Warped product manifolds

Question

Why the warped product manifolds plays an important roles in General Relativity? Are important 4d-warped product only, or every dimensions?

Answer 1

Warped products can be used to model certain spacetimes. For example, the Schwarzschild half-plane with mass $m>0$ is the set $$P_I = \{ (t,r) \in \Bbb R^2 \mid r>2m\}$$equipped with the Lorentz metric $$-h(r) \,{\rm d}t^2 + h(r)^{-1}\,{\rm d}r^2,$$where $h(r) = 1 - 2m/r$ is the so-called Schwarzschild horizon. We have the coordinate function $r: P_I \to \Bbb R$, which allows us to consider the warped product $P_I \times_r \Bbb S^2$: the Schwarzschild model. Note that as $r \to +\infty$, the metric $$-h(r)\,{\rm d}t^2 + h(r)^{-1}\,{\rm d}r^2 + r^2 {\rm d}\Omega^2$$approaches the flat Minkowski metric $-{\rm d}t^2 + {\rm d}r^2 + r^2\,{\rm d}\Omega^2$, where ${\rm d}\Omega^2$ is the metric in $\Bbb S^2$.

The FLRW model, hyperbolic space, and surfaces of revolution can also be represented as warped products.

You can consider warped products of manifolds of arbitrary dimensions from a purely geometrical viewpoint, and apply a more general theory in all of these particular cases.