curvature

Curvature of space (i.e., a curved space) is space that deviates from the well-known rules of Euclidean geometry. Curved spaces can be consistently described mathematically, and general relativity considers the space of the universe (as well as 4D spacetime) as non-flat, to help describe gravity, and cosmological theories assume some curvature. In both cases, the curvature is too small to show up in day-to-day measurements.

A curvature of space can be detected by testing the known rules of flat (Euclidean) geometry, in an analogous manner to how one might measure the curvature of the Earth's surface through measurement: for example, laying out a large triangle on the surface consisting of a given Earth altitude (e.g., sea level) will yield a shape whose angles do not add up to 180° or laying out a circle and measuring the ratio of its circumference and diameter will not yield π. Precisely these same tests can, in principle, be done in space, and given enough curvature and large enough shapes, the curvature could be measured. Sensitive trials have been carried out to see if any curvature can be detected with such methods.

A full description of curvature of space requires more than a single scalar, but one simple measure of curvature of space at a point within that space is the scalar curvature (or curvature scalar), a number representing the degree of curvature. With it, the volume of spheres surrounding the point can be related to that of spheres of identical radius but in flat space. The scalar curvature can be positive (analogous to the surface of a sphere) or negative (analogous to the surface of a saddle shape or to the inner edge of a doughnut). Again using the analogous curvature of a surface, if its scalar curvature is positive and identical throughout the surface, it is a sphere, and if the surface is instead a (non-spherical) ellipsoid, then the scalar curvature is positive throughout, but varies with a particular pattern.