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 homogeneous, isotropic expanding or contracting universe that may be simply connected or multiply connected.[1][2][3] (If multiply connected, then each event in spacetime will be represented by more than one tuple of coordinates.) The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. 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)). This model is sometimes called the Standard Model of modern cosmology.[4] It was developed independently by the named authors in the 1920s and 1930s.

Contents

General metric

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 which meets these conditions is

- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 %2B {a(t)}^2 \mathrm{d}\mathbf{\Sigma}^2

where \mathbf{\Sigma} ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. \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".

Reduced-circumference polar coordinates

In reduced-circumference polar coordinates the spatial metric has the form

\mathrm{d}\mathbf{\Sigma}^2 = \frac{\mathrm{d}r^2}{1-k r^2} %2B r^2 \mathrm{d}\mathbf{\Omega}^2, \quad \text{where } \mathrm{d}\mathbf{\Omega}^2 = \mathrm{d}\theta^2 %2B \sin^2 \theta \, \mathrm{d}\phi^2.

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.)

Hyperspherical coordinates

In hyperspherical or curvature-normalized coordinates the coordinate r is proportional to radial distance; this gives

\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}r^2 %2B S_k(r)^2 \, \mathrm{d}\mathbf{\Omega}^2

where \mathrm{d}\mathbf{\Omega} is as before and

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

As before, k may be taken as the Gaussian curvature at a(t) = 1 or as a unitless value from the set {−1,0,+1}. Note that when k = +1, r is essentially a third angle along with θ and φ. The letter χ may be used instead of r.

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

S_k(r) = \sum_{n=0}^\infty \frac{(-1)^n k^n r^{2n%2B1}}{(2n%2B1)!} = r - \frac{k r^3}{6} %2B \frac{k^2 r^5}{120} - \cdots

or as

S_k(r) = r \; \mathrm{sinc} \, (r \sqrt{k})

where sinc is the unnormalized sinc function and \sqrt{k} is either complex square root of k. These definitions are valid for all k.

Cartesian coordinates

When k = 0 one may write simply

\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}x^2 %2B \mathrm{d}y^2 %2B \mathrm{d}z^2.

This can be extended to k ≠ 0 by defining

x = r \cos \theta \,,
y = r \sin \theta \cos \phi \,, and
z = r \sin \theta \sin \phi \,,

where r is one of the radial coordinates defined above, but this is rare.

Solutions

Main article: Friedmann equations

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 the time evolution of a(t) does require Einstein's field equations together with a way of calculating the density, \rho (t), such as a cosmological equation of state.

This metric has an analytic solution to Einstein's field equations G_{\mu\nu} %2B \Lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}} T_{\mu\nu} giving the Friedmann equations when the energy-momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are[5]:

\left(\frac{\dot a}{a}\right)^{2} %2B \frac{kc^{2}}{a^2} - \frac{\Lambda c^{2}}{3} = \frac{8\pi G}{3}\rho
2\frac{\ddot a}{a} %2B \left(\frac{\dot a}{a}\right)^{2} %2B \frac{kc^{2}}{a^2} - \Lambda c^{2} = -\frac{8\pi G}{c^{2}} p.

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 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 real, lumpy 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.

Interpretation

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

{\dot \rho} = - 3 \frac{\dot a}{a}\left(\rho%2B\frac{p}{c^{2}}\right)
\frac{\ddot a}{a} = - \frac{4\pi G}{3}\left(\rho %2B \frac{3p}{c^{2}}\right) %2B \frac{\Lambda c^{2}}{3}

with k, the spatial curvature index, 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 cause 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.

The cosmological constant term

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

\rho \rightarrow \rho %2B \frac{\Lambda c^{2}}{8 \pi G}
p \rightarrow p - \frac{\Lambda c^{4}}{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.

Newtonian interpretation

The Friedmann equations are equivalent to this pair of equations:

- a^3 {\dot \rho} = 3 a^2 {\dot a} \rho %2B \frac{3 a^2 p {\dot a}}{c^2} \,
\frac{{\dot a}^2}{2} - \frac{G \frac{4 \pi a^3}{3} \rho}{a} = - \frac{k c^2}{2} \,.

The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily a) is the amount which 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.

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 a deviation from the Friedmann equations.

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, on 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 1920s, 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 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).

Because the dynamics of the FLRW model were derived by Friedmann and Lemaître, the latter two names are often omitted by scientists outside the US. Conversely, US physicists often refer to it as simply "Robertson–Walker". The full four-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 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 \rho is the density of space of this universe. The numerical value of Einstein's radius is of the order of 1010 light years.

In-line references and notes

  1. ^ For an early reference, see Robertson (1935); Robertson assumes multiple connectedness in the positive curvature case and says that "we are still free to restore" simple connectedness.
  2. ^ M. Lachieze-Rey; J.-P. Luminet (1995), "Cosmic Topology", Physics Reports 254 (3): 135–214, arXiv:gr-qc/9605010, Bibcode 1995PhR...254..135L, doi:10.1016/0370-1573(94)00085-H 
  3. ^ G. F. R. Ellis; H. van Elst (1999). "Cosmological models (Cargèse lectures 1998)". In Marc Lachièze-Rey. Theoretical and Observational Cosmology. NATO Science Series C. 541. pp. 1-116. arXiv:gr-qc/9812046. Bibcode 1999toc..conf....1E. 
  4. ^ L. Bergström, A. Goobar (2006), Cosmology and Particle Astrophysics (2nd ed.), Sprint, p. 61, ISBN 3540329242, http://books.google.com/books?id=CQYu_sutWAoC&pg=PA61 
  5. ^ P. Ojeda and H. Rosu (2006), "Supersymmetry of FRW barotropic cosmologies", International Journal of Theoretical Physics 45 (6): 1191–1196, arXiv:gr-qc/0510004, Bibcode 2006IJTP...45.1152R, doi:10.1007/s10773-006-9123-2 

Further reading