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 104 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 102 rad; m  1.4 102 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 102 rad for BS  1011T
m  0.18 rad for BS  1012 T
•
(  2 )
g  0.3rad
m  2.3102 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