Schwarzschild spacetime

5.1 Schwarzschild spacetime

The (exterior) Schwarzschild metric

has the form (69

) with

It is the unique spherically symmetric vacuum solution of Einstein’s field equation. At the same time, it is the most important and best understood spacetime in which lensing can be explicitly studied without approximations. Schwarzschild lensing beyond the weak-field approximation has astrophysical relevance in view of black holes and neutron stars. The increasing evidence that there is a supermassive black hole at the center of our Galaxy (see [106

] for background material) is a major motivation for a detailed study of Schwarzschild lensing (and of Kerr lensing; see Section 5.8). In the following we consider the Schwarzschild metric with a constant $m > 0$ and we ignore the region $r < 0$ for which the singularity at $r = 0$ is naked. The Schwarzschild metric is static on the region $2m < r < ∞$ . (The region $r < 0$ for $m > 0$ is equivalent to the region $r > 0$ for $m < 0$ . It is usually considered as unphysical but has found some recent interest in connection with lensing by wormholes; see Section 5.4.)

Historical notes.
Shortly after the discovery of the Schwarzschild metric by Schwarzschild [301 ] and independently by Droste [80 ], basic features of its lightlike geodesics were found by Flamm [113 ], Hilbert [157 ], and Weyl [347 ]. Detailed studies of its timelike and lightlike geodesics were made by Hagihara [145 ] and Darwin [72 , 73 ]. For a fairly complete list of the pre-1979 literature on Schwarzschild geodesics see Sharp [305 ]. All modern text-books on general relativity include a section on Schwarzschild geodesics, but not all of them go beyond the weak-field approximation. For a particularly detailed exposition see Chandrasekhar [54 ].

Redshift and Fermat geometry.
The redshift potential $f$ for the Schwarzschild metric is given in Equation (101 ). With the help of $f$ we can directly calculate the redshift via Equation (68 ) if observer and light source are static (i.e., $t$ -lines). If the light source or the observer does not follow a $t$ -line, a Doppler factor has to be added. Independent of the velocity of observer and light source, the redshift becomes arbitrarily large if the light source is sufficiently close to the horizon. For light source and observer freely falling, the redshift formula was discussed by Bażański and Jaranowski [24 ]. If projected to 3-space, the light rays in the Schwarzschild spacetime are the geodesics of the Fermat metric which can be read from Equation (70 ) (cf. Frankel [115 ]),

The metric coefficient $R (r)$ , as given by Equation (101

), has a strict minimum at $r = 3m$ and no other extrema (see Figure 9

). Hence, there is an unstable light sphere at this radius (recall Equation (71

)). The existence of circular light rays at $r = 3m$ was noted already by Hilbert [157 ]. The relevance of these circular light rays in view of lensing was clearly seen by Darwin [72

, 73 ] and Atkinson [13

]. They realized, in particular, that a Schwarzschild black hole produces infinitely many images of each light source, corresponding to an infinite sequence of light rays that asymptotically spiral towards a circular light ray. The circular light rays at $r = 3m$ are also associated with other physical effects such as centrifugal force reversal and “locking” of gyroscopes. These effects have been discussed with the help of the Fermat geometry (= optical reference geometry) in various articles by Abramowicz and collaborators (see, e.g., [5 , 4

, 6 , 2 ]).

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Figure 9: The function $R (r)$ for the Schwarzschild metric. Light rays that start at $rO$ with initial direction $Θ$ are confined to the region where $R (r) ≥ R (rO) sin Θ$ . The equation $R (3m ) = R (rO )sinδ$ defines for each $rO$ a critical value $δ$ . A light ray from $rO$ with $Θ = δ$ asymptotically approaches $r = 3m$ .

Index of refraction and embedding diagrams.
We know from Section 4.3 that light rays in any spherically symmetric and static spacetime can be characterized by an index of refraction. This requires introducing an isotropic radius coordinate $&tidle;r$ via Equation (72 ). In the Schwarzschild case, $&tidle;r$ is related to the Schwarzschild radius coordinate $r$ by

$&tidle;r$ ranges from $m ∕2$ to infinity if $r$ ranges from $2m$ to infinity. In terms of the isotropic coordinate, the Fermat metric (102

) takes the form

with

Hence, light propagation in the Schwarzschild metric can be mimicked by the index of refraction (105

); see Figure 10

. The index of refraction (105

) is known since Weyl [349 ]. It was employed for calculating lightlike Schwarzschild geodesics, exactly or approximately, e.g., in [13

, 232 , 105 , 205 ]. This index of refraction can be modeled by a fluid flow [289 ]. The embeddability condition (75

) is satisfied for $r > 2.25m$ (which coincides with the so-called Buchdahl limit). On this domain the Fermat geometry, if restricted to the equatorial plane $𝜗 = π∕2$ , can be represented as a surface of revolution in Euclidean 3-space (see Figure 11

). The entire region $r > 2m$ can be embedded into a space of constant negative curvature [3

].

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Figure 10: Index of refraction $n (&tidle;r)$ , given by Equation (105

), for the Schwarzschild metric as a function of the isotropic coordinate $&tidle;r$ .

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Figure 11: Fermat geometry of the equatorial plane of the Schwarzschild spacetime, embedded as a surface of revolution into Euclidean 3-space. The neck is at $r = 3m$ (i.e., $&tidle;r ≈ 1.87m$ ), the boundary of the embeddable part at $r = 2.25m$ (i.e., $&tidle;r = m$ ). The geodesics of this surface of revolution give the light rays in the Schwarzschild spacetime. A similar figure can be found in [4 ] (also cf. [159

]).

Lensing by a Schwarzschild black hole.
To get a Schwarzschild black hole, one joins at $r = 2m$ the static Schwarzschild region $2m < r < ∞$ to the non-static Schwarzschild region $0 < r < 2m$ in such a way that ingoing light rays can cross this surface but outgoing cannot. If the observation event $p O$ is at $r > 2m O$ , only the region $r > 2m$ is of relevance for lensing, because the past light cone of such an event does not intersect the black-hole horizon at $r = 2m$ . (For a Schwarzschild white hole see below.) Such a light cone is depicted in Figure 12 (cf. [182 ]). The picture was produced with the help of the representation (81 ) which requires integrating Equation (85 ) and Equation (86 ). For the Schwarzschild case, these are elliptical integrals. Their numerical evaluation is an exercise for students (see [45 ] for a MATHEMATICA program). Note that the evaluation of Equation (85 ) and Equation (86 ) requires knowledge of the turning points. In the Schwarzschild case, there is at most one turning point $rm (Θ)$ along each ray (see Figure 9 ), and it is given by the cubic equation

The representation (81

) in terms of Fermat arclength $ℓ$ (= travel time) gives us the intersections of the light cone with hypersurfaces $t = constant$ . These “instantaneous wave fronts” are depicted in Figure 13

(cf. [146

]). With the light cone explicitly known, one can analytically verify that every inextendible timelike curve in the region $r > 2m$ intersects the light cone infinitely many times, provided it is bounded away from the horizon and from (past lightlike) infinity. This shows that the observer sees infinitely many images of a light source with this worldline. The same result can be proven with the help of Morse theory (see Section 3.3), where one has to exclude the case that the worldline meets the caustic of the light cone. In the latter case the light source is seen as an Einstein ring. For static light sources (i.e., $t$ -lines), either all images are Einstein rings or none. For such light sources we can study lensing in the exact-lens-map formulation of Section 4.3 (see in particular Figure 6

). Also, Section 4.3 provides us with formulas for distance measures, brightness, and image distortion which we just have to specialize to the Schwarzschild case. For another treatment of Schwarzschild lensing with the help of the exact lens map, see [118

]. We place our static light sources at radius $rS$ . If $rO < rS$ and $3m < rS$ , only light rays with $Θ < δ$ ,

can reach the radius value $rS$ (see Figure 9

). Rays with $Θ = δ$ asymptotically spiral towards the light sphere at $r = 3m$ . $δ$ lies between 0 and $π∕2$ for $rO < 3m$ and between $π∕2$ and $π$ for $rO > 3m$ . The escape cone defined by Equation (107

) is depicted, for different values of $rO$ , in Figure 14

. It gives the domain of definition for the lens map. The lens map is graphically discussed in Figure 15

. The pictures are valid for $rO = 5m$ and $rS = 10m$ . Qualitatively, however, they look the same for all cases with $rS > rO$ and $rS > 3m$ . From the diagram one can read the position of the infinitely many images for each light source which, for the two light sources on the axis, degenerate into infinitely many Einstein rings. For each fixed source, the images are ordered by the number $i$ ( $= 1, 2,3,...$ ) which counts how often the ray has met the axis. This coincides with ordering according to travel time. With increasing order $i$ , the images come closer and closer to the rim at $Θ = δ$ (see Figure 15

). They are alternately upright and side-inverted (see Figure 16

), and their brightness rapidly decreases (see Figure 17

). These basic features of Schwarzschild lensing are known since pioneering papers by Darwin [72 ] and Atkinson [13 ] (cf. [212

, 246

, 200

]). A detailed study of Schwarzschild lensing was carried through by Virbhadra and Ellis [336

] with the help of an “almost exact lens map” (see Section 4.3). The latter assumes that observer and light source are in the asymptotic region and almost aligned, but the light rays are allowed to make arbitrarily many turns around the black hole. Various methods of how multiple imaging by a black hole could be discovered, directly or indirectly, have been discussed [212

, 200

, 15

, 14

, 274 , 76 ]. Related work has also been done for Kerr black holes (see Section 5.8). An interesting suggestion was made in [161 ]. A Schwarzschild black hole, somewhere in the universe, would send photons originating from our Sun back to the vicinity of our Sun (“boomerang photons” [315 ]). If the black hole is sufficiently close to our Solar system, this would produce images of our own Sun on the sky that could be detectable. The lensing effect of a Schwarzschild black hole has been visualized in two ways:

by showing the visual appearance of some background pattern as distorted by the black hole [66 , 295 , 234 ] (only primary images, $i = 1$ , are considered), and
by showing the visual appearence of an accretion disk around the black hole [212 , 129 , 15 , 14 ] (higher-order images are taken into account).

Numerous ray tracing programs have been developed for the Schwarzschild metric and, more generally, for the Kerr metric (see Section 5.8).

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Figure 12: Past light cone in the Schwarzschild spacetime. One sees that the light cone wraps around the horizon, then forms a tangential caustic. In the picture the caustic looks like a transverse self-intersection because one spatial dimension is suppressed. (Only the hyperplane $𝜗 = π ∕2$ is shown.) There is no radial caustic. If one follows the light rays further back in time, the light cone wraps around the horizon again and again, thereby forming infinitely many tangential caustics which alternately cover the radius line through the observer and the radius line opposite to the observer. In spacetime, each caustic is a spacelike curve along which $r$ ranges from $2m$ to $∞$ , whereas $t$ ranges from $− ∞$ to some maximal value and then back to $− ∞$ . Equal-time sections of this light cone are shown in Figure 13

.

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Figure 13: Instantaneous wave fronts of the light cone in the Schwarzschild spacetime. This picture shows intersections of the light cone in Figure 12

with hypersurfaces $t = constant$ for four $t$ -values, with $t1 > t2 > t3 > t4$ . The instantaneous wave fronts wrap around the horizon and, after reaching the first caustic, have two caustic points each. If one goes further back in time than shown in the picture, the wave fronts another time wrap around the horizon, reach the second caustic, and now have four caustic points each, and so on. In comparison to Figure 12

, the representation in terms of instantaneous wave fronts has the advantage that all three spatial dimensions are shown.

Figure 14: Escape cones in the Schwarzschild metric, for five values of $rO$ . For an observer at radius $rO$ , light sources distributed at a radius $rS$ with $rS > rO$ and $rS > 3m$ illuminate a disk whose angular radius $δ$ is given by Equation (107

). The boundary of this disk corresponds to light rays that spiral towards the light sphere at $r = 3m$ . The disk becomes smaller and smaller for $rO → 2m$ . Figure 9

illustrates that the notion of escape cones is meaningful for any spherically symmetric and static spacetime where $R$ has one minimum and no other extrema [253 ]. For the Schwarzschild spacetime, the escape cones were first mentioned in [249 , 224 ], and explicitly calculated in [319 ]. A picture similar to this one can be found, e.g., in [54

].

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Figure 15: Lens map for the Schwarzschild metric. The observer is at $rO = 5m$ , the light sources are at $rS = 10m$ . $Θ$ is the colatitude on the observer’s sky and $Φ(Θ )$ is the angle swept out by the ray (see Figure 6

). $Φ (Θ )$ was calculated with the help of Equation (87

). $Θ$ is restricted by the opening angle $δ$ of the observer’s escape cone (see Figure 14

). Rays with $Θ = δ$ asymptotically spiral towards the light sphere at $r = 3m$ . The first diagram (cf. [118

], Figure 5) shows that $Φ (Θ )$ ranges from 0 to $∞$ if $Θ$ ranges from 0 to $δ$ . So there are infinitely many Einstein rings (dashed lines) whose angular radius approaches $δ$ . One can analytically prove [212

, 246

, 39

] that the divergence of $Φ (Θ )$ for $Θ → δ$ is logarithmic. This is true whenever light rays approach an unstable light sphere [37

]. The second diagram shows $Φ (Θ)$ over a logarithmic $Θ$ -axis. The graph of $Φ$ approaches a straight line which was called the “strong-field limit” by Bozza et al. [39

, 37

]. The picture illustrates that it is a good approximation for all light rays that make at least one full turn. The third diagram shows $cosΦ (Θ )$ over a logarithmic $Θ$ -axis. For every source position $0 < 𝜗 < π$ one can read the position of the images (dotted line). There are infinitely many, numbered by their order (89

) that counts how often the light ray has crossed the axis. Images of odd order are on one side of the black hole, images of even order on the other. For the sources at $𝜗 = π$ and $𝜗 = 0$ one can read the positions of the Einstein rings.

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Figure 16: Radial angular diameter distance $D+ (Θ )$ , tangential angular diameter distance $D − (Θ )$ and travel time $T (Θ)$ in the Schwarschild spacetime. The data are the same as in Figure 15

. For the definition of $D+$ and $D −$ see Figure 7

. $D ±(Θ )$ can be calculated from $Φ (Θ)$ with the help of Equation (94

) and Equation (95

). For the Schwarzschild case, the resulting formulas are due to [84

] (cf. [85

, 118

]). Zeros of $D −$ indicate Einstein rings. If $D +$ and $D −$ have different signs, the observer sees a side-inverted image. The travel time $T (Θ)$ (= Fermat arclength) can be calculated from Equation (85

). One sees that, over the logarithmic $Θ$ -axis used here, the graph of $T$ approaches a straight line. This illustrates that $T (Θ )$ diverges logarithmically if $Θ$ approaches its limiting value $δ$ . This can be verified analytically and is characteristic of all cases where light rays approach an unstable light sphere [40

].

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Figure 17: Luminosity distance $D (Θ ) lum$ and ellipticity $𝜀(Θ )$ (image distortion) in the Schwarzschild spacetime. The data are the same as in Figures 15

and 16

. If point sources of equal bolometric luminosity are distributed at $r = rS$ , the plotted function $2.5log10(Dlum (Θ)2)$ gives their magnitude on the observer’s sky, modulo an additive constant $m0$ . For the calculation of $Dlum$ one needs $D+$ and $D −$ (see Figure 16

), and the general relations (41

) and (48

). This procedure follows [84

] (cf. [85

, 118 ]). For source and observer at large radius, related calculations can also be found in [212

, 246

, 200 , 336

]. Einstein rings have magnitude $− ∞$ in the ray-optical treatment. For a light source not on the axis, the image of order $i + 2$ is fainter than the image of order $i$ by $2.5 log10(e2π) ≈ 6.8$ magnitudes, see [212 , 246

]. (This is strictly true in the “strong-field limit”, or “strong-bending limit”, which is explained in the caption of Figure 15

.) The above picture is similar to Figure 6 in [246 ]. Note that it refers to point sources and not to a radiating spherical surface $r = rS$ of constant surface brightness; by Equation (54

), the latter would show a constant intensity. The lower part of the diagram illustrates image distortion in terms of $𝜀 = D−-− D+- D+ D−$ . Clearly, $|𝜀|$ is infinite at each Einstein ring. The double-logarithmic representation shows that beyond the second Einstein ring all images are extremely elongated in the tangential direction, $|𝜀| > 100$ . Image distortion in the Schwarzschild spacetime is also treated in [85

, 120 , 119 ], an approximation formula is derived in [241 ].

Lensing by a non-transparent Schwarzschild star.
To model a non-transparent star of radius $r∗$ one has to restrict the exterior Schwarzschild metric to the region $r > r∗$ . Lightlike geodesics terminate when they arrive at $r = r∗$ . The star’s radius cannot be smaller than $2m$ unless it is allowed to be time-dependent. The qualitative features of lensing depend on whether $r∗$ is bigger than $3m$ . Stars with $2m < r∗ ≤ 3m$ are called ultracompact [165 ]. Their existence is speculative. The lensing properties of an ultracompact star are the same as that of a Schwarzschild black hole of the same mass, for observer and light source in the region $r > r∗$ . In particular, the apparent angular radius $δ$ on the observer’s sky of an ultracompact star is given by the escape cone of Figure 14 . Also, an ultracompact star produces the same infinite sequence of images of each light source as a black hole. For $r > 3m ∗$ , only finitely many of the images survive because the other lightlike geodesics are blocked. A non-transparent star has a finite focal length $rf > 2m$ in the sense that parallel light from infinity is focused along a line that extends from radius value $rf$ to infinity. $rf$ depends on $m$ and on $r∗$ . For the values of our Sun one finds $rf = 550$ au (1 au = 1 astronomical unit = average distance from the Earth to the Sun). An observer at $r ≥ rf$ can observe strong lensing effects of the Sun on distant light sources. The idea of sending a spacecraft to $r ≥ rf$ was occasionally discussed in the literature [340 , 235 , 325 ]. The lensing properties of a non-transparent Schwarzschild star have been illustrated by showing the appearance of the star’s surface to a distant observer. For $r∗$ bigger than but of the same order of magnitude as $3m$ , this has relevance for neutron stars (see [352 , 256 , 128 , 287 , 222 , 240 ]). $r∗$ may be chosen time-dependent, e.g., to model a non-transparent collapsing star. A star starting with $r∗ > 2m$ cannot reach $r = 2m$ in finite Schwarzschild coordinate time $t$ (though in finite proper time of an observer at the star’s surface), i.e., for a collapsing star one has $r∗(t) → 2m$ for $t → ∞$ . To a distant observer, the total luminosity of a freely (geodesically) collapsing star is attenuated exponentially, $√-- L(t) ∝ exp (− t(3 3m )− 1)$ . This formula was first derived by Podurets [280 ] with an incorrect factor 2 under the exponent and corrected by Ames and Thorne [8 ]. Both papers are based on kinetic photon theory (Liouville’s equation). An alternative derivation of the luminosity formula, based on the optical scalars, was given by Dwivedi and Kantowski [84 ]. Ames and Thorne also calculated the spectral distribution of the radiation as a function of time and position on the apparent disk of the star. All these analyses considered radiation emitted at an angle $≤ π∕2$ against the normal of the star as measured by a static (Killing) observer. Actually, one has to refer not to a static observer but to an observer comoving with the star’s surface. This modification was worked out by Lake and Roeder [197 ].

Lensing by a transparent Schwarzschild star.
To model a transparent star of radius $r∗$ one has to join the exterior Schwarzschild metric at $r = r∗$ to an interior (e.g., perfect fluid) metric. Lightlike geodesics of the exterior Schwarzschild metric are to be joined to lightlike geodesics of the interior metric when they arrive at $r = r∗$ . The radius $r∗$ of the star can be time-independent only if $r∗ > 2m$ . For $2m < r∗ ≤ 3m$ (ultracompact star), the lensing properties for observer and light source in the region $r > r ∗$ differ from the black hole case only by the possible occurrence of additional images, corresponding to light rays that pass through the star. Inside such a transparent ultracompact star, there is at least one stable photon sphere, in addition to the unstable one at $r = 3m$ outside the star (cf. [152 ]). In principle, there may be arbitrarily many photon spheres [176 ]. For $r∗ > 3m$ , the lensing properties depend on whether there are light rays trapped inside the star. For a perfect fluid with constant density, this is not the case; the resulting spacetime is then asymptotically simple, i.e., all inextendible light rays come from infinity and go to infinity. General results (see Section 3.4) imply that then the number of images must be finite and odd. The light cone in this exterior-plus-interior Schwarzschild spacetime is discussed in detail by Kling and Newman [182 ]. (In this paper the authors constantly refer to their interior metric as to a “dust” where obviously a perfect fluid with constant density is meant.) Effects on light rays issuing from the star’s interior have been discussed already earlier by Lawrence [203 ]. The “escape cones”, which are shown in Figure 14 for the exterior Schwarzschild metric have been calculated by Jaffe [166 ] for points inside the star. The focal length of a transparent star with constant density is smaller than that of a non-transparent star of the same mass and radius. For the mass and the radius of our Sun, one finds 30 au for the transparent case, in contrast to the above-mentioned 550 au for the non-transparent case [235 ]. Radiation from a spherically symmetric homogeneous dust star that collapses to a black hole is calculated in [304 ], using kinetic theory. An inhomogeneous spherically symmetric dust configuration may form a naked singularity. The redshift of light rays that travel from such a naked singularity to a distant observer is discussed in [83 ].

Lensing by a Schwarzschild white hole.
To get a Schwarzschild white hole one joins at $r = 2m$ the static Schwarzschild region $2m < r < ∞$ to the non-static Schwarzschild region $0 < r < 2m$ at $r = 2m$ in such a way that outgoing light rays can cross this surface but ingoing cannot. The appearance of light sources in the region $r < 2m$ to an observer in the region $r > 2m$ is discussed in [111 , 233 , 81 , 195 , 196 ].