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59-427: The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric that meets these conditions is where Σ {\displaystyle \mathbf {\Sigma } } ranges over a 3-dimensional space of uniform curvature, that is, elliptical space , Euclidean space , or hyperbolic space . It

118-477: A {\displaystyle a} is the scale factor then ρ ∝ a − 3 ( 1 + w ) . {\displaystyle \rho \propto a^{-3(1+w)}.} If the fluid is the dominant form of matter in a flat universe , then a ∝ t 2 3 ( 1 + w ) , {\displaystyle a\propto t^{\frac {2}{3(1+w)}},} where t {\displaystyle t}

177-461: A connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature. The cosmological constant term is assumed to be treated as dark energy and thus merged into the density and pressure terms. During the Planck epoch , one cannot neglect quantum effects. So they may cause

236-524: A deviation from the Friedmann equations. The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924. Although the prestigious physics journal Zeitschrift für Physik published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein , who, on behalf of Zeitschrift für Physik , acted as

295-420: A law of physics, or to the evolution of a physical system. Fundamental laws of physics should not (explicitly) depend on position in space. That would make them quite useless. In some sense, this is also linked to the requirement that experiments should be reproducible . This principle is true for all laws of mechanics ( Newton's laws , etc.), electrodynamics, quantum mechanics, etc. In practice, this principle

354-481: A long-abandoned static model that was supposed to represent our universe in idealized form. Putting in the Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is where c {\displaystyle c} is the speed of light, G {\displaystyle G} is the Newtonian constant of gravitation , and ρ {\displaystyle \rho }

413-472: A microscope. Brass is an example of an alloy, being a homogeneous mixture of copper and zinc. Another example is steel, which is an alloy of iron with carbon and possibly other metals. The purpose of alloying is to produce desired properties in a metal that naturally lacks them. Brass, for example, is harder than copper and has a more gold-like color. Steel is harder than iron and can even be made rust proof (stainless steel). Homogeneity, in another context plays

472-542: A role in cosmology . From the perspective of 19th-century cosmology (and before), the universe was infinite , unchanging, homogeneous, and therefore filled with stars . However, German astronomer Heinrich Olbers asserted that if this were true, then the entire night sky would be filled with light and bright as day; this is known as Olbers' paradox . Olbers presented a technical paper in 1826 that attempted to answer this conundrum. The faulty premise, unknown in Olbers' time,

531-481: A similar role to that of energy (or mass) density, according to the principles of general relativity . The cosmological constant , on the other hand, causes an acceleration in the expansion of the universe. The cosmological constant term can be omitted if we make the following replacements Therefore, the cosmological constant can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its (positive) energy density: which

590-515: A sort of perfect fluid with equation of state w = 1 2 ϕ ˙ 2 − V ( ϕ ) 1 2 ϕ ˙ 2 + V ( ϕ ) , {\displaystyle w={\frac {{\frac {1}{2}}{\dot {\phi }}^{2}-V(\phi )}{{\frac {1}{2}}{\dot {\phi }}^{2}+V(\phi )}},} where ϕ ˙ {\displaystyle {\dot {\phi }}}

649-399: A valid expression for some energy . Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, E = mv could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v , and one cannot know if hc / λ should be divided or multiplied by 2 π . Nevertheless, this is

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708-579: A very powerful tool in finding characteristic units of a given problem, see dimensional analysis . Equation of state (cosmology) In cosmology , the equation of state of a perfect fluid is characterized by a dimensionless number w {\displaystyle w} , equal to the ratio of its pressure p {\displaystyle p} to its energy density ρ {\displaystyle \rho } : w ≡ p ρ . {\displaystyle w\equiv {\frac {p}{\rho }}.} It

767-454: Is w = 0 {\displaystyle w=0} , which means that its energy density decreases as ρ ∝ a − 3 = V − 1 {\displaystyle \rho \propto a^{-3}=V^{-1}} , where V {\displaystyle V} is a volume. In an expanding universe, the total energy of non-relativistic matter remains constant, with its density decreasing as

826-740: Is Newton's constant , and a ¨ {\displaystyle {\ddot {a}}} is the second proper time derivative of the scale factor. If we define (what might be called "effective") energy density and pressure as ρ ′ ≡ ρ + Λ 8 π G {\displaystyle \rho '\equiv \rho +{\frac {\Lambda }{8\pi G}}} p ′ ≡ p − Λ 8 π G {\displaystyle p'\equiv p-{\frac {\Lambda }{8\pi G}}} and p ′ = w ′ ρ ′ {\displaystyle p'=w'\rho '}

885-535: Is an equation of state of vacuum with dark energy . An attempt to generalize this to would not have general invariance without further modification. In fact, in order to get a term that causes an acceleration of the universe expansion, it is enough to have a scalar field that satisfies Such a field is sometimes called quintessence . This is due to McCrea and Milne, although sometimes incorrectly ascribed to Friedmann. The Friedmann equations are equivalent to this pair of equations: The first equation says that

944-473: Is calculating a speed , units must always combine to [length]/[time]; if one is calculating an energy , units must always combine to [mass][length] /[time] , etc. For example, the following formulae could be valid expressions for some energy: if m is a mass, v and c are velocities , p is a momentum , h is the Planck constant , λ a length. On the other hand, if the units of the right hand side do not combine to [mass][length] /[time] , it cannot be

1003-439: Is closely related to the thermodynamic equation of state and ideal gas law . The perfect gas equation of state may be written as p = ρ m R T = ρ m C 2 {\displaystyle p=\rho _{m}RT=\rho _{m}C^{2}} where ρ m {\displaystyle \rho _{m}} is the mass density, R {\displaystyle R}

1062-403: Is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. d Σ {\displaystyle \mathrm {d} \mathbf {\Sigma } } does not depend on t – all of the time dependence is in the function a ( t ), known as the " scale factor ". In reduced-circumference polar coordinates the spatial metric has

1121-489: Is the density of space of this universe. The numerical value of Einstein's radius is of the order of 10 light years , or 10 billion light years. The current standard model of cosmology, the Lambda-CDM model , uses the FLRW metric. By combining the observation data from some experiments such as WMAP and Planck with theoretical results of Ehlers–Geren–Sachs theorem and its generalization, astrophysicists now agree that

1180-559: Is the origin of the flatness and monopole problems of the Big Bang : curvature has w = − 1 / 3 {\displaystyle w=-1/3} and monopoles have w = 0 {\displaystyle w=0} , so if they were around at the time of the early Big Bang, they should still be visible today. These problems are solved by cosmic inflation which has w ≈ − 1 {\displaystyle w\approx -1} . Measuring

1239-641: Is the particular gas constant, T {\displaystyle T} is the temperature and C = R T {\displaystyle C={\sqrt {RT}}} is a characteristic thermal speed of the molecules. Thus w ≡ p ρ = ρ m C 2 ρ m c 2 = C 2 c 2 ≈ 0 {\displaystyle w\equiv {\frac {p}{\rho }}={\frac {\rho _{m}C^{2}}{\rho _{m}c^{2}}}={\frac {C^{2}}{c^{2}}}\approx 0} where c {\displaystyle c}

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1298-493: Is the proper time. In general the Friedmann acceleration equation is 3 a ¨ a = Λ − 4 π G ( ρ + 3 p ) {\displaystyle 3{\frac {\ddot {a}}{a}}=\Lambda -4\pi G(\rho +3p)} where Λ {\displaystyle \Lambda } is the cosmological constant and G {\displaystyle G}

1357-520: Is the quality of an equation having quantities of same units on both sides; homogeneity (in space) implies conservation of momentum ; and homogeneity in time implies conservation of energy . In the context of composite metals is an alloy. A blend of a metal with one or more metallic or nonmetallic materials is an alloy. The components of an alloy do not combine chemically but, rather, are very finely mixed. An alloy might be homogeneous or might contain small particles of components that can be viewed with

1416-489: Is the speed of light, ρ = ρ m c 2 {\displaystyle \rho =\rho _{m}c^{2}} and C ≪ c {\displaystyle C\ll c} for a "cold" gas. The equation of state may be used in Friedmann–Lemaître–Robertson–Walker (FLRW) equations to describe the evolution of an isotropic universe filled with a perfect fluid. If

1475-535: Is the time-derivative of ϕ {\displaystyle \phi } and V ( ϕ ) {\displaystyle V(\phi )} is the potential energy. A free ( V = 0 {\displaystyle V=0} ) scalar field has w = 1 {\displaystyle w=1} , and one with vanishing kinetic energy is equivalent to a cosmological constant: w = − 1 {\displaystyle w=-1} . Any equation of state in between, but not crossing

1534-490: Is the unnormalized sinc function and k {\displaystyle {\sqrt {k}}} is one of the imaginary, zero or real square roots of k . These definitions are valid for all  k . When k = 0 one may write simply This can be extended to k ≠ 0 by defining where r is one of the radial coordinates defined above, but this is rare. In flat ( k = 0 ) {\displaystyle (k=0)} FLRW space using Cartesian coordinates,

1593-468: Is usually violated, since one studies only a small subsystem of the universe, which of course "feels" the influence of the rest of the universe. This situation gives rise to "external fields" (electric, magnetic, gravitational, etc.) which make the description of the evolution of the system depend upon its position ( potential wells , etc.). This only stems from the fact that the objects creating these external fields are not considered as (a "dynamical") part of

1652-776: The Annales de la Société Scientifique de Bruxelles (Annals of the Scientific Society of Brussels). In the face of the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lemaître's results were noticed in particular by Arthur Eddington , and in 1930–31 Lemaître's paper was translated into English and published in the Monthly Notices of the Royal Astronomical Society . Howard P. Robertson from

1711-465: The Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are: These equations are the basis of the standard Big Bang cosmological model including the current ΛCDM model. Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of

1770-556: The electromagnetic materials domain, when interacting with a directed radiation field (light, microwave frequencies, etc.). Mathematically, homogeneity has the connotation of invariance , as all components of the equation have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition. Cumulative distribution fits this description. "The state of having identical cumulative distribution function or values". The definition of homogeneous strongly depends on

1829-413: The observable universe is well approximated by an almost FLRW model , i.e., a model that follows the FLRW metric apart from primordial density fluctuations . As of 2003, the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP . The pair of equations given above is equivalent to

Friedmann–Lemaître–Robertson–Walker metric - Misplaced Pages Continue

1888-471: The FLRW metric. Homogeneity (physics)#Translation invariance In physics, a homogeneous material or system has the same properties at every point; it is uniform without irregularities. A uniform electric field (which has the same strength and the same direction at each point) would be compatible with homogeneity (all points experience the same physics). A material constructed with different constituents can be described as effectively homogeneous in

1947-401: The FLRW metric. Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations, H 0 {\displaystyle H_{0}} = 71 ± 1 km/s/Mpc , and depending on how local determinations converge, this may point to a breakdown of the FLRW metric in the late universe, necessitating an explanation beyond

2006-460: The Robertson–Walker metric since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître" models , which are specific solutions for a ( t ) that assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant. Einstein's radius of the universe is the radius of curvature of space of Einstein's universe ,

2065-530: The US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which were always assumed by Friedmann and Lemaître). This solution, often called

2124-753: The Universe was indeed observed. According to observations, the value of equation of state of cosmological constant is near -1. Hypothetical phantom energy would have an equation of state w < − 1 {\displaystyle w<-1} , and would cause a Big Rip . Using the existing data, it is still impossible to distinguish between phantom w < − 1 {\displaystyle w<-1} and non-phantom w ≥ − 1 {\displaystyle w\geq -1} . In an expanding universe, fluids with larger equations of state disappear more quickly than those with smaller equations of state. This

2183-471: The above expression for the scale factor is not valid and a ∝ e H t {\displaystyle a\propto e^{Ht}} , where the constant H is the Hubble parameter . More generally, the expansion of the universe is accelerating for any equation of state w < − 1 / 3 {\displaystyle w<-1/3} . The accelerated expansion of

2242-564: The acceleration equation may be written as a ¨ a = − 4 3 π G ( ρ ′ + 3 p ′ ) = − 4 3 π G ( 1 + 3 w ′ ) ρ ′ {\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4}{3}}\pi G\left(\rho '+3p'\right)=-{\frac {4}{3}}\pi G(1+3w')\rho '} The equation of state for ordinary non- relativistic 'matter' (e.g. cold dust)

2301-445: The classical reference text of Landau & Lifshitz. This is a particular application of Noether's theorem . As said in the introduction, dimensional homogeneity is the quality of an equation having quantities of same units on both sides. A valid equation in physics must be homogeneous, since equality cannot apply between quantities of different nature. This can be used to spot errors in formula or calculations. For example, if one

2360-452: The context used. For example, a composite material is made up of different individual materials, known as " constituents " of the material, but may be defined as a homogeneous material when assigned a function. For example, asphalt paves our roads, but is a composite material consisting of asphalt binder and mineral aggregate, and then laid down in layers and compacted. However, homogeneity of materials does not necessarily mean isotropy . In

2419-409: The coordinate r is proportional to radial distance; this gives where d Ω {\displaystyle \mathrm {d} \mathbf {\Omega } } is as before and As before, there are two common unit conventions: Though it is usually defined piecewise as above, S is an analytic function of both k and  r . It can also be written as a power series or as where sinc

Friedmann–Lemaître–Robertson–Walker metric - Misplaced Pages Continue

2478-399: The decrease in the mass contained in a fixed cube (whose side is momentarily a ) is the amount that leaves through the sides due to the expansion of the universe plus the mass equivalent of the work done by pressure against the material being expelled. This is the conservation of mass–energy ( first law of thermodynamics ) contained within a part of the universe. The second equation says that

2537-481: The derivation of the Friedmann–Lemaître–Robertson–Walker metric). The second equation states that both the energy density and the pressure cause the expansion rate of the universe a ˙ {\displaystyle {\dot {a}}} to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a consequence of gravitation , with pressure playing

2596-503: The early universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. That being said, attempts to confirm the purely kinematic interpretation of the Cosmic Microwave Background (CMB) dipole through studies of radio galaxies and quasars show disagreement in the magnitude. Taken at face value, these observations are at odds with the Universe being described by

2655-421: The energy density of radiation decreases more quickly than the volume expansion, because its wavelength is red-shifted . Cosmic inflation and the accelerated expansion of the universe can be characterized by the equation of state of dark energy . In the simplest case, the equation of state of the cosmological constant is w = − 1 {\displaystyle w=-1} . In this case,

2714-442: The equation of state of dark energy is one of the largest efforts of observational cosmology . By accurately measuring w {\displaystyle w} , it is hoped that the cosmological constant could be distinguished from quintessence which has w ≠ − 1 {\displaystyle w\neq -1} . A scalar field ϕ {\displaystyle \phi } can be viewed as

2773-407: The following pair of equations with k {\displaystyle k} , the spatial curvature index, serving as a constant of integration for the first equation. The first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics , assuming the expansion of the universe is an adiabatic process (which is implicitly assumed in

2832-475: The form k is a constant representing the curvature of the space. There are two common unit conventions: A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical , i.e. a 3-sphere with opposite points identified.) In hyperspherical or curvature-normalized coordinates

2891-424: The kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative) gravitational potential energy (relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds

2950-451: The previous example, a composite material may not be isotropic. In another context, a material is not homogeneous in so far as it is composed of atoms and molecules . However, at the normal level of our everyday world, a pane of glass, or a sheet of metal is described as glass, or stainless steel. In other words, these are each described as a homogeneous material. A few other instances of context are: dimensional homogeneity (see below)

3009-470: The same direction at each point) would be compatible with homogeneity (at each point physics will be the same), but not with isotropy , since the field singles out one "preferred" direction. In the Lagrangian formalism, homogeneity in space implies conservation of momentum , and homogeneity in time implies conservation of energy . This is shown, using variational calculus , in standard textbooks like

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3068-513: The scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions. Friedmann died in 1925. In 1927, Georges Lemaître , a Belgian priest, astronomer and periodic professor of physics at the Catholic University of Leuven , arrived independently at results similar to those of Friedmann and published them in

3127-551: The surviving components of the Ricci tensor are and the Ricci scalar is In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are and the Ricci scalar is Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining

3186-414: The system. Translational invariance as described above is equivalent to shift invariance in system analysis , although here it is most commonly used in linear systems, whereas in physics the distinction is not usually made. The notion of isotropy , for properties independent of direction, is not a consequence of homogeneity. For example, a uniform electric field (i.e., which has the same strength and

3245-623: The time evolution of a ( t ) {\displaystyle a(t)} does require Einstein's field equations together with a way of calculating the density, ρ ( t ) , {\displaystyle \rho (t),} such as a cosmological equation of state . This metric has an analytic solution to Einstein's field equations G μ ν + Λ g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }} giving

3304-429: The universe. In a strictly FLRW model, there are no clusters of galaxies or stars, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models that calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that

3363-456: The volume increases. The equation of state for ultra-relativistic 'radiation' (including neutrinos , and in the very early universe other particles that later became non-relativistic) is w = 1 / 3 {\displaystyle w=1/3} which means that its energy density decreases as ρ ∝ a − 4 {\displaystyle \rho \propto a^{-4}} . In an expanding universe,

3422-414: The weakening is not sufficient to actually explain Olbers' paradox. Many cosmologists think that the fact that the Universe is finite in time, that is that the Universe has not been around forever, is the solution to the paradox. The fact that the night sky is dark is thus an indication for the Big Bang. By translation invariance, one means independence of (absolute) position, especially when referring to

3481-413: Was that the universe is not infinite, static, and homogeneous. The Big Bang cosmology replaced this model (expanding, finite, and inhomogeneous universe ). However, modern astronomers supply reasonable explanations to answer this question. One of at least several explanations is that distant stars and galaxies are red shifted , which weakens their apparent light and makes the night sky dark. However,

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