Gravitational lensing in a warm plasma

TL;DR

Gravitational lensing in a warm plasma extends Synge's isotropic dispersive-medium framework to a non-negligible temperature regime by introducing a warm-plasma refractive index $n^2$ that depends on the density through the plasma frequency $\omega_p$ and on temperature via $\chi$. A general Hamiltonian $\mathcal{H}=\frac{1}{2}[g^{\beta\delta}p_{\beta}p_{\delta}-(n^2-1)(p_{\gamma}V^{\gamma})^2]$ governs ray trajectories on axially symmetric spacetimes, and the authors specialize to axisymmetric stationary geometries, deriving equatorial-plane orbit equations and showing that corotating motion ($u=0$) simplifies the frequency dependence while radial flow ($u\neq 0$) yields a quartic dispersion in $p_r$. They obtain analytic expressions for deflection angles $\alpha$ and shadows in Schwarzschild and Kerr spacetimes, including the effect of the temperature gradient on the photon-sphere condition, and illustrate these with four concrete examples (static and infalling plasma on Schwarzschild and Kerr) to map the regime of validity of the warm-plasma approximation. By comparing the warm-plasma results to exact kinetic-theory expressions, the work clarifies when the approximation is trustworthy (typically $\chi\ll 1$) and highlights how plasma velocity alters lensing beyond the cold-plasma limit. The study provides practical analytic tools for modeling lensing in hot, non-magnetized plasmas and paves the way for extensions to magnetized media and more general spacetimes.

Abstract

Analytical studies of light bending in a dispersive medium near compact objects, e.g., black holes or neutron stars, are most challenged by a suitable definition of the medium. The most realistic model would be a hot magnetized plasma. In such a medium, however, an analytical description of light rays is very difficult. Therefore, usually an isotropic dispersive medium is assumed in analytical calculations. While it is possible to formulate equations for a general refractive index, which some studies do, most attention in the literature is given to the particular case of a cold, non-magnetized electron-ion plasma. Whereas this model covers many astrophysically relevant situations, there are indications that in some cases the plasma temperature is so high that the approximation of a cold plasma is no longer valid. For this reason, we consider in this paper a warm, non-magnetized electron-ion plasma, where the temperature is not set equal to zero but assumed to be small enough, such that relevant equations can be linearized with respect to it. After discussing the general equations for light rays in such a medium on a general-relativistic spacetime, we specify to the axially symmetric and stationary case which includes the spherically symmetric and static case. In particular, we calculate the influence of a warm plasma on the bending angle. In the spherically symmetric and static case, we also calculate the shadow in a warm plasma. We illustrate the general results with a static (respectively corotating) and an infalling warm plasma on Schwarzschild and Kerr spacetimes.

Figures (22)

• Figure 1: Refractive index squared of the warm plasma as a function of $\omega/\omega_p$ for different values of $\chi$. The cold plasma case ($\chi=0$) is shown by the dotted curve. The plain solid curves show situations when $\chi=$ 0.1, 0.25, 0.5, and 0.65 (from the cold plasma case bottom up).
• Figure 2: Comparison of refractive indices for a warm plasma calculated when fractional and expanded forms were used. The solid curves correspond to formula (\ref{['n_warm']}), while dashed curves shown results when expression (\ref{['n_warm_exp']}) was applied. Situations when $\chi=$ 0.1, 0.25, 0.5, and 0.65 are shown (from bottom up). Notice a logarithmic scale of the $x$-axis.
• Figure 3: Radial profiles of temperature $\chi$ for various choices of $C_\chi$ with $u(r)=0$ in the Schwarzschild spacetime. Values of $C_\chi$ span between 1 and 1.2, namely, from the bottom yellow curve up, $C_{\chi}$ equals 1, 1.02, 1.05, 1.1, 1.15, and 1.2. The dashed vertical line shows $r=2M$.
• Figure 4: Cut-off frequency $\omega _{\mathrm{co}}$ as a function of $r$, calculated with the exact formula (\ref{['eq:exactco']}) (solid, red) and with the warm-plasma approximation (\ref{['eq:frco']}) (dashed, black). We have chosen $k = 1.45$ and $C_{\chi} =1$.
• Figure 5: Index of refraction $n(r)$ for a plasma density (\ref{['eq:omegapk']}) with $k = 1.45$, temperature parameter $C_{\chi} = 1$ and for light rays with frequency constant $\omega _0^2 = \omega _c ^2 /10$ from the exact formula (solid, red) and from the warm-plasma approximation (dashed,black).