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| | Such functionality enables websites to publish the website in different languages as well as multiple brands for different regions. If you liked this article and you would like to receive more data concerning [http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html Woocommerce store] kindly stop by the website. You often see such verification on sites from Companies such as Hacker Safe, Trust Guard, and GeoTrust. Ecommerce retails sell products online, and therefore have to ship the products out to people all over the globe. One Page Checkout For Your E-commerce Site The checkout page is one of the most important parts of an e-commerce business. It should be clearly displayed horizontally across the top or in a column down one side of the page.<br><br><br><br>The ecommerce website design is the most appropriate in your case. Checking out should be done in as few clicks as possible. You should examine your budget to know what services you can afford. Create Easy Holiday Gift Packs Holiday shoppers often have no idea what specific item they wish to purchase for someone, but do have an idea as to the price range.<br><br>You can give them a reward such as write a review for a chance to receive a free gift, a discount or a gift certificate. It's not that easy as it sounds. E-commerce so that you can attract more customers and increase the demand for the products you are selling. Four: Navigation Navigation and product menus must be easy to find and organized.<br><br>Still, there are critics simply not convinced at the foolproof organization of ecommerce solutions on websites and their deployment. For the beginning ecommerce website owner, Ecrater offers a very [https://Twitter.com/search?q=simple+graphic&src=typd simple graphic] look that is easy to deal with. The same holds true for ecommerce websites. With the right solution, you will be able to improve all other aspects of your online store.<br><br>It should also be protected from water damage with a plastic bag. In addition, you should also do your best to make your special products stand out. A Brief History Of Ecommerce Ecommerce was seen as a passing fad or gimmick when the pioneers of the process began. |
| {{Cosmology}}
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| In [[Big Bang|standard cosmology]], '''comoving distance''' and '''proper distance''' are two closely related [[distance measures (cosmology)|distance measures used by cosmologists]] to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a specific moment of [[cosmological time]], which can change over time due to the [[metric expansion of space|expansion of the universe]]. ''Comoving distance'' factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors such as the motion of a galaxy within a cluster). Comoving distance and proper distance are defined to be equal at the present time; therefore, the ratio of proper distance to comoving distance now is 1. At other times, the [[Scale_factor_(cosmology)|scale factor]] differs from 1. The universe's expansion results in the proper distance changing, while the comoving distance is unchanged by this expansion because it is the proper distance divided by that scale factor.
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| ==Comoving coordinates==
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| While general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are more natural (easier to work with). Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as [[isotropic]]. Such observers are called "comoving" observers because they move along with the [[Hubble flow]].
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| <!--was: a scientist to [[formula|formulate]] the laws of physics using an arbitrary system of [[coordinates]], a scientist's job can be simplified by utilizing different coordinate systems that are easy to work with. -->
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| A comoving observer is the only observer that will perceive the universe, including the [[cosmic microwave background radiation]], to be [[isotropic]]. Non-comoving observers will see regions of the sky systematically [[blue-shift]]ed or [[red-shift]]ed. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local [[frame of reference]] called the [[comoving frame]]. The velocity of an observer relative to the local comoving frame is called the [[peculiar velocity]] of the observer.
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| Most large lumps of matter, such as galaxies, are nearly comoving, so that their peculiar velocities (owing to gravitational attraction) are low.
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| The '''comoving time''' coordinate is the elapsed time since the [[Big Bang]] according to a clock of a comoving observer and is a measure of [[cosmological time]]. The comoving spatial coordinates tell us '''where''' an event occurs while cosmological time tells us '''when''' an event occurs. Together, they form a complete [[coordinate system]], giving us both the location and time of an event.
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| Space in comoving coordinates is usually referred to as being "static", as most bodies on the scale of galaxies or larger are approximately comoving, and comoving bodies have static, unchanging comoving coordinates. So for a given pair of comoving galaxies, while the proper distance between them would have been smaller in the past and will become larger in the future due to the expansion of space, the comoving distance between them remains ''constant'' at all times.
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| The expanding Universe has an increasing [[Scale factor (cosmology)|scale factor]] which explains how constant comoving distances are reconciled with proper distances that increase with time.
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| :''See also:'' [[metric expansion of space]].
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| ==Comoving distance and proper distance==
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| ''Comoving distance'' is the distance between two points measured along a path defined at the present [[cosmological time]]. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula:
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| :<math> \chi = \int_{t_e}^t c \; {\mbox{d} t' \over a(t')} </math>
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| where ''a''(''t′'') is the [[Scale factor (cosmology)|scale factor]], ''t''<sub>e</sub> is the time of emission of the photons detected by the observer, ''t'' is the present time, and ''c'' is the [[speed of light]] in vacuum.
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| Despite being an [[time integral|integral over time]], this does give the distance that ''would'' be measured by a hypothetical tape measure at ''fixed'' time ''t'', i.e. the "proper distance" as defined below, divided by the scale factor ''a''(''t'') at that time. For a derivation see [http://arxiv.org/abs/astro-ph/0310808 (Davis and Lineweaver, 2003)] "standard relativistic definitions".
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| ;'''Definitions'''
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| * Many textbooks use the symbol <math>\! \chi</math> for the comoving distance. However, this <math>\! \chi</math> must be distinguished from the ''coordinate'' distance r in the commonly used comoving coordinate system for a [[Friedmann–Lemaître–Robertson–Walker metric|FLRW universe]] where the metric takes the form
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| :<math>\! ds^2 = -c^2 d\tau^2 = - c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1 - kr^2} + r^2 \left(d\theta^2 + \sin^2 \theta d\phi^2 \right)\right)</math>.
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| :In this case the comoving coordinate distance <math>\! r</math> is related to <math>\! \chi</math> by <math>\! \chi = r</math> if k=0 (a spatially flat universe), by <math>\! \chi = \sin^{-1} r</math> if k=1 (a positively curved 'spherical' universe), and by <math>\! \chi = \sinh^{-1} r</math> if k=-1 (a negatively curved 'hyperbolic' universe).<ref>See [http://books.google.com/books?id=3LO75VmI9BMC&lpg=PP1&pg=PA9#v=onepage&q&f=false pages 9-12] of ''The Cosmological Background Radiation'' by Marc Lachièze-Rey and Edgard Gunzig, or [http://books.google.com/books?id=ntZwxttZF-sC&lpg=PR1&pg=PA263#v=onepage&q&f=false p. 263] ''Measuring the Universe: The Cosmological Distance Ladder'' by Stephen Webb.</ref>
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| * Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them '''proper distance'''. On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted <math>\! \chi</math>, the proper distance <math>\! d(t)</math> at an arbitrary time <math>\! t</math> is simply given by <math>\! d(t) = a(t) \chi</math> where <math>\! a(t)</math> is the [[Scale factor (cosmology)|scale factor]]. (e.g. [http://arxiv.org/abs/astro-ph/0310808 Davis and Lineweaver, 2003]) The proper distance <math>\! d(t)</math> between two galaxies at time ''t'' is just the distance that would be measured by rulers between them at that time.<ref>see p. 4 of [http://arxiv.org/pdf/astro-ph/9905116v4 Distance Measures in Cosmology] by David W. Hogg.</ref>
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| ===Uses of the proper distance===
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| [[Cosmological time]] is identical to locally measured time for an observer at a fixed comoving spatial position, that is, in the local [[comoving frame]]. Proper distance is also equal to the locally measured distance in the comoving frame for ''nearby'' objects. To measure the proper distance between two ''distant'' objects, one imagines that one has many comoving observers in a straight line between the two objects, so that all of the observers are close to each other, and form a chain between the two distant objects. All of these observers must have the same cosmological time. Each observer measures their distance to the nearest observer in the chain, and the length of the chain, the sum of distances between nearby observers, is the total proper distance.<ref>Steven Weinberg, ''Gravitation and Cosmology'' (1972), p. 415</ref> <!-- a diagram would probably be helpful -->
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| It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to [[proper length]] in [[special relativity]]) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or [[Spacetime#Spacetime in general relativity|spacelike]] [[geodesic]] between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own [[world line]]s, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the [[Friedmann–Lemaître–Robertson–Walker metric|FLRW metric]] is set to zero (an empty '[[Milne universe]]'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the flat [[Minkowski space|Minkowski spacetime]] of special relativity, one where surfaces of constant time-coordinate appear as [[hyperbola]]s when drawn in a [[Minkowski diagram]] from the perspective of an [[inertial frame of reference]].<ref>See the diagram on [http://books.google.com/books?id=1TXO7GmwZFgC&lpg=PP1&pg=PA28#v=onepage&q&f=false p. 28] of ''Physical Foundations of Cosmology'' by V. F. Mukhanov, along with the accompanying discussion.</ref> In this case, for two events which are simultaneous according the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the [[proper length]] between these same events,[http://www.astro.ucla.edu/~wright/cosmo_02.htm (Wright)] which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a [[geodesic]] in flat Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are [[relativity of simultaneity|simultaneous]].
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| If one divides a change in proper distance by the interval of cosmological time where the change was measured (or takes the [[derivative]] of proper distance with respect to cosmological time) and calls this a "velocity", then the resulting "velocities" of galaxies or quasars can be above the speed of light, ''c''. This apparent superluminal expansion is not in conflict with special or general relativity, and is a consequence of the particular definitions used in cosmology. Even light itself does not have a "velocity" of ''c'' in this sense; the total velocity of any object can be expressed as the sum <math>\! v_{tot} = v_{rec} + v_{pec}</math> where <math>\! v_{rec}</math> is the recession velocity due to the expansion of the universe (the velocity given by [[Hubble's law]]) and <math>\! v_{pec}</math> is the "peculiar velocity" measured by local observers (with <math>\! v_{rec} = \dot{a}(t) \chi(t)</math> and <math>\! v_{pec} = a(t) \dot{\chi}(t)</math>, the dots indicating a first [[derivative]]), so for light <math>\! v_{pec}</math> is equal to ''c'' (-''c'' if the light is emitted towards our position at the origin and +''c'' if emitted away from us) but the total velocity <math>\! v_{tot}</math> is generally different than ''c''.[http://arxiv.org/abs/astro-ph/0310808 (Davis and Lineweaver 2003, p. 19)] Even in special relativity the coordinate speed of light is only guaranteed to be ''c'' in an [[inertial frame of reference|inertial frame]], in a non-inertial frame the coordinate speed may be different than ''c'';<ref>see [http://books.google.com/books?id=AzfFo6A94WEC&lpg=PR1&pg=PA219#v=onepage&q&f=false p. 219] of ''Relativity and the Nature of Spacetime'' by Vesselin Petkov</ref> in general relativity no coordinate system on a large region of curved spacetime is "inertial", but in the local neighborhood of any point in curved spacetime we can define a "local inertial frame" and the local speed of light will be ''c'' in this frame,<ref>see [http://books.google.com/books?id=RK8qDGKSTPwC&lpg=PR1&pg=PA94#v=onepage&q&f=false p. 94] of ''An Introduction to the Science of Cosmology'' by Derek J. Raine, Edwin George Thomas, and E. G. Thomas</ref> with massive objects such as stars and galaxies always having a local speed smaller than ''c''. The cosmological definitions used to define the velocities of distant objects are coordinate-dependent - there is no general coordinate-independent definition of velocity between distant objects in general relativity [http://math.ucr.edu/home/baez/einstein/node2.html (Baez and Bunn, 2006)]. The issue of how best to describe and popularize the apparent superluminal expansion of the universe has caused a minor amount of controversy. One viewpoint is presented in [http://arxiv.org/abs/astro-ph/0310808 (Davis and Lineweaver, 2003)].
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| ==Proper distance vs. comoving distance from small galaxies to galaxy clusters==
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| Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (that is, the comoving distance measured using the scale factor of the universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.
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| ==See also==
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| *[[Distance measures (cosmology)]] for comparison with other distance measures
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| *[[Friedmann–Lemaître–Robertson–Walker metric]] for information about the most popular cosmological model
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| *[[Shape of the universe]]
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| ==References==
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| {{Reflist|30em}}
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| ==External links==
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| *[http://arxiv.org/abs/astro-ph/9905116 Distance measures in cosmology]
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| *[http://www.astro.ucla.edu/~wright/cosmo_01.htm Ned Wright's cosmology tutorial]
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| *[http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )]
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| *[http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1972gcpa.book.....W&db_key=AST Weinberg, Steven (1972)]
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| *[http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1993ppc..book.....P&db_key=AST Peebles (1993)]
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| *[http://arxiv.org/abs/astro-ph/0310808 Davis and Lineweaver, Expanding Confusion]
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| *[http://www.astro.ucla.edu/~wright/CosmoCalc.html Ned Wright's Javascript cosmology calculator]
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| *[http://arXiv.org/abs/astro-ph/9603028 General method, including locally inhomogeneous case] and [[Fortran 77]] software
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| *[http://cosmo.torun.pl/GPLdownload/dodec/cosmdist-0.2.0.tar.gz cosmdist-0.2.0] - command line and/or [[C (programming language)|C]] or [[Fortran]] library, based on [[GNU Scientific Library]], for <math>d_p, d_{pm}, t</math> as functions of ''z'' and their inverses
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| [[Category:Physical cosmology]]
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| [[Category:Coordinate charts in general relativity]]
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| [[Category:Physical quantities]]
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Such functionality enables websites to publish the website in different languages as well as multiple brands for different regions. If you liked this article and you would like to receive more data concerning Woocommerce store kindly stop by the website. You often see such verification on sites from Companies such as Hacker Safe, Trust Guard, and GeoTrust. Ecommerce retails sell products online, and therefore have to ship the products out to people all over the globe. One Page Checkout For Your E-commerce Site The checkout page is one of the most important parts of an e-commerce business. It should be clearly displayed horizontally across the top or in a column down one side of the page.
The ecommerce website design is the most appropriate in your case. Checking out should be done in as few clicks as possible. You should examine your budget to know what services you can afford. Create Easy Holiday Gift Packs Holiday shoppers often have no idea what specific item they wish to purchase for someone, but do have an idea as to the price range.
You can give them a reward such as write a review for a chance to receive a free gift, a discount or a gift certificate. It's not that easy as it sounds. E-commerce so that you can attract more customers and increase the demand for the products you are selling. Four: Navigation Navigation and product menus must be easy to find and organized.
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