Friedmann-Lemaître-Robertson-Walker metric

The Friedmann-Lemaître-Robertson-Walker metric , or FLRW metric for short , is an exact solution to Einstein's field equations of general relativity and describes a homogeneous , isotropic ( cosmological principle ) expansion or contraction of the universe . It is known under different combinations of the names of the four scientists Alexander Friedmann , Georges Lemaître , Howard P. Robertson and Arthur Geoffrey Walker , e.g. B. Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW) .

Because it is so easy to calculate, the FLRW metric is used as a first approximation for the standard cosmological Big Bang model of the universe . Since the FLRW assumes homogeneity, it is often wrongly claimed that the Big Bang model cannot explain the lumpiness of the universe. Models that will calculate the lumpiness of the universe extend the FLRW. In 2003, the theoretical consequences of the various extensions to the FLRW already seemed well understood. The aim was to reconcile this with the observations made in the COBE and WMAP projects .

formulation

The Robertson-Walker line element is obtained through the requirement for isotropy

{\ displaystyle \ mathrm {d} s ^ {2} = c ^ {2} \ mathrm {d} t ^ {2} -a (t) ^ {2} R _ {\ mathrm {C}} ^ {2} \ left ({\ frac {\ mathrm {d} x ^ {2}} {1-k \ x ^ {2}}} + x ^ {2} \ mathrm {d} \ Omega ^ {2} \ right) \,}

where is the curvature parameter and . ${\ displaystyle k = + 1.0, -1}$ ${\ displaystyle x = r / R _ {\ mathrm {C}}}$

The metric can be represented in a simplified manner depending on the value of the curvature parameter:

{\ displaystyle \ mathrm {d} s ^ {2} = c ^ {2} \ mathrm {d} t ^ {2} -a (t) ^ {2} \ cdot (\ mathrm {d} r ^ {2 } + {\ bar {r}} ^ {2} \ mathrm {d} \ Omega ^ {2})}

in which

${\ displaystyle c}$ the speed of light ,
${\ displaystyle a (t)}$ the scale factor of the universe at the time , ${\ displaystyle t}$
${\ displaystyle r}$ the distance from the moving observer,
${\ displaystyle {\ bar {r}}}$ the covariant distance:

{\ displaystyle {\ bar {r}} = {\ begin {cases} R _ {\ mathrm {C}} \ sin (r / R _ {\ mathrm {C}}) & {\ text {for positive curvature}} \ \ r & {\ text {for curvature}} 0 \\ R _ {\ mathrm {C}} \ sinh (r / R _ {\ mathrm {C}}) & {\ text {for negative curvature}} \ end {cases} }}

${\ displaystyle R _ {\ mathrm {C}}}$ the absolute value of the radius of curvature ,

and are.

{\ displaystyle \ mathrm {d} \ Omega ^ {2} = \ mathrm {d} \ theta ^ {2} + \ sin ^ {2} \ theta \ cdot \ mathrm {d} \ phi ^ {2}}

If one assumes the FLRW metric and a suitable energy-momentum tensor , Einstein's field equations are reduced to Friedmann's equations . The solution to the Friedmann equations is the time course of the scale factor of the FLRW metric.

Fast FLRW models

All observations in the universe on sufficiently large length scales (namely larger than the largest identifiable objects in the universe, the galaxy clusters ) can be easily explained by an almost FLRW model. A fast FLRW model follows the FLRW metric, whereby the development of the matter distribution from primordial fluctuations can be calculated as a small perturbation . In an exact FLRW model, there are no galaxy clusters, stars, or people because these objects are denser than the average of the universe. Nevertheless , for the sake of brevity , a fast FLRW model is referred to as a FLRW model (or FRW model).

literature

HP Robertson: Kinematics and world structure , Astrophysical Journal, Vol 82, 1935, pp 284-301, Vol 83, 1936, pp 187-201, pp 257-271
AG Walker: On Milne's theory of world-structure , Proc. Lond. Math. Soc. (2), Vol. 42, 1937, pp. 90-127
George FR Ellis , Henk van Elst: Cosmological Models (Cargèse lectures 1998). (English: arxiv.org )

Ray d'Inverno: Introducing Einstein's Relativity . Oxford University Press, Oxford 1992, ISBN 0-19-859686-3 (Chapter 23 provides a brief introduction to the FLRW models).