Curved space
Encyclopedia
Curved space often refers to a spatial geometry which is not “flat” where a flat space is described by Euclidean geometry
. Curved spaces can generally be described by Riemannian geometry
though some simple cases can be described in other ways. Curved spaces play an essential role in General Relativity
where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.
One of the defining characteristics of a curved space is its departure with the Pythagorean theorem
. In a curved space
.
The Pythagorean relationship can often be restored by describing the space with an extra dimension.
Suppose we have a non-euclidean three dimensional space with coordinates
. Because it is not flat
.
But if we now describe the three dimensional space with four dimensions (
) we can choose coordinates such that
.
Note that the coordinate
is not the same as the coordinate 
.
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
. Since four coordinates have four degrees of freedom it must have a constraint placed on it. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. That is
.
The constant can be positive or negative. For convenience we can choose the constant to be
where 
now is positive and 
.
We can now use this constraint to eliminate the artificial fourth coordinate
. The differential of the constraining equation is
leading to 
.
Plugging
into the original equation gives
.
This form is usually not particularly appealing and so a coordinate transform is often applied:
, 
, 
. With this coordinate transformation
.
. An isotropic and homogenous space can be described by the metric:
.
This reduces to Euclidean space
when
. 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
(
, not to be confused with the R of the previous section). That is 
. Calculation of the these components from the metric gives that
where 
.
This gives the metric:
.
where
can be zero, positive, or negative and is not limited to ±1.
.
In the limit that the constant of curvature (
) becomes infinitely large, a flat, Euclidean space
is returned. It is essentially the same as setting
to zero. If 
is not zero the space is not Euclidean. When 
the space is said to be closed or elliptic
. When
the space is said to be open or hyperbolic
.
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°. The volume, however, is not
.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. Curved spaces can generally be described by Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
though some simple cases can be described in other ways. Curved spaces play an essential role in General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.
Simple two-dimensional example
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. The surface of a sphere can be completely described by two dimensions.Embedding
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
. In a curved space
.The Pythagorean relationship can often be restored by describing the space with an extra dimension.
Suppose we have a non-euclidean three dimensional space with coordinates
. Because it is not flat.But if we now describe the three dimensional space with four dimensions (
) we can choose coordinates such that.Note that the coordinate
is not the same as the coordinate .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
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
. Since four coordinates have four degrees of freedom it must have a constraint placed on it. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. That is
.The constant can be positive or negative. For convenience we can choose the constant to be
where now is positive and .We can now use this constraint to eliminate the artificial fourth coordinate
. The differential of the constraining equation is leading to .Plugging
into the original equation gives.This form is usually not particularly appealing and so a coordinate transform is often applied:
, , . With this coordinate transformation.Without embedding
The geometry of a n-dimensional space can also be described with Riemannian geometryRiemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
. An isotropic and homogenous space can be described by the metric:
.This reduces to Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
when
. But a space can be said to be “flatConformally flat
A Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.More formally, let be a pseudo-Riemannian manifold...
” when the Weyl Tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
has all zero components. In three dimensions this condition is met when the Ricci Tensor
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
(
) is equal to the metric times the Ricci ScalarScalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
(
, not to be confused with the R of the previous section). That is . Calculation of the these components from the metric gives that where .This gives the metric:
.where
can be zero, positive, or negative and is not limited to ±1.Open, flat, closed
An isotropic and homogenous space can be described by the metric:.In the limit that the constant of curvature (
) becomes infinitely large, a flat, Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is returned. It is essentially the same as setting
to zero. If is not zero the space is not Euclidean. When the space is said to be closed or ellipticElliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
. When
the space is said to be open or hyperbolicHyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
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°. The volume, however, is not
.See also
- CAT(k) spaceCAT(k) spaceIn mathematics, a CAT space is a specific type of metric space. Intuitively, triangles in a CAT space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a CAT space, the curvature is bounded from above by k...
- Non-positive curvatureNon-positive curvatureIn mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero...