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Friedmann-Robertson-Walker metric.


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The Friedmann-Lemaītre-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of General relativity and which describes a homogeneous, isotropic expanding/contracting universe. Depending on geographical/historical preferences, this may be referred to under the names of a preferred subset of the four scientists Alexander Friedmann, Georges Lemaītre, Howard Percy Robertson and Arthur Geoffrey Walker, e.g. Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW).

The FRLW 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:

ds^2 = dt^2 - {a(t)}^2 \left( \frac{dr^2}{1-k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \right)

wherek describes the curvature and is constant in time anda(t) is the scale factor and is explicity time dependent.

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 coordinater carries dimensionality as doesk. In this choicek does not equal ±1 or 0 but k = H_0^2 \left( \Omega_0 - 1 \right).

Another choice is to specify thatk 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 coordinater 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

ds^2 = dt^2 - a(t)^2 \left[ d\chi^2 + S^2_k(\chi) \left(d\theta^2 + \sin^2\theta 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) fork greater than, equal to, and less than 0 respectively. This normalization assumes the scale factor is dimensionless but it can be easily converted to normalizedk. The Comoving distance is distance to an object with zero peculiar velocity. In the curvature normalized coordinates it is?. The proper distance is the physical distance to a point in space at an instant in time. The proper distance isa | t(?).

Solutions

This metric has an analytic solution to the Einstein field equationsGµ? - ?gµ? = 8pTµ? 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 ?CDM 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.

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.



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