Curved space

Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry.

[2] The Friedmann–Lemaître–Robertson–Walker metric is a curved metric which forms the current foundation for the description of the expansion of the universe and the shape of the universe.

[citation needed] The fact that photons have no mass yet are distorted by gravity, means that the explanation would have to be something besides photonic mass.

Hence, the belief that large bodies curve space and so light, traveling on the curved space will, appear as being subject to gravity.

A very familiar example of a curved space is the surface of a sphere.

While to our familiar outlook the sphere looks three-dimensional, if an object is constrained to lie on the surface, it only has two dimensions that it can move in.

Even the surface of the Earth, which is fractal in complexity, is still only a two-dimensional boundary along the outside of a volume.

[3] One of the defining characteristics of a curved space is its departure from the Pythagorean theorem.

Suppose we have a three-dimensional non-Euclidean space with coordinates

Because it is not flat But if we now describe the three-dimensional space with four dimensions (

For the choice of the 4D coordinates to be valid descriptors of the original 3D space it must have the same number of degrees of freedom.

We can choose a constraint such that Pythagorean theorem holds in the new 4D space.

For convenience we can choose the constant to be We can now use this constraint to eliminate the artificial fourth coordinate

into the original equation gives This form is usually not particularly appealing and so a coordinate transform is often applied:

But a space can be said to be "flat" when the Weyl tensor has all zero components.

In three dimensions this condition is met when the Ricci tensor (

) is equal to the metric times the Ricci scalar (

An isotropic and homogeneous space can be described by the metric:[citation needed] In the limit that the constant of curvature (

) becomes infinitely large, a flat, Euclidean space is returned.

Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°.

Triangles which lie on the surface of a closed space will have a sum of angles which is greater than 180°.