section_path: Cosmological models/Model of the universe based on general relativity
section_name: Model of the universe based on general relativity
paragraph_text: Model of the universe based on general relativity
  
  General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. It is the basis of current cosmological models of the universe. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.
  
  The relation is specified by the Einstein field equations, a system of partial differential equations. In general relativity, the distribution of matter and energy determines the geometry of spacetime, which in turn describes the acceleration of matter. Therefore, solutions of the Einstein field equations describe the evolution of the universe. Combined with measurements of the amount, type, and distribution of matter in the universe, the equations of general relativity describe the evolution of the universe over time.
  
  With the assumption of the cosmological principle that the universe is homogeneous and isotropic everywhere, a specific solution of the field equations that describes the universe is the metric tensor called the Friedmann–Lemaître–Robertson–Walker metric,
  
  where (r, θ, φ) correspond to a spherical coordinate system. This metric has only two undetermined parameters. An overall dimensionless length scale factor R describes the size scale of the universe as a function of time (an increase in R is the expansion of the universe), and a curvature index k describes the geometry. The index k is defined so that it can take only one of three values: 0, corresponding to flat Euclidean geometry; 1, corresponding to a space of positive curvature; or −1, corresponding to a space of positive or negative curvature. The value of R as a function of time t depends upon k and the cosmological constant Λ. The cosmological constant represents the energy density of the vacuum of space and could be related to dark energy. The equation describing how R varies with time is known as the Friedmann equation after its inventor, Alexander Friedmann.
  
  The solutions for R(t) depend on k and Λ, but some qualitative features of such solutions are general. First and most importantly, the length scale R of the universe can remain constant only if the universe is perfectly isotropic with positive curvature (k = 1) and has one precise value of density everywhere, as first noted by Albert Einstein.
  
  Second, all solutions suggest that there was a gravitational singularity in the past, when R went to zero and matter and energy were infinitely dense. It may seem that this conclusion is uncertain because it is based on the questionable assumptions of perfect homogeneity and isotropy (the cosmological principle) and that only the gravitational interaction is significant. However, the Penrose–Hawking singularity theorems show that a singularity should exist for very general conditions. Hence, according to Einstein's field equations, R grew rapidly from an unimaginably hot, dense state that existed immediately following this singularity (when R had a small, finite value); this is the essence of the Big Bang model of the universe. Understanding the singularity of the Big Bang likely requires a quantum theory of gravity, which has not yet been formulated.
  
  Third, the curvature index k determines the sign of the curvature of constant-time spatial surfaces averaged over sufficiently large length scales (greater than about a billion light-years). If k = 1, the curvature is positive and the universe has a finite volume. A universe with positive curvature is often visualized as a three-dimensional sphere embedded in a four-dimensional space. Conversely, if k is zero or negative, the universe has an infinite volume. It may seem counter-intuitive that an infinite and yet infinitely dense universe could be created in a single instant when R = 0, but exactly that is predicted mathematically when k is nonpositive and the cosmological principle is satisfied. By analogy, an infinite plane has zero curvature but infinite area, whereas an infinite cylinder is finite in one direction and a torus is finite in both.
  
  The ultimate fate of the universe is still unknown because it depends critically on the curvature index k and the cosmological constant Λ. If the universe were sufficiently dense, k would equal +1, meaning that its average curvature throughout is positive and the universe will eventually recollapse in a Big Crunch, possibly starting a new universe in a Big Bounce. Conversely, if the universe were insufficiently dense, k would equal 0 or −1 and the universe would expand forever, cooling off and eventually reaching the Big Freeze and the heat death of the universe. Modern data suggests that the expansion of the universe is accelerating; if this acceleration is sufficiently rapid, the universe may eventually reach a Big Rip. Observationally, the universe appears to be flat (k = 0), with an overall density that is very close to the critical value between recollapse and eternal expansion.