference hydrogen maser is also reported to BIPM as a frequency standard for TAI (international atomic time).
The measurement of time is overseen by BIPM (Bureau International des Poids et Mesures), located in Sèvres, France, which ensures uniformity of measurements and their traceability to the International System of Units (SI) worldwide. BIPM operates under authority of the Metre Convention, a diplomatic treaty between fiftyone nations, the Member States of the Convention, through a series of Consultative Committees, whose members are the respective national metrology laboratories.
The equations of general relativity predict a nonstatic universe. However, Einstein accepted only a static universe, and modified the Einstein field equation to reflect this by adding the cosmological constant, which he later described as the biggest mistake of his life. But in 1927, Georges LeMaître (1894–1966) argued, on the basis of general relativity, that the universe originated in a primordial explosion. At the fifth Solvay conference, that year, Einstein brushed him off with "Vos calculs sont corrects, mais votre physique est abominable."^{[37]} (“Your math is correct, but your physics is abominable”). In 1929, Edwin Hubble (1889–1953) announced his discovery of the expanding universe. The current generally accepted cosmological model, the LambdaCDM model, has a positive cosmological constant and thus not only an expanding universe but an accelerating expanding universe.
If the universe were expanding, then it must have been much smaller and therefore hotter and denser in the past. George Gamow (1904–1968) hypothesized that the abundance of the elements in the Periodic Table of the Elements, might be accounted for by nuclear reactions in a hot dense universe. He was disputed by Fred Hoyle (1915–2001), who invented the term 'Big Bang' to disparage it. Fermi and others noted that this process would have stopped after only the light elements were created, and thus did not account for the abundance of heavier elements.
Gamow's prediction was a 5–10 kelvin black body radiation temperature for the universe, after it cooled during the expansion. This was corroborated by Penzias and Wilson in 1965. Subsequent experiments arrived at a 2.7 kelvin temperature, corresponding to an age of the universe of 13.8 billion years after the Big Bang.
This dramatic result has raised issues: what happened between the singularity of the Big Bang and the Planck time, which, after all, is the smallest observable time. When might have time separated out from the spacetime foam;^{[39]} there are only hints based on broken symmetries (see Spontaneous symmetry breaking, Timeline of the Big Bang, and the articles in Category:Physical cosmology).
General relativity gave us our modern notion of the expanding universe that started in the Big Bang. Using relativity and quantum theory we have been able to roughly reconstruct the history of the universe. In our epoch, during which electromagnetic waves can propagate without being disturbed by conductors or charges, we can see the stars, at great distances from us, in the night sky. (Before this epoch, there was a time, 300,000 years after the big bang, during which starlight would not have been visible.)
Ilya Prigogine's reprise is "Time precedes existence". He contrasts the views of Newton, Einstein and quantum physics which offer a symmetric view of time (as discussed above) with his own views, which point out that statistical and thermodynamic physics can explain irreversible phenomena^{[40]} as well as the arrow of time and the Big Bang.
Suppose the beginning of the universe was like the South Pole of the earth, with degrees of latitude playing the role of time. The universe would start as a point at the South Pole. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. To ask what happened before the beginning of the universe would become a meaningless question because there is nothing south of the South Pole.'

