More about the Schwarzschild Geometry

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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 \[ r_s = {2 G M \over c^2} \ . \] where \(G\) is Newton’s gravitational constant, and \(c\) is the speed of light. For a 30 solar mass object, like the black hole in the fictional star system here, the Schwarzschild radius is about 100 kilometers. 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 \(\frac{1}{2} v^2 = G M / r\) 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, \(c = 1\)) \[ d s^2 = - \, (1 - r_s / r ) d t^2 + {d r^2 \over 1 - r_s / r} + r^2 d o^2 \ . \] The quantity \(d s\) denotes the invariant spacetime interval, an absolute measure of the distance between two events in space and time, \(t\) is a ‘universal’ time coordinate, \(r\) is the circumferential radius, defined so that the circumference of a sphere at radius \(r\) is \(2\pi r\), and \(d o\) is an interval of spherical solid angle.

Embedding diagram The Schwarzschild geometry is illustrated in the embedding diagram at the top of the page, which shows a 2-dimensional representation of the 3-dimensional spatial geometry at a particular instant of universal time t. One should imagine that objects are confined to move only on the 2-dimensional 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 \(d r\) of circumferential radius \(r\) is \(d r / \sqrt{1 - r_s/r}\), which is larger than the radial interval \(d r\) 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 ‘space-like’: 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 ‘time-like’: 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 \(d t\) of universal time is \(\sqrt{1 - r_s/r} \, d t\), which is less than the universal time interval \(d t\). 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 \[ \sqrt{1 - r_s / r} \ . \] This time dilation factor tends to zero as \(r\) approaches the Schwarzschild radius \(r_s\), which means that someone at the Schwarzschild radius will appear to freeze to a stop, as seen by anyone outside the Schwarzschild radius.

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 \[ {1 \over \sqrt{1 - r_s/r}} \] 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 3-dimensional 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 ill-behaved at the horizon, the Schwarzschild radius (vertical red line). However, the pathology is an artefact of the Schwarzschild coordinate system. Spacetime itself is well-behaved 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.

Free-fall spacetime diagram Free-fall coordinates reveal that the Schwarzschild geometry looks like ordinary flat space, with the distinctive feature that space itself is flowing radially inwards (hence the \(-\) sign) at the Newtonian escape velocity, \[ v = - \sqrt{2 G M \over r} \ . \] The infall velocity \(v\) passes the speed of light \(c\) at the horizon. 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, photon-canoes 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 free-fall metric expresses mathematically the above physical assertions. The free-fall metric is \[ d s^2 = - d t_\textrm{ff}^2 + (d r - v \, d t_\textrm{ff})^2 + r^2 d o^2 \ , \] where \(r\) is the usual Schwarzschild radial coordinate, and the free-fall time coordinate \(t_\textrm{ff}\) is the proper time experienced by persons who free-fall radially inward, at velocity \(d r / d t_\textrm{ff} = v\), from zero velocity at infinity: \[ t_\textrm{ff} = t + 2 \sqrt{r} + \ln \left| {\sqrt{r} - 1 \over \sqrt{r} + 1} \right| \] in units where the speed of light and the Schwarzschild radius are both unity, \(c = 1\) and \(r_s = 1\). The free-fall metric shows that the spatial geometry is flat, having spatial metric \(d r^2 + r^2 d o^2\), on hypersurfaces of fixed free-fall time, \(d t_\textrm{ff} = 0\). The colouring of lines in the free-fall 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 free-fall time \(t_\textrm{ff}\). Watch Schwarzschild morph into free-fall (41K GIF); or same morph, double-size on screen (same 41K GIF).

Eddington-Finkelstein spacetime diagram Eddington-Finkelstein 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 \(45^\circ\) in the spacetime diagram. Finkelstein time \(t_\textrm{F}\) is related to Schwarzschild time \(t\) by \[ t_\textrm{F} = t + \ln | r - 1 | \] 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, double-size on screen (same 28K GIF). Watch Finkelstein morph into free-fall (38K GIF); or same morph, double-size on screen (same 38K GIF).

Kruskal-Szekeres spacetime diagram Kruskal-Szekeres coordinates show transparently the causal structure of the Schwarzschild geometry. By construction, radially infalling (yellow) or outgoing (ochre) light rays move at \(45^\circ\) leftward or rightward in the Kruskal-Szekeres spacetime diagram. Watch Finkelstein morph into Kruskal (50K GIF); or same morph, double-size on screen (same 50K GIF). In addition to the normal horizon (pink-red 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 Kruskal-Szekeres 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 pink-red 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 pink-red horizon and red antihorizon cross. What lies beyond the antihorizon of the Schwarzschild geometry? The complete Kruskal-Szekeres 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.

Kruskal-Szekeres metric Kruskal time \(t_\textrm{K}\) and radial coordinate \(r_\textrm{K}\) (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 \(r^\ast\) denote what Misner, Thorne & Wheeler (1973, “Gravitation”) call the ‘tortoise coordinate’ \[ r^\ast = r + \ln | r - 1 | \] (in units where the speed of light and the Schwarzschild radius are both unity, \(c = 1\) and \(r_s = 1\)). The tortoise coordinate \(r^\ast\) has the property that radially infalling and outgoing light rays satisfy \[ r^\ast + t = \mbox{constant} \ , \\ r^\ast - t = \mbox{constant} \ , \] respectively. Kruskal time \(t_\textrm{K}\) and Kruskal radius \(r_\textrm{K}\) are then defined by \[ r_\textrm{K} + t_\textrm{K} = 2 \, e^{( r^\ast + t ) / 2} \ , \\ r_\textrm{K} - t_\textrm{K} = \pm 2 \, e^{( r^\ast - t ) / 2} \ , \] where the overall sign in the last equation is positive (\(+\)) outside the Schwarzschild radius, \(r > 1\), and negative (\(-\)) inside the Schwarzschild radius, \(r < 1\). The Kruskal metric is \[ d s^2 = r^{-1} e^{-r} \left( - \, d t_\textrm{K}^2 + d r_\textrm{K}^2 \right) + r^2 d o^2 \ . \] The Schwarzschild radial coordinate \(r\), which appears in the factors \(r^{-1} e^{-r}\) and \(r^2\) in the Kruskal metric, is to be understood as an implicit function of the Kruskal coordinates \(t_\textrm{K}\) and \(r_\textrm{K}\). The Kruskal metric shows explicitly that the Schwarzschild geometry is well-behaved 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 \(45^\circ\) 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 \(t_\textrm{P}\) and radial \(r_\textrm{P}\) coordinate are related to the Kruskal time \(t_\textrm{K}\) and radial \(r_\textrm{K}\) coordinate by \[ r_\textrm{P} + t_\textrm{P} = {r_\textrm{K} + t_\textrm{K} \over 2 + | r_\textrm{K} + t_\textrm{K} |} \ , \\ r_\textrm{P} - t_\textrm{P} = {r_\textrm{K} - t_\textrm{K} \over 2 + | r_\textrm{K} - t_\textrm{K} |} \ , \] which have the property that the singularity is horizontal in these coordinates. Watch Kruskal morph into Penrose (41K GIF); or same morph, double-size on screen (same 41K GIF).

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Updated 28 Feb 2006; converted to mathjax 3 Feb 2018