Vasicek model - Revision history

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	~~{{General relativity}}~~		Greetings! I am Marvella and I really feel comfortable when individuals use the complete title. Doing ceramics is what my family and I enjoy. He utilized to be unemployed but now he is a meter reader. Puerto Rico is where he's usually been living but she requirements to transfer because of her family members.<br><br>Here is my webpage; over the counter std test - [http://nuvem.tk/altergalactica/AliceedMaurermy click to read],

	In the [[theory of relativity]], it is convenient to express results in terms of a [[spacetime]] [[coordinate system]] relative to an implied [[Observer_(special_relativity)\|observer]]. In many (but not all) coordinate systems, an [[event (relativity)\|event]] is specified by one [[time]] coordinate and ~~three [[Three-dimensional space\|spatial]] coordinates. The time specified by~~ the ~~time coordinate is referred to as '''coordinate time''' to distinguish it from [[proper time]]~~.

	~~In the special case of an [[inertial]] observer in [[special relativity]], by convention the coordinate time at an event~~ is the same as the [[proper time]] measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the [[Einstein synchronisation]] convention.

	~~== Coordinate time, proper time, and clock synchronization ==~~

	~~Fuller explanation of the concept of coordinate time arises from its relationships with proper time~~ and ~~with clock synchronization~~. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of [[General theory of relativity\|general relativity theory]], because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed. Specific [[Einstein synchronization\|clock synchronization procedures]] were defined by [[Albert Einstein\|Einstein]] and give rise to a limited concept of [[Relativity of simultaneity\|simultaneity]].<ref name=klnr1992>S A Klioner (1992), [http://articles.adsabs.harvard.edu/full/1992CeMDA..53...81K "The problem of clock synchronization - A relativistic approach"], ''Celestial Mechanics and Dynamical Astronomy'', vol.53 (1992), pp. 81-109.</ref>

	Two events are called simultaneous in a chosen reference frame if and only if the chosen coordinate time has the same value for both of them;<ref>S A Klioner (2008), [http://www.aanda.org/index.php?option=article&access=bibcode&bibcode=2008A%2526A...478..951KPDF "Relativistic scaling of astronomical quantities and the system of astronomical units"], ''Astronomy and Astrophysics'', vol.478 (2008), pp.951-958, at section 5: "On the concept of coordinate time scales", esp. p.955.</ref> and this condition allows for the physical possibility and likelihood that they will not be simultaneous from the standpoint of another reference frame.<ref name=klnr1992 />

	~~But the coordinate time~~ is not a time that could be measured by a clock located at the place that nominally defines the reference frame, e.g. a clock located at the solar system barycenter would not measure the coordinate time of the barycentric reference frame, and a clock located at the geocenter would not measure the coordinate time of a geocentric reference frame.<ref>S A Klioner (2008), cited above, at page 954.</ref>

	~~==Mathematics==~~

	~~For non-inertial observers, and in [[general relativity]], coordinate systems can be chosen more freely. For~~ a ~~clock whose spatial coordinates are constant, the relationship between proper time τ ([[Greek alphabet\|Greek]] lowercase [[tau]]) and coordinate time ''t'', i~~.~~e. the rate of [[time dilation]],~~ is ~~given by~~

	~~{{NumBlk\|:\|~~
	~~<math>\frac{d\tau}{dt} = \sqrt{-g_{00}}</math>~~
	~~\|{{EquationRef\|1}}}}~~

	where ~~''g''<sub>00</sub> is a component of the [[metric tensor]], which incorporates [[gravitational time dilation]] (under the assumption that the zeroth component is [[timelike]]).~~

	An alternative formulation, correct to the order of terms in 1/''c''<sup>2</sup>, gives the relation between proper and coordinate time in terms of more-easily recognizable quantities in dynamics:<ref>This is for example equation (6) at page 36 of T D Moyer (1981), "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference", ''Celestial Mechanics'', vol.23 (1981), pages 33-56.)</ref>

	~~{{NumBlk\|:\|~~
	~~<math>\frac{d\tau}{dt} = 1 - \frac{U}{c^2} - \frac{v^2}{2c^2} </math>~~
	~~\|{{EquationRef\|2}}}}~~

	~~in which:~~
	~~:<math>U = \sum_i \frac{GM_i}{r_i}</math>~~

	is a [[summation\|sum]] of [[gravitational potential]]s due to the [[mass]]es in the neighborhood, based on their distances ''r<sub>i</sub>'' from the clock. This sum of the terms ''GM<sub>i</sub>/r<sub>i</sub>'' is evaluated approximately, as a sum of [[Newton's law of universal gravitation#Problems with Newton's theory\|Newtonian gravitational potentials]] (plus any tidal potentials considered), and is represented using the positive astronomical sign convention for gravitational potentials.

	~~Also ''c'' is the [[speed of light]], and ''v'' is the [[speed]] of the clock (in the coordinates of the chosen [[Frame of reference\|reference frame]]) defined by:~~

	~~{{NumBlk\|:\|~~
	~~<math>v^2 = (dx^2 + dy^2 + dz^2)/(dt_c)^2 </math>~~
	~~\|{{EquationRef\|3}}}}~~

	where ''dx, dy, dz'' and ''dt<sub>c</sub>'' are [[Differential (infinitesimal)\|small increments]] in three orthogonal [[spacelike]] coordinates ''x, y, z'' and in the coordinate time ''t<sub>c</sub>'' of the clock's position in the chosen reference frame.

	Equation ({{EquationNote\|2}}) is a fundamental and much-quoted [[differential equation]] for the relation between proper time and coordinate time, i.e. for time dilation. A derivation, starting from the [[Schwarzschild metric]], with further reference sources, is given in [[Time dilation#Time dilation due to gravitation and motion together\|Time dilation due to gravitation and motion together]].

	~~==Measurement==~~

	~~The coordinate times cannot be measured,~~ but ~~only computed from the (proper-time) readings of real clocks with the aid of the time dilation relationship shown in equation ({{EquationNote\|2}}) (or some alternative or refined form of it).~~

	~~Only for explanatory purposes it is possible~~ to ~~conceive a hypothetical observer and trajectory on which the proper time~~ of the clock would coincide with coordinate time: such an observer and clock have to be conceived at rest with respect to the chosen reference frame (''v'' = 0 in ({{EquationNote\|2}}) above) but also (in an unattainably hypothetical situation) infinitely far away from its gravitational masses (also ''U'' = 0 in ({{EquationNote\|2}}) above).<~~ref~~>~~S A Klioner (2008), cited above, at page 955.~~<~~/ref~~> ~~Even such an illustration~~ is ~~of limited use because the coordinate time is defined everywhere in the reference frame, while the hypothetical observer and clock chosen to illustrate it has only a limited choice of trajectory.~~

	~~== Coordinate time scales ==~~

	A '''coordinate time scale''' (or '''coordinate time standard''') is a [[time standard]] designed for use as the time coordinate in calculations that need to take account of relativistic effects. The choice of a time coordinate implies the choice of an entire frame of reference.

	As described above, a time coordinate can to a limited extent be illustrated by the proper time of a clock that is notionally infinitely far away from the objects of interest and at rest with respect to the chosen reference frame. This notional clock, because it is outside all [[gravity well]]s, is not influenced by [[gravitational time dilation]]. The proper time of objects within a gravity well will pass more slowly than the coordinate time even when they are at rest with respect to the coordinate reference frame. Gravitational as well as motional time dilation must be considered for each object of interest, and the effects are functions of the velocity relative to the reference frame and of the [[gravitational potential]] as indicated in ({{EquationNote\|2}}).

	There are four purpose-designed coordinate time scales defined by the [[IAU]] for use in [[astronomy]]. [[Barycentric Coordinate Time]] (TCB) is based on a reference frame comoving with the [[Barycentric coordinates (astronomy)\|barycenter]] of the [[Solar system]], and has been defined for use in calculating motion of bodies within the Solar system. However, from the standpoint of Earth-based observers, general time dilation including gravitational time dilation causes Barycentric Coordinate Time, which is based on the [[Second#International second\|SI second]], to appear when observed from the Earth to have time units that pass more quickly than SI seconds measured by an Earth-based clock, with a rate of divergence of about 0.5 seconds per year.<ref name=timgraf>[http://www.ucolick.org/~sla/leapsecs/deltat.png A graph giving an overview of the rate differences (when observed from the Earth's surface) and offsets between various standard time scales], present and past, defined by the IAU: for description see Fig. 1 (at p.835) in P K Seidelmann & T Fukushima (1992), [http://articles.adsabs.harvard.edu/full/1992A%26A...265..833S "Why new time scales?"], ''Astronomy & Astrophysics'' vol.265 (1992), pages 833-838.</ref> Accordingly, for many practical astronomical purposes, a scaled modification of TCB has been defined, called for historical reasons [[Barycentric Dynamical Time]] (TDB), with a time unit that evaluates to SI seconds when observed from the Earth's surface, thus assuring that at least for several millennia TDB will remain within 2 milliseconds of [[Terrestrial Time]] (TT),<ref>[http://www.iau.org/static/resolutions/IAU2006_Resol3.pdf IAU 2006 resolution 3], see Recommendation and footnotes, note 3.</ref><ref>These differences between coordinate time scales are mainly periodic, the basis for them explained in G M Clemence & V Szebehely, [http://articles.adsabs.harvard.edu/abs/1967AJ.....72.1324C "Annual variation of an atomic clock"], Astronomical Journal, Vol.72 (1967), p.1324-6.</ref> albeit that the time unit of TDB, if measured by the hypothetical observer described above, at rest in the reference frame and at infinite distance, would be very slightly slower than the SI second (by 1 part in 1/L<sub>B</sub> = 1 part in 10<sup>8</sup>/1.550519768).<ref>Scaling defined in [http://~~www.iau~~.~~org/static~~/~~resolutions~~/~~IAU2006_Resol3.pdf IAU 2006 resolution 3].</ref>~~

	[[Geocentric Coordinate Time]] (TCG) is based on a reference frame comoving with the geocenter (the center of the Earth), and is defined in principle for use for calculations concerning phenomena on or in the region of the Earth, such as planetary rotation and [[satellite]] motions. To a much smaller extent than with TCB compared with TDB, but for a corresponding reason, the SI second of TCG when observed from the Earth's surface shows a slight acceleration on the SI seconds realized by Earth-surface-based clocks. Accordingly, [[Terrestrial Time]] (TT) has also been defined as a scaled version of TCG, with the scaling such that on the defined geoid the unit rate is equal to the SI second, albeit that in terms of TCG the SI second of TT is a very little slower (this time by 1 part in 1/L<sub>G</sub> = 1 part in 10<sup>10</sup>/6.969290134).<ref>Scaling defined in [http://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm Resolutions of the IAU 2000 24th General Assembly (Manchester)], ~~see Resolution B1.9.</ref>~~

	~~==See also==~~

	*[[Absolute time and space]]
	*[[Introduction to special relativity]]
	*[[Introduction to the mathematics of general relativity]]

	~~==References==~~
	~~<references/>~~

	~~{{Time Topics}}~~
	~~{{Time measurement and standards}}~~

	~~[[Category:Theory of relativity]]~~
	~~[[Category:Time scales]]~~

en>Peter9002: /* External links */

2014-01-19T22:39:38Z

External links

← Older revision		Revision as of 00:39, 20 January 2014
Line 1:		Line 1:
	~~Greetings~~. The ~~writer~~'s name is ~~Phebe~~ and ~~she feels comfy when people use~~ the ~~full title~~. ~~What I love doing~~ is ~~doing ceramics but I haven~~'t ~~produced~~ a ~~dime~~ with it. ~~For many years he~~'s been ~~operating as~~ a ~~receptionist~~. ~~Years in~~ the past ~~we moved~~ to ~~North Dakota~~.<br><br>~~Look~~ at ~~my web site .~~.. [http://www.~~neweracinema~~.~~com~~/~~tube~~/~~user~~/~~KOPR~~ http://~~www~~.~~neweracinema~~.~~com~~/~~tube~~/~~user~~/~~KOPR~~]		{{General relativity}}

			In the [[theory of relativity]], it is convenient to express results in terms of a [[spacetime]] [[coordinate system]] relative to an implied [[Observer_(special_relativity)\|observer]]. In many (but not all) coordinate systems, an [[event (relativity)\|event]] is specified by one [[time]] coordinate and three [[Three-dimensional space\|spatial]] coordinates. The time specified by the time coordinate is referred to as '''coordinate time''' to distinguish it from [[proper time]].

			In the special case of an [[inertial]] observer in [[special relativity]], by convention the coordinate time at an event is the same as the [[proper time]] measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the [[Einstein synchronisation]] convention.

			== Coordinate time, proper time, and clock synchronization ==

			Fuller explanation of the concept of coordinate time arises from its relationships with proper time and with clock synchronization. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of [[General theory of relativity\|general relativity theory]], because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed. Specific [[Einstein synchronization\|clock synchronization procedures]] were defined by [[Albert Einstein\|Einstein]] and give rise to a limited concept of [[Relativity of simultaneity\|simultaneity]].<ref name=klnr1992>S A Klioner (1992), [http://articles.adsabs.harvard.edu/full/1992CeMDA..53...81K "The problem of clock synchronization - A relativistic approach"], ''Celestial Mechanics and Dynamical Astronomy'', vol.53 (1992), pp. 81-109.</ref>

			Two events are called simultaneous in a chosen reference frame if and only if the chosen coordinate time has the same value for both of them;<ref>S A Klioner (2008), [http://www.aanda.org/index.php?option=article&access=bibcode&bibcode=2008A%2526A...478..951KPDF "Relativistic scaling of astronomical quantities and the system of astronomical units"], ''Astronomy and Astrophysics'', vol.478 (2008), pp.951-958, at section 5: "On the concept of coordinate time scales", esp. p.955.</ref> and this condition allows for the physical possibility and likelihood that they will not be simultaneous from the standpoint of another reference frame.<ref name=klnr1992 />

			But the coordinate time is not a time that could be measured by a clock located at the place that nominally defines the reference frame, e.g. a clock located at the solar system barycenter would not measure the coordinate time of the barycentric reference frame, and a clock located at the geocenter would not measure the coordinate time of a geocentric reference frame.<ref>S A Klioner (2008), cited above, at page 954.</ref>

			==Mathematics==

			For non-inertial observers, and in [[general relativity]], coordinate systems can be chosen more freely. For a clock whose spatial coordinates are constant, the relationship between proper time τ ([[Greek alphabet\|Greek]] lowercase [[tau]]) and coordinate time ''t'', i.e. the rate of [[time dilation]], is given by

			{{NumBlk\|:\|
			<math>\frac{d\tau}{dt} = \sqrt{-g_{00}}</math>
			\|{{EquationRef\|1}}}}

			where ''g''<sub>00</sub> is a component of the [[metric tensor]], which incorporates [[gravitational time dilation]] (under the assumption that the zeroth component is [[timelike]]).

			An alternative formulation, correct to the order of terms in 1/''c''<sup>2</sup>, gives the relation between proper and coordinate time in terms of more-easily recognizable quantities in dynamics:<ref>This is for example equation (6) at page 36 of T D Moyer (1981), "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference", ''Celestial Mechanics'', vol.23 (1981), pages 33-56.)</ref>

			{{NumBlk\|:\|
			<math>\frac{d\tau}{dt} = 1 - \frac{U}{c^2} - \frac{v^2}{2c^2} </math>
			\|{{EquationRef\|2}}}}

			in which:
			:<math>U = \sum_i \frac{GM_i}{r_i}</math>

			is a [[summation\|sum]] of [[gravitational potential]]s due to the [[mass]]es in the neighborhood, based on their distances ''r<sub>i</sub>'' from the clock. This sum of the terms ''GM<sub>i</sub>/r<sub>i</sub>'' is evaluated approximately, as a sum of [[Newton's law of universal gravitation#Problems with Newton's theory\|Newtonian gravitational potentials]] (plus any tidal potentials considered), and is represented using the positive astronomical sign convention for gravitational potentials.

			Also ''c'' is the [[speed of light]], and ''v'' is the [[speed]] of the clock (in the coordinates of the chosen [[Frame of reference\|reference frame]]) defined by:

			{{NumBlk\|:\|
			<math>v^2 = (dx^2 + dy^2 + dz^2)/(dt_c)^2 </math>
			\|{{EquationRef\|3}}}}

			where ''dx, dy, dz'' and ''dt<sub>c</sub>'' are [[Differential (infinitesimal)\|small increments]] in three orthogonal [[spacelike]] coordinates ''x, y, z'' and in the coordinate time ''t<sub>c</sub>'' of the clock's position in the chosen reference frame.

			Equation ({{EquationNote\|2}}) is a fundamental and much-quoted [[differential equation]] for the relation between proper time and coordinate time, i.e. for time dilation. A derivation, starting from the [[Schwarzschild metric]], with further reference sources, is given in [[Time dilation#Time dilation due to gravitation and motion together\|Time dilation due to gravitation and motion together]].

			==Measurement==

			The coordinate times cannot be measured, but only computed from the (proper-time) readings of real clocks with the aid of the time dilation relationship shown in equation ({{EquationNote\|2}}) (or some alternative or refined form of it).

			Only for explanatory purposes it is possible to conceive a hypothetical observer and trajectory on which the proper time of the clock would coincide with coordinate time: such an observer and clock have to be conceived at rest with respect to the chosen reference frame (''v'' = 0 in ({{EquationNote\|2}}) above) but also (in an unattainably hypothetical situation) infinitely far away from its gravitational masses (also ''U'' = 0 in ({{EquationNote\|2}}) above).<ref>S A Klioner (2008), cited above, at page 955.</ref> Even such an illustration is of limited use because the coordinate time is defined everywhere in the reference frame, while the hypothetical observer and clock chosen to illustrate it has only a limited choice of trajectory.

			== Coordinate time scales ==

			A '''coordinate time scale''' (or '''coordinate time standard''') is a [[time standard]] designed for use as the time coordinate in calculations that need to take account of relativistic effects. The choice of a time coordinate implies the choice of an entire frame of reference.

			As described above, a time coordinate can to a limited extent be illustrated by the proper time of a clock that is notionally infinitely far away from the objects of interest and at rest with respect to the chosen reference frame. This notional clock, because it is outside all [[gravity well]]s, is not influenced by [[gravitational time dilation]]. The proper time of objects within a gravity well will pass more slowly than the coordinate time even when they are at rest with respect to the coordinate reference frame. Gravitational as well as motional time dilation must be considered for each object of interest, and the effects are functions of the velocity relative to the reference frame and of the [[gravitational potential]] as indicated in ({{EquationNote\|2}}).

			There are four purpose-designed coordinate time scales defined by the [[IAU]] for use in [[astronomy]]. [[Barycentric Coordinate Time]] (TCB) is based on a reference frame comoving with the [[Barycentric coordinates (astronomy)\|barycenter]] of the [[Solar system]], and has been defined for use in calculating motion of bodies within the Solar system. However, from the standpoint of Earth-based observers, general time dilation including gravitational time dilation causes Barycentric Coordinate Time, which is based on the [[Second#International second\|SI second]], to appear when observed from the Earth to have time units that pass more quickly than SI seconds measured by an Earth-based clock, with a rate of divergence of about 0.5 seconds per year.<ref name=timgraf>[http://www.ucolick.org/~sla/leapsecs/deltat.png A graph giving an overview of the rate differences (when observed from the Earth's surface) and offsets between various standard time scales], present and past, defined by the IAU: for description see Fig. 1 (at p.835) in P K Seidelmann & T Fukushima (1992), [http://articles.adsabs.harvard.edu/full/1992A%26A...265..833S "Why new time scales?"], ''Astronomy & Astrophysics'' vol.265 (1992), pages 833-838.</ref> Accordingly, for many practical astronomical purposes, a scaled modification of TCB has been defined, called for historical reasons [[Barycentric Dynamical Time]] (TDB), with a time unit that evaluates to SI seconds when observed from the Earth's surface, thus assuring that at least for several millennia TDB will remain within 2 milliseconds of [[Terrestrial Time]] (TT),<ref>[http://www.iau.org/static/resolutions/IAU2006_Resol3.pdf IAU 2006 resolution 3], see Recommendation and footnotes, note 3.</ref><ref>These differences between coordinate time scales are mainly periodic, the basis for them explained in G M Clemence & V Szebehely, [http://articles.adsabs.harvard.edu/abs/1967AJ.....72.1324C "Annual variation of an atomic clock"], Astronomical Journal, Vol.72 (1967), p.1324-6.</ref> albeit that the time unit of TDB, if measured by the hypothetical observer described above, at rest in the reference frame and at infinite distance, would be very slightly slower than the SI second (by 1 part in 1/L<sub>B</sub> = 1 part in 10<sup>8</sup>/1.550519768).<ref>Scaling defined in [http://www.iau.org/static/resolutions/IAU2006_Resol3.pdf IAU 2006 resolution 3].</ref>

			[[Geocentric Coordinate Time]] (TCG) is based on a reference frame comoving with the geocenter (the center of the Earth), and is defined in principle for use for calculations concerning phenomena on or in the region of the Earth, such as planetary rotation and [[satellite]] motions. To a much smaller extent than with TCB compared with TDB, but for a corresponding reason, the SI second of TCG when observed from the Earth's surface shows a slight acceleration on the SI seconds realized by Earth-surface-based clocks. Accordingly, [[Terrestrial Time]] (TT) has also been defined as a scaled version of TCG, with the scaling such that on the defined geoid the unit rate is equal to the SI second, albeit that in terms of TCG the SI second of TT is a very little slower (this time by 1 part in 1/L<sub>G</sub> = 1 part in 10<sup>10</sup>/6.969290134).<ref>Scaling defined in [http://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm Resolutions of the IAU 2000 24th General Assembly (Manchester)], see Resolution B1.9.</ref>

			==See also==

			*[[Absolute time and space]]
			*[[Introduction to special relativity]]
			*[[Introduction to the mathematics of general relativity]]

			==References==
			<references/>

			{{Time Topics}}
			{{Time measurement and standards}}

			[[Category:Theory of relativity]]
			[[Category:Time scales]]

208.54.83.193: Changed link so that it directly links to ornstein uhlenbeck process

2012-08-29T18:09:25Z

Changed link so that it directly links to ornstein uhlenbeck process

New page

Greetings. The writer's name is Phebe and she feels comfy when people use the full title. What I love doing is doing ceramics but I haven't produced a dime with it. For many years he's been operating as a receptionist. Years in the past we moved to North Dakota.<br><br>Look at my web site ... [http://www.neweracinema.com/tube/user/KOPR http://www.neweracinema.com/tube/user/KOPR]