Warped product
of two
Riemannian (or
pseudo-Riemannian
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
) manifolds
and
with respect to a function
is the product space
with the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
.
Warped geometries are useful in that
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary differential equation, ordinary and partial differential equations, in which algebra allows one to rewrite an equation so tha ...
can be used when solving
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s over them.
Examples
Warped geometries acquire their full meaning when we substitute the variable ''y'' for ''t'', time and ''x'', for ''s'', space. Then the ''f''(''y'') factor of the spatial dimension becomes the effect of time that in words of Einstein "curves space". How it curves space will define one or other solution to a space-time world. For that reason, different models of space-time use warped geometries.
Many basic solutions of the
Einstein field equations
In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
are warped geometries, for example, the
Schwarzschild solution
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
and the
Friedmann–Lemaitre–Robertson–Walker models.
Also, warped geometries are the key building block of
Randall–Sundrum model
In physics, Randall–Sundrum models (RS) (also called 5-dimensional warped geometry theory) are models that describe the world in terms of a warped-geometry higher-dimensional universe, or more concretely as a 5-dimensional anti-de Sitter space ...
s in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
.
See also
*
Metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
*
Exact solutions in general relativity
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be a ...
*
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with co ...
References
Differential geometry
General relativity
String theory
{{Relativity-stub