Friedmann-Lemaître-Robertson-Walker metric

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The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a simply connected, 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)).

1 General Metric
2 Normalization
- 2.1 Hyperspherical coordinates
- 2.2 Cartesian coordinates
3 Solutions
4 Name and History
5 Einstein's radius of the Universe
6 References
7 External links

[edit] General Metric

The 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 (in units where $c = 1$ )

$\mathrm{d}\tau^2 = \mathrm{d}t^2 - {a(t)}^2 \left( \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\theta^2 + r^2 \sin^2 \theta \, \mathrm{d}\phi^2 \right)$

where $k$ describes the spatial curvature and is constant in time, and $a (t)$ is the scale factor and is explicitly time dependent. The speed of light at r = 0 is the reciprocal of a(t).

Einstein's field equations are not used in this derivation: the metric follows from the geometric properties of homogeneity and isotropy. The specific form of $a (t)$ does require Einstein's field equations, and also the definition of the cosmological equation of state, $ρ(a)$ .

[edit] Normalization

The metric leaves some choice of normalization. One common choice is to say that the scale factor is 1 today ( $a(t_0) \equiv 1$ ). In this choice the radial coordinate $r$ carries dimensionality as does $k$ . Then $k$ does not equal ±1 or 0 but $k = H_0^2 \left( \Omega_0 - 1 \right)$ .

Another choice is to specify that $k$ is ± 1 or 0. This choice makes $k/a(t_0)^2 = H_0^2 \left( \Omega_0 - 1 \right)$ where the scale factor now carries the dimensionality and the radial coordinate $r$ is dimensionless.

[edit] Hyperspherical coordinates

The metric is often written in a curvature normalized way via the transformation

$\chi = \begin{cases} \sqrt{k}^{-1} \sin^{-1} \left( \sqrt{k} r \right), &k > 0 \\ r, &k = 0 \\ \sqrt{|k|}^{-1} \sinh^{-1} \left( \sqrt{|k|} r \right), &k < 0. \end{cases}$

In curvature normalized coordinates the metric becomes

$\mathrm{d}\tau^2 = \mathrm{d}t^2 - a(t)^2 \left[ \mathrm{d}\chi^2 + S^2_k(\chi) \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right) \right]$

where $S_k(\chi) \equiv \sqrt{k}^{-1} \sin\left( \sqrt{k} \chi \right), \chi, \textrm{and} \sqrt{|k|}^{-1} \sinh \left( \sqrt{|k|} \chi \right)$ for $k$ 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 $k$ .

The comoving distance is the distance to an object with zero peculiar velocity. In the curvature normalized coordinates it is $χ$ . The proper distance is the physical distance to a point in space at an instant in time. The proper distance is $a(t)\, \chi$ .

[edit] Cartesian coordinates

If we let

$x = r \cos [\theta] \,$ ,

$y = r \sin [\theta] \cos [\phi] \,$ , and

$z = r \sin [\theta] \sin [\phi] \,$ ,

then the metric becomes

$- c^2 (\mathrm{d}\tau)^2 = - c^2 (\mathrm{d}t)^2 + {(a[t])}^2 \left( \frac {k (x (\mathrm{d}x )+ y (\mathrm{d}y) + z (\mathrm{d}z))^2} {1 - k (x^2 + y^2 + z^2)} + (\mathrm{d}x)^2 + (\mathrm{d}y)^2 + (\mathrm{d}z)^2 \right)$

for $k (x^2 + y^2 + z^2) < 1 \,$ . When k > 0, this coordinate patch only covers the nearer half of the universe.

[edit] Solutions

Main article: Friedmann equations

This metric has an analytic solution to Einstein's field equations $G μν - Λ g μν = 8π G T μν$ giving the Friedmann equations when the energy-momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are:

$\frac{{\dot a}^2}{a^2} + \frac{k}{a^2} - \frac{\Lambda}{3} = \frac{8\pi G}{3}\rho$ $2\frac{\ddot a}{a} + \frac{{\dot a}^2}{a^2} + \frac{k}{a^2} - \Lambda = -8\pi G p.$

These equations serve as a first approximation 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 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 model 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 the FLRW models as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, i.e., a model which 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.

[edit] Interpretation

The pair of equations given above is equivalent to the following pair of equations

${\dot \rho} = - 3 \frac{\dot a}{a} (p + \rho)$ $\frac{\ddot a}{a} = - \frac{4\pi G}{3}(\rho + 3p) + \frac{\Lambda}{3}$

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 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 causes the expansion rate of the universe ${\dot a}$ 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.

[edit] The cosmological constant term

The cosmological constant term can be omitted if we make the following replacement

$\rho \rightarrow \rho + \frac{\Lambda}{8 \pi G}$

$p \rightarrow p - \frac{\Lambda}{8 \pi G}.$

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:

$p = - \rho c^2. \,$

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

$p < - \frac {\rho c^2} {3}. \,$

Such a field is sometimes called quintessence.

[edit] Newtonian approximation

In a certain limit, the above equations can be approximated by classical mechanics.

Early in history of the universe when $a$ is small enough, the spatial curvature of the universe, $k a - 2$ , is negligible compared to the density term (proportional to $a - 3$ for pressure-free matter (also called "dust" or "cold matter") or $a - 4$ for radiation). The cosmological constant term is also relatively small. Then one may neglect the terms involving $k$ and $Λ$ in the equations above, i.e. treat the universe as approximately spatially flat.

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 $k$ and $Λ$ are negligible. It can then be brought to the following form

${1\over 2} {{\dot a}^2} \propto {a^2} 8\pi G \rho.$

This can be interpreted naively as an energy conservation equation: the universe has a mass $M$ proportional to $a 3 ρ$ , and thus its potential energy is proportional to $-\frac{GM^2}{a}\propto -GM\rho a^2$ . Its kinetic energy, on the other hand, is proportional to ${1\over 2} M {{\dot a}^2}$ . Conservation of energy is thus ${1\over 2} M {{\dot a}^2} - c M a^2 G\rho = 0$ , 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.

[edit] Name and History

The main results of the FLRW model were first derived by the Soviet mathematician Alexander Friedmann in 1922 and 1924. Although his work was published in the prestigious physics journal Zeitschrift für Physik, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, an behalf of Zeitschrift für Physik, acted as 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 astronomy student and a part-time lecturer at the University of Leuven, arrived independently at similar results as Friedmann and published them in 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 1920's, Lemaître's results were noticed in particular by Arthur Eddington, and in 1930–31 his paper was translated into English and published in the Monthly Notices of the Royal Astronomical Society.

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 Friedmann 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.

[edit] Einstein's radius of the Universe

Einstein's radius of the universe is the 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 $\dot{a} = \ddot{a} = 0$ in the Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is $R_E=c/\sqrt {4\pi G\rho}$ , where $c$ is the speed of light, $G$ is the Newtonian gravitational constant, and $ρ$ is the density of space of this universe. The numerical value of Einstein's radius is of order of 10¹⁰ light years.