An Evolving Universe
On the Curvature of Space (1922)
The true scale of the cosmos is enough to beggar even the greatest mind.
Take a second and consider from your own experiences the vastness of the Earth—its numerous continents, oceans, and nation-states. Over a million Earths can fit into the sun alone and that is to say nothing of the gaping expanse of the solar system out to the Oort Cloud. There are approximately 100 billion stars in the Milky Way, a moderately sized galaxy; there are as many as 2 trillion galaxies in the observable universe.
The immensity of the cosmos is only understood when one fails to understand it. Nonetheless, it is possible to mathematically describe the evolution of the universe as a whole. In 1922 Russian physicist, Alexander Friedmann, published a pioneering paper titled On the Curvature of Space which laid the foundation for modern cosmology.
At a time when the prevailing cosmological models envisioned a static universe, Friedmann introduced an alternative picture: a dynamic spacetime, the fate of which is determined by its matter and energy content.
Competing Cosmological Models
Friedmann's work emerged against the backdrop of Albert Einstein’s and Willem de Sitter’s provisional models of the universe. Both scientists had applied general relativity to cosmology, but with a few conceptual differences. Einstein’s approach proposed a cylindrical geometry where the size of the universe remained constant over time. This required the addition of a cosmological constant, Λ, to counteract gravitational collapse.
The de Sitter model, on the other hand, described a universe dominated by a cosmological constant, but nearly empty in terms of matter. This results in a universe with constant size but with no significant gravitational influence from matter.
Friedmann radicalized these ideas by introducing dynamic solutions of Einstein field equations. To do this he invoked two basic assumptions for the sake of simplicity: that the universe is isotropic (the same in all directions) and homogeneous (distributed uniformly on large scales). The derived metric describes a three-dimensional spatial hypersurface in which the distance between galaxies varies as a function of time:
where R(t) is the scale factor describing the universe's size at time t, and k is the spatial curvature constant (k = 1 for spherical geometry, k = 0 for flat geometry, and k = −1 for hyperbolic geometry).
From this setup, Friedmann was able to formulate two key equations governing the dynamics of R(t). The first Friedmann equation is:
where R˙ is the rate of change of the scale factor, G is the gravitational constant, and ρ is the matter density. The second Friedman equation is a second order acceleration equation:
where p is the pressure. According to these equations, the way the universe evolves depends on its initial conditions, matter density, and the value of Λ.
End Game
In his landmark paper, Friedmann speculates on three possible scenarios:
The Monotonic Universe: In cases where Λ is large and positive, or where the total mass in the universe is sufficiently small, the universe expands indefinitely. Friedmann terms this a "monotonic world of the first kind," where R grows continuously with time.
The Oscillating Universe: If the matter density and Λ is finely balanced, Friedmann showed that the universe could undergo periodic cycles of expansion and contraction. This "periodic world" was particularly intriguing, suggesting a universe with a finite lifetime between each oscillation.
The Closed Universe: For certain values of Λ and a high matter density, Friedmann described a universe that expands from a singularity (a point of infinite density) and eventually reaches a maximum size before contracting back to a singularity.
Determining which of these scenarios will play out in the long-term required careful experimental observation. Even today, there is much debate on how to interpret large-scale astrophysical phenomena in light of Friedman’s equations.
What we know for sure, however, is that the universe is no where near static. Friedmann’s work initially met resistance because the notion of a static, eternal universe was so deeply ingrained in the psyche of the scientific community. Einstein most notably dismissed it, suggesting an error in Friedmann’s calculations. Upon reviewing the work more closely, Einstein was forced to acknowledged its validity.
It wasn't until Edwin Hubble's 1929 observation of galactic redshifts—indicating that galaxies are receding from us—that Friedmann's ideas gained empirical support. More recent evidence indicates that this expansion is accelerating though it is unclear whether this will continue indefinitely. Billions of years after the Big Bang event, the end of reality remains just as mysterious as its origin.
Related Papers:
Einstein, A. (1917). Cosmological considerations in the general theory of relativity. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 142–152.
Friedmann builds upon Einstein’s 1917 work, which introduced the idea of a static universe by adding the cosmological constant (Λ) to the field equations. Friedmann explicitly refers to Einstein’s solution for a "cylindrical world" where the universe has constant spatial curvature and the curvature radius depends on the total matter density. Friedmann relaxes Einstein’s assumption of a static universe and reinterprets the cosmological constant as a term that permits dynamic solutions, including expanding and contracting universes.de Sitter, W. (1917). On Einstein’s theory of gravitation and its astronomical consequences. Monthly Notices of the Royal Astronomical Society, 78(1), 3–28.
Friedmann references de Sitter's solution, which describes a "spherical world" with no matter (T_μν=0) and a nonzero cosmological constant. De Sitter’s model presented a universe of constant spatial curvature dominated entirely by the cosmological constant. Friedmann extends de Sitter's ideas by introducing time-dependent curvature in universes where matter and energy densities are nonzero, deriving solutions that generalize both de Sitter's and Einstein’s static models.Klein, F. (1918). On the integral form of the conservation theorems and the theory of the spatially closed world. Göttinger Nachrichten, 171–189.
Friedmann cites Klein’s work on spatially closed universes and conservation laws in General Relativity as a foundation for considering curved spaces with specific boundary conditions. Klein's exploration of closed geometries informs Friedmann's analysis of the "stationary world," particularly the interpretation of a static cylindrical or spherical geometry as special cases of a broader, time-dependent framework.