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Transcript
Light bending by a black body radiation Jin Young Kim (Kunsan National Univ.) 10th CosPA Meeting, Hawaii J.Y. Kim and T. Lee, arXiv:1310.6800[hep-ph] Outline of the talk • Nonlinear property of QED vacuum • Velocity shift of light in radiation background • Trajectory equation based on geometric optics • Calculate the bending angle of a light ray when the energy density of radiation emitted by a black body dilutes spherically and cylindrically. • Assuming a neutron as an isothermal black body, estimate the order of magnitude for the bending angle and compare it with the bending by other sources. Nontrivial QED vacua • In classical electrodynamics vacuum is defined as the absence of charged matter. • In QED vacuum is defined as the absence of external currents. • VEV of electromagnetic current can be nonzero in the presence of non-charge-like sources. electric or magnetic field, temperature, … • nontrivial vacua = QED vacua in presence of noncharge-like sources • If the propagating light is coupled to this current, the light cone condition is altered. • The velocity shift can be described as the index of refraction in geometric optics. Nonlinear Properties of QED Vacuum • Strong electric or magnetic field can cause a materiallike behavior by quantum correction. • Euler-Heisenberg Lagrangian: low-energy effective action of multiple photon interactions Speed of light under electric and magnetic field • In the presence of a background EM field, the nonlinear interaction modifies the dispersion relation and results in a change of speed of light. c n c 1 • the correction to the speed of light a 14 : perpendicular mode (photon polarizati on plane (u, E)) a 8 : parallel mode (photon polarizati on plane (u, E)) Speed of light in general nontrivial vacua • Light cone condition for photons traveling in general nontrivial QED vacua [Dittrich and Gies (1998)] effective action charge • For small correction, the propagation direction , and average over • For EM field, two-loop corrected velocity shift agrees with the result from Euler-Heisenberg lagrangian Light velocity in radiation background • Light cone condition for non-trivial vacuum induced by the energy density of electromagnetic radiation null propagation vector U (1,1,0,0) in sphericalpolar coordinatesystem • Velocity shift averaged over polarization Differential bending by non-uniform refractive index • When the index of refraction is non-uniform, light ray can be bent by the gradient of index of refraction. • Calculate the bending by geometrical optics. 1 n1 n2 2 sin 1 n2 v1 : Snell' s law sin 2 n1 v2 1 2 , n2 n1 n sin( 2 ) n 1 cot 2 1 sin 2 n1 n dr 1 tan tan | n dr | n n n n Trajectory equation u0 n u u • When the correction to the index of refraction is small, approximate the trajectory equation to the leading order. (photon from x to x ) ds dx : leading order Bending by electric field [Kim and Lee, MPLA (2010)] y b Q x • Total bending angle can be obtained by integration with boundary condition Bending by magnetic field [Kim and Lee, JCAP(2011)] • Contrary to Coulomb case, the bending by a magnetic dipole depends on the orientation of dipole relative to the direction of the incoming photon. • Maximal bending for a ray passing the pole y B r b x z Bending by a spherical BB • As a source of lens, consider a spherical BB emitting energy in steady state. • In general the temperature of an astronomical object may different for different surface points. • For example, the temperature of a magnetized neutron star on the pole is higher than the equator. • For simplicity, consider the mean effective surface temperature as a function of radius assuming that the neutron star is emitting energy isotropically as a black body in steady state. Index of refraction as a function of radius • Energy density of free photons emitted by a BB at temperature T (Stefan’s law) • Dilution of energy density: • Index of refraction, to the leading order, • can be replaced by (critical temperature of QED) Trajectory equation • Take the direction of incoming ray as +x axis on the xy-plane. • Index of refraction: • Trajectory equation: • Boundary condition: Bending angle • Leading order solution with • Bending angle from y b x Bending by a cylindrical BB • Take the axis of cylinder as z-axis. • Energy density: • Index of refraction: • Trajectory equation: • Solution: • Bending angle: Dependence on the impact parameter • Dependence on impact parameter is imprinted by the dilution of energy density Order-of-magnitude estimation for a neutron star • Mass: • Surface magnetic field: • Surface temperature: • The magnetic bending is bigger than the thermal bending for , while the thermal bending is bigger than the magnetic bending for . • However, both the magnetic and thermal bending angles are still small compared with the gravitational bending.