Non-Euclidean Geometry and the Fate of the Universe
Abstract
Non-Euclidean geometry provides the mathematical framework required to describe
the large-scale structure of the universe. This paper discusses the non-Euclidean geom-
etry and explains how each geometry is related to cosmological models and the possible
fate of the universe.
1 Introduction
Non-Euclidean geometry originated as a mathematical branch but has become fundamental
in understanding the structure of the universe. Geometry can be classified according to
the sum of the angles of a triangle. In spherical geometry the sum is greater than 180◦ , in
Euclidean geometry it is exactly 180◦ , and in hyperbolic geometry it is less than 180◦ . In
cosmology, these geometries correspond to different spatial curvatures and play a key role in
determining the fate of the universe.
2 Spacetime Geometry
Einstein’s general theory of relativity describes gravity as a manifestation of spacetime cur-
vature rather than a force. Matter and energy determine this curvature, which governs the
motion of matter and light. On cosmological scales, the universe is well approximated as
homogeneous and isotropic. These assumptions lead to the Friedmann–Lemaı̂tre–Robertson–
Walker (FLRW) metric, which permits only three types of spatial geometry: positive, zero,
or negative curvature.
3 The Friedmann Equation
Applying Einstein’s field equations to the FLRW metric yields the Friedmann equation,
2
ȧ 8πG kc2 Λc2
= ρ− 2 + ,
a 3 a 3
where a(t) is the scale factor, ρ is the total energy density, k = +1, 0, −1 represents spher-
ical, flat, and hyperbolic geometry, Λ is the cosmological constant, and G and c are the
gravitational constant and the speed of light.
1
Abstract
Non-Euclidean geometry provides the mathematical framework required to describe
the large-scale structure of the universe. This paper discusses the non-Euclidean geom-
etry and explains how each geometry is related to cosmological models and the possible
fate of the universe.
1 Introduction
Non-Euclidean geometry originated as a mathematical branch but has become fundamental
in understanding the structure of the universe. Geometry can be classified according to
the sum of the angles of a triangle. In spherical geometry the sum is greater than 180◦ , in
Euclidean geometry it is exactly 180◦ , and in hyperbolic geometry it is less than 180◦ . In
cosmology, these geometries correspond to different spatial curvatures and play a key role in
determining the fate of the universe.
2 Spacetime Geometry
Einstein’s general theory of relativity describes gravity as a manifestation of spacetime cur-
vature rather than a force. Matter and energy determine this curvature, which governs the
motion of matter and light. On cosmological scales, the universe is well approximated as
homogeneous and isotropic. These assumptions lead to the Friedmann–Lemaı̂tre–Robertson–
Walker (FLRW) metric, which permits only three types of spatial geometry: positive, zero,
or negative curvature.
3 The Friedmann Equation
Applying Einstein’s field equations to the FLRW metric yields the Friedmann equation,
2
ȧ 8πG kc2 Λc2
= ρ− 2 + ,
a 3 a 3
where a(t) is the scale factor, ρ is the total energy density, k = +1, 0, −1 represents spher-
ical, flat, and hyperbolic geometry, Λ is the cosmological constant, and G and c are the
gravitational constant and the speed of light.
1
