Characterisation of circular light rays in a plasma

TL;DR

This work develops a dispersive-plasma generalisation of circular light-ray analysis in axially symmetric, stationary spacetimes by introducing two wave-front potentials $\mathcal{R}_{\pm}$ whose critical points locate circular rays. Light in a non-magnetised plasma obeys a Hamiltonian $\mathcal{H} = \tfrac12\big(g^{\rho\sigma}p_\rho p_\sigma + \omega_p^2\big)$, making rays timelike geodesics of the conformally rescaled metric $\omega_p^2 g_{\mu\nu}$ and introducing frequency dependence via the plasma frequency $\omega_p$. The authors derive domain, transformation, and limiting properties of $\mathcal{R}_{\pm}$ (near horizons, axes, and infinity) and prove existence theorems for circular rays using Brouwer degree, showing that adding a plasma perturbs vacuum circular rays by an even number of extra extrema and saddles; in many cases, at least one circular ray persists if the vacuum count is odd. They illustrate the formalism with Minkowski, Schwarzschild, Kerr, and NUT spacetimes, demonstrating how a plasma can create or destroy circular rays and thereby influence lensing and shadows. The framework generalises vacuum topological results to realistic media and remains valid whether the plasma’s self-gravity is important or negligible, providing a practical tool for predicting plasma-modified light paths near ultracompact objects.

Abstract

It is the purpose of this paper to give a characterisation of circular light rays in a plasma on an axially symmetric and stationary spacetime. We restrict to the case of an unmagnetised, pressure-free electron-ion plasma and we assume that the plasma shares the symmetry of the spacetime. As a main tool we use two potentials, one for prograde and one for retrograde light rays, whose critical points are exactly the circular light rays in the plasma. In the case that the plasma density vanishes, the corresponding equipotential surfaces reduce to the relativistic Von Zeipel cylinders which have been discussed in many papers since the 1970s. In a plasma, the gradients of the potentials give the centrifugal and the Coriolis forces experienced by a light ray, where the plasma has an influence only on the centrifugal force. The introduction of these potentials allows us to generalise topological methods that have been successfully used for proving the existence or non-existence of circular vacuum light rays to the plasma case. The general results are illustrated with examples on Minkowski, Schwarzschild, Kerr and NUT spacetimes.

Figures (6)

• Figure 1: This picture shows the equipotential lines $\mathcal{R}{}_+ = \mathcal{R}{}_- = \mathrm{constant}$ for the plasma density (\ref{['eq:Minkomegap']}) on Minkowski spacetime for $\omega _0 = \sqrt{4/3} \, \omega _c$ in the $( r \, \mathrm{sin} \, \vartheta , r \, \mathrm{cos} \, \vartheta )$-halfplane $\mathcal{U}$. $r_0$ is chosen as the unit on the axes. The circular light rays are marked by black dots. There are infinitely many saddles with local maxima in between. The saddles are minima (stable) in $r$ direction and maxima (unstable) in $\vartheta$ direction. The potential goes, in an oscillatory fashion, to $- \infty$ for $r \, \mathrm{sin} \, \vartheta \to \infty$.
• Figure 2: This picture shows the equipotential lines $\mathcal{R}{}_+ = \mathcal{R}{}_- = \mathrm{constant}$ for the plasma density (\ref{['eq:Monkomegap3']}) on Minkowski spacetime in the $(\rho , z )$-half-plane $\mathcal{U}$, with $\rho _0$ used as the units on the axes. We have chosen $\omega _0$ equal to $\omega _c$ which implies that the strict inequality $\omega _0 > \omega _p (\rho , z )$ holds for all $0 < \rho < \infty$ but not in the limit $\rho \to \infty$. The picture shows that there is exactly one circular light ray, namely a saddle, marked by a black dot. In this case the assumptions of Proposition \ref{['prop:Brouwer']} are violated because it is impossible to find a one-parameter family of compact sets such that the critical points of $\mathcal{R}{}_{\pm}^{\varepsilon}$ are in their interior for all $\varepsilon \in [0, 1]$.
• Figure 3: The picture on the left shows the equipotential lines $\mathcal{R}{}_+ = \mathcal{R}{}_- = \mathrm{constant}$ for the plasma density (\ref{['eq:Schwomegap']}) on Schwarzschild spacetime with $\omega _0 = \sqrt{1.001} \, \omega _c$. We plot $\rho = r \, \mathrm{sin} \, \vartheta$ on the horizontal axis and $z = r \, \mathrm{cos} \, \vartheta$ on the vertical axis, choosing $m$ as the unit on both axes. The region inside the horizon is shown as a black disc. The potential goes to $- \infty$ at the horizon and at infinity, while it goes to zero on the axis; this confirms our general results of Sections \ref{['subsec:hor']}, \ref{['subsec:ax']} and \ref{['subsec:inf']}. There are three circular light rays, marked by black dots: a local maximum of the potential in the equatorial plane and two saddles off the equatorial plane. They all lie on the circle $r = 4 m$ which is marked by a dashed line in the picture on the left. In the picture on the right, this sphere with its three circular light rays is shown in three-space. The two off-equatorial light rays are unstable in $r$-direction and stable in $\vartheta$-direction, while the light ray in the equatorial plane is unstable in both directions. Light rays from the asymptotic region can spiral asymptotically towards these three circular light rays. This is relevant for the construction of the shadow in this spacetime which turns out to be a circular disc, cf. Perlick and Tsupko PerlickTsupko2017, Section VII.
• Figure 4: These pictures show the equipotential lines $\mathcal{R}{}_+ = \mathrm{constant}$ (on the left) and $\mathcal{R}{}_- = \mathrm{constant}$ (on the right) for the plasma density (\ref{['eq:Kerr1omegap']}) on Kerr spacetime with $a = 3m/4$ and $\omega _0 = \sqrt{1.001} \,\omega _c$. On the horizontal axis we plot $\rho = r \, \mathrm{sin} \, \vartheta$ and on the vertical axis we plot $z = r \, \mathrm{cos} \, \vartheta$, using $m$ as the unit on both axes. The region inside the horizon is shown as a black disc, in the picture on the right the ergoregion is shown in grey. There are three corotating light rays in the domain of outer communication, marked by black dots in the picture on the left, a local maximum of the potential in the equatorial plane and two saddles off the equatorial plane. The saddles are maxima (unstable) in the $r$ direction and minima (stable) in the $\vartheta$ direction. This is similar to our Schwarzschild example shown in Fig. \ref{['fig:Schwarzschild']}. For the potential $\mathcal{R}{}_-$ we show in the picture on the right the equipotential lines only on $\mathcal{U}{}^{\mathrm{out}}$. There are no circular light rays for this particular plasma density in the ergoregion, so we left $\mathcal{U}{}^{\mathrm{erg}}$ in the picture grey. Outside of the ergoregion, we read from the picture that there is exactly one circular light ray, and that it is a saddle, very similar to the vacuum case.
• Figure 5: This picture shows the equipotential lines $\mathcal{R}{}_- = \mathrm{constant}$ for the plasma density (\ref{['eq:Kerr2omegap']}) on Kerr spacetime with $a = m$ and $\omega _0 = 0$. Again, we plot $\rho = r \, \mathrm{sin} \, \vartheta$ on the horizontal axis and $z = r \, \mathrm{cos} \, \vartheta$ on the vertical axis, choosing $m$ as the unit on both axes. The regions below the (degenerate) horizon and outside of the ergoregion are shown in black, i.e., only the ergoregion is displayed. There are two circular light rays in the ergoregion, marked by black dots in the picture. The inner one is a local maximum, the outer one is a saddle which is a minimum (stable) in $r$ direction and a maximum (unstable) in the $\vartheta$ direction. Note that in comparison to the vacuum light rays in (Schwarzschild or) Kerr spacetime the stability properties of the saddle are reversed. In this case the assumptions of Proposition \ref{['prop:Brouwer']} are indeed satisfied, but this was not implied by our general results because the horizon is degenerate.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2