Friedmann-Lemaître-Robertson-Walker metric

From Wikipedia, the free encyclopedia

Physical Cosmology

Physical Cosmology
Universe · Big Bang Age of the universe Timeline of the Big Bang... Ultimate fate of the Universe
Early universe
Inflation · Nucleosynthesis Cosmic microwave background
Expanding universe
Redshift · Hubble's law Metric expansion of space Friedmann equations · FLRW metric
Structure formation
Shape of the universe Structure formation Galaxy formation Large-scale structure
Components
Lambda-CDM model Dark energy · Dark matter
History
Timeline of cosmology...
Cosmology experiments
Observational cosmology 2dF · SDSS CoBE · BOOMERanG · WMAP
Scientists
Einstein · Lemaître · Friedman Hubble · Penzias · Wilson Gamow · Dicke · Zel'dovich Mather · Smoot · others
This box: view • talk • edit

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)).

1 General Metric
2 Normalization
3 Solutions
4 External links
5 References

[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

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

where $k$ describes the curvature and is constant in time, and $a (t)$ is the scale factor and is explicitly time dependent.

[edit] Normalization

The metric leaves some choice of normalization. One common choice is to say that scale factor is 1 today ( $a(t_0) \equiv 1$ ). In this choice the coordinate $r$ carries dimensionality as does $k$ . In this choice $k$ 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 coordinate $r$ is dimensionless.

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

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

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 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 $a | t (χ)$ .

[edit] Solutions

This metric has an analytic solution to the Einstein field equations $G μν - Λ g μν = 8π T μν$ giving the Friedmann equations when the energy-momentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are:

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

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.

[edit] External links

Ellis, George F. R., van Elst, Henk. Cosmological models (Cargèse lectures 1998). arXiv.org eprint archive. Retrieved on July 30, 2005.

[edit] References

d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0-19-859686-3. . See chapter 23 for a particularly clear and concise introduction to the FLRW models.

Retrieved from "http://en.wikipedia.org../../../f/r/i/Friedmann-Lema%C3%AEtre-Robertson-Walker_metric_f2d1.html"

Categories: Coordinate charts in general relativity | Exact solutions in general relativity