Charged Black Holes: The Reissner-Nordström Geometry

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Reissner-Nordström geometry

The Reissner-Nordström geometry describes the geometry of empty space surrounding a charged black hole.

If the charge of the black hole is less than its mass (measured in geometric units G = c = 1), then the geometry contains two horizons, an outer horizon and an inner horizon. Between the two horizons space is like a waterfall, falling faster than the speed of light, carrying everything with it. Upstream and downstream of the waterfall, space moves slower than the speed of light, and relative calm prevails.

Fundamental charged particles like electrons and quarks are not black holes: their charge is much greater than their mass, and they do not contain horizons.

If the geometry is continued all the way to the centre of the black hole, then there is a gravitationally repulsive, negative-mass singularity there. Uncharged persons who fall into the charged black hole are repelled by the singularity, and do not fall into it.

The diagram at left is an embedding diagram of the Reissner-Nordström geometry, a 2-dimensional representation of the 3-dimensional spatial geometry at an instant of Reissner-Nordström time. Between the horizons, radial lines at fixed Reissner-Nordström time are time-like rather than space-like, which is to say that they are possible wordlines of radially infalling (albeit not freely falling) observers. The animated dashes follow the positions of such infalling observers as a function of their own proper time.

Caveats
The Universe at large appears to be electrically neutral, or close to it. Thus real black holes are unlikely to be charged. If a black hole did somehow become charged, it would quickly neutralize itself by accreting charge of the opposite sign.
It is not clear how a gravitationally repulsive, negative-mass singularity could form. If it did, it is likely that the singularity would spontaneously destroy itself by popping charged particle-antiparticle pairs out of the vacuum inside the inner horizon. By swallowing particles of charge opposite to itself, the singularity would tend to neutralize both its charge and its negative mass, redistributing the charge over space inside the inner horizon.
In these pages I have somewhat arbitrarily replaced the Reissner-Nordström geometry near the singularity with flat space. Specifically, the inward rush of space into the black hole slows to a halt at the turnaround point r₀ inside the inner horizon (see the discussion in the section below on the Free-fall spacetime diagram), and I have replaced the space interior to r₀ with flat space. This is equivalent to concentrating all the charge of the black hole into a thin shell at the turnaround point r₀.

Reissner-Nordström metric
The Reissner-Nordström metric is

ds² = - B(r) dt² + B(r)^-1 dr² + r² do²
where the metric coefficient B(r) is

B(r) = 1 - 2 M
r
+ Q²
r²
.
This expression is in geometric units, where the speed of light and Newton's gravitational constant are both one, c = G = 1. In standard units, B(r) = 1 - 2 G M / (c² r) + G Q² / (c⁴ r²).
Horizons occur where the metric coefficient B(r) is zero, which happens at outer and inner horizons r₊ and r_-:

r₊ = M + (M² - Q²)^1/2 ,
r_- = M - (M² - Q²)^1/2 .

In terms of the horizon locations r_±, the metric coefficient B(r) is

B(r) = (r - r_-)(r - r₊)
r²
.

Reissner-Nordström spacetime diagram

This is a spacetime diagram of the Reissner-Nordström geometry. The horizontal axis represents radial distance, while the vertical axis represents time.
The two vertical red lines are the inner and outer horizons, at radial positions r_- and r₊. Yellow and ochre lines are the worldlines of light rays moving radially inward and outward respectively. Each point at radius r in the spacetime diagram represents a 3-dimensional spatial sphere of circumference 2 p r, as measured by observers at rest in the Reissner-Nordström geometry. The dark purple lines are lines of constant Reissner-Nordström time, while the vertical dark blue lines are lines of constant circumferential radius r. The bright blue line marks zero radius, r = 0.
Like the Schwarzschild geometry, the Reissner-Nordström geometry appears ill-behaved at its horizons, with light rays appearing to asymptote to the horizons without passing through. Again, the pathology is an artefact of the static coordinate system. Infalling light rays do in fact pass through the horizons, and there is no singularity at either horizon. The components of the Riemann curvature tensor remain finite at both horizons.
As in the Schwarzschild geometry, there are coordinate systems which behave better at the horizons, and which reveal more clearly the physics of the Reissner-Nordström geometry. Some of these coordinate systems are illustrated below.

Free-fall spacetime diagram for the Reissner-Nordström geometry

The picture of a black hole as a region into which space is flowing inward at the Newtonian escape velocity

v = [2 G M(r) / r]^1/2 = (1 - B)^1/2
works also for the Reissner-Nordström geometry. The mass M(r) at radial position r is the effective mass interior to r, which is the total mass M at infinity, less the mass Q² / (2 r) contained in the electromagnetic field outside r:

M(r) = M - Q²
2 r
.
The electromagnetic mass Q² / (2 r) is the mass outside r associated with the energy density E² / (8 p) of the electric field E = Q / r² surrounding a charge Q.
The infall velocity v of space passes the speed of light c at the outer horizon r₊, but slows back down to less than the speed of light at the inner horizon r_-. The velocity slows all the way to zero at the turnaround point r₀ inside the inner horizon

r₀ = Q²
2 M
.

The free-fall metric for the Reissner-Nordström geometry takes the same form as for Schwarzschild

ds² = - dt_ff² + (dr + v dt_ff)² + r² do²

with the free-fall velocity v given above. The free-fall time coordinate t_ff is the proper time experienced by persons who free-fall radially inward, at velocity dr/dt_ff = - v, from zero velocity at infinity:

t_ff = t + (2 M)^1/2 æ
ç
è 2 x^1/2 - r₊ x₊^1/2
r₊ - r_-
ln ô
ô
ô x^1/2 + x₊^1/2
x^1/2 - x₊^1/2
ô
ô
ô + r_- x_-^1/2
r₊ - r_-
ln ô
ô
ô x^1/2 + x_-^1/2
x^1/2 - x_-^1/2
ô
ô
ô ö
÷
ø .
where the coordinate x is the radial position relative to the turnaround point r₀

x = r - r₀
and x_± = r_± - r₀ are the values of x at the horizons r_±.
The free-fall metric shows that the spatial geometry is flat, having spatial metric dr² + r² do², on hypersurfaces of fixed free-fall time, dt_ff = 0.
The colouring of lines in the free-fall spacetime diagram is as in the Reissner-Nordström spacetime diagram, with the addition of green lines which are worldlines of observers who free fall radially from zero velocity at infinity, and horizontal dark green lines which are lines of constant free-fall time t_ff.
Watch Reissner-Nordström morph into free-fall (41K GIF); or same morph, double-size on screen (same 41K GIF).

Finkelstein spacetime diagram of the Reissner-Nordström geometry

As usual, the Finkelstein radial coordinate r is the circumferential radius, defined so that the proper circumference of a sphere at radius r is 2 p r, while the Finkelstein time coordinate is defined so that radially infalling light rays (yellow lines) move at 45^o in the spacetime diagram.
Finkelstein time t_F is related to Reissner-Nordström time t by

t_F = t + 1
2 g₊
ln ô
ô
ô r - r₊
r₀ - r₊
ô
ô
ô + 1
2 g_-
ln ô
ô
ô r - r_-
r₀ - r_-
ô
ô
ô ,
where g_± = g(r_±) are the surface gravities at the two horizons

g_± = ± r₊ - r_-
2 r_±²
.
The gravity g(r) at radial position r is the inward acceleration

g(r) = dv
dt_ff
= - v dv
dr
= 1
2
dB
dr
.

The colouring of lines is as in the Schwarzschild case: the red line is the horizon, the cyan line at zero radius is the singularity, yellow and ochre lines are respectively the wordlines of radially infalling and outgoing light rays, while dark purple and blue lines are respectively lines of constant Schwarzschild time and constant circumferential radius.
Watch Reissner-Nordström morph into Finkelstein (37K GIF); or same morph, double-size on screen (same 37K GIF).
Watch Finkelstein morph into free-fall (38K GIF); or same morph, double-size on screen (same 38K GIF).

Penrose diagram of the Reissner-Nordström geometry

Coordinates of Penrose diagram constructed so that the metric is well-behaved across both outer and inner horizons. Given this restriction, it's impossible to make the zero radius part vertical.
Watch Finkelstein morph into Penrose (51K GIF); or same morph, double-size on screen (same 51K GIF).

Penrose diagram of the complete Reissner-Nordström geometry

Suppose that you fall into a charged black hole. At the moment that you cross the inner horizon, you see an infinitely blueshifted point of light appear directly ahead, in the direction of the black hole. This infinitely blueshifted point of light is a record of the entire past history of the Universe, condensed into an instant.
Inside the inner horizon, the gravitational repulsion of the central singularity slows you down and turns you around, accelerating you back out through the inner horizon of a white hole. As you approach the inner horizon of the white hole, this time looking outward directly away from the black hole, part of the image of the outside Universe seems to break away from the rest. As you pass through the inner horizon this breakaway image concentrates into another infinitely blueshifted point of light, which disappears in a blazing flash. This time the infinitely blueshifted point of light contains the entire future of the Universe, condensed into an instant.
The white hole spews you out into a new Universe. Since light cannot fall into the white hole from the new Universe, you do not see the new Universe until you pass through the outer horizon of the white hole. At the instant you pass through the outer horizon, you witness once again an infinitely blueshifted point of light appear directly ahead, away from the white hole. The infinitely blueshifted point of light contains the entire past of the new Universe concentrated into an instant. The point of light opens up to reveal the new Universe, which you join.
Looking back into the white hole, you can see the Universe from which you came, but to which you cannot return.

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