http://www.amazon.ca/Introduction-Black-Physics-Valeri-Frolov/dp/0199692297

(Based on V.F. & A.Shoom, Phys.Rev. D84, 044026 (2011); and V.F. & A.Shoom gr-qc/1205.4479 (2012) )

XIV International Conference Geometry, Integrability and Quantization

June 8 - 13, 2012 Varna

Main goal is to study how the spin of a photon affects its

motion in the gravitational field.

(WKB approach for a massless field with spin)

An example of gravitational spin-spin interaction: Asymmetry of Hawking radiation for polarized light

Gravito-electromagnetism

Weak field limit:

2 2 2 2

2 2

3

12

4(1 2 ) ( ) (1 2 ) ,

, ,

1Transverse gauge condition: 0

ds c dt A dx dt dxc c c

GM G J xA

r c r

Ac t

12

1 12 2

1 12 2

1Define: ,

Then one has:

1, 0,

1 44 , ,

0

E A B Ac t

E B Bc t

GE G B E j

c t c

jt

For a particle mo

2

tion:

,d p v

F F E Bdt c

g B

Particle with spin Particle with magnetic dipole moment

Maxwell equations Dirac (Pauli) equation

Geometric optics (WKB) approximation

GRAVITY Electromagnetism

Stern-Gerlach Experiment

2 2

1

2

00

2 2 2 3

0 1 0 10

0 0 0

( )( , ; ) exp

2 2

( ) ( )( , ; )

( , ;

exp

) ( , ;

2 2

) (

2 24

, ;

2

)

z

x zz x z

m x xmK x x t

i t i t

m

K

z z B z z t B tmK z z t

i

x x t K x x t K z

p pH

t i t i

H H B z

z t

m

i m

Hsu, Berrondo, Van Huele “Stern-Gerlach dynamics of quantum propagators” Phys. Rev. A 83, 012109 (2011)

0

2 2 2 3

0 1 0

1

1

2

0

, ,2

/ , ,

( , ; ) exp

( ) ( )( ) ,

2 2 12

( / )

z

z

z

z

z

z z

z

m z z B z z t B tS p dz Hd

pH z B

m

z p m p

K z

t

z t

m

iS

t

2

00 0 0

Magnetic moment of the electron: ,2 2

By applying formal WKB to the Pau

( )

li equation with this , one

( , ; ) exp(

would g

2

et

/ ),

B B

e

m z zK z z t iS S

m

t

g e

Lessons

(i) In the exact solution for a wave packet there exists correlation between orientation of spin and spatial trajectory of electron;

(ii) At late time the up and down spin wave packets are moving along classical trajectories;

(iii) Formal WKB solution represents the motion of the `center of mass’ of two packets

2

10

2

0 1

2

1

Condition when terms with become important

c

; ;2

2 (

an be written as

)

or, equivalently, /( )

B tL x x Vt z

m

z L m x x

L mV

B t

B

To obtain a correct long time asymptotic behavior of the wave packet one needs: (i) to `diagonalize’ the field equations; (ii) to `enhance’ spin-dependent term (iii) include it in the eikonal function

Spinoptics in gravitational field

(i) Spin induced effects (ii) Many-component field (iii) Helicity states (iv) Massless field (v) Gauge invariance

Riemann-Silberstein vector: F E iH

3( , ) ( ) ( ) ( )

( ) are the amplitudes of right and left circularly

polarized EM waves with vector

Consider purely right polarized monochromatic wave

i t ikr i t ikrF t r d k e k a k e a k e

a k

k

F

( , ) ( )i tt r e r +F

Consider complex Maxwell field .

(Anti-)Self-dual field * .

For monochromatic wave ~ .i t

i

e

F

F F

F F

In a curved ST there is a unique splitting of a complex

Maxwell field into its self-dual and anti-self-dual parts.

As a result the right and left polarized photons are well

defined and the helicity is preserved.

Maxwell equations in a stationary ST

2 2

( )

2

2

2

- ultrasta

, ,

tionary metric

( )

t

ji ii ij

hds hdS

dS dt g dx dx dx

( )i

i i iit t t q x g g qg

Since Maxwell eqns are conformally invariant it is convenient to perform calculations in the ultrastationary metric

3+1 form of Maxwell equations 2 0 2

0

i i ij ij

i i ij ijE F B F D h F H h F

j ij ij i j j i

i i ijD E H g B H E g E g

k ij ijk

ij ijk kB e B H e H

[ ] [ ] [ ]i ijk

j k DC E g HA B C B EB H ge A

div 0 curl , div 0 curlB E B D H D

1 1E [( ) ( )] [ ], E div 0

8 4E D B H V E H V

Master equation for c-polarized light

Riemann-Silberstein vectors: F E iH G D iB

*, *

* *

i t i t i t i t

i t i t i t i t

E e e H e e

D e e B e e

E E H H

D D B B

i i E H D BGF

div 0 curl [ ]i g G G GF F F

curl [ ]i g F F F

“Standard” geometric optics

-1Small dimensionless parameter: =( )

is characteristic length scale of the problem

Geometric optics ansatz i SefF

( )

There is a phase factor ambiguity

, ( ) /i xf e f S S x

1Exact equation: curlLf f

[ ]

, ,

Lf f i n f

n p g p S

Standard Geometric Optics

1 2

0 1 2... f f f f

0

1

1 0

2

2 1

[ curl ]

[ curl ] 0

L f

L f f

L …f f

2

0

2

is a degenerate operator. Condition of existence of solutions

of eqn implies the ei

Effective Hamiltonian is:

1 1

konal equation (=

( ) ( ) ( )( )2 2

10 )

i ij

i i i j jH x p p g p g

L f

L

S

p g

g

Characteristic scale : | | 2

4 / | |

F F

F

L L g

L g

(i) Light ray is a 4D null geodesic

(ii) Vector of linear polarization is 4D

parallel transported

4 - D point of view :

To fix an ambiguity in the choice of the phase,

we require that vectors of the basis ( , , *) along rays

are Fermi transported;

( , ) ( , ) ,

As a result the lowest order polar

n n n

n m m

a a n a w w a n w n F

ization dependent

correction is included in the phase.

0

( )

0 ( ) 1 ,

curl2

i S xx dx

f e S x g dmx d

g g g

F

Modified Geometric Optics

Modified geometric optics

cur

curl ,

l [curl

2

[

]

] 0,

2

n p g p S g g g

L f f

iL f

i n f

f g f

2

det 0 ( , ) 1

1 1( ) ( ) ( )( ),

2

( ) 1,

2

i ij

i i ji jH x p p g p pg

n g

g

L n S

2

2

1

, curl curlcurl ,

1 ( , ), (2 )

D x dxf f g g

d d

d dxg M

d d

2

2

Null curves are solutions of these equations. For =0,

null geodesics

4D form of the effective equations (in the ultrastationary metric)

.

D x DxF

d d

Polarized Photon Scattering in Kerr ST

(i) How does the photon bending angle depend on its polarization?

(ii) How does the position of the image of a photon arriving to an observer depend on its polarization? (iii) How does the arrival time of such photons depend on their polarization?

All calculations are for an extremal Kerr BH (M=a=1)

Capture Domain (equatorial plane)

Shift in the bending angle: (angle) /

1, cos /10, / 6, / 3, / 2, | | /( ) 7.0a M L M

Image splitting

Time Delay

Effect of the second order in ;

Fermat principle in gravitational field

[Landau & Lifshits "Classical Field Theory";

Brill in "Relativity, Astrophysics and Cosmology,

1973]

Rainbow effect for BH shadow

(1) Frequency dependence of the shadow position for circular polarized light; (2) For given frequency shadow position depends on the polarization

SUMMARY

(1) Standard GO picture: In the Kerr ST a linearly polarized photon moves a null geodesic and its polarization vector is parallel propagated.

(2) Modified GO picture: Linear polarized photon beam splits into two circular polarized beams.

(3) Right and left polarized photons have different trajectories.

(4) In a stationary ST their motion can be obtained by introducing frequency dependent effective metric.

(5) Effects: Shift in bending angle, shift of images, time delay.