powerpoint presentation · 2012. 6. 6. · frolov/dp/0199692297 (based on v.f. & a.shoom,...
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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.