Report

Light bending in radiation background Jin Young Kim (Kunsan National University) Based on Kim and T. Lee, JCAP 01 (2014) 002 (arXiv:1310.6800); Kim, JCAP 10 (2012) 056 (arXiv:1208.1319); Kim and T. Lee, JCAP 11 (2011) 017 (arXiv:1101.3433); Kim and T. Lee, MPLA 26, 1481 (2011) (arXiv:1012.1134). Outline • Nonlinear property of QED vacuum • Trajectory equation • Bending by electric field • Bending by magnetic field • Bending in radiation background • Summary Motivation • Light bending by massive object is a useful tool in astrophysics : Gravitational lensing • Can Light be bent by electromagnetic field? • At classical level, bending is prohibited by the linearity of electrodynamics. • Light bending by EM field must involve a nonlinear interaction from quantum correction. • The box diagram of QED gives such a nonlinear interaction : Euler-Heisenberg interaction (1936) Non-trivial 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 • Euler-Heisenberg Lagrangian: low-energy effective action of multiple photon interactions • 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 c n 1 • Strong electric or magnetic field can cause a materiallike behavior by quantum correction. Velocity shift and index of refraction • In the presence of electric field, the correction to the speed of light is given by a 14 : perpendicular mode (photonpolarization plane(u, E)) a 8 : parallelmode (photonpolarization plane(u, E)) • For magnetic field, E cB • Index of refraction • If the index of refraction is non-uniform, light ray can be bent by the gradient of index of refraction. Light bending by sugar solution • Place sugar at the bottom of container and pour water. • As the sugar dissolve a continually varying index of refraction develops. • A laser beam in the sugar solution bends toward the bottom. Snell’s law 1 sin 1 n2 v1 sin 2 n1 v2 n1 n2 2 n1 n2 n1 n2 n3 n1 n2 n3 Differential bending by non-uniform refractive index • In the presence of a continually varying refractive index, the light ray bends. • Calculate the bending by differential calculus in geometric optics 1 n1 n 2 n2 sin 1 n2 1 : Snell's law sin 2 n1 2 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 index of refraction is small, approximate the trajectory equation to the leading order (photon from x to x ) ds dx : leading order Bending by spherical symmetric electric charge photonincomingfrom x with impactparameterb • Total bending angle can be obtained by integration with boundary condition Bending by charged black hole • Consider a charged non-rotating black hole 1 b • Constraint on black hole • Restore the physical constants • Parameterize the charge as 1 b4 Order-of-magnitude estimation • Black hole with ten solar mass • Since the calculation is based on flat space time, impact parameter should be large enough ( 1, a 14: mode) ( 0.1, a 14: mode) • Ratio of bending angles at (for heavier BH, the relative bending becomes weaker ) Light bending by electrically charged BHs seems not negligible compared to the gravitational bending. Bending by magnetic dipole • Contrary to Coulomb case, the bending by a magnetic dipole depends on the orientation of dipole relative to the direction of the incoming photon. • Locate the dipole at origin. • Take the direction of incoming photon as +x axis. • Define the direction cosines of dipole relative to the incoming photon. z M x Mˆ y Bending by magnetic dipole y b B x r h z v Bending angles y b B x r h v z boundaryconditions: y() b, z() 0 ; y() z() 0 bending angles: h y(), v z() M2 6 b Special cases i) z direction, passing the equator z b 0, 1 r B y z x M Mˆ x y Special cases ii) -x direction (parallel or anti-parallel) y B b r x 1, 0 Special cases iii) axis along +y direction, light passing the north pole y B r b x 0, 1 z • The gradient of index of refraction is maximal along this direction, giving the maximal bending Order-of-magnitude estimation • Parameterize the impact parameter • Maximal possible bending angles ( magnetized NS with solar mass BS 109 T, a 14, 1) for strongly g 0.59rad; m 1.4 104 rad • Up to BS 109 T , the bending by magnetic field can not dominate the gravitational bending. • 10 BS 1013 T , a 14, g 5.9 102 rad; m 1.4 102 rad Validity of Euler-Heisenberg action • Critical values for vacuum polarization BC 2 3 m 2c 2 m c 4.4 109 T ; EC 1.3 1018 V/m e e • Screening by electron-positron pair creation above the critical field strength • Since the Euler-Heisenberg effective action is represented as an asymptotic series, its application is confined to weak field limits. • When the magnetic field is above the critical limit, the calculation is not valid. Light bending under ultra-strong EM field • Analytic series representation for one-loop effective action from Schwinger’s integral form [Cho et al, 2006] • Index of refraction Upper limit on the magnetic field • No significant change of index of refraction by ultrastrong electric field. • B-field on the surface of magnetar: 1011 T • Physical limit to the B-field of neutron star: 1012 1014 T • To be consistent with one-loop B / BC / 430 • Up to the order of 1012 T (B / BC 200) , the index of refraction is close to one Light bending under ultra-strong magnetic field • Photon passing the equator of the dipole • Index of refraction z b • Trajectory equation r x • Bending angle B y Order-of-magnitude estimation • Power dependence • Maximal possible bending angles ( magnetized NS of solar mass g 0.59rad 1) for strongly m 1.8 102 rad for BS 1011T m 0.18 rad for BS 1012 T • ( 2 ) g 0.3rad m 2.3102 rad for BS 1012 T Speed of light in general non-trivial vacua • Light cone condition for photons traveling in general non-trivial 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 sphericalpolarcoordinatesystem • Velocity shift averaged over polarization Bending by a spherical black body • 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: Order-of-magnitude estimation • 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. Dependence on the impact parameter • Dependence on impact parameter is imprinted by the dilution of energy density How to observe? Power dependence • Measure the total bending angles for different values of the impact parameter (may be possible by extraterrestrial observational facilities) • Check the power dependence by fitting to Birefringence • The bending of perpendicular polarization is 1.75(14/8) times larger than the bending of parallel polarization. • Even in the region where the bending by magnetic field is weak, by eliminating the overall gravitational bending, the polarization dependence can be tested if the allowed precision is sufficient enough. How to observe? Neutron star binary system 0 T / 2 • Use the neutron star binary system with nondegenerate star (<100). • Assume the two have the same mass. • Bending angles at time t=0 and t=T/2 are the same if we consider only the gravitational bending. • The bending angle will be different by magnetic field