# More about the Schwarzschild Geometry

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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 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 $$t_\textrm{ff}$$.

 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.

 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. 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.

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Updated 28 Feb 2006; converted to mathjax 3 Feb 2018