Warped geometry

In mathematics and physics, in particular differential geometry and general relativity, a warped geometry is a Riemannian or Lorentzian manifold whose metric tensor can be written in form

${\displaystyle ds^{2}=g_{ab}(y)\,dy^{a}\,dy^{b}+f(y)g_{ij}(x)\,dx^{i}\,dx^{j}.}$

The geometry almost decomposes into a Cartesian product of the y geometry and the x geometry – except that the x part is warped, i.e. it is rescaled by a scalar function of the other coordinates y. For this reason, the metric of a warped geometry is often called a warped product metric.[1][2]

Warped geometries are useful in that separation of variables can be used when solving partial differential equations 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 d(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 are warped geometries, for example, the Schwarzschild solution and the Friedmann–Lemaitre–Robertson–Walker models.

Also, warped geometries are the key building block of Randall–Sundrum models in string theory.

See also

• Metric tensor
• Exact solutions in general relativity
• Poincaré half-plane model

References

1. Chen, Bang-Yen (2011). Pseudo-Riemannian geometry, [delta]-invariants and applications. World Scientific. ISBN 978-981-4329-63-7.
2. O'Neill, Barrett (1983). Semi-Riemannian geometry. Academic Press. ISBN 0-12-526740-1.

3. Chen, Bang-Yen (2017). Differential geometry of warped product manifolds and submanifolds. World Scientific. ISBN 978-981-3208-92-6.