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The greatest symmetry in the universe: what is n-dimensional de Sitter space?
In mathematical physics, an n-dimensional de Sitter space (usually denoted by dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentz analysis analogue of the n-dimensional sphere (n-sphere) and can be regarded as a simple yet profound mathematical model describing the structure of the universe. The main application of de Sitter space in general relativity is that it provides a mathematical basis consistent with the observed accelerating expansion of the universe.
De Sitter space is the vacuum solution of Einstein's field equation under the positive cosmological constant, corresponding to positive vacuum energy density and negative pressure.
De Sitter space and anti-de Sitter space are also named after Willem de Sitter. He is a professor of astronomy at Leiden University and worked closely with Albert Einstein in the 1920s to study the space-time structure of our universe. The independent discovery of de Sitter space is also attributed to Tullio Levi-Civita.
Definition and properties of de Sitter space
A de Sitter space can be defined as a submanifold embedded in a generalized leapfrog space with standard metrics. More specifically, n-dimensional de Sitter space describes a manifold of one layer of hyperboloids, and the standard leapk space is defined as:
ds^2 = -dx_0^2 + \sum_{i=1}^{n} dx_i^2
Here, the so-called hyperboloid satisfies the following equation:
-x_0^2 + \sum_{i=1}^{n} x_i^2 = \alpha^2
Among them, α is a non-zero constant, and the unit is length. The induced metric of de Sitter space is introduced from the ambient leapk metric, has Lorentzian signature and is not degenerate.
The isometric transformation group of de Sitter space is the Lorentz group O(1, n), which means that it has n(n + 1)/2 independent Kiel stars.
Constant curvature is an intrinsic property of every maximally symmetric space. The Riemannian curvature tensor possessed by de Sitter space can be expressed as:
R_{ρσμν} = \frac{1}{\alpha^2}(g_{ρμ}g_{σν} - g_{ρν}g_{σμ})
This shows that de Sitter space is an Einsteinian manifold because its Riemannian curvature tensor is metrically related. This means that de Sitter space is a vacuum solution to Einstein's equations, and the specific value of the cosmological constant varies based on the dimension in which it is located.
Coordinate systems and their applications
De Sitter space can be expressed in a static coordinate system, and such expressions can be used to study effective dynamics:
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)
Under such a coordinate system, the form of the de Sitter metric shows the franchise of the expansion of the universe:
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
It should be noted that there is a cosmic horizon located at r = α.
Summary
De Sitter space, as a mathematical model that explains the structure of the universe, not only allows us to understand the properties of the expanding universe, but also paves the way for future cosmological research. Its symmetry and physical properties reflect the profound insights of today's physics. In what way will it affect our understanding of the universe is still a question worth thinking about.
