Reissner–Nordström and charged horizons

Define mass length m as before and a nonnegative charge length q through Q squared, including the SI vacuum permittivity. The metric retains the spherical Schwarzschild form but gains a positive inverse square radius term in f. Reversing the sign of the electric charge reverses the electric field without changing this metric, because the electromagnetic stress depends quadratically on the field. A charged test particle additionally feels a Lorentz force; a neutral geodesic is a different trajectory problem.

The classical Maxwell stress is trace-free in four dimensions. The traced Einstein equation with zero Lambda therefore gives zero Ricci scalar outside the source. Yet its Ricci tensor does not vanish when the electromagnetic field is nonzero. This is a concrete warning against using a single scalar as a complete vacuum or flatness test. The exact static exterior also idealizes away electromagnetic radiation, external plasma, discharge processes, and any material interior required for a different spacetime completion.

Multiplying f by radius squared turns its roots into an elementary quadratic. When q lies strictly between zero and m, there are two distinct positive roots, an outer event horizon in the standard black-hole completion and an inner Cauchy horizon in the ideal extended geometry. At q equal to m they coincide as a double root. When q exceeds m, the ideal metric has no real horizon root. This algebra classifies an exact family; it does not establish which members can form from a specified set of regular initial data.

The outer static region lies beyond the larger root. At distinct roots, regular crossing coordinates remove the static chart singularities. The classical inner horizon has a different causal role: continuation beyond it is not fixed by the exterior Cauchy data in the same way. Perturbations can be strongly amplified there, so the perfectly analytic interior is not a robust prediction for a realistic collapsing system. Such instability statements must not be promoted into an unrestricted proof of every version of cosmic censorship.

For the asymptotically normalized static time, half the radial derivative of f at the outer horizon defines a geometric surface gravity with inverse length units. Multiplication by c squared converts it to acceleration units. Its expression through the root separation makes the extremal limit transparent. The vanishing separation makes this quantity zero, even though the horizon area and mass stay nonzero. Zero surface gravity therefore does not mean absence of a horizon or absence of curvature.

As a nonextremal outer horizon is approached, static proper acceleration diverges while the horizon-normalized quantity stays finite; the redshift factor connecting them is essential. At an extremal charged horizon, the exterior static-acceleration limit is instead the finite value $c^2/m$, while $\kappa_{\mathrm{geo}}=0$. No Hawking-temperature formula is needed for the classical calculation here. If quantum emission is discussed later, it requires additional semiclassical assumptions. The square-root lab can reproduce selected discriminants within its positive control range, but it does not evaluate the Maxwell stress, locate a Cauchy development, or determine whether a horizon is physically formed.