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The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. [8]
The cosmological principle implies that the metric of the universe must be of the form = where ds 3 2 is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. This metric is called the Friedmann–Lemaître–Robertson–Walker (FLRW ...
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. [citation needed] The fact that photons have no mass yet are distorted by gravity, means that the explanation would have to be something besides photonic ...
In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds a connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature.
In the case of the flatness problem, the parameter which appears fine-tuned is the density of matter and energy in the universe. This value affects the curvature of space-time, with a very specific critical value being required for a flat universe. The current density of the universe is observed to be very close to this critical value.
Manifolds of constant curvature are most familiar in the case of two dimensions, where the elliptic plane or surface of a sphere is a surface of constant positive curvature, a flat (i.e., Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature. Einstein's general theory of ...
Einstein's static universe is closed (i.e. has hyperspherical topology and positive spatial curvature), and contains uniform dust and a positive cosmological constant with value precisely = /, where is Newtonian gravitational constant, is the energy density of the matter in the universe and is the speed of light.
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).