Optical Medium Approach: Simplifying General Relativity and Nonlinear Electrodynamics for Educational Purposes

Abstract

This paper explores the optical approach to simplifying complex concepts in general relativity (GR) and nonlinear vacuum electrodynamics. The focus is on using optical analogies to simplify the understanding of spacetime curvature and interactions in strong gravitational and magnetic fields. We demonstrate how applying concepts of effective refractive index can facilitate the teaching and comprehension of GR optical effects, such as gravitational lensing and the behavior of light around massive objects. Additionally, the paper covers the application of optical analogies in the context of nonlinear vacuum electrodynamics, showing how strong magnetic fields affect light propagation. This interdisciplinary approach provides a more natural understanding and modeling of complex physical phenomena, making them better accessible for study and teaching.

1. Introduction

Studying the effects of general relativity (GR), particularly its optical effects, presents a challenging task, primarily due to the need to explain spacetime curvature and utilize advanced mathematical tools [1]. It is possible to accurately describe the physical phenomena of spacetime using the field equations of GR. However, these equations possess a complex mathematical structure, and even the simplest geodesic equations generally lack analytical solutions. These complexities often become significant barriers in the educational process, especially when lacking enough knowledge in higher mathematics.

However, despite these challenges, there is a growing interest in including GR in school and university curricula, given its importance in modern physics and its practical applications in areas such as GPS (Global Positioning System) technology and astronomy [2]. Recent studies indicate that even high school students can develop a qualitative understanding of key GR concepts, provided the students have access to appropriate educational resources and support from the teachers [3,4].

There are various methods for teaching GR, including the use of numerical modeling and computer simulations [1,5]. However, many of these approaches remain challenging to grasp at a natural level, which limits their effectiveness for educational purposes. Research suggests that using more visual methods, such as optical analogies, can significantly enhance students’ understanding of complex GR concepts by making them more accessible [6].

The purpose of using optical analogies, such as the refractive index, in the context of GR was first proposed by Igor Tamm [7], then by Nándor Balazs [8], and has been further developed in subsequent research [9,10,11,12,13,14,15]. Building on this studies, we demonstrate how the concept of an effective refractive index can be applied to model GR effects, such as the behavior of light near stationary and rotating massive objects, as well as in the case of nonlinear vacuum electrodynamics [16,17]. This underscores the value of this approach for modeling and teaching complex physical phenomena.

We show that using these analogies not only makes it more straightforward to understand complex GR concepts but also makes those concepts more accessible for teaching, potentially simplifying the learning process and increasing student interest in the topics. To this end, in this paper, we demonstrate how the examples and methods compiled can facilitate the educational process and make complex theoretical concepts more accessible to students and young researchers.

The current paper is organized as follows. Section 2 reviews the concept of the optical medium approach in the GR theory. In Section 3, we consider the effective refractive index in the case of spherical symmetric gravitational field. In Section 4, we study the refractive index and deflection angle for the rotating compact objects. In Section 5, we calculate the effective refractive index in nonlinear vacuum electrodynamics. The application of the considered method to magnetars is shown in Section 6. Finally, Section 7 contains a summary of the results.

2. The Concept of the Optical Medium Approach

The concept of the optical medium approach involves discarding some of the internal paradigms of a given theory and replacing them with a medium that has a corresponding refractive index n [18]. The refractive index can reproduce the optical effects derived from the theory. This concept allows for natural and visual representation of complex physical phenomena, making them more accessible for understanding and analysis.

In the case of GR, the internal paradigm is the curvature of spacetime. The curvature caused by massive objects affects the trajectories of light and matter, creating effects such as the behavior of light near massive objects; see Figure 1, right. In the optical medium approach, one can represent this curved geometry as a medium with a variable refractive index, which often depends on mass and distance [19]; see Figure 1, left. This allows using of optical analogies to understand and model the effects of GR without delving into the complex mathematical aspects of the theory. Many studies have been conducted using the optical medium approach, which enables the study of motion with high accuracy without the need to solve geodesic equations [9,11]. This method applies not only to homogeneous media in spherically symmetric spacetime but also to the various scenarios of gravitational fields of compact objects with quadrupole moments [14,15,20,21].

Similarly, in the case of the nonlinear theory of vacuum electrodynamics, the internal paradigm involves the formation of electron–positron pairs in the presence of a strong magnetic field [22]. In this context, the optical medium approach allows us to represent the vacuum as a nonlinear medium with a refractive index that depends on the intensity of the magnetic field [17]. This helps us straightforwardly understand how strong fields affect the properties of the vacuum and particle interactions.

One of the key advantages of the optical medium approach is the ability to simulate gravitational fields in laboratory conditions using optically non-uniform media. This allows for the visual reproduction of effects similar to those described by the theory of gravitational lensing.

In Ref. [23], methods for forming a spherical layered structure through the adjustment of the temperature regime of a metal sphere placed in a gas environment are proposed. Since the refractive index of the gas changes with temperature, heating or cooling the sphere allows for the creation of the needed temperature distribution and the achievement of the necessary refractive index profile $n\left(r\right)$, with r denoting the distance. However, the main challenge of this model lies in accurately reproducing the required temperature distribution in the gas.

Instead of gas, it is more convenient to use a transparent solid material with a pronounced temperature dependence of the refractive index; see Figure 2. For example, in Ref. [24], plexiglass was used. Using such a material is more straightforward than gas environments; however, even in this case, only a qualitative reproduction of the optical characteristics of gravitational lenses can be achieved, as it is complicated to ensure an accurate temperature distribution throughout the volume of the lens. These complications can be overcomed by avoiding the creation of a non-uniform medium and instead reproducing only the required deflection angle, $\theta \left(r\right)$, using a lens made of homogeneous transparent material with a specially designed surface geometry, the shape of which was calculated in Refs. [23,24].

The shape of a lens that provides a specified refraction law $\theta \left(y\right)$, where y is the parameter that associated to the impact parameter b. For simplicity, one of the lens surfaces is made flat, allowing rays traveling parallel to the x-axis to pass through the entry surface without deviation. The shape of the refracting (exit) surface $x=x_{\mathrm{S}}\left(y\right)$ is determined based on Snell’s law: $nsin\alpha =sin\beta$, where the angles $\alpha$ and $\beta$ are measured relative to the surface normal N. Given that $\beta =\alpha +\theta \left(y\right)$ and $tan\alpha =d{x}_{\mathrm{S}}/dy$, the wanted equation for the surface can be obtained as

$$x_{\mathrm{S}}\left(y\right)=x_0-{\int }_0^y{\displaystyle \frac{sin\theta \left(y^{\prime }\right)}{n-cos\theta \left(y^{\prime }\right)}}d{y}^{\prime }.$$

(1)

Expression (1) remains valid not only in the case of axial symmetry, but also in the case when the deflection angle $\theta$ depends not just on the impact parameter y, but on the azimuthal angle $\varphi$ as well. However, in that case, the refractive surface of the lens $x=x_{\mathrm{S}}(y,\varphi )$ is no longer a surface of revolution.

In what follows, we analytically consider the methods for calculating the effective refractive index and deflection angle for some cases, as these quantities play a key role in conducting demonstrative experiments.

3. The Effective Refractive Index in the Case of the Schwarzschild Solution

The solution to the Einstein field equation for a static and spherically symmetric gravitational field, known as the Schwarzschild solution [25], has the following form:

$$d{s}^2=\left(1-r_g/r\right)c^{2}d{t}^2-r^2\left({\sin }^2\theta d{\varphi }^2+d{\theta }^2\right)-{\displaystyle \frac{d{r}^2}{1-r_g/r}},$$

(2)

where $r_g=2GM/c^2$ is the so-called Schwarzschild radius with G the Newtonian gravitational constant, M the object mass, and c the speed of light, and t denotes the time. Throughout the paper, the $d{a}^2$ denotes the differential squared, ${\left(da\right)}^2$, not the differential of the squared variable. The line element (2) can be expressed in isotropic form $d{s}^2=f\left(\rho \right)d{t}^2-d{\mathit{\rho }}^2$ by introducing a new radius coordinate $\rho$ using the transformation

$$r=\rho {\left(1+{\displaystyle \frac{r_g}{4\rho }}\right)}^2$$

(3)

since in the isotropic form, the metric has the same coefficient for all spatial coordinates ($\rho ,\theta ,\phi$) [14,25].

Upon the transformation (3), the metric reads

$$d{s}^2={\left({\displaystyle \frac{1-r_g/\left(4\rho \right)}{1+r_g/\left(4\rho \right)}}\right)}^2{c}^{2}d{t}^2-{\left(1+{\displaystyle \frac{r_g}{4\rho }}\right)}^4\left(d{\rho }^2+{\rho }^2({sin}^2\theta d{\varphi }^2+d{\theta }^2)\right),$$

(4)

where $d{\rho }^2+{\rho }^2({sin}^2\theta d{\varphi }^2+d{\theta }^2)$ has the dimension of the square of the infinitesimal length vector, $d{\mathit{\rho }}^2$. The speed of light in isotropic coordinates can be determined as follows:

$$v\left(\rho \right)=\left|{\displaystyle \frac{d\overrightarrow{\rho }}{dt}}\right|={\displaystyle \frac{\left(1-r_g/\left(4\rho \right)\right)c}{{\left(1+r_g/\left(4\rho \right)\right)}^3}}.$$

(5)

Using inverse coordinate transformation, $\rho ={\displaystyle \frac{1}{2}}\left((r-r_g/2)+\sqrt{r(r-r_g)}\right)$, building on the work of [14] one can express the velocity of light as a function of r:

$$v\left(r\right)=v\left(\rho \right){\displaystyle \frac{dr}{d\rho }}={\displaystyle \frac{c(r-r_g)}{r}}.$$

(6)

Defining $n\left(r\right)=c/v\left(r\right)$, the resulting equation provides the effective refractive index for light in a Schwarzschild gravitational field:

$$n\left(r\right)={\displaystyle \frac{r}{r-r_g}}.$$

(7)

The effective refractive index $n\left(r\right)$ (7) from the Schwarzschild solution reveals how gravitational fields influence light propagation. As r approaches the Schwarzschild radius $r_g$, $n\left(r\right)$ increases dramatically, indicating stronger gravitational effects. Under the far-field approximation, the expression (7) can be expressed as

$$n\left(r\right)=1+r_g/r.$$

(8)

Using the effective refractive index, one can derive characteristics of GR effects, such as the deflection angle, straightforwardly compared to solving geodesic equations, as demonstrated in Appendix A.

Having established the effective refractive index in the Schwarzschild metric, one can extend this approach to other solutions of the Einstein field equations.

4. The Effective Refractive Index in the Case of a Rotating Object

For practical purposes, the linearized Kerr metric [26] satisfactorily describes the gravitational field around a rotating star or planet. To simplify the problem, one can consider the deflection in the equatorial plane using this linearized Kerr metric:

$$d{s}^2\approx -\left[\left(1-{\displaystyle \frac{r_g}{r}}\right)+{\displaystyle \frac{2r_{g}a}{r}}{\displaystyle \frac{d\phi }{cdt}}\right]c^{2}d{t}^2+{\displaystyle \frac{1}{(1-r_g/r)}}d{r}^2+r^2(d{\theta }^2+d{\phi }^2),$$

(9)

where $a=J/m$ represents the spin parameter of the rotating object, which is a measure of the angular momentum per unit mass m. The multiplier $d\phi /(c·dt)$ denotes a frame-dragging parameter due to the rotating body.

By using the same transformation (3) to convert to the isotropic form and performing the same operations as in the case of the Schwarzschild solution, one derives expressions for the effective refractive index in the case of a rotating object [10]:

$$n(r,\theta )={\displaystyle \frac{r}{r-r_g}}·{\left[1+{\displaystyle \frac{a}{c}}{\displaystyle \frac{r_g}{r-r_g}}{\displaystyle \frac{d\phi }{dt}}\right]}^{-1/2}.$$

(10)

The frame-dragging parameter for a slowly rotating mass in the equatorial plane (at $\theta$ = $\pi /2$) can be obtained by applying the principle of conservation of momentum and energy for photons. These conserved quantities can be expressed in terms of the covariant components of the metric $g_{\mu \nu }$ (with the Greek letter indices taking the time and space coordinates):

$$E=-g_{tt}{\displaystyle \frac{dt}{d\lambda }}-g_{t\phi }{\displaystyle \frac{d\phi }{d\lambda }},$$

(11)

$$L=g_{\phi \phi }{\displaystyle \frac{d\phi }{d\lambda }}+g_{t\phi }{\displaystyle \frac{dt}{d\lambda }}.$$

(12)

where $\lambda$ is the parameter by which the trajectory of light is parametrized. Expressing $d\phi /d\lambda$ and $dt/d\lambda$ from Equations (11) and (12), one derives the relationship

$${\displaystyle \frac{d\phi }{dt}}={\displaystyle \frac{g_{t\phi }E+g_{tt}L}{g_{\phi \phi }E+g_{t\phi }L}}.$$

(13)

By utilizing the relationship between E and L through the impact parameter $b=L/E$, and substituting the components of the Kerr metric, one obtains expressions for the frame-dragging parameter:

$${\displaystyle \frac{d\phi }{dt}}={\displaystyle \frac{r_{g}a+(r-r_g)b}{r^3-r_{g}ab}}c.$$

(14)

Then, up to the first order in a, the refractive index in the equatorial plane becomes

$$n\left(r\right)=1+{\displaystyle \frac{r_g}{r}}+{\displaystyle \frac{r_{g}ab}{r^3}}.$$

(15)

The result (15) shows that the refractive index depends on the spin parameter a. When $a=0$, it implies that the central gravitational body is non-rotating. In such a scenario, the refractive index reduces to the Schwarzschild form (8). Comparing Equations (8) and (15), one deduces that

$$n\left(r\right)=n_{\mathrm{Sch}}\left(r\right)·[1+\delta n_{\mathrm{rot}}\left(r\right)],$$

(16)

where $n_{\mathrm{Sch}}\left(r\right)=1+r_g/r$ is the main term corresponding to the static gravitational field, and $\delta n_{\mathrm{rot}}\left(r\right)=ab/\left[r^2\left(1+r/r_g\right)\right]$ is the correction taking into account the influence of angular momentum within the weak field. The angle of gravitational light deflection corresponding to this refractive index is provided in Appendix B.

5. The Effective Refractive Index in Nonlinear Vacuum Electrodynamics

In classical electrodynamics, the propagation of light through a vacuum is unaffected by external magnetic fields. This is due to the linear nature of Maxwell’s equations [27], which do not predict any interaction between the electromagnetic waves (light) and a static magnetic field. However, when considering the nonlinear electrodynamics of vacuum [16,17,28,29,30,31,32,33,34,35,36,37,38,39,40,41], the interaction between propagating light and external magnetic fields becomes possible due to higher-order corrections, leading to novel phenomena such as vacuum birefringence. Nonlinear electrodynamics extends the classical theory by incorporating higher-order corrections that become significant in strong electromagnetic fields. One such framework is generalized Born–Infeld (BI) electrodynamics [17,35], which introduces nonlinear terms in the Lagrangian to account for these effects. This theory predicts that the vacuum behaves like a nonlinear optical medium with properties dependent on the external field strength.

To understand how light propagates in such a nonlinear medium, consider the Lagrangian density of let us generalized BI electrodynamics:

$$\mathcal{L}={\xi }^2\left(1-\sqrt{1+{\displaystyle \frac{2S}{{\xi }^2}}-{\displaystyle \frac{P^2}{{\xi }^2{\eta }^2}}}\right),$$

(17)

where $S=\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$ and $P=\frac{1}{4}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }$ with $F_{\mu \nu }$ the electromagnetic tensor. The parameters $\xi$ and $\eta$ are constants characterizing the nonlinear properties of the vacuum. Specifically, $\xi$ is related to the maximum possible field strength in BI electrodynamics, while $\eta$ influences the birefringence effect in the presence of an external magnetic field.

In further calculations, we consider the background magnetic field ${\mathbf{B}}_{\mathbf{0}}$ caused by a magnetic dipole. We focus on a simplified case when the light ray travels through the equatorial plane of the magnetic dipole, as illustrated in Figure 3.

The refractive indices for two polarization modes (perpendicular and parallel to the background magnetic field) were derived in Ref. [17] from the eigenvalues of the polarization tensors:

$$n_{\perp }=\sqrt{1+B_0^2/{\xi }^2},$$

(18)

$$n_{\Vert }=\sqrt{1+B_0^2/{\eta }^2}.$$

(19)

For relatively small values of the parameters $B_0^2/{\xi }^2$ and $B_0^2/{\eta }^2$, the effective refractive indices can be approximated as

$$n_{\perp }\approx 1+{\displaystyle \frac{B_0^2}{2{\xi }^2}},$$

(20)

$$n_{\Vert }\approx 1+{\displaystyle \frac{B_0^2}{2{\eta }^2}}.$$

(21)

The components of effective refractive indices, $n_{\perp }$ and $n_{\Vert }$, derived from the nonlinear electrodynamics of vacuum illustrate how strong magnetic fields can alter light propagation. These indices show that the vacuum behaves as a nonlinear optical medium, leading to phenomena such as light deflection in Appendix C.

This approach not only transparentizes complex electromagnetic interactions in extreme conditions but also aligns with the broader purpose of using optical analogies to model and comprehend various physical phenomena.

6. Application on Magnetars

Magnetars are a type of neutron star known for their extremely strong magnetic fields, which can reach up to ${10}^{11}$T on their surface [42,43]. Such fields are sufficient to exhibit the effects predicted by the nonlinear theory of vacuum electrodynamics. In addition to their intense gravitational fields, magnetars present a unique environment where both gravitational and electromagnetic influences must be considered to fully understand the behavior of light in the vicinity of magnetars [44]. The strong magnetic fields of magnetars are capable of inducing significant light bending, birefringence, and other nonlinear optical effects in the vacuum surrounding them. That is, the propagation of light near a magnetar is affected not only by the curvature of spacetime due to gravity but by the nonlinear interactions with the magnetic field as well [32].

By applying the optical medium approach, one can represent the effective refractive index around a magnetar as a combination of contributions from gravity and nonlinear electrodynamics [35]. Specifically, the total effective refractive index $n_{\mathrm{eff}}$ near the surface of a magnetar on the equatorial plane can be expressed as a product of the gravitational influence $n_{\mathrm{gr}}$, and a correction factor $\delta n_{\mathrm{mag}}$ [33] as:

$$n_{\mathrm{eff}}\left(r\right)=n_{\mathrm{gr}}\left(r\right)·[1+\delta n_{\mathrm{mag}}\left(r\right)],$$

(22)

where $n_{\mathrm{gr}}=1+r_{\mathrm{g}}/r$ corresponds to the Schwarzschild refractive index and $\delta n_{\mathrm{mag}}=\left[1+B_0^2/\left(2{\xi }^2\right)\right]r/(r+r_{\mathrm{g}})$ is the correction factor due to the magnetic field. This form emphasizes the non-additive nature of the refractive index while providing a comprehensive view of light propagation near magnetars, considering both spacetime curvature and strong magnetic field effects.

7. Discussion

This paper has demonstrated the utility of the optical medium approach for understanding complex physical phenomena in Einstein’s theory of gravity and nonlinear vacuum electrodynamics. Using optical analogies simplifies the representation and analysis of these phenomena, making them more accessible for understanding and teaching. Additionally, the use of materials such as plexiglass to model gravitational fields in laboratory conditions allows for a qualitative reproduction of gravitational lensing effects making this approach particularly useful for demonstrative experiments.

The optical medium approach is particularly useful for explaining the behavior of light around magnetars, which possess strong magnetic fields. Magnetars create unique conditions where both gravitational and electromagnetic effects are at play. Representing spacetime curvature and nonlinear vacuum effects through refractive indices provides a straightforward description of light behavior in such extreme environments. This approach enhances the understanding of magnetar physics and serves as an effective tool for educating students and young researchers.