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