Abstract
1. INTRODUCTION
Einstein’s general theory of relativity is one of the cornerstones in modern physics. It holds the best understanding of gravity so far, explaining that the nature of universal gravitation originates from matter-energy, resulting in curved spacetime. To date, general relativity (GR) has passed all the experimental tests with flying colors, such as the precession of Mercury [1], gravitational redshift [2], the observations of solar-mass pulsars in binary systems [3], and the gravitational waves from several stellar mass, black hole candidate in-spirals [4]. Recently, the report of first detection of the Schwarzschild precession in S2’s orbit around the nearest massive black hole (candidate) in the Galaxy Center has received attention, being more prominent evidence of GR [5]. In optics, in analogy to the equality between matter-energy and curved spacetime, macroscopic Maxwell’s equations in complex inhomogeneous media can be mapped into free-space Maxwell’s equations of an arbitrary spacetime metric [6–10], leading to lots of transformation optical applications like invisibility cloaks [11–17], field rotators [18,19], and illusion devices [20]. Moreover, despite the functionality of controlling the flow of light, this analogy has also been utilized to mimic some exciting gravitational effects related to GR in return, for instance, black holes [21–27], Einstein’s ring [28], de Sitter space [29,30], and cosmic strings [31,32].
For the above mimicking, most works are from the perspective of transformation optics. However, in classical aspect, the analogy has another inherent correspondence, revealed through the least action principle in mechanics (Maupertuis’s principle) and Fermat’s principle in geometrical optics. This correspondence is also called optical-mechanical analogy [33–36]. It means that we can observe some gravitational phenomena in the regime of geometrical optics using light rays. Comparing the forms of Hamilton equations in these two fields, one can obtain the closed connection between classical mechanics and geometrical optics, i.e., the potential and total energy (of unit mass) can construct a gradient refractive index profile , with the same shape of trajectories in both mechanics and geometrical optics [36]. Here, based on the optical-mechanical analogy, in the geometrical optics regime, we use a gradient index lens to mimic the star S2’s Schwarzschild precession near the Galactic Center massive black hole SgrA* mentioned above. With the help of light ray trajectories in the gradient index lens, the difference between universal gravitation and GR is clearly shown. The extra cubic term in the effective potential of the Schwarzschild spacetime makes the path in precession and causes Newton’s second law to fail [37]. Moreover, inspired by such mimicking, we propose two other types of gradient index lenses stemming from the Newton and Hooke potentials, each with an additional quadratic perturbation . The orbits in those lenses will not always be in precession. It can be closed under certain conditions or be trapped by the center. Such gravitational effects of additional quadratic term systems in astronomy are possibly related to the charged Reissner–Nordström metric [38,39] of GR and the dark matter particle motions in galaxies [40].
2. SCHWARZSCHILD PRECESSION MIMICKING
The nearest black hole candidate SgrA* is located at the center of the Milky Way, surrounded by a very dense cluster of stars. One of the closest stars is S2, whose distance from SgrA* at orbit’s perihelion is about 120Au (Au, atomic units). Due to the extreme gravity environment, the system of SgrA* provides a wonderful “laboratory” for GR testing. For the star S2, the orbit obeys the Schwarzschild metric of Einstein’s field equation:
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Now let us treat the mechanical problem in the geometrical optics regime using a gradient index lens. To begin, we employ the closed relationship between classical mechanics and geometrical optics based on the optical-mechanical analogy. The shape of particle trajectories in mechanics is the same as the trajectories of light rays in a lens with a refractive index distribution [36]
For star S2, the per unit mass energy and per unit orbital angular momentum are and , respectively. Here is semi-major axis and is the eccentricity of the orbit. From Ref. [5], , , and thus the perihelion . The mass of the nearest black hole candidate SgrA* is ( is the solar mass). The Schwarzschild precession of per S2’s orbit can be obtained from GR as , where is the Schwarzschild radius. With the help of the optical-mechanical analogy, this precession of star S2 can be visualized vividly in a gradient index lens. Substitute and modified universal gravitational potential into Eq. (10), and the refractive index profile mimicking Schwarzschild precession induced by the black hole SgrA* is . In fact, there are several other mimickings about static black holes, such as [22,23,26]. However, the gradient lens we propose based on the optical-mechanical analogy is very different from those in the previous works. The lens here is a general formula for interactions between two massive celestial bodies, not limited to black holes. Our method involves the astronomical quantities, which is more reasonable in revealing the actual celestial motions in the universe, while the mimicking is emphasized on the trapping effect, and it is more suitable for a light absorber, or we can call it an “optical black hole.” In Fig. 1(a), we draw the light ray trajectories of both universal gravitation analogy and Schwarzschild precession analogy. The massive black hole is placed in the origin, and the trajectories start at the perihelion indicated by the cyan spot below the origin. The direction of velocity is perpendicular to the radius vector at that point. The universal gravitation analogy is shown in a red dashed elliptically closed curve, while the blue curve represents the mimicking of Schwarzschild precession. From Fig. 1(a), it can be found that the precession accumulates and becomes obvious with the evolution of the “time” . At the beginning, the two curves are very close to each other, and after 15 periods the difference is gradually revealed. Although does not correspond to the physical time , the shapes of the trajectories in mechanics as well as the precession per orbit () are conserved in the geometrical optics [33]. Therefore, if we care less about the physical time, this mimicking is fast and effective to make some useful predictions at first, by avoiding the complex mathematical calculations of the metric of curved spacetime.
Figure 1.(a) Light ray trajectories of universal gravitation analogy and Schwarzschild precession analogy of the star S2’s orbit around the nearest massive black hole SgrA* candidate (in the origin). Trajectories start from the perihelion
In Fig. 1(a) we enlarge a small area (the white dashed box) to show the precession clearly. The precession looks very small as depicted in the inset (in the right corner). However, it is much bigger compared with the precession of Mercury around the Sun. We also extract the and components of universal gravitation analogy (red curve) and Schwarzschild precession analogy (blue curve). The results are presented in Figs. 1(a) and 1(b), respectively. We show the last three periods and find that the separation between universal gravitation and Schwarzschild precession is obvious in the plot of . The separation is getting larger as the “time” increases. However, the curves of remain almost the same due to the high eccentricity of the orbit.
3. PERTURBATION OF QUADRATIC TERM
In the previous section, the Schwarzschild precession is successfully reproduced in geometrical optics based on the optical-mechanical analogy. The modified universal gravitational potential is the key role in connecting mechanics with optics. In the vast universe, there are various kinds of effective potentials that describe the interaction between celestial bodies. Some effective potentials can be adopted to represent the influence of GR if the spacetime is static, like the Schwarzschild metric and the Reissner–Nordström metric. In this section, we are interested in whether there exist other potentials in astronomy that can make a different trajectory, not always in the elliptical orbit or in precession. In this way, the novel gravitational effects in celestial mechanics may enlighten us to meet some new ideas for optics based on the optical-mechanical analogy. In Schwarzschild precession, it is the cubic term that yields the precession of the perihelion and causes the orbit to no longer be closed. However, if the additional term in the modified potential is quadratic, i.e., , will the phenomenon in the corresponding gradient index lens be the same as Schwarzschild precession? The answer here is no. For convenience, we consider a more concise and basic potential form in Newton mechanics, i.e., , to manifest gravity. Now we add the extra quadratic term with coefficient and make up a new modified Newton potential . In mechanics, the trajectory of a particle (set the mass as unit) under such a central potential field can be precisely solved, and the expression is [37]
Figure 2.Light ray trajectories in the lens
In Fig. 2, the light rays start from point (1, 0) with different launching angle , stating their various orbit angular momentum . The first row is the case of , with and in Figs. 2(a) and 2(b), respectively. The trajectory in Fig. 2(a) is trapped by the center, with a pure imaginary value of (), while the trajectory in Fig. 2(b) is closed. It has finished passing through the origin three times, and then it finally joins into the starting point. The value of is . However, the situations in the case of the negative are different, as shown in Figs. 2(c) and 2(d). For , the trajectories of the light rays have clear boundaries. They are stably restricted in an annular region with the positive refractive index. The ray in Fig. 2(c) with a launching angle is in precession and keeps dancing around the center. Its is equal to . The ray in Fig. 2(d) with the initial angle is closed in a petaloid shape, and the equals . The closed trajectory has seven petals and passes through the origin five times. The reason why the trajectories are in various shapes can be dated back to Eq. (11) (the path solution in mechanics). The value of factor before the polar angle is essential. If is rational, taking the form of (, are coprime integers), the trajectory will be repeated and closed after the period . Recalling the closed trajectories in this part [ in Fig. 2(b) and in Fig. 2(d)], we find that light rays will have petals and travel the origin times for . We call it the general rule of closed orbits. Nevertheless, if is nonrational, there is no finite period, and it will keep dancing around the center as illustrated in Fig. 2(c) (). And as for the nonreal , the term in Eq. (11) will change into hyperbolic cosine form, which is further away from the period function. Thus the ray will collapse to straight the center if the launching direction is toward the center [Fig. 2(a)], or it will travel for some distance and turn back to the center due to the reflection resulting from the gradually decreasing (in mechanics the turning point is due to the vanishing velocity) if the launching direction is opposite to the center.
In fact, the phenomena for the case in quadratic modified Newton potential of Figs. 2(c) and 2(d) are related to another exact metric of Einstein’s field equation, i.e., the Reissner–Nordström metric. This metric describes the spacetime around a spherically symmetric nonrotating massive body with an electric charge . If is zero, it reduces to the Schwarzschild metric. However, if the electric charge is nonzero, the charged particle motion has additional quadratic item , along with the fourth power item in Eq. (8), whereas the higher power term can be neglected in some ways, and thus the situation is fundamentally similar to Figs. 2(c) and 2(d).
4. MODEL EXTENSION
The above Newton profile is a basic potential in universe. Another common potential, the Hooke potential (also named the harmonic potential) also supports closed elliptical trajectories, which can describe the gravitation of dark matter particles within the spherical isotropic homogeneous galaxies [40]. With the optical-mechanical analogy, the Hooke potential is associated with the Luneburg lens [43], with the energy constant . The Luneburg lens is also a perfect imaging lens and has extensive applications in optical antennas. Now this begs the question: if the quadratic term is introduced into the Hooke potential, how does the light ray behave in its induced gradient index lens?
To solve this problem, we can first search its orbit in mechanics, and then transform the quantities in the language of optics based on the analogy. By the laws of conservation of energy and momentum, the trajectory of the particle in the modified Hooke potential can be solved as [37]
Figure 3.Light ray trajectories in the gradient lens
There are two closed trajectories in Figs. 3(b) and 3(d). Their shapes are exactly consistent with the general rule of closed orbits. In Fig. 3(b), the value of is , and thus the trajectory has seven petals and passes the origin four times, while for in Fig. 3(d), the trajectory has eight petals after traveling around the origin one time. Besides, the analysis of the trapping effect in Fig. 3(a) and the precession in Fig. 3(c) is the same as that in the previous section. It should be noted that the parameters of and can be arbitrary, and those we chose are to better evidence our consequences. The trajectories in these two types of lenses (induced by the modified Newton potential and Hooke potential) are pretty interesting. The lenses can perform a “selective imaging (self-imaging)” in optics, as only certain sets of the light rays following the general rule can be closed. Thus it can be regarded as a special kind of optical absolute instrument, as their conditions for imaging are restricted. Moreover, the trapping effect and the precessional motion may also have potential applications in light absorbing and confining (cavity), and these effects are more common in such lenses.
5. CONCLUSION
In this paper, we propose a gradient refractive index lens to mimic the Schwarzschild precession in the orbit of star S2 near the Galactic Center massive black hole SgrA* for the first time to our knowledge, based on the optical-mechanical analogy in the field of geometrical optics. The optical-mechanical analogy involves potential and energy of the system, and thus we use a modified universal gravitational potential to represent the influence of GR. Inspired by the Schwarzschild precession mimicking, we propose two types of gradient lenses stemming from the modified Newton potential and modified Hooke potential, which are possibly related to the Reissner–Nordström metric and dark matter motions in astronomy. With the help of ray tracing, we find the light trajectories in these lenses will be closed, governed by a general rule, otherwise they will be trapped by the center or be continuously in precession. These interesting properties can be utilized in designing selective optical imaging devices and light absorbers. The optical-mechanical analogy builds a useful bridge to connect mechanics with optics, and our work shows some of its capabilities. Such interdisciplinary work of celestial mechanics and optics promises to potentially enlighten us with some new thoughts and ideas for both fields, for example, the Eaton lens, the Luneburg lens, and the Morse lens in optics [44].
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