Conformally flat manifold

File:Conformal map.svg

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the geodesics maintain in all points of M the angles by moving from one to the other, as well as keeping the null geodesics unchanged,{{Cite book|author=Ray D'Inverno|title=Introducing Einstein's Relativity|pages=88–89|chapter=6.13 The Weyl tensor}} that means there exists a function \lambda(x) such that g(x) = \lambda^2(x)\, \eta, where \lambda(x) is known as the conformal factor and x is a point on the manifold.

More formally, let (M,g) be a pseudo-Riemannian manifold. Then (M,g) is conformally flat if for each point x in M, there exists a neighborhood U of x and a smooth function f defined on U such that (U,e^{2f} g) is flat (i.e. the curvature of e^{2f} g vanishes on U). The function f need not be defined on all of M.

Some authors use the definition of locally conformally flat when referred to just some point x on M and reserve the definition of conformally flat for the case in which the relation is valid for all x on M.

Examples

  • Every manifold with constant sectional curvature is conformally flat.
  • Every 2-dimensional pseudo-Riemannian manifold is conformally flat.
  • The line element of the two dimensional spherical coordinates, like the one used in the geographic coordinate system,
  • : ds^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 \,,Spherical coordinate system - Integration and differentiation in spherical coordinates has metric tensor g_{ik} = \begin{bmatrix} 1 & 0 \\ 0 & sin^2 \theta \end{bmatrix} and is not flat but with the stereographic projection can be mapped to a flat space using the conformal factor 2 \over (1+r^2), where r is the distance from the origin of the flat space,Stereographic projection - Properties. The Riemann's formula obtaining
  • :ds^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 \, = \frac{4}{(1+r^2)^2}(dx^2 +dy^2) .
  • A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes.
  • An n-dimensional pseudo-Riemannian manifold for n ≥ 4 is conformally flat if and only if the Weyl tensor vanishes.
  • Every compact, simply connected, conformally Euclidean Riemannian manifold is conformally equivalent to the round sphere.{{cite journal|last1=Kuiper|first1=N. H.|title=On conformally flat spaces in the large|journal=Annals of Mathematics|date=1949|volume=50|issue=4|pages=916–924|doi=10.2307/1969587|jstor=1969587|ref=Kuiper}}

:* The stereographic projection provides a coordinate system for the sphere in which conformal flatness is explicit, as the metric is proportional to the flat one.

  • In general relativity conformally flat manifolds can often be used, for example to describe Friedmann–Lemaître–Robertson–Walker metric.{{Cite journal|last=Garecki|first=Janusz|year=2008|title=On Energy of the Friedman Universes in Conformally Flat Coordinates|journal=Acta Physica Polonica B|volume=39|issue=4|pages=781–797|arxiv=0708.2783|bibcode=2008AcPPB..39..781G}} However it was also shown that there are no conformally flat slices of the Kerr spacetime.{{Cite journal|last1=Garat|first1=Alcides|last2=Price|first2=Richard H.|date=2000-05-18|title=Nonexistence of conformally flat slices of the Kerr spacetime|journal=Physical Review D|language=en|volume=61|issue=12|pages=124011|doi=10.1103/PhysRevD.61.124011|arxiv=gr-qc/0002013|bibcode=2000PhRvD..61l4011G|s2cid=119452751|issn=0556-2821}}

: For example, the Kruskal-Szekeres coordinates have line element

: ds^2 = \left(1-\frac{2GM}{r} \right) dv \, du with metric tensor g_{ik} = \begin{bmatrix} 0 & 1-\frac{2GM}{r} \\ 1-\frac{2GM}{r} & 0 \end{bmatrix} and so is not flat. But with the transformations t = (v + u)/2 and x = (v - u)/2

:becomes

: ds^2 = \left(1-\frac{2GM}{r} \right) (dt^2 - dx^2) with metric tensor g_{ik} = \begin{bmatrix} 1-\frac{2GM}{r} & 0 \\ 0 & -1+\frac{2GM}{r} \end{bmatrix},

: which is the flat metric times the conformal factor 1-\frac{2GM}{r}.{{Cite book|author=Ray D'Inverno|title=Introducing Einstein's Relativity|pages=230–231|chapter=17.2 The Kruskal solution}}

See also

References