Relativistic Hall Effect Research Articles

Spin Hall effects and the localization of massless spinning particles

The spin Hall effects of light represent a diverse class of polarization-dependent physical phenomena involving the dynamics of electromagnetic wave packets. In a medium with an inhomogeneous refractive index, wave packets can be effectively described by massless spinning particles following polarization-dependent trajectories. Similarly, in curved spacetime the gravitational spin Hall effect of light is represented by polarization-dependent deviations from null geodesics. In this paper, we analyze the equations of motion describing the gravitational spin Hall effect of light. We show that these equations are a special case of the Mathisson-Papapetrou equations for spinning objects in general relativity. This allows us to use several known results for the Mathisson-Papapetrou equations, and apply them to the study of electromagnetic wave packets. We derive conservation laws, we discuss the limits of validity of the spin Hall equations, and we study how the energy centroids of wave packets, effectively described as massless spinning particles, depend on the external choice of a timelike vector field, representing a family of observers. In flat spacetime, the relativistic Hall effect and the Wigner(-Souriau) translations are recovered, while our equations also provide a generalization of these effects in arbitrary spacetimes. We construct a large class of wave packets that can be described by the spin Hall equations, but also find its limits by giving examples of wave packets which are more general and are not described by the spin Hall equations. Lastly, we examine the assumption that electromagnetic wave packets are massless. While this is approximately true in many contexts, it is not exact. We show that failing to carefully account for the limitations of the massless approximation results in the appearance of unphysical ``centroids'' which are nowhere near the wave packet itself.

Graphene gets bent

Graphene gets bent

Modification of magnetohydrodynamic waves by the relativistic Hall effect.

This study shows that a relativistic Hall effect significantly changes the properties of wave propagation by deriving a linear dispersion relation for relativistic Hall magnetohydrodynamics (HMHD). Whereas, in nonrelativistic HMHD, the phase and group velocities of fast magnetosonic wave become anisotropic with an increasing Hall effect, the relativistic Hall effect brings upper bounds to the anisotropies. The Alfvén wave group velocity with strong Hall effect also becomes less anisotropic than the nonrelativistic case. Moreover, the group velocity surfaces of Alfvén and fast waves coalesce into a single surface in the direction other than near perpendicular to the ambient magnetic field. It is also remarkable that a characteristic scale length of the relativistic HMHD depends on ion temperature, magnetic field strength, and density while the nonrelativistic HMHD scale length, i.e., ion skin depth, depends only on density. The modified characteristic scale length increases as the ion temperature increases and decreases as the magnetic field strength increases.

Transverse relativistic effects in paraxial wave interference

We consider relativistic deformations of interfering paraxial waves moving in the transverse direction. Owing to superluminal transverse phase velocities, noticeable deformations of the interference patterns arise when the waves move with respect to each other with non-relativistic velocities. Similar distortions also appear on a mutual tilt of the interfering waves, which causes a phase delay analogous to the relativistic time delay. We illustrate these observations by the interference between a vortex wave beam and a plane wave, which exhibits a pronounced deformation of the radial fringes into a fork-like pattern (relativistic Hall effect). Furthermore, we describe an additional relativistic motion of the interference fringes (a counter-rotation in the vortex case), which becomes noticeable at the same non-relativistic velocities.

Relativistic Hall Effect

We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsic angular momentum (AM). When observed in a moving reference frame, the centroid of the object undergoes an AM-dependent transverse shift. This is the relativistic analogue of the spin-Hall effect, which occurs in free space without any external fields. Remarkably, the shifts of the geometric and energy centroids differ by a factor of 2, and both centroids are crucial for the Lorentz transformations of the AM tensor. We examine manifestations of the relativistic Hall effect in quantum vortices and mechanical flywheels and also discuss various fundamental aspects of this phenomenon. The perfect agreement of quantum and relativistic approaches allows applications at strikingly different scales, from elementary spinning particles, through classical light, to rotating black holes.

GENERALIZED RELATIVISTIC MAGNETOHYDRODYNAMIC EQUATIONS FOR PAIR AND ELECTRON-ION PLASMAS

We derived one-fluid equations based on a relativistic two-fluid approximation of e$^\pm$ pair plasma and electron-ion plasma to reveal the specific relativistic nature of their behavior. Assuming simple condition on the relativistic one-fluid equations, we propose generalized relativistic magnetohydrodynamic (RMHD) equations which satisfy causality. We show the linear analyses of these equations regarding various plasma waves to show the validity of the generalized RMHD equations derived here and to reveal the distinct properties of the pair plasma and electron-ion plasma. The distinct properties relate to (i) the inertia effect of electric charge, (ii) the momentum of electric current, (iii) the relativistic Hall effect, (iv) the thermal electromotive force, and (v) the thermalized energy exchange between the two fluids. Using the generalized RMHD equations, we also clarify the condition that we can use standard RMHD equations and that we need the distinct RMHD equations of pair and electron-ion plasmas. The standard RMHD is available only when the relative velocity of the two fluids is nonrelativistic, a difference of the enthalpy densities of the two fluids is much smaller than the total enthalpy density, and the above distinct properties of the pair/electron-ion plasma are negligible. We discuss a general relativistic version of the equations applicable to the pair and electron-ion plasmas in black hole magnetospheres. We find the effective resistivity due to shear of frame ragging around a rotating black hole.