Back to Collapse to a Black Hole
Forward to The Extremal ReissnerNordström Geometry
Other Relativity and Black Hole links
index  movies  approach  orbit  singularity  dive  Schwarzschild  wormhole  collapse  ReissnerNordström  extremal RN  Hawking  quiz  home  links 
ReissnerNordström geometry
The ReissnerNordström 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, negativemass 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 ReissnerNordström geometry, a 2dimensional representation of the 3dimensional spatial geometry at an instant of ReissnerNordström time. Between the horizons, radial lines at fixed ReissnerNordström time are timelike rather than spacelike, 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, negativemass singularity could form. If it did, it is likely that the singularity would spontaneously destroy itself by popping charged particleantiparticle 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 ReissnerNordström 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 r_{0} inside the inner horizon (see the discussion in the section below on the Freefall spacetime diagram), and I have replaced the space interior to r_{0} with flat space. This is equivalent to concentrating all the charge of the black hole into a thin shell at the turnaround point r_{0}. 
ReissnerNordström metric
The ReissnerNordström metric is
Horizons occur where the metric coefficient B(r) is zero,
which happens at outer and inner horizons
r_{+}
and
r_{}:

ReissnerNordström spacetime diagram
This is a spacetime diagram of the ReissnerNordström 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 3dimensional spatial sphere of circumference 2 p r, as measured by observers at rest in the ReissnerNordström geometry. The dark purple lines are lines of constant ReissnerNordström 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 ReissnerNordström geometry appears illbehaved 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 ReissnerNordström geometry. Some of these coordinate systems are illustrated below. 
Freefall spacetime diagram for the ReissnerNordström geometry
The picture of a black hole as a region into which
space is flowing inward at the Newtonian escape velocity
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
r_{0}
inside the inner horizon
The freefall metric for the ReissnerNordström geometry takes the
same form as for Schwarzschild
The freefall metric shows that the spatial geometry is flat, having spatial metric dr^{2} + r^{2} do^{2}, on hypersurfaces of fixed freefall time, dt_{ff} = 0. The colouring of lines in the freefall spacetime diagram is as in the ReissnerNordström 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 freefall time t_{ff}. Watch ReissnerNordström morph into freefall (41K GIF); or same morph, doublesize on screen (same 41K GIF). 
Finkelstein spacetime diagram of the ReissnerNordström 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 ReissnerNordström time
t by
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 ReissnerNordström morph into Finkelstein (37K GIF); or same morph, doublesize on screen (same 37K GIF). Watch Finkelstein morph into freefall (38K GIF); or same morph, doublesize on screen (same 38K GIF). 
Penrose diagram of the ReissnerNordström geometry
Coordinates of Penrose diagram constructed so that the metric is wellbehaved 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, doublesize on screen (same 51K GIF). 
Penrose diagram of the complete ReissnerNordström 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. 
Back to Collapse to a Black Hole
Forward to The Extremal ReissnerNordström Geometry
Other Relativity and Black Hole links
index  movies  approach  orbit  singularity  dive  Schwarzschild  wormhole  collapse  ReissnerNordström  extremal RN  Hawking  quiz  home  links 
Updated 19 Apr 2001