De Sitter space

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In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dSn, is the Lorentzian manifold analog of an n-sphere (with its canonical Riemannian metric); it is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. The de Sitter space, as well as the anti-de Sitter space is named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked in the 1920s in Leiden closely together on the spacetime structure of our universe.

In the language of general relativity, de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equations with a positive (repulsive) cosmological constant Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda } (corresponding to a positive vacuum energy density and negative pressure). When n = 4 (3 space dimensions plus time), it is a cosmological model for the physical universe; see de Sitter universe.

De Sitter space was discovered by Willem de Sitter, and, at the same time, independently by Tullio Levi-Civita.

More recently it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity.

Definition

De Sitter space can be defined as a submanifold of a Minkowski space of one higher dimension. Take Minkowski space R1,n with the standard metric:

De Sitter space is the submanifold described by the hyperboloid of one sheet

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } is some positive constant with dimensions 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha ^{2}} with 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.)

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 vectors and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by

De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{\mu \nu }={\frac {n-1}{\alpha ^{2}}}g_{\mu \nu }}

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2}}}.}

The scalar curvature of de Sitter space is given by

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

Static coordinates

We can introduce static coordinates for de Sitter as follows:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): x_{0}={\sqrt {\alpha ^{2}-r^{2}}}\sinh(t/\alpha )
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}={\sqrt {\alpha ^{2}-r^{2}}}\cosh(t/\alpha )}

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}

Note that there is a cosmological horizon at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): r=\alpha .

Flat slicing

Let

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i}=e^{t/\alpha }y_{i},\qquad 2\leq i\leq n}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^{2}=\sum _{i}y_{i}^{2}} . Then in the coordinates metric reads:

where is the flat metric on 's.

Open slicing

Let

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i}=\alpha z_{i}\sinh(t/\alpha )\sinh \xi ,\qquad 2\leq i\leq n}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \sum _{i}z_{i}^{2}=1 forming a with the standard metric . Then the metric of the de Sitter space reads

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}(t/\alpha )dH_{n-1}^{2},}

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}\xi d\Omega _{n-2}^{2}}

is the metric of a Euclidean hyperbolic space.

Closed slicing

Let

where s describe a . Then the metric reads:

Changing the time variable to the conformal time via we obtain a metric conformally equivalent to Einstein static universe:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ds^{2}={\frac {\alpha ^{2}}{\cos ^{2}\eta }}(-d\eta ^{2}+d\Omega _{n-1}^{2}).}

This serves to find the Penrose diagram of de Sitter space.Template:Clarify

dS slicing

Let

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}=\alpha \sin(\chi /\alpha )\sinh(t/\alpha )\cosh \xi ,}
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i}=\alpha z_{i}\sin(\chi /\alpha )\sinh(t/\alpha )\sinh \xi ,\qquad 3\leq i\leq n}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{i}} s describe a Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{n-3}} . Then the metric reads:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ds^{2}=d\chi ^{2}+\sin ^{2}(\chi /\alpha )ds_{dS,\alpha ,n-1}^{2},}

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}(t/\alpha )dH_{n-2}^{2}}

is the metric of an Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-1} dimensional de Sitter space with radius of curvature in open slicing coordinates. The hyperbolic metric is given by:

This is the analytic continuation of the open slicing coordinates under and also switching and because they change their timelike/spacelike nature.

See also

References

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