Friedman–Lemaître–Robertson–Walker metric

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The Friedmann-Lemaître-Robertson-Walker metric or FLRW model is an exact solution of the Einstein field equations of general relativity. It describes an expanding (or contracting), homogeneous and isotropic universe. Depending on geographic or historical preferences, some subset of the names of scientists Alexander Friedmann, Georges Lemaître, Howard Percy Robertson (Howard P. Robertson), and Arthur Geoffrey Walker are used in the name of this metric.

Metric form

The FLRW metric starts with the assumption of homogeneity and isotropy. It also assumes that the spatial component of the metric can be dependent on the time. In units where c = 1, the general metric that satisfies these conditions is:

(1) ${displaystyle mathrm {d} s^{2}=-mathrm {d} t^{2}+{a(t)}^{2}left({frac {mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}mathrm {d} theta ^{2}+r^{2}sin ^{2}theta ,mathrm {d} phi ^{2}right)}$

Where $k$ describes the curvature and is constant in time and $a(t)$ It's him. scale factor and is explicitly dependent on time and natural units are used by establishing the speed of light to unity. Einstein field equations are not used in this solution: the metric is obtained from the geometric properties of homogeneity and isotropy. The specific form of $a(t)$ needs to know the equations of the field and the definition of the state density equation, ${displaystyle rho (a)}$ .

Normalization

The metric leaves some possibility of normalization. A common choice is to consider the current scale factor as unity ( ${displaystyle a(t_{0})equiv 1}$ ). In this election the coordinate $r$ It's dimensional just like $k$ . In this approach $k$ No. is equal to ±1 or 0 but ${displaystyle k=H_{0}^{2}left(Omega _{0}-1right)}$ .

Another possibility is to specify that $k$ is ± 1 or 0. From this you get that ${displaystyle k/a(t_{0})^{2}=H_{0}^{2}left(Omega _{0}-1right)}$ where the scale factor is now dimensional and coordinated $r$ It's dimensional.

The metric is often written in a normalized curvature manner by the transformation

$0\r,&k=0\{sqrt {|k|}}^{,-1}sinh ^{-1}left({sqrt {|k|}},rright),&kχ χ ={k− − 1without− − 1 (kr),k▪0r,k=0日本語k日本語− − 1sinh− − 1 (日本語k日本語r),k.0.{displaystyle chi ={begin{cases}{sqrt {k}}}^{,-1}sin ^{-1}left({sqrt {k},rright}{k}{right}{, pretendk ==0}{{sqrt}{excluding}{k}{cHFFFFFFFFFFFFFFFFFFFFFF}}{cHFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFFFFFFFFFFFFFFFFFF}{cH00}{cHFFFFFFFFFFFFFF}{cH00}{cHFFFFFFFFFFFFFFFFFF}}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cHFF}{cHFFFF}{cHFFFFFF}{cHFF0\r,&k=0\{sqrt {|k|}}^{,-1}sinh ^{-1}left({sqrt {|k|}},rright),&k$

In curvature normalized coordinates the metric becomes:

(2) ${displaystyle mathrm {d} s^{2}=-mathrm {d} t^{2}+a(t)^{2}left[mathrm {d} chi ^{2}+S_{k}^{2}(chi)left(mathrm {d} theta ^{2}+sin ^{2}theta ,mathrm {d} phi ^{2}right)right]}$

Where:

$0\chi &k=0\{sqrt {|k|}}^{,-1}sinh left({sqrt {|k|}},chi right)&kSk(χ χ )={k− − 1without (kχ χ )k▪0χ χ k=0日本語k日本語− − 1sinh (日本語k日本語χ χ )k.0{displaystyle S_{k}(chi)={begin{cases}{sqrt {k}}}{,-1}sin left({sqrt {k}},chi right right}{chi >}{chi &k=0{{sqrt}{cHFFFF00}{cH00}{cHFFFFFFFF00}{cH00}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFF00}{cH00}{cH00FFFF00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH00}{cH00FFFF00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0\chi &k=0\{sqrt {|k|}}^{,-1}sinh left({sqrt {|k|}},chi right)&k$

This choice assumes that the scale factor is dimensional but can easily become the $k,$ normalized.

The combo distance is the distance to an object with zero peculiar speed. In the normalized curvature the coordinate is $chi$ . The distance itself is the physical distance to a point in space in an instant of time. The distance itself is ${displaystyle a(t),chi }$ .

General space-time properties of FLRW

Material content

The solution given by the FLRW metric, describes a universe full of an ideal fluid with density and pressure given by the Friedmann equations. It's a solution to Einstein's field equations. ${displaystyle G_{mu nu }-Lambda g_{mu nu }=8pi T_{mu nu }}$ giving the equations of Friedmann when the tensor moment energy is assumed in the same way as isotropo and homogeneous. In units in which c = 1, the resulting equations are:

(3) ${displaystyle {frac {{dot {a}}^{2}}{a^{2}}}+{frac {k}{a^{2}}}-{frac {Lambda }{3}}={frac {8pi G}{3}}rho }$

(4) ${displaystyle 2{frac {ddot {a}}{a}}+{frac {{dot {a}}^{2}}{a^{2}}}+{frac {k}{a^{2}}}-Lambda =-8pi Gp}$

Where:

{displaystyle kin {-1,0,1}}

is the sign of the spatial curvature.

a,

is the scale factor, from which the size of the observable universe can be calculated.

Lambda ,

is the cosmological constant

G,

is the constant of universal gravitation

{displaystyle rhop,}

are the density and pressure of interstellar matter.

These equations serve as a first approximation of the conventional cosmological model of the Big Bang including the current Lambda-CDM model.

Because the FLRW metric accurately describes a perfectly homogeneous universe, some sources erroneously claim that the Big Bang model based on the FLRW metric cannot account for the observed lumpiness of the universe. In a strict FLRW model, there are no galaxy clusters or clumps of stars, since those structures constitute inhomogeneities. However, the FLRW is used as a first approximation for the evolution of the universe because it is simple and the models that calculate the lumpiness of the universe are added to the FLRW as extensions. Many cosmologists agree that the observable universe closely approximates a quasi-FLRW model, that is, a model that uses the FLRW metric from primordial density fluctuations. Until 2003, the theoretical implications of the various extensions of the FLRW seemed to be well understood and the goal is to make these consistent with the COBE and WMAP observations.

Geodesics

The free movement of particles in a universe, that is, the trajectories they follow as the entire space-time evolves, are given by the geodesic lines calculable from the metric:

{displaystyle {begin{matrix}{ddot {t}}+{cfrac {{a'}_{t}}{2c^{2}}}left[{dot {r}}^{2}+{bar {r}}^{2}left({dot {theta }}^{2}+sin ^{2}theta {dot {varphi }}^{2}right)right]=0&qquad {ddot {r}}+{cfrac {{{dot {a}}'}_{t}}{a}}+{dot {r}}{dot {t}}-{bar {r}}{{bar {r}}'}_{r}left[{dot {theta }}^{2}+sin ^{2}theta {dot {varphi }}^{2}right]=0\\{ddot {theta }}+{cfrac {{{dot {a}}'}_{t}}{a}}{dot {theta }}{dot {t}}+{cfrac {{bar {r}}_{r}}{2{bar {r}}}}{dot {theta }}{dot {r}}-sin theta cos theta {dot {varphi }}^{2}=0&{ddot {varphi }}+{cfrac {{{dot {a}}'}_{t}}{a}}{dot {varphi }}{dot {t}}+{cfrac {{bar {r}}_{r}}{2{bar {r}}}}{dot {varphi }}{dot {r}}+2tan theta {dot {varphi }}{dot {theta }}=0end{matrix}}}

It can be verified that the so-called galactic observers that move together with the matter that causes the curvature of space-time given by:

{displaystyle t(tau)=tauquad r(tau)=r_{0}quad theta (tau)=theta _{0},quad varphi (tau)=varphi _{0}}

They are geodesic lines.

Riemann Tensor

Of the potentially 55 independent components of the Riemann tensor, in the same coordinates used in the metric (2), the Riemann tensor can be write from at most six non-zero components:

${displaystyle {begin{matrix}R_{0101}=a{ddot {a}}&R_{0202}=a{ddot {a}}S_{k}^{2}&R_{0303}=a{ddot {a}}Sk^{2}sin ^{2}theta \R_{1212}=a^{2}S_{k}sin ^{2}theta (S_{k}''-S{dot {a}}^{2})&R_{1313}=a^{2}S_{k}(S_{k}''-S{dot {a}}^{2})&R_{2323}=a^{2}S_{k}^{2}sin ^{2}theta (1+S_{k}^{2}{dot {a}}^{2}-{S_{k}'}^{2})end{matrix}}}$

Isometry Group

For any value of the parameters the FLRW metric defines a spatially isotropous and homogeneous universe, although there is no symmetry regarding time, that makes the isometry group precisely the symmetry group of an isotrope and homogeneous space of uniform curvature. That group is a Lie group of dimension 6, for the case of a flat space (k = 0) that group is precisely: ${displaystyle mathbb {R} ^{3}times SO(3)}$

Cosmological models based on the FLRW metric

The FLRW metric is used as a first approximation for the cosmological model of the universe since the big bang. Since FLRW assumes homogeneity, it has been wrongly speculated that the big bang model cannot explain the temperature variations of the universe on different scales. Currently, the FLRW is used as a first approximation for the evolution of the universe because it is simple to calculate and can be extended to model the temperature variations of the universe on different scales. Since 2003, the theoretical implications of different extensions of the FLRW metric have been known and work is underway to make them consistent with the observational evidence obtained from COBE and WMAP.

Interpretation

The equations (3) and (4) are equivalent to the following pair of equations:

${displaystyle {dot {rho }}=-3{frac {dot {a}}{a}}(p+rho)}$

${displaystyle {frac {ddot {a}}{a}}=-4pi G({1 over 3}rho +p)+{1 over 3}Lambda }$

with ${displaystyle k;}$ making constant integration for the second equation.

The first equation can be obtained from thermodynamic considerations and is equivalent to the first law of thermodynamics, assuming that the expansion of the universe is an adiabatic process (which is implicitly assumed in obtaining the Friedmann-Lemaître- Robertson-Walker).

The second equation says that energy density and pressure cause the expansion rate of the universe ${displaystyle {dot {a}}}$ do not diminish, e.g. both cause a deceleration in the expansion of the universe. This is a consequence of gravity, with pressure playing a role similar to that energy density (mass), 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

The cosmological constant term can be omitted if we substitute the following terms:

${displaystyle rho rightarrow rho +{frac {Lambda }{8pi G}}qquad prightarrow p-{frac {Lambda }{8pi G}}}$

Therefore, the cosmological constant can be interpreted as being a form of energy that has a negative pressure, equal in magnitude to this (positive) energy density:

${displaystyle p=-u;}$

Such a form of energy, a generalization of the notion of a cosmological constant, is known as dark energy.

In fact, to obtain a term that causes an acceleration of the expansion of the Universe, it is enough to have a scalar field that satisfies

${displaystyle p=-{frac {u}{3}}}$

Such a field is sometimes called quintessence.

Newtonian approximation

To a certain extent, the previous equations ((3) and (4) can be approached using classic mechanics. For scale factor values a(t) large enough, the universe is approximately flat in the sense that the term density (proportional to ${displaystyle a^{-3}}$ for dark matter or matter ${displaystyle a^{-4}}$ for radiation) much higher than the term curvature ${displaystyle {k over a^{2}}}$ and this one can be despised. The term of the cosmological constant is also relatively small and can be despised and then the first of the equations becomes simply:

(♪) ${displaystyle {1 over 2}{{dot {a}}^{2}}approx {frac {8Gpi }{3}}a^{2}rho }$

This equation can in fact be interpreted as the classical Newtonian law of conservation of energy:

The universe has a mass $M$ proportional to ${displaystyle a^{3}rho ;}$ and, therefore, its potential energy is proportional to ${displaystyle -GM^{2}/a=-GMrho a^{2};}$ .
The kinetic energy of the universe on the other hand is proportional to ${displaystyle M{dot {a}}^{2}/2}$

The sum of kinetic energy plus potential energy multiplied by a certain constant is precisely the equation (*):

${displaystyle {1 over 2}M{{dot {a}}^{2}}-CMa^{2}Grho =0}$

Being C a certain constant proportionality that must be taken equal to ${displaystyle 8pi /3}$ to be consistent with the result of the equation (♪).

Note that in the very early stages of the universe, this approach is not adequate for several reasons. For example, during cosmic inflation the cosmological constant term dominates the equations of motion. Even earlier, during Planck's time, quantum effects cannot be neglected.

Name and History

The main results of the FLRW model were first obtained by the Soviet physicist Alexander Friedmann between 1922–1924. Although his work was published in the prestigious physics journal Zeitschrift für Physik, it went relatively unnoticed by his contemporaries. Friedmann communicated his results directly to Einstein, who confirmed that the model was mathematically correct but failed to appreciate the physical significance of Friedmann's predictions.

Friedmann died in 1925. In 1927, the priest Georges Lemaître, a Belgian mathematician, physicist and astronomer, part-time professor at the Catholic University of Louvain, arrived at similar results independently of Friedmann and published them in the Annals of the Scientific Society of Brussels. Faced with the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lemaître's results were noted and in 1930–1931 his paper was translated into English and published in Nature.

Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem deeply in the 1930s. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a Lorentzian band that is homogeneous and isotropic (as stated above, ie a geometric result and not specifically tied to the equations of general relativity, which were always assumed to be true by Friedman and Lemaître).

Due to the fact that the dynamics of the FLRW model was obtained by Friedmann and Lemaître, the following two names are sometimes omitted by scientists outside the US. On the contrary, US physicists frequently they refer to it simply as the "Robertson-Walker" metric. The full title with the four names is more democratic and is used frequently. Often the metric "Robertson-Walker" is so called because they proved its generic properties, it is distinguished from "Friedmann-Lemaître" dynamical models, specific solutions for a(t) which assume that only the Stress energy contributions are cold matter, radiation, and a cosmological constant.

The radius of Einstein's universe

The Radio of the Universe of Einstein It is the radius of curvature of the space of the static universe of Einstein, an enormously abandoned static model that supposedly represented our universe in an idealized way. Putting ${displaystyle {dot {a}}={ddot {a}}=0}$ in the equation of Friedman, the radius of curvature of the space of this Universe (the radius of Einstein) is:

${displaystyle R_{E}=c/{sqrt {4pi Grho }}}$

where $c$ is the speed of light, $G$ is the newtonian gravitational constant and $rho$ is the density of the space of the universe. The numerical value of Einstein's radio is of the order of 10¹⁰ Light years.

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