nLab Reissner-Nordström wormhole

Idea

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

Formulation

Let $M$ be the mass and $Q$ be the charge, then the Reissner-Nordström spacetime is given by:

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

The event horizons lie at the radii $r_\pm=M\pm\sqrt{M^2-Q^2}$. 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:

$$u^2 =r-r_+.$$

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

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

$$\mathrm{d}s^2 =-\frac{u^2}{(u^2+r_+)^2}\left(u^2+2\sqrt{M^2-Q^2}\right)c^2\mathrm{d}t^2+4(u^2+r_+)^2\left(u^2+2\sqrt{M^2-Q^2}\right)^{-1}\mathrm{d}u^2 +(u^2+r_+)\mathrm{d}\Omega^2$$

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