Friedmann-Lemaître-Robertson-Walker metric
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"FRW" redirects here. For other uses, see FRW (disambiguation).
The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity; it describes a homogeneous, isotropic expanding or contracting universe. Depending on geographical or historical preferences, a subset of the four scientists -- Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker -- may be named (e.g., Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW) or Friedmann-Lemaître (FL)).
General MetricThe FLRW metric starts with the assumption of homogeneity and isotropy. It also assumes that the spatial component of the metric can be time dependent. The generic metric which meets these conditions is where Failed to parse (Missing texvc executable; please see math/README to configure.): k describes the curvature and is constant in time, and Failed to parse (Missing texvc executable; please see math/README to configure.): a(t) is the scale factor and is explicitly time dependent. Arbitrary units are used, and the speed of light at r =0 is the reciprocal of a(t). The Einstein field equations are not used in this derivation: the metric follows from the geometric properties of homogeneity and isotropy. The specific form of Failed to parse (Missing texvc executable; please see math/README to configure.): a(t) does require the Einstein field equations, and also the definition of the density equation of state, Failed to parse (Missing texvc executable; please see math/README to configure.): \rho(a) . NormalizationThe metric leaves some choice of normalization. One common choice is to say that scale factor is 1 today (Failed to parse (Missing texvc executable; please see math/README to configure.): a(t_0) \equiv 1 ). In this choice the coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): r carries dimensionality as does Failed to parse (Missing texvc executable; please see math/README to configure.): k . In this choice Failed to parse (Missing texvc executable; please see math/README to configure.): k does not equal ±1 or 0 but Failed to parse (Missing texvc executable; please see math/README to configure.): k = H_0^2 \left( \Omega_0 - 1 \right) . Another choice is to specify that Failed to parse (Missing texvc executable; please see math/README to configure.): k is ± 1 or 0. This choice makes Failed to parse (Missing texvc executable; please see math/README to configure.): k/a(t_0)^2 = H_0^2 \left( \Omega_0 - 1 \right) where the scale factor now carries the dimensionality and the coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): r is dimensionless. The metric is often written in a curvature normalized way via the transformation In curvature normalized coordinates the metric becomes Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\tau^2 = \mathrm{d}t^2 - a(t)^2 \left[ \mathrm{d}\chi^2 + S^2_k(\chi) \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right) \right] where Failed to parse (Missing texvc executable; please see math/README to configure.): S_k(\chi) \equiv \sqrt{k}^{-1} \sin\left( \sqrt{k} \chi \right), \chi, \textrm{and} \sqrt{|k|}^{-1} \sinh \left( \sqrt{|k|} \chi \right) for Failed to parse (Missing texvc executable; please see math/README to configure.): k greater than, equal to, and less than 0 respectively. This normalization assumes the scale factor is dimensionless but it can be easily converted to normalized Failed to parse (Missing texvc executable; please see math/README to configure.): k . The comoving distance is distance to an object with zero peculiar velocity. In the curvature normalized coordinates it is Failed to parse (Missing texvc executable; please see math/README to configure.): \chi . The proper distance is the physical distance to a point in space at an instant in time. The proper distance is Failed to parse (Missing texvc executable; please see math/README to configure.): a(t)\, \chi . SolutionsThis metric has an analytic solution to the Einstein field equations Failed to parse (Missing texvc executable; please see math/README to configure.): G_{\mu\nu} - \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu} giving the Friedmann equations when the energy-momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are: These equations serve as a first approximation of the standard big bang cosmological model including the current ΛCDM model. Because the FLRW 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, stars or people, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW is used as a first approximation for the evolution of the universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto FLRW as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, that is, a model which follows the FLRW metric apart from primordial density fluctuations. As of 2003, the theoretical implications of the various extensions to FLRW appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP. InterpretationThe above pair of equations is equivalent to the following pair of equations with k serving as a constant of integration for the second equation. The first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics, assuming the universe expansion 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 causes the universe expansion rate Failed to parse (Missing texvc executable; please see math/README to configure.): {\dot a} to decrease, i.e. both cause a decceleration in the expansion of the universe. This is a consequence of gravity, 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. The cosmological constant termThe cosmological constant term can be omitted if we make the following replacement Therefore the cosmological constant can be interpreted as arising from a form of energy which has negative pressure, equal in magnitude to its (positive) energy density:
. Such form of energy - a generalization of the notion of a cosmological constant - is known as dark energy. In fact, in order to get a term which causes an acceleration of the universe expansion, it is enough to have a scalar field which satisfies
. Such a field is sometimes called quintessence. Newtonian approximationIn a certain limit, the above equations can be approximated by Classical mechanics. For small enough Failed to parse (Missing texvc executable; please see math/README to configure.): a
(which is always the case early enough in the universe), the universe is approximately flat in the sense that the density term (proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): a^{-3}
for cold matter or Failed to parse (Missing texvc executable; please see math/README to configure.): a^{-4}
for radiation) is much larger than the curvature term Failed to parse (Missing texvc executable; please see math/README to configure.): {k \over a^2}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda in the equations above. As discussed above, by using the first law of thermodynamics, the pair of equations of motion can be reduced to a single equation. Let us then observe the first equation above, in the limit where both Failed to parse (Missing texvc executable; please see math/README to configure.): k and Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda are negligible. It can then be brought to the following form This can be interpreted naively as an energy conservation equation: the universe has a mass Failed to parse (Missing texvc executable; please see math/README to configure.): M proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): a^3\rho , and thus its potential energy is proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): -\frac{GM^2}{a}\propto -GM\rho a^2 . Its kinetic energy, on the other hand, is proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): {1\over 2} M {{\dot a}^2} . Conservation of energy is thus Failed to parse (Missing texvc executable; please see math/README to configure.): {1\over 2} M {{\dot a}^2} - c M a^2 G\rho = 0 , with c some constant. Note that too early in the universe, this approximation cannot be trusted. For example, during cosmic inflation a cosmological constant-like term dominates the equations of motion. Even earlier, during the Planck epoch, one cannot neglect quantum effects. Name and HistoryThe main results of FLRW model were first derived by Soviet physicist Alexander Friedmann in 1922–24. Although his work was published in a prestigious physical journal Zeitschrift für Physik, it remained relatively unnoticed by his contemporaries. Friedmann communicated his results directly to Einstein, who acknowledged correctness of his math but failed to appreciate the physical significance of Friedmann's predictions. Friedmann died in 1925. In 1927, Georges Lemaître, a Belgian astronomy student and a part-time lecturer at the University of Leuven, arrived at similar results independently of Friedmann and published them in Annals of the Scientific Society of Brussels. In the face of observational evidence of expansion of the universe obtained by Edwin Hubble in the late 1920's, Lemaître's results were noticed and in 1930–31 his paper was translated into English and published in Nature. Howard Percy Robertson from the United States (US) and Arthur Geoffrey Walker from Great Britain explored the problem further during the 1930's. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a Lorentzian manifold that is both 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 Friedman and Lemaître). Due to the fact that the dynamics of the FLRW model were derived by Friedmann and Lemaître, the latter two names are often omitted by scientists outside the United States. Conversely, US physicists often refer to it as simply "Robertson-Walker". The full 4-name title is the most democratic and it is frequently used. Often the "Robertson-Walker" metric, so-called since they proved its generic properties, is distinguished from the dynamical "Friedmann-Lemaître" models, specific solutions for a(t) which assume that the only contributions to stress-energy are cold matter ("dust"), radiation, and a cosmological constant. Einstein's radius of the UniverseEinstein's radius of the universe is radius of curvature of space of Einstein's universe, a long-abandoned static model that was supposed to represent our universe in idealized form. Putting Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{a} = \ddot{a} = 0
in the Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is Failed to parse (Missing texvc executable; please see math/README to configure.): R_E=c/\sqrt {4\pi G\rho}
, where Failed to parse (Missing texvc executable; please see math/README to configure.): c is the speed of light, Failed to parse (Missing texvc executable; please see math/README to configure.): G is the Newtonian gravitational constant, and Failed to parse (Missing texvc executable; please see math/README to configure.): \rho is the density of space of this universe. The numerical value of Einstein's radius is of order of 1010 light years. References
External links
ca:Mètrica FLRW de:Friedmann-Lemaître-Robertson-Walker-Metrik es:Métrica de Friedman-Lemaître-Robertson-Walker fr:Friedmann-Lemaître-Robertson-Walker ia:Metrica de Friedmann-Lemaître-Robertson-Walker it:Metrica di Friedmann - Lemaître - Robertson - Walker ja:フリードマン・ルメートル・ロバートソン・ウォーカー計量 pl:Metryka Friedmana-Lemaître'a-Robertsona-Walkera pt:Métrica de Friedman-Lemaître-Robertson-Walker ru:Вселенная Фридмана fi:Fridmanin–Lemaîtren–Robertsonin–Walkerin metriikka ur:رابرٹسن والکر متناسقات |
