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The Friedmann–Lemaître–Robertson–Walker (FLRW) model using Friedmann equations is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of fluid dynamics, that is, modeling the matter within the universe as a perfect fluid. Although stars and structures of mass can be introduced ...
If k = +1, then a is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time. If k = −1, then (loosely speaking) one can say that i · a is the radius of curvature of the universe. a is the scale factor which is taken to be 1 at the present time.
The ΛCDM model assumes that the shape of the universe is of zero curvature (is flat) and has an undetermined topology. In 2019, interpretation of Planck data suggested that the curvature of the universe might be positive (often called "closed"), which would contradict the ΛCDM model.
Alternatively, as before, k may be taken to belong to the set {−1 ,0, +1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = ±1, a(t) is the radius of curvature of the space, and may also be written R(t). Note that when k = +1, r is essentially a third angle along with θ and φ.
In mathematical physics, n-dimensional de Sitter space (often denoted dS n) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature.It is the Lorentzian [further explanation needed] analogue of an n-sphere (with its canonical Riemannian metric).
General relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from the presence of matter or energy. Energy and mass are equivalent (as expressed in the equation E = mc 2). Space and time values can be related respectively to time and space units by multiplying or dividing ...
The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship.
Curved spaces play an essential role in general relativity, where gravity is often visualized as curved spacetime. [2] The Friedmann–Lemaître–Robertson–Walker metric is a curved metric which forms the current foundation for the description of the expansion of the universe and the shape of the universe.