1. Introduction

Recently, the discovery of gravitational waves [1] and the “pictures” of black holes [2] have drawn great attention, which once again verify Einstein’s general relativity [3,4]. In 1915, Karl Schwarzschild published the first exact solution [5], a point mass case in the vacuum (known as an exterior solution) just two months after Einstein’s field equations. It has been extensively used to explain the behaviors of black holes, which were thereby named Schwarzschild black holes. One month later, he proposed the interior solution of a ball of perfect fluid [6]. By matching the boundary of the interior and exterior solutions, it could be used to explain the behaviors of relativistic massive stars. The Schwarzschild solutions play the important role in discussing general relativity and beyond, such as space-time singularity, quantum field theory, gravitation collapse, and so on [7].

On the other hand, transformation optics (TO) [8–10], has been proposed a decade ago, which is very similar to general relativity [3]. General relativity (GR) sets up an equivalence of matter-energy density and curved space, while TO follows an equivalence of refractive index profile (in general, anisotropic) and curved space [11]. Therefore, it is flexible to mimic many metrics (even not solutions of Einstein’s field equations) and demonstrate similar light behaviors by using TO. For example, there are many works on optical black holes, both from theories [12–14] and experiments [15–17]. In addition, the effects of cosmic strings [18,19], and even wormholes [20,21] have also been proposed. Recently, the author and colleagues have proposed a method, “transformation cosmology”, and found various lenses that are related to the Schwarzschild metric, de Siter metric, and Anti-de Siter metric [22].

To note, the equivalence of the refractive index profile and curved space also opens another research interest, i.e., the correspondence between the gradient media and the curved surface. That surface is usually a two-dimensional (2D) manifold with a homogenous refractive index embedded in three-dimensional (3D) Euclidean space. In TO framework, that curved surface is often called the geodesic lens [23]. Under such correspondence, many effects on the surface are revealed through their 2D counterparts. For example, the function of deflecting incident beam [23,24], the imaging property [25], and the spectrum distribution on the curved surface [26]. Recently, the chaotic dynamics on curved surfaces of rational symmetry have been studied, in which the corresponding nonuniform 2D table billiard is employed to analyze the chaoticity [27,28]. It is also believed that the curved surface can be regarded as an embedding diagram of curved space-time, such as the hyperbolic surface for the Morris-Thorne wormhole [29] and the Flamm paraboloid for the Schwarzschild black hole [30].

In this work, based on the “transformation cosmology” method, we propose an analytical solution of an optical lens for the full version of the Schwarzschild metric (both interior and exterior). It is known that in the previous analogy of metrics in curved space, the exterior solution has been investigated a lot, while studies on interior metrics or full metrics remain exclusive. Here we are going to demonstrate the light-bending properties of the full Schwarzschild metric. By inspecting the equivalent gradient lens, we get two opposite topologies when changing the interior Schwarzschild metric. Our work will help to explore the phenomena of massive stars with optical settings in the laboratory and may inspire some optical applications such as energy-harvesters.

2. Theory

Let us start with the exterior Schwarzschild metric [5]

The exterior Schwarzschild metric is often used to describe the space-time around a static black hole without charge and rotation. The Schwarzschild radius ${r_s}$ is also the event horizon for making ${g_{00}} = 0$ and ${r_s} = 2GM$(for c = 1). The exterior geometry has been extremely successful in explaining the precession of the star or planet’s perihelion, and the bending of light where the exterior Schwarzschild gravitational field acts as a gravitational lens.

Following the transformation cosmology method [22], with the mapping of $r = \frac{{{{(R + \frac{{{r_s}}}{4})}^2}}}{R} = \frac{{{{(R + {R_s})}^2}}}{R}$, a simple gradient index lens with

Now, let us introduce the interior Schwarzschild metric written as [6]

By making the isotropic Schwarzschild space-time [22]

In Eq. (6), $a$ is a constant of integration to be determined. Therefore, the related refractive index profile could be calculated as [22],

In addition, the inverse mapping of the optical Schwarzschild black hole is

Letting $t = \frac{{{r_s}}}{{{r_g}}}$, at $r = {r_g}$, Eq. (6) and (8) should be matched to each other to share the same refractive index at ${R_g}$. Therefore, we obtain,

For different values of t, i.e., different ratios of ${r_s}$ to ${r_g}$, a corresponding value $a$ can be obtained for a continuous refractive index profile at ${R_g}$. It should be noted that there is a critical value of t in the real celestial systems, i.e. $t = \frac{8}{9}$. If $t \ge \frac{8}{9}$, ${r_g} \le \frac{9}{8}{r_s}$, i.e., $M \ge \frac{{4{r_g}}}{{9G}}$, the pressure at the center of the star will be infinite, therefore there is no static solution in general relativity. Usually, a star with $M \ge \frac{{4{r_g}}}{{9G}}$ will inevitably keep shrinking, eventually forming a black hole [6,31]. We will use this critical condition $t = \frac{8}{9}$ to distinguish the star and black hole situations of the full Schwarzschild metric.

3. Simulation results

Firstly, let us consider a massive star with $t = \frac{3}{4}$, and it can be calculated that $a = \frac{{27}}{{32}}$. From Eq. (6), ${R_g} = \frac{2}{3}a{r_g} = \frac{3}{4}{r_s} = 3{R_s}$. We thereby obtain the index profile for $R \le {R_g}$,

The exterior region ($R > {R_g}$) still follows Eq. (2). We can check that the refractive index at ${R_g}$ is $n({R_g} = 3{R_s}) = \frac{{32}}{9}$. Figure 1 shows the refractive index profile for the lens related to such a star (the red curve). The refractive index at the origin is maximal and less than 10 ($n(0) = \frac{{256}}{{27}} \approx 9.48$). We then explore the light behavior near such a star. A point source is placed at the photon sphere ${R_{ph}} = (2 + \sqrt 3 ){R_s}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (x = {R_{ph}},y = 0)$. Figure 2(a) shows the ray-tracing simulation of this case. It is clear that outside the photon sphere with the blue rays, the light bending effect is the same as the previous optical Schwarzschild black hole [22]. And light propagates in a circle at the photon sphere, indicated by the yellow ray. While for the interior region, the light will no longer be trapped. Excitingly, it almost forms an image inside the star (not a perfect image). In fact, if we check with Eq. (10), we can find that it exactly follows the form of Maxwell’s fish-eye lens [32,33]. Meanwhile, some light can be seriously deflected by nearly 360 degrees. The corresponding wave simulation of the point source is depicted in Fig. 2(c). It can be seen that the light field (${E_z}$) inside the star participates in the focusing (approximately imaging in rays), while the outside field makes an apparent interference pattern. We also elaborated the light behavior illuminated by the parallel beam in Figs. 2(b) and (d) with ray and wave simulation, respectively. The rays are bent explicitly, and it performs a caustic image when getting into the star, and then leave away from it to infinity. To better visualize, we use the Gaussian beam incident on the lens, it is found that there is a small focusing point inside the star, which also demonstrates similar phenomena in Maxwell’s fish-eye lens.

 

Fig. 1. The refractive index profiles for a “star” (in red), a “critical black hole” (in blue), and a “singular black hole” (in green). The refractive index of “star” is finite at the origin ($n(0) = \frac{{256}}{{27}}$), and the “critical black hole” is infinite, while the “singular black hole” is negative, respectively.

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Fig. 2. Ray and wave simulation near the Schwarzschild “star”. (a) Ray trajectories when a point source locates at the photon sphere (yellow circle). Blue color indicates that the rays are emitted from the right side, i.e., outside the photon sphere. Red color indicates the rays inside the photon sphere. (b) Ray trajectories when parallel beam illuminates. (c) (d) Corresponding wave simulations (E_z field) of (a), and (b) respectively. Black circle represents the transformed star radius ${R_g}$. R_s is set as 1 m. ${R_g} = 3{R_s}$. Background colormap in (a) and (b) is the refractive index distribution. The frequency for incident wave is set as 0.18 GHz. The simulations are carried out by commercial software COMSOL Multiphysics.

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Secondly, let us consider an interesting situation when $t = \frac{8}{9}$. This is the critical condition for a massive star to collapse into a black hole in Schwarzschild’s paper [6,31]. At this time, $a = \frac{{27}}{{16}}$. From Eq. (6), ${R_g} = \frac{3}{4}a{r_g} = \frac{1}{2}{r_s} = 2{R_s}$. We thereby obtain the index profile for $R \le {R_g}$,

The exterior region ($R > {R_g}$) still follows Eq. (2). We can check that the refractive index at ${R_g}$ is $n({R_g} = 2{R_s}) = \frac{{27}}{4}$. The blue curve in Fig. 1 depicts the refractive index profile for the lens related to such a critical black hole. The refractive index goes to infinity at the origin. It is in proportion to $\frac{1}{{{R^2}}}$, which takes the form of optical black holes in Ref. [14]. The $t = \frac{8}{9}$ is indeed a critical value, it makes the coefficient before $R_s^2$ in the denominator of Eq. (7) become zero. We will find in the next section that for a larger t, that coefficient will become negative. Figure 3 displays the light behavior near the “critical black hole”. For example, in Fig. 3(a), the blue rays on the right part of the photon sphere can escape from the central black hole, similar to the exterior Schwarzschild black hole. While the red ray inside the photon sphere will be trapped in the center. When it is approaching the origin, the phase velocity of the light is much slower. In ray simulation, it needs infinite time to fall on the singularity, therefore we only show the result in a finite time. The corresponding wave simulation is demonstrated in Fig. 3(c). To avoid the central singularity and better perform the field pattern, we set a tiny area with the refractive index profile of $n(R)(1 + i)$ to absorb the energy, as indicated by the dark gray disk. It can be seen that the field distribution is accorded with ray-tracing in Fig. 3(a). In addition, we also present the results by parallel beam illumination in Figs. 3(b) and (d). Note that although most incident rays are finally attracted to the origin, some rays still can escape from it. When the beam width is larger, this phenomenon is more obvious. The wave pattern for incident Gaussian beam clearly shows the effect of this black hole. Under such conditions, major energy is swallowed, and little survives on the right side.

 

Fig. 3. Ray and wave simulation near the Schwarzschild “critical black hole”. (a) Ray trajectories when a point source locates at the photon sphere (yellow circle). (b) Ray trajectories when parallel beam illuminates. (c) (d) Corresponding wave simulations of (a), (b) respectively. Background colormap in (a) and (b) shows the refractive index distribution. R_s is set as 1 m. The interior boundary is ${R_g} = 2{R_s}$. The frequency for wave is set as 0.06 GHz, and the radius of the dark gray disk in wave simulation is 0.5 m.

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Finally, we consider a larger $t = \frac{{15}}{{16}}$, which might be no longer possible in real universe. The related value of a is $\frac{{125}}{{256}}$. From Eq. (6), ${R_g} = \frac{4}{5}a{r_g} = \frac{5}{{12}}{r_s} = \frac{5}{3}{R_s}$. We thereby obtain the index profile for $R \le {R_g}$,

The exterior region ($R > {R_g}$) still follows Eq. (2). We can check that the refractive index at ${R_g}$ is $n({R_g} = \frac{5}{3}{R_s}) = \frac{{256}}{{25}}$. Figure 1 shows the refractive index profile for the lens related to such a black hole with the green curve. We would like to call it a “singular black hole” for two reasons. One is that it might not be related to a real astronomical effect. The other is that, the refractive index goes to infinity when $R = \sqrt {\frac{{125}}{{189}}} {R_s} \equiv {R_c}$. In addition, inside the region ($R < {R_c}$), Eq. (12) becomes a Poincaré disk form [22,34], yet with a negative refractive index. Besides, from Fig. 1, it can be also concluded that the refractive index distributions for the three cases are very different, especially when approaching the origin. The three refractive index profiles reflect three distinct properties of the interior curved space-time. By inspecting Eq. (7), we can find the time component g₀₀ determines the behavior of the n(R). When n(R) is finite in the origin, it means that g₀₀ has no singularity which corresponds to a regular Schwarzschild star with finite pressure in the center. When n(R) is infinite, it reveals g₀₀ has a zero point at the origin, which indicates the critical condition for the star collapsing into a black hole. While when n(R) is negative at the origin and then becomes singular at R_c, which reflects g₀₀ has a zero point at R_c. Figure 4(a) and (b) show the ray-tracing simulations for a point source at the photon sphere as well as parallel beam incidence in the case of singular black hole. One could discover that the behavior of the rays is very similar to the case of “critical black hole”. Note that for all the cases, the refractive index outside R_g is the same, although they are with different values. In particular, $R = {R_c}$ is a new horizon to trap light for that R_c is the zero point of g₀₀. Therefore, outside rays are similar to the former black hole for the same gradient distribution, and inside rays are all absorbed, which accounts for the nearly similar behaviors to the critical black hole. Wave simulations are depicted in Fig. 4(c) and (d), in which the smaller black circle represents the new horizon. It is a little bigger than ${R_c}$ and we also set the refractive index of $n(R)(1 + i)$ in the inner core to absorb light. The wave simulations are in good agreement with the ray-tracing results.

 

Fig. 4. Ray and wave simulation near the Schwarzschild “singular black hole”. Ray trajectories when a point source locates at the photon sphere (yellow circle) (a) and when a parallel beam illuminates (b). (c) (d) Corresponding wave simulations of (a) and (b) respectively. R_s is set as 1 m. The interior boundary is ${R_g} = \frac{5}{3}{R_s}$. Background colormap in (a) and (b) shows the refractive index distribution, an infinite singularity lies on R_c. The frequency for wave is set as 0.1 GHz, and the inner core is about 1.2R_c.

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Before getting into the conclusion, let’s briefly discuss the choice of t in the article. Inspired by the gravitational collapse phenomena, the critical condition of $t = \frac{8}{9}$ must be studied. In order to gradually show the behavior of light under the parameter evolution from regular stars to gravitational collapse, we display the case of $t < \frac{8}{9}$ in the beginning and show $t > \frac{8}{9}$ in the end. Actually, the choice of t is arbitrary, and it will not alter the consequence. Namely, if we chose any other t that is stratified $0 < t < \frac{8}{9}$, it still results in a Maxwell’s fisheye-like profile. And if $\frac{8}{9} < t < 1$, there will be a singularity larger than R_s and a Poincaré disk-like profile within. Thus, the topology of the Schwarzschild star and Schwarzschild singular black hole will remain unchanged. In the singular black case, there is a negative refractive index in the region of $R < {R_c}$, which may be not seen in the previous analogy. As it describes a black hole, it can define on the region that $R > {R_c}$. In addition, for the case that $t > 1$, there will exist an imaginary component in interior Schwarzschild metric which may not make sense in the space-time metric, since the metric is used to describe a four-dimensional spacetime interval. We can also find that it will lead to a pure imaginary refractive index in Eq. (7).

4. Conclusion

To summarize, we have found a perfect optical solution, i.e., a refractive index profile to analogize the full Schwarzschild metric. The table-top model is enough to reproduce the null geodesics and it could be useful to explore astronomical phenomena for massive stars. In particular, we find that light behavior Schwarzschild star will form a nearly perfect image and such a star is like Maxwell’s fish-eye lens. We also discover that when the ratio of the massive star radius to the Schwarzschild radius is a critical value, the related refractive index profile will be changed back to those of traditional optical black holes. In GR, it is related to the condition for a massive star to collapse into a real black hole. If this ratio becomes further bigger, the index profile becomes another black hole with a new horizon. Whereas inside the black hole, it is a Poincaré disk with a negative index, which is actually related to de Siter metric. The transformation cosmology method allows us to visualize the essential curved space-time from an optical perspective and also provides a new degree to revisit the inherent topology of the gradient index lenses. All of these lenses could be realized and further explored using the technique of geodesic lens in the future [25,26].