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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.