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
(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:
![{\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60a2f98c1a3d6da5b4142fc4c832450e1a221a1b)
de Sitter space is the submanifold described by the hyperboloid of one sheet
![{\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b10cc418b2866908f82ed2999f1de4be1f7802a6)
where
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
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. 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]
![{\displaystyle R_{\rho \sigma \mu \nu }={1 \over \alpha ^{2))\left(g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d10ca765173e6a344a83940cf0f433cce38735)
(using the sign convention
for the Riemann curvature tensor). de Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:
![{\displaystyle R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }={\frac {n-1}{\alpha ^{2))}g_{\mu \nu ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/296aedc3591818e656648f5487365656dcf8e7dd)
This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
![{\displaystyle \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fde480d3503cead2a43fe1b3595124e767f9a4c)
The scalar curvature of de Sitter space is given by[4]
![{\displaystyle R={\frac {n(n-1)}{\alpha ^{2))}={\frac {2n}{n-2))\Lambda .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6535e0395f7ecfd03c30aad7e7cd9475f79d2f98)
For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.
Coordinates
Static coordinates
We can introduce static coordinates
for de Sitter as follows:
![{\displaystyle {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2))}\sinh \left({\frac {1}{\alpha ))t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2))}\cosh \left({\frac {1}{\alpha ))t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ba32b16481a89fc3e82dc3f4675a34a9a1af4)
where
gives the standard embedding the (n − 2)-sphere in Rn−1. In these coordinates the de Sitter metric takes the form:
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/846e157859cf53d3bb2c1549b0b44830a0e859ee)
Note that there is a cosmological horizon at
.
Flat slicing
Let
![{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha ))t\right)+{\frac {1}{2\alpha ))r^{2}e^((\frac {1}{\alpha ))t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha ))t\right)-{\frac {1}{2\alpha ))r^{2}e^((\frac {1}{\alpha ))t},\\x_{i}&=e^((\frac {1}{\alpha ))t}y_{i},\qquad 2\leq i\leq n\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/349a3603f826f690930778ef5b13171604bd0e8c)
where
. Then in the
coordinates metric reads:
![{\displaystyle ds^{2}=-dt^{2}+e^{2{\frac {1}{\alpha ))t}dy^{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/de241bd7100a85251b1062ae303df0a1b6771663)
where
is the flat metric on
's.
Setting
, we obtain the conformally flat metric:
![{\displaystyle ds^{2}={\frac {\alpha ^{2)){(\zeta _{\infty }-\zeta )^{2))}\left(dy^{2}-d\zeta ^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e53b06dfb2ea8474c52cffbb00b4be2d86ca2c7)
Open slicing
Let
![{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha ))t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha ))t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha ))t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da32cf7fa2b73c3e5083336aa1766e27db198994)
where
forming a
with the standard metric
. Then the metric of the de Sitter space reads
![{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha ))t\right)dH_{n-1}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98268adb83c510a11971a202d348d9067b8a225)
where
![{\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-2}^{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b17a53842cde125476531343676d998a25b699)
is the standard hyperbolic metric.
Closed slicing
Let
![{\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha ))t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha ))t\right)z_{i},\qquad 1\leq i\leq n\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c47d0dfba421e730e34101b86abb003e06e614)
where
s describe a
. Then the metric reads:
![{\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\cosh ^{2}\left({\frac {1}{\alpha ))t\right)d\Omega _{n-1}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0e00852fb01ab49ba2a9ba942a0c0eb1c5a17b)
Changing the time variable to the conformal time via
we obtain a metric conformally equivalent to Einstein static universe:
![{\displaystyle ds^{2}={\frac {\alpha ^{2)){\cos ^{2}\eta ))\left(-d\eta ^{2}+d\Omega _{n-1}^{2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32266687ff58bb723adc898cd0baa1f92ff3e66f)
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
![{\displaystyle {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha ))\chi \right)\sinh \left({\frac {1}{\alpha ))t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha ))\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha ))\chi \right)\cosh \left({\frac {1}{\alpha ))t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha ))\chi \right)\sinh \left({\frac {1}{\alpha ))t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf0ea6478be4fcd7437df3ba797574578ca2435)
where
s describe a
. Then the metric reads:
![{\displaystyle ds^{2}=d\chi ^{2}+\sin ^{2}\left({\frac {1}{\alpha ))\chi \right)ds_{dS,\alpha ,n-1}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b358741e354d21fa4d27ae81c5c122d2f8f43eea)
where
![{\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha ))t\right)dH_{n-2}^{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/231af2b10c76d026a8f46fa5ecf7d4a6afd86967)
is the metric of an
dimensional de Sitter space with radius of curvature
in open slicing coordinates. The hyperbolic metric is given by:
![{\displaystyle dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb06642a9baa8b6eedf440f4d78d2515bd44a11)
This is the analytic continuation of the open slicing coordinates under
and also switching
and
because they change their timelike/spacelike nature.