nLab Schwarzschild wormhole

Idea

A Schwarzschild wormhole is an Einstein-Rosen bridge constructed from the Schwarzschild spacetime of a black hole.

Formulation

Let $M$ be the mass, then the Schwarzschild spacetime is given by:

$$\mathrm{d}^2s =\left( 1 -\frac{2M}{r} -\frac{\Lambda}{3}r^2 \right)\mathrm{d}^2t +\left( 1 -\frac{2M}{r} -\frac{\Lambda}{3}r^2 \right)^{-1}\mathrm{d}^2r +r^2\mathrm{d}\Omega.$$

The event horizon lies at the Schwarzschild radius $r_S=2M$. The idea of an Einstein-Rosen bridge is now to transform the parameter $r\in\mathbb{R}^+$ into a parameter $u\in\mathbb{R}$ with:

$$u^2 =r-r_S.$$

Since there is no $u$ for $r\lt r_S$, one $u$ for $r=r_S$ and two $u$ for $r\gt r_S$, the transformation topologically describes glueing the exterior part of the Schwarzschild spacetime to itself along the event horizon. One has the transformation for the differential and the radial component:

$$2u\mathrm{d}u =\mathrm{d}r;$$

$$1-\frac{2M}{r} =\frac{r-2M}{r} =\frac{u^2}{u^2+2M}$$

This results in the wormhole metric:

$$\mathrm{d}s^2 =-\frac{u^2}{u^2+2M}c^2\mathrm{d} t^2 +4(u^2+2M)\mathrm{d}u^2 +(u^2+2M)^2\mathrm{d}\Omega^2.$$

Related concepts

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

black hole spacetimes with dark energy	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild-de Sitter spacetime	Kerr-de Sitter spacetime
positive charge	Reissner-Nordström-de Sitter spacetime	Kerr-Newman-de Sitter spacetime

wormhole spacetimes	vanishing angular momentum
vanishing charge	Schwarzschild wormhole
positive charge	Reissner-Nordström wormhole