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tion 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 the friedmann equations when the energy momentum tensor is similarly assumed to be isotropic and homogeneous the resulting equations are 13 a a 2 k c 2 a 2 λ c 2 3 κ c 4 3 ρ 2 a a a a 2 k c 2 a 2 λ c 2 κ c 2 p displaystyle begin aligned left frac dot a a right 2 frac kc 2 a 2 frac lambda c 2 3 frac kappa c 4 3 rho 4pt 2 frac ddot a a left frac dot a a right 2 frac kc 2 a 2 lambda c 2 kappa c 2 p end aligned because the flrw model assumes homogeneity some popular accounts mistakenly assert that the big bang model cannot account for the observed lumpiness of 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 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 update 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 interpretation edit the pair of equations given above is equivalent to the following pair of equations ρ 3 a a ρ p c 2 a a κ c 4 6 ρ 3 p c 2 λ c 2 3 displaystyle begin aligned dot rho 3 frac dot a a left rho frac p c 2 right 1ex frac ddot a a frac kappa c 4 6 left rho frac 3p c 2 right frac lambda c 2 3 end aligned 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 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 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 cosmological constant edit the cosmological constant term can be omitted if we make the following replacements ρ ρ λ κ c 2 p p λ κ displaystyle begin aligned rho to rho frac lambda kappa c 2 p to p frac lambda kappa end aligned therefore the cosmological constant can be interpreted as arising from a form of energy that has negative pressure equal in magnitude to its positive mass energy density p ρ c 2 displaystyle p rho c 2 which is an equation of state of vacuum with dark energy an attempt to generalize this to p w ρ c 2 displaystyle p w rho c 2 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 p ρ c 2 3 displaystyle p frac rho c 2 3 such a field is sometimes called quintessence dust models edit setting the pressure of the perfect fluid in the friedmann equations to zero p 0 displaystyle p 0 gives a cosmological dust model 14 231 newtonian analog edit in 1934 mccrea and milne 15 showed that the friedmann equations in the case of a pressureless fluid can be derived with non relativistic newtonian dynamics 14 231 a 3 ρ 3 a 2 a ρ 3 a 2 p a c 2 a 2 2 κ c 4 a 3 ρ 6 a k c 2 2 displaystyle begin aligned a 3 dot rho 3a 2 dot a rho frac 3a 2 p dot a c 2 1ex frac dot a 2 2 frac kappa c 4 a 3 rho 6a frac kc 2 2 end aligned the first equation says that 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 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 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 useful solutions edit the friedmann equations can be solved exactly in presence of a perfect fluid with equation of state p w ρ c 2 displaystyle p w rho c 2 where p is the pressure ρ is the mass density of the fluid in the comoving frame and w is some constant in spatially flat case k 0 the solution for the scale factor is a t a 0 t 2 3 w 1 displaystyle a t a_ 0 t frac 2 3 w 1 where a 0 is some integration constant to be fixed by the choice of initial conditions this family of solutions labelled by w is extremely important for cosmology for example w 0 describes a matter dominated universe where the pressure is negligible with respect to the mass density from the generic solution one easily sees that in a matter dominated universe the scale factor goes as a t t 2 3 matter dominated displaystyle a t propto t 2 3 qquad text matter dominated another important example is the case of a radiation dominated universe namely when w 1 3 this leads to a t t 1 2 radiation dominated displaystyle a t propto t 1 2 qquad text radiation dominated note that this solution is not valid for domination of the cosmological constant which corresponds to an w 1 in this case the energy density is constant and the scale factor grows exponentially solutions for other values of k can be found at tersic balsa lecture notes on astrophysics retrieved 24 february 2022 mixtures edit if the matter is a mixture of two or more non interacting fluids each with such an equation of state then ρ f 3 h ρ f p f c 2 displaystyle dot rho _ f 3h left rho _ f frac p_ f c 2 right holds separately for each such fluid f in each case ρ f 3 h ρ f w f ρ f displaystyle dot rho _ f 3h left rho _ f w_ f rho _ f right from which we get ρ f a 3 1 w f displaystyle rho _ f propto a 3 left 1 w_ f right for example one can form a linear combination of such terms ρ a a 3 b a 4 c a 0 displaystyle rho aa 3 ba 4 ca 0 where a is the density of dust ordinary matter w 0 when a 1 b is the density of radiation w 1 3 when a 1 and c is the density of dark energy w 1 one then substitutes this into a a 2 8 π g 3 ρ k c 2 a 2 displaystyle left frac dot a a right 2 frac 8 pi g 3 rho frac kc 2 a 2 and solves for a as a function of time history edit alexander friedmann friedmann published two cosmology papers in the 1922 1923 time frame he adopted the same homogeneity and isotropy assumptions used by albert einstein and by willem de sitter in their papers both published in 1917 both of the earlier works also assumed the universe was static eternally unchanging einstein postulated an additional term to his equations of general relativity to ensure this stability in his paper de sitter showed that spacetime had curvature even in the absence of matter the new equations of general relativity implied that a vacuum had properties that altered spacetime 16 152 the universe being static was a fundamental assumption of philosophy and science however friedmann abandoned the idea in his first paper on the curvature of space starting with einstein s 10 equations of relativity friedmann applies the symmetry of an isotropic universe and a simple model for mass energy density to derive a relationship between that density and the curvature of spacetime he demonstrates that in addition to a single static solution many time dependent solutions also exist 16 157 friedmann s second paper on the possibility of a world with constant negative curvature published in 1924 explored more complex geometrical ideas this paper established the idea that the finiteness of spacetime was not a property that could be established based on the equations of general relativity alone both finite and infinite geometries could be used to give solutions friedmann used two concepts of a three dimensional sphere as analogy a trip at constant latitude could return to the starting point or the sphere might have an infinite number of sheets and the trip never repeats 16 167 friedmann s papers were largely ignored except initially by einstein who actively dismissed them however once edwin hubble published astronomical evidence that the universe was expanding einstein became convinced independently of friedmann georges lemaître discovered some aspects of the same solutions and wrote persuasively about the concept of a universe born from a primordial atom lemaître was credited with the big bang concept but friedmann s ideas were deeper and ultimately more influential for scientists 17 in popular culture edit several students at tsinghua university ccp leader xi jinping s alma mater participating in the 2022 covid 19 protests in china carried placards with friedmann equations scrawled on them interpreted by some as a play on the words free man 18 19 others have interpreted the use of the equations as a call to open up china and stop its zero covid policy as the friedmann equations relate to the expansion or opening of the universe citation needed see also edit mathematics of general relativity solutions of the einstein field equations sources edit friedman a 29 may 1922 über die krümmung des raumes pdf z phys in german 10 1 petrograd springer verlag gmbh co kg 377 386 bibcode 1922zphy 10 377f doi 10 1007 bf01332580 s2cid 125190902 via michael kachelrieß norges teknisk naturvitenskapelige universitet english translation friedman a 1999 on the curvature of space general relativity and gravitation 31 12 translated by f r ellis h van elst springer verlag gmbh co kg 1991 2000 bibcode 1999gregr 31 1991f doi 10 1023 a 1026751225741 s2cid 122950995 the original russian manuscript of this paper is preserved in the ehrenfest archive friedmann a 1924 über die möglichkeit einer welt mit konstanter negativer krümmung des raumes z phys in german 21 1 326 332 bibcode 1924zphy 21 326f doi 10 1007 bf01328280 s2cid 120551579 english translation friedmann a 1999 on the possibility of a world with constant negative curvature of space general relativity and gravitation 31 12 2001 2008 bibcode 1999gregr 31 2001f doi 10 1023 a 1026755309811 s2cid 123512351 1 2 3 4 5 6 7 navas s et al particle data group 2024 review of particle physics physical review d 110 3 1 708 doi 10 1103 physrevd 110 030001 hdl 20 500 11850 695340 22 1 3 the friedmann equations of motion 1 2 3 4 5 6 7 8 peacock j a 1998 12 28 cosmological physics 1 ed cambridge university press doi 10 1017 cbo9780511804533 isbn 978 0 521 41072 4 1 2 3 d inverno ray 2008 introducing einstein s relativity repr ed oxford clarendon press isbn 978 0 19 859686 8 1 2 3 john a peacock 1999 3 the isotropic universe 3 1 the robertson walker metric cosmological physics cambridge university press nasa ipac extragalactic database california institute of technology archived 8 september 2025 at the wayback machine dodelson scott 2003 modern cosmology san diego calif academic press isbn 978 0 12 219141 1 rees martin 2001 just six numbers the deep forces that shape the universe astronomy science repr ed new york ny basic books isbn 978 0 465 03673 8 universe 101 nasa retrieved september 9 2015 the actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters scolnic daniel riess adam g murakami yukei s peterson erik r brout dillon acevedo maria carreres bastien jones david o said khaled howlett cullan anand gagandeep s 2025 01 15 the hubble tension in our own backyard desi and the nearness of the coma cluster the astrophysical journal letters 979 1 l9 arxiv 2409 14546 bibcode 2025apj 979l 9s doi 10 3847 2041 8213 ada0bd issn 2041 8205 nemiroff robert j patla bijunath 2008 adventures in friedmann cosmology a detailed expansion of the cosmological friedmann equations american journal of physics 76 3 265 276 arxiv astro ph 0703739 bibcode 2008amjph 76 265n doi 10 1119 1 2830536 s2cid 51782808 bergström lars goobar ariel 2008 cosmology and particle astrophysics springer praxis books in astronomy and planetary science 2 ed reprinted ed chichester uk praxis publ p 61 isbn 978 3 540 32924 4 rosu h c ojeda may p june 2006 supersymmetry of frw barotropic cosmologies international journal of theoretical physics 45 6 1152 1157 arxiv gr qc 0510004 bibcode 2006ijtp 45 1152r doi 10 1007 s10773 006 9123 2 issn 0020 7748 s2cid 119496918 1 2 longair malcolm s 2023 galaxy formation astronomy and astrophysics library berlin heidelberg springer berlin heidelberg doi 10 1007 978 3 662 65891 8 isbn 978 3 662 65890 1 mccrea w h milne e a 1934 newtonian universes and the curvature of space quarterly journal of mathematics 5 73 80 bibcode 1934qjmat 5 73m doi 10 1093 qmath os 5 1 73 1 2 3 tropp eduard a frenkel viktor ya chernin artur d 1993 06 03 alexander a friedmann the man who made the universe expand translated by dron alexander burov michael 1 ed cambridge university press doi 10 1017 cbo9780511608131 isbn 978 0 521 38470 4 belenkiy ari 2012 10 01 alexander friedmann and the origins of modern cosmology physics today 65 10 38 43 bibcode 2012pht 65j 38b doi 10 1063 pt 3 1750 issn 0031 9228 it s clear that in adumbrating big bang cosmology friedmann went much further than his predecessors or early successors like lemaître murphy matt november 28 2022 china s protests blank paper becomes the symbol of rare demonstrations bbc news sullivan helen 2022 11 29 blank paper equations and alpacas the symbols of china s zero covid protests the guardian issn 0261 3077 retrieved 2026 04 24 further reading edit liebscher dierck ekkehard 2005 expansion cosmology berlin springer pp 53 77 isbn 3 540 23261 3 v t e relativity special relativity background principle of relativity galilean relativity galilean transformation special relativity doubly special relativity fundamental concepts frame of reference speed of light hyperbolic orthogonality rapidity maxwell s equations proper length proper time proper acceleration relativistic mass formulation lorentz transformation textbooks phenomena time dilation mass energy equivalence e mc 2 length contraction relativity of simultaneity relativistic doppler effect tho...
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