Einstein's static universe
Einstein's static universe is a cosmological model proposed by Albert Einstein in 1917, representing the first application of general relativity to the large-scale structure of the cosmos.[1] It describes a finite, closed, and eternally static universe with uniform matter distribution and positive spatial curvature, maintained in equilibrium through the introduction of a cosmological constant term in the field equations.[1] This model assumed that the universe has no expansion or contraction, aligning with the then-prevailing astronomical observations of relatively low stellar velocities and a fixed cosmic structure.[1] The model emerged from Einstein's efforts to reconcile general relativity with a static cosmos, influenced by Mach's principle that inertia arises from distant matter and the need for a finite universe to avoid paradoxes in Newtonian gravity.[1] To achieve staticity, Einstein modified the Einstein field equations by adding a term \lambda g_{\mu\nu}, where \lambda is the cosmological constant, providing a repulsive effect that balances the attractive pull of matter density \rho.[1] The resulting geometry is that of a three-sphere with radius R = 1/\sqrt{\lambda}, and the relation \lambda = \kappa \rho / 2 ensures equilibrium, where \kappa is the gravitational constant.[1] Despite its initial appeal as the only relativistic static model with non-zero matter density, the Einstein static universe was later shown to be unstable to perturbations, as demonstrated by Eddington in 1930, meaning small fluctuations in density would lead to either collapse or expansion.[1] Furthermore, Edwin Hubble's 1929 observations of galactic redshifts provided empirical evidence for an expanding universe, prompting Einstein to abandon the model in 1931 and famously call the cosmological constant his "biggest blunder."[1] Nonetheless, the introduction of the cosmological constant proved influential, re-emerging in modern cosmology to explain accelerated expansion via dark energy.[1]Historical Context
General Relativity and Early Cosmology
Albert Einstein completed the formulation of general relativity in November 1915, presenting the final form of the field equations during a series of four papers to the Prussian Academy of Sciences. The equations, known as the Einstein field equations, relate the geometry of spacetime to the distribution of matter and energy, expressed as R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, where R_{\mu\nu} is the Ricci curvature tensor, R is the Ricci scalar, g_{\mu\nu} is the metric tensor, T_{\mu\nu} is the stress-energy tensor, G is the gravitational constant, and c is the speed of light.[2] These equations provided a covariant framework for gravity, resolving inconsistencies in earlier attempts and enabling applications beyond solar system scales.[3] Early cosmological efforts using general relativity emerged shortly after, with Willem de Sitter proposing in 1917 a solution describing an empty universe dominated by a positive cosmological constant. This de Sitter universe featured hyperbolic spatial geometry and represented a vacuum solution to the modified field equations, expanding exponentially without matter.[4] De Sitter's model highlighted the potential for non-static spacetimes in relativity, though it lacked matter and was initially seen as a mathematical curiosity rather than a physical description of the cosmos.[5] At the time, the dominant philosophical perspective in cosmology favored a static and eternal universe, rooted in Newtonian traditions where the cosmos was envisioned as infinite, unchanging, and in equilibrium. This view was reinforced by Ernst Mach's principle, which emphasized that inertial frames derive from the global distribution of matter in the universe, influencing Einstein's development of general relativity to align with relational notions of space rather than absolute structures.[6] Such ideas shaped early relativistic cosmology, prioritizing models that preserved a timeless, finite yet unbounded universe over dynamic alternatives.[7] In 1922, Alexander Friedmann derived dynamic solutions to Einstein's field equations, demonstrating that homogeneous and isotropic universes could expand or contract depending on initial conditions and curvature. Friedmann's equations permitted parabolic, hyperbolic, or elliptic geometries, with the possibility of a finite lifetime for closed universes, challenging the static paradigm.[8] These solutions were initially overlooked by the scientific community, including Einstein, who critiqued them as mathematical errors before later acknowledging their validity.[9]Einstein's Proposal in 1917
In 1917, Albert Einstein published his seminal paper "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie" (translated as "Cosmological Considerations in the General Theory of Relativity") in the Sitzungsberichte der Preußischen Akademie der Wissenschaften, marking the first application of general relativity to cosmology. The paper was presented on February 8 and appeared in print on February 15, just 11 months after Einstein's foundational 1916 work on general relativity.[10] Einstein sought to resolve foundational issues in the theory, particularly the determination of inertial frames, by extending the field equations to the entire universe rather than isolated systems. Einstein's primary motivation was to construct a static cosmological model compatible with Mach's principle, which posits that inertia arises from the distribution of matter in the universe, not an absolute space.[10] He argued that an infinite, empty space at infinity would violate this principle by allowing inertia relative to "space" rather than to distant masses, necessitating a finite, closed universe filled with matter to define inertia properly. In Einstein's view, the universe must be a self-contained continuum where all physical properties, including the metric, are determined solely by matter.[10] To achieve this static solution, Einstein introduced a new term, the cosmological constant \Lambda, modifying the general relativity field equations to: R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} This addition allowed for a balance between gravitational attraction and a repulsive effect, enabling a finite, matter-filled universe in equilibrium. Einstein described the spatial geometry as hyperspherical—a three-dimensional analog of a sphere embedded in four-dimensional space—with uniform matter distribution on large scales and no expansion or contraction.[10] In contemporaneous correspondence, he estimated the universe's radius at approximately $10^7 light-years based on an average matter density of about $10^{-22} g/cm³, envisioning it as eternal and unchanging.[10][11]Mathematical Formulation
Key Assumptions and Modifications
Einstein's static universe model relied on several foundational physical and mathematical assumptions to reconcile general relativity with the prevailing view of a non-evolving cosmos. Central to the model was the assumption of spatial homogeneity and isotropy on large scales, embodying the cosmological principle that the universe appears the same from any point and in any direction. This led Einstein to adopt a metric describing a closed, finite spatial geometry, expressed in modern notation as the Friedmann-Lemaître-Robertson-Walker (FLRW) form with positive curvature:ds^2 = -c^2 dt^2 + a^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2 d\Omega^2 \right]
where k = +1 for a closed universe and the scale factor a(t) is constant to ensure staticity.[12][1] The matter content was modeled as pressureless dust (p = [0](/page/0)), representing a uniform distribution of non-relativistic matter with positive average density \rho, justified by the observed low velocities of stars relative to their mean positions. This assumption simplified the stress-energy tensor while capturing the dominant gravitational effects of stellar matter in a static configuration. To prevent the gravitational collapse implied by a matter-filled closed universe or the expansion suggested by general relativity alone, Einstein introduced a positive cosmological constant [\Lambda](/page/Lambda) > 0 into the field equations, modifying them to G_{\mu\nu} + [\Lambda](/page/Lambda) g_{\mu\nu} = \kappa T_{\mu\nu}. This term provided a repulsive force that balanced the attractive pull of matter, enabling a finite, eternal static solution with a fixed cosmic radius. Finally, strict staticity was enforced by the boundary condition of zero spatial velocity for all matter elements in the chosen reference frame, ensuring no expansion or contraction over time and aligning with the era's observational data on stellar motions.