Reissner-Nordstrom spacetime ($r_{\pm }=M\pm \sqrt{M^2-Q^2}$)

The four-dimensional charged spherically symmetric electrovacuum solution has radial function $f=1-2M/r+Q^2/r^2$. For $0<|Q|<M$, its outer and inner horizon radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. A complete two-ended spatial bridge has a maximal Cauchy development ending at smoothly extendible inner Cauchy horizons in the exact solution.

For $0<|Q|<M$, the charged-particle radial equation in Reissner-Nordstrom spacetime implies that any radial turning point obeysSince $1+l^2/r^2>0$, it follows that $F\ge 0$. Thus no massive test particle, regardless of its electric charge, can have a radial turning point in $r_{-}<r<r_{+}$, where $F<0$. At a regular horizon crossing, a timelike vector cannot lie in the tangent space of a null hypersurface, so a massive particle has $\dot{r}\ne 0$ there as well. The algebraic inequalities $r\ge r_{+}$ or $r\le r_{-}$ are necessary conditions, not an existence theorem for radial turning points.

For a massive test particle with electric charge $e$, choose the electromagnetic four-potential $A_t=-Q/r$ and proper time $\tau$. The cyclic coordinates $t,\varphi$ in the equatorial Lagrangian give the conserved quantities $E=mF\dot{t}+eQ/r$ and $l=r^2\dot{\varphi }$. Here $l$ is angular momentum per unit mass. Substituting these quantities into the four-velocity normalization $g(U,U)=-1$ givesThe energy includes the gauge-dependent electrostatic term; the displayed convention fixes that gauge by taking the potential to vanish at infinity.

For the Reissner-Nordstrom metric with $F=1-2M/r+Q^2/r^2$, the Kretschmann scalar isThe first expression follows by squaring the independent Riemann curvature tensor components in an orthonormal frame. This Kretschmann scalar diverges at $r=0$ for any nonzero $M$ or $Q$, but is finite at every positive root of $F$. Regular Eddington-Finkelstein coordinates show that those positive roots are coordinate singularities of the static chart. In the extremal case $Q^2=M^2>0$, its horizon value is $8/M^4$.

For $f=(r-r_{+})(r-r_{-})/r^2$, near-horizon Euclidean geometry has polar form $d{\rho }^2+[f^{\prime }(r_{+})/2{]}^2{\rho }^{2}d{\tau }^2$. Smoothness gives inverse temperature $4\pi /f^{\prime }(r_{+})$ and the displayed formula. This assumes a nondegenerate Killing horizon and time normalized at infinity. Its nonextremal limit is zero at $M=|Q|$.

In the ideal thermal-cutoff model where charged particles of mass $m$ are emitted only for $T>m$, a sufficiently large initial mass keeps charge fixed throughout evaporation exactly when $m$ is at least the fixed-charge Reissner-Nordstrom temperature maximum. Neutral radiation then approaches an extremal charged remnant of area $4\pi Q^2$. This conclusion does not include nonthermal electric-field discharge, nor establish finite-time attainment.

Set $y=r_{+}/|Q|\ge 1$. Then $M/|Q|=(y+y^{-1})/2$ and $|Q|T=(y^{-1}-y^{-3})/(4\pi )$. Differentiation gives a unique maximum at $y=\sqrt{3}$, proving the displayed mass and temperature. The curve is zero at extremality and asymptotic to $1/(8\pi M)$ at large mass.

A neutral radial timelike geodesic of an extremal black hole has conserved specific energy $E=(1-M/r{)}^2\dot{t}$, giving the displayed radial equation. An infalling geodesic reaches $r=M$ with finite nonzero $\dot{r}=-E$ and finite proper time. In ingoing coordinates, outgoing radial null curves have $dr/dv=(1-M/r{)}^2/2$; those starting inside approach $r=M$ only at infinite $v$ and cannot return to the same exterior future null infinity. This distinguishes causal access from length on a static slice, and supplies an operational event horizon test even without strictly trapped round spheres.

For $M=|Q|>0$ and the exterior areal radius $r>M$, integrate $d\rho /\rho =dr/(r-M)$ and choose $\rho =r-M$. The Reissner-Nordstrom metric becomes the displayed form. Constant-time proper distance to $\rho =0$ diverges, but these exterior coordinates do not cover the horizon; ingoing null coordinates are regular there. Infinite spatial throat length therefore does not establish absence of an event horizon.

For $M>0$ and $M\ge |Q|$, put $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The stationary Killing vector field normalized at infinity generates the outer Killing horizon. In an ingoing null chart its covector is $dr$ on that horizon; the Killing equation then gives ${\nabla }_{k}k=\frac{1}{2}{f}^{\prime }(r_{+})k$, where $f=(r-r_{+})(r-r_{-})/r^2$. Thus the positive nonextremal surface gravity is $(r_{+}-r_{-})/(2r_{+}^2)$. It tends to zero when $M=|Q|$, where the horizon becomes a degenerate Killing horizon. The past horizon branch has the opposite signed nonaffinity for the same time-translation field.

For $0<|Q|<M$, a Reissner-Nordstrom spacetime has an outer event horizon and an inner Cauchy horizon. Between them $r$ is timelike; inside the inner horizon $r$ is spacelike and the $r=0$ curvature singularity is timelike. The exact analytic extension contains repeated exterior blocks beyond the Cauchy horizons. Small perturbations produce severe inner-horizon blueshift and mass inflation, so that exact continuation is not a robust prediction for collapse. At $|Q|=M$ the horizons merge into a degenerate Killing horizon; for $|Q|>M$ the spherical electrovacuum solution instead has a naked singularity.

In ingoing Reissner-Nordstrom spacetime, orient $-{\partial }_r$ to the future and normalize the other radial null vector to ${\partial }_v+(f/2){\partial }_r$. The spherical null expansion in ingoing coordinates gives $-2/r$ and $f/r$. Hence the strictly future trapped surfaces among the round spheres occur exactly between the distinct horizons. At either horizon the outgoing null expansion vanishes. Inside the inner horizon the same round spheres are not future trapped.