De Sitter space

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Short description: Maximally symmetric Lorentzian manifold with a positive cosmological constant

In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric).

The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant Λ (corresponding to a positive vacuum energy density and negative pressure).

de Sitter space and anti-de Sitter space are named after Willem de Sitter (1872–1934),[1][2] professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked closely together in Leiden in the 1920s on the spacetime structure of our universe. de Sitter space was also discovered, independently, and about the same time, by Tullio Levi-Civita.[3]

Definition

de Sitter space can be defined as a submanifold of a generalized Minkowski space of one higher dimension. Take Minkowski space R1,n with the standard metric: ds2=dx02+i=1ndxi2.

de Sitter space is the submanifold described by the hyperboloid of one sheet x02+i=1nxi2=α2, where α is some nonzero constant with its dimension being that of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces α2 with α2 in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space. For a detailed proof, see Minkowski space § Geometry.)

de Sitter space can also be defined as the quotient O(1, n) / O(1, n − 1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.

Topologically, de Sitter space is R × Sn−1 (so that if n ≥ 3 then de Sitter space is simply connected).

Properties

The isometry group of de Sitter space is the Lorentz group O(1, n). The metric therefore then has n(n + 1)/2 independent Killing vector fields and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by[4]

Rρσμν=1α2(gρμgσνgρνgσμ)

(using the sign convention Rρσμν=μΓνσρνΓμσρ+ΓμλρΓνσλΓνλρΓμσλ for the Riemann curvature tensor). de Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

Rμν=Rλμλν=n1α2gμν

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

Λ=(n1)(n2)2α2.

The scalar curvature of de Sitter space is given by[4]

R=n(n1)α2=2nn2Λ.

For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.

Coordinates

Static coordinates

We can introduce static coordinates (t,r,) for de Sitter as follows:

x0=α2r2sinh(1αt)x1=α2r2cosh(1αt)xi=rzi2in.

where zi gives the standard embedding the (n − 2)-sphere in Rn−1. In these coordinates the de Sitter metric takes the form:

ds2=(1r2α2)dt2+(1r2α2)1dr2+r2dΩn22.

Note that there is a cosmological horizon at r=α.

Flat slicing

Let

x0=αsinh(1αt)+12αr2e1αt,x1=αcosh(1αt)12αr2e1αt,xi=e1αtyi,2in

where r2=iyi2. Then in the (t,yi) coordinates metric reads:

ds2=dt2+e21αtdy2

where dy2=idyi2 is the flat metric on yi's.

Setting ζ=ζαe1αt, we obtain the conformally flat metric:

ds2=α2(ζζ)2(dy2dζ2)

Open slicing

Let

x0=αsinh(1αt)coshξ,x1=αcosh(1αt),xi=αzisinh(1αt)sinhξ,2in

where izi2=1 forming a Sn2 with the standard metric idzi2=dΩn22. Then the metric of the de Sitter space reads

ds2=dt2+α2sinh2(1αt)dHn12,

where

dHn12=dξ2+sinh2(ξ)dΩn22

is the standard hyperbolic metric.

Closed slicing

Let

x0=αsinh(1αt),xi=αcosh(1αt)zi,1in

where zis describe a Sn1. Then the metric reads:

ds2=dt2+α2cosh2(1αt)dΩn12.

Changing the time variable to the conformal time via tan(12η)=tanh(12αt) we obtain a metric conformally equivalent to Einstein static universe:

ds2=α2cos2η(dη2+dΩn12).

These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its Penrose diagram.[5]

dS slicing

Let

x0=αsin(1αχ)sinh(1αt)coshξ,x1=αcos(1αχ),x2=αsin(1αχ)cosh(1αt),xi=αzisin(1αχ)sinh(1αt)sinhξ,3in

where zis describe a Sn3. Then the metric reads:

ds2=dχ2+sin2(1αχ)dsdS,α,n12,

where

dsdS,α,n12=dt2+α2sinh2(1αt)dHn22

is the metric of an n1 dimensional de Sitter space with radius of curvature α in open slicing coordinates. The hyperbolic metric is given by:

dHn22=dξ2+sinh2(ξ)dΩn32.

This is the analytic continuation of the open slicing coordinates under (t,ξ,θ,ϕ1,ϕ2,,ϕn3)(iχ,ξ,it,θ,ϕ1,,ϕn4) and also switching x0 and x2 because they change their timelike/spacelike nature.

See also

References

  1. de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis", Proc. Kon. Ned. Acad. Wet. 19: 1217–1225, Bibcode1917KNAB...19.1217D, https://www.dwc.knaw.nl/DL/publications/PU00012455.pdf 
  2. de Sitter, W. (1917), "On the curvature of space", Proc. Kon. Ned. Acad. Wet. 20: 229–243, https://www.dwc.knaw.nl/DL/publications/PU00012216.pdf 
  3. Levi-Civita, Tullio (1917), "Realtà fisica di alcuni spazî normali del Bianchi", Rendiconti, Reale Accademia dei Lincei 26: 519–31 
  4. 4.0 4.1 Zee 2013, p. 626
  5. Hawking & Ellis. The large scale structure of space–time. Cambridge Univ. Press. 
  • Zee, Anthony (2013). Einstein Gravity in a Nutshell. Princeton University Press. ISBN 9780691145587. 

Further reading