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| [[File:Pendule de Foucault.jpg|thumb|right|[[Léon Foucault|Foucault]]'s [[Foucault pendulum|pendulum]] in the [[Panthéon, Paris|Panthéon]] of [[Paris]] can measure [[time]] as well as demonstrate the [[rotation]] of [[Earth]]. ]]
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| '''Time in physics''' is defined by its [[operational definition|measurement]]: time is what a [[clock]] reads.<ref>{{cite book
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| |title=Process instruments and controls handbook
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| |edition=3
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| |first1=Douglas M.
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| |last1=Considine
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| |first2=Glenn D.
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| |last2=Considine
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| |publisher=McGraw-Hill
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| |year=1985
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| |isbn=0-07-012436-1
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| |pages=18–61
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| |url=http://books.google.com/books?id=kt1UAAAAMAAJ}}</ref> <!-- Physicists use [[theory|theories]] to predict measurements of time. What exactly time "is" and how it works is still largely undefined, except in relation to the other fundamental quantities. --> In classical, non-relativistic physics it is a [[scalar (physics)|scalar]] quantity and, like [[length]], [[mass]], and [[electric charge|charge]], is usually described as a [[fundamental quantity]]. Time can be combined mathematically with other [[physical quantities]] to [[Formal proof|derive]] other concepts such as [[motion (physics)|motion]], [[kinetic energy]] and time-dependent [[Field (physics)|fields]]. ''[[:category:Timekeeping|Timekeeping]]'' is a complex of technological and scientific issues, and part of the foundation of ''[[recordkeeping]]''.
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| | |
| ==Markers of time==
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| | |
| {{main|History of timekeeping devices}}
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| Before there were clocks, time was measured by those physical processes<ref>For example, [[Galileo]] measured the period of a [[simple harmonic oscillator]] with his [[pulse]].</ref> which were understandable to each epoch of civilization:<ref name=OttoN/>
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| *the first appearance (see: [[heliacal rising]]) of [[Sirius]] to mark the [[flooding of the Nile]] each year<ref name=OttoN>[[Otto Neugebauer]] ''The Exact Sciences in Antiquity''. Princeton: Princeton University Press, 1952; 2nd edition, Brown University Press, 1957; reprint, New York: Dover publications, 1969. Page 82.</ref>
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| *the periodic succession of [[night]] and [[day]], one after the other, in seemingly eternal succession<ref>See, for example [[William Shakespeare]] ''Hamlet'': " ... to thine own self be true,
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| And it must follow, as the night the day,
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| Thou canst not then be false to any man."</ref>
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| *the position on the horizon of the first appearance of the sun at dawn<ref>{{cite web|url=http://solar-center.stanford.edu/AO/dawn-rising.html |title=Heliacal/Dawn Risings |publisher=Solar-center.stanford.edu |date= |accessdate=2012-08-17}}</ref>
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| *the position of the sun in the sky<ref>[http://eo.nso.edu/MrSunspot/answerbook/sundial.html Farmers have used the sun to mark time for thousands of years, as the most ancient method of telling time.]</ref>
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| *the marking of the moment of [[noon]]time during the day<ref>[[Eratosthenes]] used this criterion in his measurement of the circumference of Earth</ref>
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| *the length of the shadow cast by a [[gnomon]]<ref>[[Fred Hoyle]] (1962), ''Astronomy: A history of man's investigation of the universe'', Crescent Books, Inc., London LC 62-14108, p.31''</ref>
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| Eventually,<ref>The Mesopotamian (modern-day Iraq) astronomers recorded astronomical observations with the naked eye, more than 3500 years ago. [[P. W. Bridgman]] defined his [[operational definition]] in the twentieth c.</ref><ref>[[Naked-eye stars|Naked eye astronomy]] became obsolete in 1609 with Galileo's observations with a telescope. Galileo Galilei Linceo, [http://www.rarebookroom.org/Control/galsid/index.html ''Sidereus Nuncius''] (''[[Starry Messenger]]'') 1610.</ref> it became possible to characterize the passage of time with instrumentation, using [[operational definition]]s. Simultaneously, our conception of time has evolved, as shown below.<ref>http://tycho.usno.navy.mil/gpstt.html http://www.phys.lsu.edu/mog/mog9/node9.html Today, automated astronomical observations from satellites and spacecraft require relativistic corrections of the reported positions.</ref> <!--This sentence sounds a little like POV, unless it is backed up by references. Thanks. Please read Neugebauer. Fred Hoyle also gives some of these examples. Added a reference which requires reading the article below. -->
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| | |
| ==The unit of measurement of time: the second==
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| In the [[International System of Units]] (SI), the unit of time is the [[second]] (symbol: <math>\mathrm{s}</math>). It is a [[SI base unit]], and it is currently defined as "the duration of {{nowrap|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." <ref>{{cite web
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| |url=http://www.bipm.org/en/si/si_brochure/chapter2/2-1/second.html
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| |title=Unit of time (second)
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| |accessdate=2008-06-08
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| |work=SI brochure
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| |pages=Section 2.1.1.3
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| |publisher=[[International Bureau of Weights and Measures]] (BIPM)
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| }}
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| </ref> This definition is based on the operation of a cesium [[atomic clock]].
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| ===The state of the art in timekeeping===
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| {| class="infobox plainlist"
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| |-
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| ! Prerequisites
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| |-
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| * [[Measurement]]
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| * [[Scientific notation]]
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| * [[Natural units]]
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| |}
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| The [[Coordinated Universal Time|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<sup>−15</sup><ref>[http://tf.nist.gov/general/pdf/1823.pdf S. R. Jefferts et al., "Accuracy evaluation of NIST-F1".]</ref> (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<sup>−44</sup> seconds - many orders of magnitude below the resolution of current time standards.
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| ==Conceptions of time==
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| {{main|Time}}
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| [[File:Andromeda galaxy Ssc2005-20a1.jpg|thumb|left|300px|Andromeda galaxy ([[Andromeda Galaxy|M31]]) is two million [[light-year]]s away. Thus we are viewing M31's light from two million years ago,<ref>Fred Adams and Greg Laughlin (1999), ''Five Ages of the Universe'' ISBN 0-684-86576-9 p.35.</ref> a time before [[human]]s existed on Earth.]]
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| Both [[Galileo]] and [[Isaac Newton|Newton]] and most people up until the 20th century thought that time was the same for everyone everywhere. This is the basis for [[:category:timelines|timeline]]s, where time is a [[parameter]]. Our modern conception of time is based on [[Albert Einstein|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<ref>See [[Planck epoch]] for the smallest physical timestep. Also see [[Time#Time and the Big Bang]]. {{cite web
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| |url=http://www.admin.ox.ac.uk/po/news/2005-06/feb/27.shtml
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| |title=Professor Stephen Hawking lectures on the origin of the universe
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| |first=Stephen
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| |last=Hawking
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| |date=2006-02-27
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| |publisher=University of Oxford
| |
| |quote=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.'
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| |accessdate=2008-01-10}}</ref>{{dubious|date=August 2010|reason=Hawking has since refined his view}} began about 13.8 billion years ago in the [[big bang]]. (See [[#Time in cosmology|Time in Cosmology]] below) Whether it will ever come to an end is an open question. (See [[philosophy of physics]].)
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| | |
| ===Regularities in nature===
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| {{Main|History of science}}
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| In order to measure time, one can record the number of occurrences (events) of some [[periodic function|periodic]] [[phenomenon]]. The regular recurrences of the [[seasons]], the [[motion (physics)|motion]]s of the [[sun]], [[moon]] and [[star]]s 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 [[millennium|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]].<ref>Charles Hose and William McDougall (1912) ''The Pagan Tribes of Borneo'', [http://books.google.com/books?id=phTSAAAAMAAJ&pg=PA108-IA1&dq=Pagan+Tribes+of+Borneo+1912+aso+do+plate+60&hl=en&ei=zd2ATOu5K8P68AaF4dlS&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q&f=false Plate 60.] Kenyahs measuring the Length of the Shadow at Noon to determine the Time for sowing PADI p. 108. This photograph is reproduced as plate B in Fred Hoyle (1962), ''Astronomy: A history of man's investigation of the universe'', Crescent Books, Inc., London LC 62-14108, p.31. The measurement process is explained by: Gene Ammarell (1997), "Astronomy in the Indo-Malay Archipelago", p.119, ''Encyclopaedia of the history of science, technology, and medicine in non-western cultures'', [[Helaine Selin]], ed., which describes Kenyah Tribesmen of Borneo measuring the shadow cast by a gnomon, or ''tukar do'' with a measuring scale, or ''aso do''.</ref>
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| 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.
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| At first, [[timekeeping]] was done by hand by priests, and then for commerce, with watchmen to note time as part of their duties.
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| The tabulation of the [[equinox]]es, the [[Marine sandglass|sandglass]], and the [[water clock]] became more and more accurate, and finally reliable. For ships at sea, boys were used to turn the [[Marine sandglass|sandglass]]es and to call the hours.
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| | |
| ====Mechanical clocks====
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| [[Richard of Wallingford]] (1292–1336), abbot of St. Alban's abbey, famously built a [[Clock#Early mechanical clocks|mechanical clock]] as an astronomical [[orrery]] about 1330.<ref>North, J. (2004) ''God's Clockmaker: Richard of Wallingford and the Invention of Time''. Oxbow Books. ISBN 1-85285-451-0</ref><ref>Watson, E (1979) "The St Albans Clock of Richard of Wallingford". ''Antiquarian Horology'' 372-384.</ref>
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| By the time of Richard of Wallingford, the use of [[ratchet (device)|ratchet]]s and [[gear]]s 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 clock]]s were widely used in the 18th and 19th century. They have largely been replaced in general use by quartz and [[digital clock]]s. [[Atomic clocks]] can theoretically keep accurate time for millions of years. They are appropriate for [[Standardization|standard]]s and scientific use.
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| | |
| === Galileo: the flow of time ===
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| {{main|reproducibility}}
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| In 1583, [[Galileo Galilei]] (1564–1642) discovered that a [[harmonic oscillator|pendulum's harmonic motion]] has a constant period, which he learned by timing the motion of a swaying lamp in [[Simple harmonic motion|harmonic motion]] at [[Mass (liturgy)|mass]] at the cathedral of [[Pisa]], with his [[pulse]].<ref>Jo Ellen Barnett, ''Time's Pendulum'' ISBN 0-306-45787-3 p.99.</ref>
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| | |
| In his ''[[Two New Sciences]]'' (1638), [[Galileo Galilei|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
| |
| :"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results."<ref name="galileo">[[Galileo]] 1638 ''Discorsi e dimostrazioni matematiche, intorno á due nuoue scienze'' '''213''', Leida, Appresso gli Elsevirii (Louis Elsevier), or ''Mathematical discourses and demonstrations, relating to [[Two New Sciences]]'', English translation by Henry Crew and Alfonso de Salvio 1914. Section '''213''' is reprinted on pages 534-535 of ''On the Shoulders of Giants'':The Great Works of Physics and Astronomy (works by [[Copernicus]], [[Johannes Kepler|Kepler]], [[Galileo]], [[Isaac Newton|Newton]], and [[Albert Einstein|Einstein]]). [[Stephen Hawking]], ed. 2002 ISBN 0-7624-1348-4</ref>
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| Galileo's experimental setup to measure the literal ''[[Two New Sciences#The flow of time|flow of time]]'', in order to describe the motion of a ball, preceded [[Isaac Newton]]'s statement in his [[Philosophiæ Naturalis Principia Mathematica|Principia]]:
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| :''I do not define [[time]], [[space]], [[Location (geography)|place]] and [[motion (physics)|motion]], as being well known to all.''<ref name="galileo2">[[Isaac Newton|Newton]] 1687 ''[[Philosophiae Naturalis Principia Mathematica]]'', Londini, Jussu Societatis Regiae ac Typis J. Streater, or '''''[[The Mathematical Principles of Natural Philosophy]]''''', [[London]], English translation by [[Andrew Motte]] 1700s. From part of the Scholium, reprinted on page 737 of ''On the Shoulders of Giants'':The Great Works of Physics and Astronomy (works by [[Copernicus]], [[Johannes Kepler|Kepler]], [[Galileo]], [[Isaac Newton|Newton]], and [[Albert Einstein|Einstein]]). [[Stephen Hawking]], ed. 2002 ISBN 0-7624-1348-4</ref>
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| | |
| The [[Galilean transformation]]s assume that time is the same for all [[Frame of reference|reference frames]].
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| ===Newton's physics: linear time===
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| {{main|classical physics}}
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| 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''.
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| | |
| :''Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.''<ref name="newton">[[Isaac Newton|Newton]] 1687 page 738.</ref>
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| 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''.
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| | |
| 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 [[fluent (mathematics)|fluent]]s 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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| | |
| {| class="infobox plainlist"
| |
| |-
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| ! Prerequisites
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| |-
| |
| |
| |
| * [[differential equations]]
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| * [[partial differential equations]]
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| |}
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| | |
| [[Joseph Louis Lagrange|Lagrange]] (1736–1813) would aid in the formulation of a simpler version<ref>"Dynamics is a four-dimensional geometry." --Lagrange (1796), ''Thèorie des fonctions analytiques'', as quoted by Ilya Prigogine (1996), ''The End of Certainty'' ISBN 0-684-83705-6 p.58</ref> of Newton's equations. He started with an energy term, L, named the ''Lagrangian'' in his honor, and formulated ''[[Lagrange's equations]]'':
| |
| :<math>
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| \frac{d}{dt}
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| \frac{\partial L}{\partial \dot{\theta}}
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| - \frac{\partial L}{\partial \theta} = 0.
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| </math>
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| The dotted quantities, <math>{\dot{\theta}}</math> denote a function which corresponds to a Newtonian [[Method of Fluxions|fluxion]], whereas <math>{{\theta}}</math> denote a function which corresponds to a Newtonian [[fluent (mathematics)|fluent]]. But linear time is the parameter for the relationship between the <math>{\dot{\theta}}</math> and the <math>{{\theta}}</math> of the physical system under consideration.
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| 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.
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| | |
| Lagrange's equations can be transformed, under a [[Legendre transformation]], to ''[[Hamilton's equations]]''; the [[Hamiltonian mechanics|Hamiltonian]] formulation for the equations of motion of some conjugate variables p,q (for example, momentum p and position q) is:
| |
| {| class="infobox plainlist"
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| |-
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| ! Prerequisites
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| |-
| |
| |
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| * [[operator (physics)|Operator]]s
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| * [[Poisson bracket]]s
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| |}
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| :<math>\dot p = -\frac{\partial H}{\partial q} = \{p,H\} = -\{H,p\} </math>
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| :<math>\dot q =~~\frac{\partial H}{\partial p} = \{q,H\} = -\{H,q\} </math>
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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.
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| This relationship, it was to be found, also has [[Commutator|corresponding forms]] in [[quantum mechanics]] as well as in the [[classical mechanics]] shown above. These relationships bespeak a conception of time which is reversible.
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| | |
| ===Thermodynamics and the paradox of irreversibility===
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| {{main|arrow of time}}
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| 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
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| *[[Conservation of energy|1st law of thermodynamics]]
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| In 1824 [[Nicolas Léonard Sadi Carnot|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:
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| *[[Entropy|2nd law of thermodynamics]]
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| <!--
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| :<math>E =\, \cdots</math> (thermal energy)
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| | |
| :<math>ds =\, \cdots</math>
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| :<math>\frac{\partial ^2T}{\partial t^2} =\frac{\partial T}{\partial x}</math>
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| | |
| -->
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| 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:<ref name="einstein2">pp. 182-195. [[Stephen Hawking]] 1996. ''The Illustrated Brief History of Time'': updated and expanded edition ISBN 0-553-10374-1</ref>
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| *Psychological arrow of time - our perception of an inexorable flow.
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| *Thermodynamic arrow of time - distinguished by the growth of [[entropy]].
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| *Cosmological arrow of time - distinguished by the expansion of the universe.
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| | |
| 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"''.<ref>[[Erwin Schrödinger]] (1945) ''What is Life?''</ref> [[Ilya Prigogine]] (1917–2003) stated that other thermodynamic systems which, like life, are also far from equilibrium, can also exhibit stable spatio-temporal structures. Soon afterward, the [[Belousov-Zhabotinsky reaction]]s<ref>G. Nicolis and I. Prigogine (1989), ''Exploring Complexity''</ref> were reported, which demonstrate oscillating colors in a chemical solution.<ref>R. Kapral and K. Showalter, eds. (1995), ''Chemical Waves and Patterns''</ref> These nonequilibrium thermodynamic branches reach a ''[[Bifurcation theory|bifurcation point]]'', which is unstable, and another thermodynamic branch becomes stable in its stead.<ref>Ilya Prigogine (1996) ''The End of Certainty'' pp. 63-71</ref>
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| ===Electromagnetism and the speed of light===
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| {{main|Maxwell's equations}}
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| 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|del operator]] (<math>\nabla</math>) are known as [[Maxwell's equations]] for [[electromagnetism]].
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| In free space (that is, space not containing [[electric charge]]s), the equations take the form (using [[International System of Units|SI units]]):<ref>{{cite book
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| |title=An introduction to electromagnetic theory
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| |first1=P. C.
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| |last1=Clemmow
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| |publisher=CUP Archive
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| |year=1973
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| |isbn=0-521-09815-7
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| |pages=56–57
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| |url=http://books.google.com/books?id=ahQ7AAAAIAAJ}}, [http://books.google.com/books?id=ahQ7AAAAIAAJ&pg=PA56 Extract of pages 56, 57]
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| </ref>
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| {| class="infobox plainlist"
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| |-
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| ! Prerequisites
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| |-
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| |
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| * [[vector notation]]
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| * [[partial differential equations]]
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| |}
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| :<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}</math>
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| :<math>\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}</math>
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| | |
| :<math>\nabla \cdot \mathbf{E} = 0</math>
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| | |
| :<math>\nabla \cdot \mathbf{B} = 0</math>
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| | |
| where
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| :''ε''<sub>0</sub> and ''μ''<sub>0</sub> are the [[vacuum permittivity|electric permittivity]] and the [[vacuum permeability|magnetic permeability of free space]];
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| :''c'' = <math>1/\sqrt{\epsilon_0 \mu_0}</math> is the [[speed of light]] in free space, 299 792 458 [[metre|m]]/[[second|s]];
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| :'''E''' is the electric field;
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| :'''B''' is the magnetic field.
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| | |
| 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.
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| 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.
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| 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.
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| The [[Michelson-Morley 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 transformation]]s, 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'',<ref>Henri Poincaré, (1902). ''Science and Hypothesis'' [http://spartan.ac.brocku.ca/~lward/Poincare/Poincare_1905_toc.html Eprint]</ref> which was published before Einstein's articles of 1905.
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| | |
| 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|date=May 2008}}
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| ===Einstein's physics: spacetime===
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| :''Main articles: [[special relativity]]'' (1905), ''[[general relativity]]'' (1915).
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| | |
| [[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:
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| {{Quote|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.
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| | |
| 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."
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| | |
| 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''<sub>A</sub> from A towards B, let it at the "B time" ''t''<sub>B</sub> be reflected at B in the direction of A, and arrive again at A at the “A time” ''t''′<sub>A</sub>.
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| | |
| In accordance with definition the two clocks synchronize if
| |
| : <math>t_\text{B} - t_\text{A} = t'_\text{A} - t_\text{B}\text{.}\,\!</math>
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| | |
| 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:—
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| #If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
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| #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" <ref name="arrow_of_time">[[Albert Einstein|Einstein]] 1905, ''Zur Elektrodynamik bewegter Körper'' [On the electrodynamics of moving bodies] reprinted 1922 in ''[[Special relativity|Das Relativitätsprinzip]]'', B.G. Teubner, Leipzig. '''''[[Special relativity|The Principles of Relativity]]: A Collection of Original Papers on the Special Theory of Relativity''''', by H.A. Lorentz, A. Einstein, H. Minkowski, and W. H. Weyl, is part of ''Fortschritte der mathematischen Wissenschaften in Monographien, Heft 2''. The English translation is by W. Perrett and G.B. Jeffrey, reprinted on page 1169 of ''On the Shoulders of Giants'':The Great Works of Physics and Astronomy (works by [[Copernicus]], [[Johannes Kepler|Kepler]], [[Galileo]], [[Isaac Newton|Newton]], and [[Albert Einstein|Einstein]]). [[Stephen Hawking]], ed. 2002 ISBN 0-7624-1348-4</ref>}} 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:
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| :<math>\mathbf{v}={d\mathbf{r}\over dt} \text{,}</math> where '''r''' is position and ''t'' is time.
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| | |
| 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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| {| class="infobox plainlist"
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| |-
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| ! Prerequisites
| |
| |-
| |
| |
| |
| * [[algebra]]
| |
| * [[trigonometry]]
| |
| |}
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| :<math>\begin{cases}
| |
| t' &= \gamma(t - vx/c^2) \text{ where } \gamma = 1/\sqrt{1-v^2/c^2} \\
| |
| x' &= \gamma(x - vt)\\
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| y' &= y \\
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| z' &= z
| |
| \end{cases}</math>
| |
| 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]]. [[File:Relativity of Simultaneity.svg|thumb|Event B is simultaneous with A in the green reference frame, but it occurred
| |
| before in the blue frame, and will occur later in the red frame.]] More specifically, the Lorentz transformation is a hyperbolic rotation <math>
| |
| \begin{pmatrix}
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| ct' \\
| |
| x'
| |
| \end{pmatrix}
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| =
| |
| \begin{pmatrix}
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| \cosh \phi & - \sinh \phi \\
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| - \sinh \phi & \cosh \phi
| |
| \end{pmatrix}
| |
| | |
| \begin{pmatrix}
| |
| ct \\
| |
| x
| |
| \end{pmatrix} \text{ where } \phi = \operatorname{artanh}\,\frac{v}{c} \text{,}
| |
| </math> which is a change of coordinates in the four-dimensional [[Minkowski space]], a dimension of which is ''ct''. (In [[Euclidean space]] an ordinary rotation <math>
| |
| \begin{pmatrix}
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| x' \\
| |
| y'
| |
| \end{pmatrix}
| |
| =
| |
| \begin{pmatrix}
| |
| \cos \theta & - \sin \theta \\
| |
| \sin \theta & \cos \theta
| |
| \end{pmatrix}
| |
| | |
| \begin{pmatrix}
| |
| x \\
| |
| y
| |
| \end{pmatrix}
| |
| </math> 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 {{nowrap|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 [[natural units|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''′):
| |
| :<math>\Delta t= {{\Delta \tau}\over\sqrt{1 - v^2/c^2}}</math>
| |
| where:
| |
| *∆''τ'' is the time between two events as measured in the moving reference frame in which they occur at the same place (e.g. two ticks on a moving clock); it is called the [[proper time]] between the two events;
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| *∆''t'' is the time between these same two events, but as measured in the stationary reference frame;
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| *''v'' is the speed of the moving reference frame relative to the stationary one;
| |
| *''c'' is the [[speed of light]].
| |
| | |
| 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 [[paradox]]es as the [[twin paradox]].
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| | |
| 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]]:
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| | |
| :<math>\left[(dx^1)^2+(dx^2)^2+(dx^3)^2-c(dt)^2)\right],</math>
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| | |
| Einstein developed a geometric solution to Lorentz's transformation that preserves [[Maxwell's equations]]. His [[Einstein's field equations|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.
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| | |
| Einstein's equations predict that time should be altered by the presence of [[gravitational fields]] (see the [[Schwarzschild metric]]):
| |
| | |
| :<math>T=\frac{dt}{\sqrt{\left( 1 - \frac{2GM}{rc^2} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2GM}{rc^2} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2}}</math>
| |
| | |
| Where:
| |
| | |
| :<math>T</math> is the [[gravitational time dilation]] of an object at a distance of <math>r</math>.
| |
| | |
| :<math>dt</math> is the change in coordinate time, or the interval of coordinate time.
| |
| | |
| :<math>G</math> is the [[gravitational constant]]
| |
| | |
| :<math>M</math> is the [[mass]] generating the field
| |
| | |
| :<math>\sqrt{\left( 1 - \frac{2GM}{rc^2} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2GM}{rc^2} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2}</math> is the change in [[proper time]] <math>d\tau</math>, or the interval of [[proper time]].
| |
| | |
| Or one could use the following simpler approximation:
| |
| | |
| :<math>\frac{dt}{d\tau} = \frac{1}{ \sqrt{1 - \left( \frac{2GM}{rc^2} \right)}}. </math>
| |
| | |
| Time runs slower the stronger the gravitational field, and hence [[acceleration]], is. The predictions of time dilation are confirmed by [[particle accelerator|particle acceleration]] experiments and [[cosmic ray]] evidence, where moving particles [[particle decay|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 [[Edwin F. Taylor|Taylor]] and [[John Archibald Wheeler|Wheeler]] as:<ref>{{cite web
| |
| |url=http://www.eftaylor.com/pub/chapter1.pdf
| |
| |last=Taylor
| |
| |authorlink=Edwin F. Taylor
| |
| |year=2000
| |
| |title=Exploring Black Holes: Introduction to General Relativity
| |
| |publisher=Addison Wesley Longman.
| |
| }}
| |
| </ref>
| |
| ::''"Principle of Extremal Aging: The path a free object takes between two events in spacetime is the path for which the time lapse between these events, recorded on the object's wristwatch, is an extremum."''
| |
| | |
| 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.
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| | |
| === Time in quantum mechanics ===
| |
| {{see also|quantum mechanics}}
| |
| | |
| There is a time parameter in the equations of [[quantum mechanics]]. The [[Schrödinger equation]]<ref name="schrodinger">E. Schrödinger, [[Phys. Rev.]] '''28''' 1049 (1926)</ref> is
| |
| | |
| {| class="infobox plainlist"
| |
| |-
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| ! Prerequisites
| |
| |-
| |
| |
| |
| * [[physics]]
| |
| * [[quantum mechanics]]
| |
| |}
| |
| :<math> H(t) \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle</math>
| |
| One solution can be
| |
| :<math> | \psi_e(t) \rangle = e^{-iHt / \hbar} | \psi_e(0) \rangle </math>.
| |
| where <math> e^{-iHt / \hbar} </math>
| |
| is called the [[time evolution operator]], and ''H'' is the [[Hamiltonian (quantum mechanics)|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 bracket]]s are superseded by a nonzero [[commutator]], say [H,A] for [[observable]] A, and Hamiltonian H:
| |
| | |
| :<math>\frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_\mathrm{classical}.</math>
| |
| | |
| This equation denotes an [[uncertainty principle|uncertainty relation]] in quantum physics. For example, with ''time'' (the observable A), the ''energy'' E (from the Hamiltonian H) gives:
| |
| | |
| :<math>\Delta E \Delta T \ge \frac{\hbar}{2} </math>
| |
| :where
| |
| :<math>\Delta E</math> is the uncertainty in energy
| |
| :<math>\Delta T</math> is the uncertainty in time
| |
| :<math>\hbar</math> is [[Planck's constant]]
| |
| The more [[Accuracy and precision|precisely]] one measures the duration of a [[Phenomenon|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 (physics)|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 [[chemical element|elements]]. Starting with [[Otto Stern]]'s and [[Walter Gerlach]]'s experiment with [[molecular beam]]s in a magnetic field, [[Isidor Rabi]] (1898–1988), was able to [[modulation|modulate]] the magnetic resonance of the beam. In 1945 Rabi then suggested that this technique be the basis of a clock<ref>[http://tf.nist.gov/timefreq/cesium/atomichistory.htm A Brief History of Atomic Clocks at NIST]</ref> using the [[resonant frequency]] of an atomic beam.
| |
| | |
| [[John G. Cramer|John Cramer]] [http://www.seattlepi.com/local/292378_timeguy15.html is preparing an experiment] to determine whether [[quantum entanglement]] is also nonlocal in [[time]] as it is in [[space]]. This can also be stated as 'sending a signal back in time'. Cramer has recently published an [http://faculty.washington.edu/jcramer/Nonlocal_2007.pdf update] indicating that the final experiment will take more time to prepare.
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| | |
| ==Dynamical systems==
| |
| 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 theory|chaos]]'' (i.e. [[nonlinearity]]/[[irreversibility]]): the [[characteristic time]], or rate of [[information entropy]] production, of a [[system]]. [[Benoît Mandelbrot|Mandelbrot]] introduces [[intrinsic time]] in his book ''Multifractals and [[1/f noise]]''.
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| ==Signalling==
| |
| {| class="infobox plainlist"
| |
| |-
| |
| ! Prerequisites
| |
| |-
| |
| |
| |
| * [[electrical engineering]]
| |
| * [[signal processing]]
| |
| |}
| |
| | |
| Signalling is one application of the [[electromagnetic wave]]s 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 [[Communications protocol|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]].
| |
| [[File:Lorentz transform of world line.gif|thumb|left|Evolution of a [[world line]] of an accelerated massive particle. This worldline is restricted to the [[timelike]] top and bottom sections of this [[spacetime]] figure and can not cross the top ([[future]]) nor the bottom ([[past]]) [[light cone]]. The left and right sections, outside the light cones are [[spacelike]].]]
| |
| 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.
| |
| <!--
| |
| Messages and signals were still local and based upon common cultures, languages and religions. If parties had to cross cultures, a ''lingua franca'' had to be relied upon, as well as upon a medium of exchange. But the scarcity of goods could still create common motivation for trade, legal or illegal. And eventually illegal or immoral trade was ceased due to public outcry of messages. Theft could not be condoned when the cultures could no longer hide the crimes or injustice.
| |
| -->In 19th century [[telegraphy]], [[electrical circuit]]s, some spanning [[continent]]s and [[ocean]]s, could transmit [[code]]s - 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 [[Synchronization|synchronized]], a [[plesiochronous]] condition, from which [[jitter]] need be eliminated.
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| | |
| That said, [[system]]s ''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: [[Self-clocking signal]].
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| | |
| ==Technology for timekeeping standards==
| |
| The [[primary time standard]] in the [[U.S.]] is currently [[NIST-F1]], a [[laser]]-cooled [[Caesium|Cs]] fountain,<ref>D. M. Meekhof, S. R. Jefferts, M. Stepanovíc, and T. E. Parker (2001) "Accuracy Evaluation of a Cesium Fountain Primary Frequency Standard at NIST", ''IEEE Transactions on Instrumentation and Measurement''. '''50''', no. 2, (April 2001) pp. 507-509</ref> the latest in a series of time and frequency standards, from the [[ammonia]]-based atomic clock (1949) to the [[caesium]]-based NBS-1 (1952) to NIST-7 (1993). The respective clock uncertainty declined from 10,000 nanoseconds per day to 0.5 nanoseconds per day in 5 decades.<ref>James Jespersen and Jane Fitz-Randolph (1999). ''From sundials to atomic clocks : understanding time and frequency''. Washington, D.C. : U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology. 308 p. : ill. ; 28 cm.
| |
| ISBN 0-16-050010-9</ref> In 2001 the clock uncertainty for NIST-F1 was 0.1 nanoseconds/day. Development of increasingly accurate frequency standards is underway.
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| | |
| In this time and frequency standard, a population of caesium atoms is laser-cooled 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 NIST-F1 clock. These results are reported to [[BIPM]].
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| | |
| Additionally, a reference [[Maser#Hydrogen maser|hydrogen maser]] is also reported to BIPM as a frequency standard for [[International Atomic Time|TAI]] ([[International Atomic Time|international atomic time]]).
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| | |
| 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 fifty-one nations, the Member States of the Convention, through a series of Consultative Committees, whose members are the respective national metrology laboratories.
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| | |
| ==Time in cosmology==
| |
| {{main|physical cosmology}}
| |
| The equations of general relativity predict a non-static 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 "{{lang|fr|Vos calculs sont corrects, mais votre physique est abominable.}}"<ref>John C. Mather and John Boslough (1996), ''The Very First Light'' ISBN 0-465-01575-1 p.41.</ref> (“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 [[Lambda-CDM 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. [[Enrico Fermi|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.
| |
| [[File:WMAP.jpg|thumb|right|[[WMAP]] fluctuations of the [[cosmic microwave background radiation]].<ref>[[George Smoot]] and Keay Davidson (1993) ''Wrinkles in Time'' ISBN 0-688-12330-9 A memoir of the experiment program for detecting the predicted fluctuations in the [[cosmic microwave background radiation]]</ref>]]
| |
| 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]];<ref>[[Martin Rees]] (1997), ''Before the Beginning'' ISBN 0-201-15142-1 p.210</ref> there are only hints based on broken symmetries (see [[Spontaneous symmetry breaking]], [[Timeline of the Big Bang]], and the articles in [[:Category:Physical cosmology]]).
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| | |
| [[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 [[age of the universe|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.)
| |
| <!--
| |
| [[Gabriele Veneziano]] suggests that [[space]] and [[time]] might switch roles within a black hole,<ref>[[Gabriele Veneziano]], "The myth of the beginning of time", ''A Matter of Time''. ''Scientific American'', '''16''' no. 1, 2006. pp.72-81. </ref> and that the big bang occurred when physical conditions dictated by string theory initiate a bounce. [[Roger Penrose]] suggests that the transition from a previous universe to a next universe satisfy the conditions imposed by the [[Weyl curvature hypothesis]] in a [[conformal transformation]].<ref>[[Roger Penrose]], public lecture at University of British Columbia, August 18, 2006</ref>
| |
| -->
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| | |
| ==Reprise==
| |
| [[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 process|irreversible phenomena]]<ref>Prigogine, Ilya (1996), ''The End of Certainty: Time, Chaos and the New Laws of Nature''. ISBN 0-684-83705-6 On pages 163 and 182.</ref> as well as the [[arrow of time]] and the [[Big Bang]].
| |
| | |
| ==See also==
| |
| * [[Relativistic dynamics]]
| |
| * [[:Category:systems of units]]
| |
| | |
| ==References==
| |
| {{reflist|2}}
| |
| | |
| ==Further reading==
| |
| * Boorstein, Daniel J., ''The Discoverers''. Vintage. February 12, 1985. ISBN 0-394-72625-1
| |
| * [[H. Dieter Zeh|Dieter Zeh, H.]], ''The physical basis of the direction of time''. Springer. ISBN 978-3-540-42081-1
| |
| * [[Thomas S. Kuhn|Kuhn, Thomas S.]], ''The Structure of Scientific Revolutions''. ISBN 0-226-45808-3
| |
| * [[Benoît Mandelbrot|Mandelbrot, Benoît]], ''Multifractals and 1/f noise''. Springer Verlag. February 1999. ISBN 0-387-98539-5
| |
| * [[Ilya Prigogine|Prigogine, Ilya]] (1984), ''Order out of Chaos''. ISBN 0-394-54204-5
| |
| * [[Michel Serres|Serres, Michel]], et al., "''Conversations on Science, Culture, and Time (Studies in Literature and Science)''". March, 1995. ISBN 0-472-06548-3
| |
| * Stengers, Isabelle, and Ilya Prigogine, ''Theory Out of Bounds''. University of Minnesota Press. November 1997. ISBN 0-8166-2517-4
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|
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| {{Time topics}}
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| {{Time measurement and standards}}
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| | |
| [[Category:Time|Physics]]
| |
| [[Category:Timekeeping]]
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| [[Category:Philosophy of physics]]
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