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Schwarzschild geometry
The Schwarzschild geometry describes the spacetime geometry of empty space surrounding any spherical mass. Karl Schwarzschild derived this geometry at the close of 1915, within a few weeks of Albert Einstein publishing his fundamental paper on the Theory of General Relativity. The history of this discovery and much more is wonderfully recounted in Kip Thorne's book ``Black Holes & Time Warps: Einstein's Outrageous Legacy''. A description of this embedding diagram appears below. Try John Walker's Orbit's in Strongly Curved Spacetime for a Java applet which allows you to play around with orbits in the Schwarzschild geometry. 
Schwarzschild radius
One of the remarkable predictions of Schwarzschild's geometry
was that if a mass M were compressed inside a critical radius
r_{s},
nowadays called the Schwarzschild radius,
then its gravity would become so strong that not even light could escape.
The Schwarzschild radius r_{s} of a mass M
is given by
Curiously, the Schwarzschild radius had already been derived (with the correct result, but an incorrect theory) by John Michell in 1783 in the context of Newtonian gravity and the corpuscular theory of light. Michel derived the critical radius by setting the gravitational escape velocity v equal to the speed of light c in the Newtonian formula v^{2} / 2 = G M / r for the escape velocity v from the surface of a star of mass M and radius r. 
Horizon
The Schwarzschild surface, the sphere at 1 Schwarzschild radius, is also called the horizon of a black hole, since an outside observer, even one just outside the Schwarzschild surface, can see nothing beyond the horizon. 
Schwarzschild metric
Schwarzschild's geometry is described by the metric
(in units where the speed of light is one, c = 1)

Embedding diagram
The Schwarzschild geometry is illustrated in the embedding diagram at the top of the page, which shows a 2dimensional representation of the 3dimensional spatial geometry at a particular instant of universal time t. One should imagine that objects are confined to move only on the 2dimensional surface. Each circle actually represents a sphere, of circumference 2 pi r. According to the Schwarzschild metric, the proper radial distance, the actual distance measured by an observer at rest at radius r, between two spheres separated by an interval dr of circumferential radius r is (1  r_{s}/r)^{1/2} dr, which is larger than the radial interval dr expected in a flat, Euclidean geometry. Thus the geometry is `stretched' in the radial direction, as shown in the embedding diagram. Outside the horizon, the lines in the embedding diagram are `spacelike': they would be measured by some actual observer (in this case an observer at rest in the Schwarzschild geometry) as being intervals of space at some instant of the observer's proper time (an observer's proper time is the time actually measured by the observer, as experienced by the observer's brain or recorded by a watch on the observer's wrist). Inside the horizon, lines in the Schwarzschild embedding diagram change to being `timelike': they represent intervals of time measured at the position of some observer, rather than intervals of space at an instant of some observer's time. That is to say, the lines in the embedding diagram inside the horizon represent possible trajectories of infalling (though not necessarily freely falling) observers. The shape of the embedding diagram inside the horizon, as drawn at the top of the page, is somewhat arbitrary. The animated dashes do however show correctly intervals of proper time as experienced by an observer infalling along a line of constant Schwarzschild time t. 
Gravitational slowing of time
In general relativity, clocks at rest run slower inside a gravitational potential than outside.
In the case of the Schwarzschild metric,
the proper time,
the actual time measured by an observer at rest at radius r,
during an interval dt of universal time is
(1  r_{s}/r)^{1/2} dt,
which is less than the universal time interval dt.
Thus a distant observer at rest will observe the clock of an observer at rest
at radius r to run more slowly than the distant observer's own clock,
by a factor

Gravitational redshift
The gravitational slowing of time produces a gravitational redshift of photons. That is, an outside observer will observe photons emitted from within a gravitational potential to be redshifted to lower frequencies, or equivalently to longer wavelengths. Conversely, an observer at rest in a gravitational potential will observe photons from outside to be blueshifted to higher frequencies, shorter wavelengths. In the case of the Schwarzschild metric, a distant observer at rest will observe photons emitted by a source at rest at radius r to be redshifted so that the observed wavelength is larger by a factor ( 1  r_{s} / r )^{1/2} than the emitted wavelength. The redshift factor tends to infinity as r approaches the Schwarzschild radius r_{s}, which means that someone at the Schwarzschild radius will appear infinitely redshifted, as seen by anyone outside the Schwarzschild radius.That the redshift factor is the same as the time dilation factor (well, so one's the reciprocal of the other, but that's just because the redshift factor is, conventionally, a ratio of wavelengths rather than a ratio of frequencies) is no coincidence. Photons are a good clocks. When a photon is redshifted, its frequency, the rate at which it ticks, slows down. In the illustration shown, a source at rest at 1.18 Schwarzschild radii emits light rays with the same initial wavelength in 6 equally spaced directions. The light ray going out is redshifted, while the rays falling in become blueshifted, from the point of view of observers at rest in the Schwarzschild geometry. Five of the 6 rays end up falling into the black hole (the two yellow rays would fall in, but I cut them off so they wouldn't block the view). 
No stationary frames inside the Schwarzschild radius
According to the Schwarzschild metric, at the Schwarzschild radius r_{s}, proper radial distance intervals become infinite, and proper time passes infinitely slowly. Inside the Schwarzschild radius, proper radial distances and proper times appear to become imaginary (that is, the square root of a negative number). Historically, it took decades before this strange behaviour was understood properly (see again Kip Thorne's book ``Black Holes and Time Warps'' for an account). The problem with the Schwarzschild metric is that it describes the geometry as measured by observers at rest. It is now realized that once inside the Schwarzschild radius, there can be no observers at rest: everything plunges inevitably to the central singularity. In effect, the very fabric of spacetime falls to the singularity, carrying everything with it. No pressure can withstand the inexorable collapse. To paraphrase Misner, Thorne & Wheeler (1973, ``Gravitation'', p. 823), that same unseen power of the world which impels everyone from age 20 to 40, and from 40 to 80, impels objects inside the horizon irresistably towards the singularity. Answer to the quiz question 8: False. The Schwarzschild metric remains valid inside the Schwarzschild radius. It is fine to perform mathematical calculations using the Schwarzschild metric. Inside the Schwarzschild radius, if you transform to frames of reference which fall inward (or outward, for a white hole!) faster than the speed of light, then the geometry becomes `normal' again. 
Schwarzschild spacetime diagram
This spacetime diagram illustrates the temporal geometry of the Schwarzschild metric, at the expense of suppressing information about the spatial geometry. By comparison, the embedding diagram at the top of the page illustrated the spatial geometry, while suppressing information about the temporal geometry. The horizontal axis represents radial distance, while the vertical axis represents time. The cyan vertical line is the central singularity, at zero radius, while the red vertical line is the horizon, at one Schwarzschild radius. 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 pi r. Dark purple and blue lines are respectively lines of constant Schwarzschild time and constant circumferential radius. The Schwarzschild spacetime geometry appears illbehaved at the horizon, the Schwarzschild radius (vertical red line). However, the pathology is an artefact of the Schwarzschild coordinate system. Spacetime itself is wellbehaved at the Schwarzschild radius, as can be ascertained by computing the components of the Riemann curvature tensor, all of whose components remain finite at the Schwarzschild radius. The curious change in the character of the Schwarzschild geometry inside versus outside the horizon can be seen in the spacetime diagram. Whereas outside the horizon infalling and outgoing light rays move generally upward, in the direction of increasing Schwarzschild time, inside the horizon infalling and outgoing light rays move generally leftward, toward the singularity. General Relativity permits an arbitrary relabelling of coordinates. Some coordinate systems which behave better at the Schwarzschild radius are illustrated below. 
Freefall spacetime diagram
Freefall coordinates reveal that the Schwarzschild geometry
looks like ordinary flat space,
with the distinctive feature
that space itself is flowing radially inwards at the Newtonian escape velocity
Picture space as flowing like a river into the black hole. Imagine light rays, photons, as canoes paddling fiercely in the current. Outside the horizon, photoncanoes paddling upstream can make way against the flow. But inside the horizon, the space river is flowing inward so fast that it beats all canoes, carrying them inevitably towards their ultimate fate, the central singularity. Does the notion that space inside the horizon of a black hole falls faster than the speed of light violate Einstein's law that nothing can move faster than light? No. Einstein's law applies to the velocity of objects moving in spacetime as measured with respect to locally inertial frames. Here it is space itself that is moving.
The freefall metric expresses mathematically the above physical assertions.
The freefall metric is
The colouring of lines in the freefall spacetime diagram is as in the Schwarzschild case, 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 Schwarzschild morph into freefall (41K GIF); or same morph, doublesize on screen (same 41K GIF). 
EddingtonFinkelstein spacetime diagram
EddingtonFinkelstein coordinates differ from Schwarzschild coordinates only in the relabelling of the time. The relabelling is arranged so that radially infalling light rays (yellow lines) move at 45^{o} in the spacetime diagram. Finkelstein time t_{F} is related to Schwarzschild time t by t_{F} = t + lnr  1 in units where the speed of light and the Schwarzschild radius are one, c = 1 and r_{s} = 1.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 Schwarzschild morph into Finkelstein (28K GIF); or same morph, doublesize on screen (same 28K GIF). Watch Finkelstein morph into freefall (38K GIF); or same morph, doublesize on screen (same 38K GIF). 
KruskalSzekeres spacetime diagram
KruskalSzekeres coordinates show transparently the causal structure of the Schwarzschild geometry. By construction, radially infalling (yellow) or outgoing (ochre) light rays move at 45^{o} leftward or rightward in the KruskalSzekeres spacetime diagram. Watch Finkelstein morph into Kruskal (50K GIF); or same morph, doublesize on screen (same 50K GIF). In addition to the normal horizon (pinkred line from centre to top right), through which light rays (yellow lines) and people can fall, there appears in the Kruskal diagram to be a second horizon, a `past' horizon or antihorizon (red line from bottom right to top left). In the Schwarzschild or Finkelstein coordinate systems, this antihorizon existed only in the infinite past. As it happens, lines of constant Schwarzschild time (dark purple) correspond to straight lines passing through the origin (where the horizon and the antihorizon cross) in the KruskalSzekeres coordinate system. How does the Kruskal diagram relate to what happened in the Falling into a Black Hole movie? The red grid on the surface of the black hole in the movie corresponds to the red antihorizon in the Kruskal diagram. When we fell through the horizon in the movie, it appeared that the Schwarzschild surface split into two, and we found ourselves inside the Schwarzschild bubble. The upper Schwarzschild surface of the bubble, coloured white in the movie, is the normal pinkred horizon in the Kruskal diagram. The lower Schwarzschild surface of the bubble, coloured red in the movie, is the red antihorizon in the Kruskal diagram. The place where the upper (white) Schwarzschild surface joined the lower (red) Schwarzschild surface in the movie corresponds to the origin in the Kruskal diagram, where the pinkred horizon and red antihorizon cross. What lies beyond the antihorizon of the Schwarzschild geometry? The complete KruskalSzekeres spacetime diagram, discussed in the section on White Holes and Wormholes, reveals the suprising answer that beyond the antihorizon is another Universe, a second copy of the Schwarzschild geometry, connected to this Universe by a wormhole. 
KruskalSzekeres metric
Kruskal time
t_{K}
and radial coordinate
r_{K}
(respectively the vertical and horizontal coordinate in the Kruskal
spacetime diagram)
are related to Schwarzschild time t and radial coordinate
r, the circumferential radius,
by the following transformation.
Let R denote what Misner, Thorne & Wheeler (1973, ``Gravitation'')
call the `tortoise coordinate'
The Kruskal metric shows explicitly that the Schwarzschild geometry is wellbehaved at the Schwarzschild radius, r = 1. 
Penrose diagram of the Schwarzschild geometry
Penrose invented his diagrams as a device for depicting the complete causal structure of any given geometry. Penrose diagrams map everything in the geometry on to a finite diagram, including points at infinite distance and in the infinite past and future. Light rays (null geodesics) are arranged so that they always point at 45^{o} from the upward vertical.
Penrose diagrams
are spacetime diagrams
in which the metric takes a certain generic, although not unique, form.
In the Penrose diagram of the Schwarzschild geometry at left,
the Penrose time
t_{P}
and radial
r_{P}
coordinate are related to the Kruskal time
t_{K}
and radial
r_{K}
coordinate by
Watch Kruskal morph into Penrose (41K GIF); or same morph, doublesize on screen (same 41K GIF). 
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Updated 28 Feb 2006