Light bending in radiation background Based on Kim and T. Lee, JCAP 01 (2014) 002 (arXiv:1310.6800);...
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Transcript of Light bending in radiation background Based on Kim and T. Lee, JCAP 01 (2014) 002 (arXiv:1310.6800);...
Light bending in radiation background
Based onKim 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).
Jin Young Kim (Kunsan National Univer-sity)
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 lin-earity of electrodynamics.
• Light bending by EM field must involve a nonlinear interaction from quantum correction.
• The box diagram of QED gives such a nonlinear in-teraction : 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 exter-nal 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 non-
charge-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 non-linear interaction modifies the dispersion relation and results in a change of speed of light.
• Strong electric or magnetic field can cause a mate-rial-like behavior by quantum correction.
1
cnc
Velocity shift and index of refraction
• In the presence of electric field, the correction to the speed of light is given by
B E c
E))(u, planeonpolarizati(photon modelar perpendicu :14 a
• For magnetic field,
• Index of refraction
• If the index of refraction is non-uniform, light ray can be bent by the gradient of index of refrac-tion.
E))(u, planeonpolarizati(photon mode parallel :8 a
Light bending by sugar solution
• Place sugar at the bottom of container and pour wa-ter.
• 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
1n
2n
3n
321 nnn
sin
sin
2
1
1
2
2
1
v
v
n
n
1n
2n
21 nn
1
2
Differential bending by non-uniform refractive in-dex
• In the presence of a continually varying refrac-tive index, the light ray bends.
• Calculate the bending by differential calculus in geometric optics
1cot1sin
)sin(
12
2
2
n
n
||1
tantan rdnnn
rdn
n
n
law sSnell' : sin
sin
2
1
1
2
2
1
n
n
nnn 1221 ,
1
2
1n
2nn
Trajectory equation
• When the index of refraction is small, approxi-mate the trajectory equation to the leading order
order leading : dxds
) to from(photon xx
0un
uu
Bending by spherical symmetric electric charge
bx parameter impact with from incomingphoton
• Total bending angle can be obtained by integra-tion with boundary condition
Bending by charged black hole
• Consider a charged non-rotating black hole
b
1 4
1
b
• Constraint on black hole
• Restore the physical constants
• Parameterize the charge as
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
mode) : 14,1( a
• Ratio of bending angles at
Light bending by electrically charged BHs seems not negligible compared to the gravitational bend-ing.
mode) : 14,1.0( a
(for heavier BH, the relative bend-ing becomes weaker )
Bending by magnetic dipole
• Contrary to Coulomb case, the bending by a mag-netic 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.
M̂
y
x
z
M
Bending by magnetic dipole
x
z
y
vh
br
B
Bending angles
0)()( ; 0)( ,)( :conditionsboundary zyzby
x
z
y
vh
br
B
)( ),( :angles bending zy vh
6
2
b
M
Special cases
b
y
x
Br
z
1 ,0
i) z direction, passing the equa-tor
M̂
y
x
z
M
Special cases
0 ,1
ii) -x direction (parallel or anti-parallel)
b
y
x
rB
Special cases
1 ,0
y
x
B
r
z
b
iii) axis along +y direction, light passing the north pole
• The gradient of index of refraction is maximal along this direction, giving the maximal bending
Order-of-magnitude estimation
• Maximal possible bending angles for strongly magnetized NS with solar mass
• Parameterize the impact param-eter
rad 104.1 rad;59.0 ,14 T,10 4mg
9S
aB
) 1(
• Up to , the bending by magnetic field can not dominate the gravitational bending.
T109S B
rad 104.1 rad; 109.5 ,14 T,10 2m
2g
13S
aB
•
10
Validity of Euler-Heisenberg action
• Critical values for vacuum polarization
• Screening by electron-positron pair creation above the critical field strength
V/m103.1E T;104.4B 1832
C9
22
C e
cm
e
cm
• Since the Euler-Heisenberg effective action is rep-resented 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 ultra-strong electric field.
• Physical limit to the B-field of neutron star:
T1010 1412
• B-field on the surface of magnetar:
T1011
• Up to the order of , the index of refraction is close to one
)200/( T1012 CBB
• To be consistent with one-loop
430// CBB
Light bending under ultra-strong magnetic field
• Photon passing the equator of the dipole • Index of refraction
• Trajectory equation
b
y
x
Br
z
• Bending angle
Order-of-magnitude estimation
• Maximal possible bending angles for strongly magnetized NS of solar mass
• Power dependence
T10for rad 108.1 11S
2m B
) 1(
rad59.0 g
T10for rad 18.0 12Sm B
•
) 2(
rad3.0 g T10for rad 103.2 12S
2m B
Speed of light in general non-trivial vacua
• Light cone condition for photons traveling in general non-trivial QED vacua
effective action charge
[Dittrich and Gies (1998)]
• For small correction, , and average over the propagation direction
• For EM field, two-loop corrected velocity shift agrees with the result from Euler-Heisenberg la-grangian
Light velocity in radiation background
• Light cone condition for non-trivial vacuum in-duced by the energy density of electromagnetic radiation
null propagation vec-tor
system coordinatepolar sphericalin )0,0,1,1(U
• 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 ob-ject may different for different surface points.
• For example, the temperature of a magnetized neu-tron 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 den-sity:
• 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
b
y
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
• Surface temperature:
• Surface magnetic field:
• Mass:
• 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 gravita-tional bending.
Dependence on the impact parameter
• Dependence on impact parameter is imprinted by the dilution of energy density
How to observe?
• The bending of perpendicular polarization is 1.75(14/8) times larger than the bending of par-allel polarization.
• Even in the region where the bending by mag-netic field is weak, by eliminating the overall gravitational bending, the polarization depen-dence can be tested if the allowed precision is sufficient enough.
Birefringence
Power dependence
• Measure the total bending angles for different values of the impact parameter (may be possi-ble by extraterrestrial observational facilities)
• Check the power dependence by fitting to
How to observe?
• Use the neutron star binary system with nonde-generate 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
Neutron star binary system
0
2/T