The Reissner-Nordström Black Hole

Why the Reissner-Nordström black hole matters

You will almost certainly never observe a Reissner-Nordström (RN) black hole in the sky, and yet it is one of the most studied objects in theoretical physics. The reason is that it is the simplest black hole complicated enough to teach you almost everything that goes wrong — and right — with the interiors of realistic rotating black holes, but without the ferocious algebra of the Kerr solution. It is spherically symmetric, so the mathematics is one-dimensional in the radial coordinate, yet it already exhibits two horizons, a timelike (avoidable) singularity, a Cauchy horizon where predictability fails, an extremal limit of zero temperature, and the threat of a naked singularity.

• The no-hair theorem made concrete. RN is the electrically charged, non-spinning corner of the three-parameter (M, J, Q) black-hole family. It shows that even electromagnetic complexity collapses to a single number, Q.
• Cosmic censorship's test case. The overcharged Q > M solution is the cleanest example of a would-be naked singularity, and the question "can you push a black hole past extremality?" has driven decades of thought experiments.
• The inner horizon and predictability. RN's Cauchy horizon is where Einstein's equations lose the ability to predict the future, and mass inflation — first worked out for RN — is why that boundary is expected to become singular.
• String-theory entropy. Extremal charged black holes are the ones whose Bekenstein-Hawking entropy Strominger and Vafa reproduced from microscopic state counting in 1996, a landmark for quantum gravity.
• A clean thermodynamic laboratory. Because you can dial Q at fixed M, RN black holes have rich thermodynamics — including a temperature that peaks and then falls to zero at extremality, and even phase transitions in anti-de Sitter space.

How it works, step by step

Everything about the RN black hole follows from one function, the metric function f(r). Start from Schwarzschild and add the charge term:

1. Write the metric. In geometric units (G = c = 1, and absorbing the Coulomb constant into Q), the line element is ds² = −f(r) dt² + f(r)⁻¹ dr² + r²dΩ², with f(r) = 1 − 2M/r + Q²/r². The final +Q²/r² term is the electromagnetic stress-energy pushing outward — it is repulsive, unlike the attractive −2M/r mass term.
2. Find the horizons. Horizons live where f(r) = 0, i.e. where r² − 2Mr + Q² = 0. Solving the quadratic gives r± = M ± √(M² − Q²). If Q < M there are two real roots: the outer event horizon r₊ and the inner Cauchy horizon r₋.
3. Cross the outer horizon. Between r₋ and r₊ the sign of f flips, so r becomes a timelike coordinate. An infalling observer is dragged to smaller r inexorably — just like inside Schwarzschild. The outer horizon is a true point of no return.
4. Cross the inner horizon. Below r₋, f becomes positive again and r is spacelike once more. Now the observer is no longer forced onto the singularity — the r = 0 singularity is timelike and can, in the idealized geometry, be avoided or even circled around.
5. Meet the repulsive core. Very close to r = 0 the +Q²/r² term dominates over −2M/r, so gravity effectively becomes repulsive and can turn infalling geodesics back outward. In the maximally extended diagram this connects to new asymptotic regions — "other universes" — though this is a mathematical artifact, not a travel guide.
6. Approach extremality. As Q → M the two horizons squeeze together. At Q = M they coincide at r = M as a single degenerate horizon, and the surface gravity — hence the Hawking temperature — drops to zero.
7. Overcharge and lose the horizon. If you could set Q > M, the discriminant M² − Q² goes negative, the roots become complex, and there is no horizon anywhere. The r = 0 singularity is exposed to the outside — a naked singularity.

Restoring SI units, the charge term in the metric function is GQ²/(4πε₀r²c⁴), and the extremality condition Q = M becomes Q = √(4πε₀G)·M ≈ 8.6 × 10⁻¹¹ coulombs per kilogram of mass. That still corresponds to an utterly tiny charge imbalance — only about one excess proton for every 10¹⁸ nucleons — which is exactly why real black holes are never charged: any surrounding plasma erases such a minute imbalance almost instantly.

The key equation, with every symbol defined

The heart of the solution is the metric function and its roots:

f(r) = 1 − 2GM/(rc²) + GQ²/(4πε₀ r²c⁴)  →  horizons at   r± = M ± √(M² − Q²)  (geometric units)

Symbol	Meaning	Units (SI)
f(r)	Metric function; horizons are its zeros	dimensionless
r	Areal (Schwarzschild-type) radial coordinate	metres (m)
M	Gravitational mass of the black hole	kilograms (kg)
Q	Total electric charge	coulombs (C)
G	Newton's gravitational constant, 6.674 × 10⁻¹¹	m³ kg⁻¹ s⁻²
c	Speed of light, 2.998 × 10⁸	m s⁻¹
ε₀	Vacuum permittivity, 8.854 × 10⁻¹²	F m⁻¹
r₊ , r₋	Outer event horizon and inner Cauchy horizon	metres (m)

In geometric units one writes the charge in geometrized form (so M and Q have the same dimension of length), which is why the tidy expression r± = M ± √(M² − Q²) hides all the constants. The three regimes fall straight out of the discriminant M² − Q²:

Regime	Condition	Horizons	Character
Sub-extremal	Q < M	Two: r₊ > r₋ > 0	Ordinary charged black hole; timelike singularity hidden
Extremal	Q = M	One (degenerate) at r = M	Zero temperature; string-theory microstate benchmark
Overcharged	Q > M	None	Naked singularity — forbidden by cosmic censorship
Uncharged limit	Q = 0	One at r = 2M	Recovers Schwarzschild exactly

Compared to the other members of the black-hole family, RN sits at a well-defined address:

Solution	Spin J	Charge Q	Horizons
Schwarzschild (1916)	0	0	One
Reissner-Nordström (1916–18)	0	≠ 0	Two (or one if extremal)
Kerr (1963)	≠ 0	0	Two
Kerr-Newman (1965)	≠ 0	≠ 0	Two

A worked example and a little history

Hans Reissner, a German aeronautical engineer and physicist, published the charged point-mass solution in 1916, only months after Karl Schwarzschild's original vacuum result. Gunnar Nordström, a Finnish theorist better known for his own scalar theory of gravity, extended and completed the analysis in 1918; George Barker Jeffery arrived at similar results in 1921, which is why the metric occasionally carries extra names. For decades it was treated as a curiosity, until the 1960s golden age of relativity — when Roy Kerr's rotating solution, the singularity theorems of Penrose and Hawking, and the maximal Kruskal-type extensions of RN's own diagram made the interior structure a live subject.

Worked example — how much charge is "a lot"? Take a ten-solar-mass black hole, M ≈ 2.0 × 10³¹ kg. The extremal charge is Q ≈ (8.6 × 10⁻¹¹ C/kg) × M ≈ 1.7 × 10²¹ C. That sounds staggering — more than a billion trillion coulombs — but relative to the number of particles involved it is minuscule: it corresponds to an excess of only about one proton for every 10¹⁸ nucleons. Any surrounding ionized gas would supply the opposite charges and wipe out that tiny imbalance almost instantly. This is the quantitative reason astrophysicists set Q = 0 and use Kerr: the surviving charge-to-mass ratio is vanishingly small.

Where RN still bites. The 1990 discovery of mass inflation by Eric Poisson and Werner Israel — the runaway growth of the effective mass at the inner horizon under infalling radiation — was worked out first in the RN geometry precisely because it is spherically symmetric. It showed that the inner Cauchy horizon, and with it the tantalizing "other universes" of the maximal diagram, is generically destroyed by a curvature singularity. That result, refined by the strong cosmic censorship program, is one of RN's most enduring legacies.

Common misconceptions

• "Charged black holes are common." They are not — any net charge is neutralized by ambient plasma within milliseconds, so essentially every real black hole is uncharged Kerr.
• "The extra term makes gravity stronger." The opposite: +Q²/r² is repulsive. Charge shrinks the event horizon and, near the core, can reverse gravity entirely.
• "Two horizons means two points of no return." Only the outer horizon is a true event horizon. The inner one is a Cauchy horizon — a predictability boundary — and is unstable to perturbations.
• "You can travel to another universe through it." The idealized maximal diagram suggests this, but mass inflation turns the inner horizon into a real singularity, closing the tunnel.
• "An extremal black hole is the hottest." It is the coldest: extremal RN has exactly zero Hawking temperature and cannot evaporate in the usual thermal way.
• "You can overcharge a black hole to expose the singularity." Attempts fail — electrostatic repulsion prevents the extra like-charge from crossing the horizon, protecting cosmic censorship.