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March 10, 2005 13:37 WSPC/146-MPLA 01615
Modern Physics Letters AVol. 20, No. 8 (2005) 597–603c© World Scientific Publishing Company
GRAVITATIONAL FARADAY EFFECT INDUCED BY
HIGH-POWER LASERS
PEIYONG JI∗, HUA ZHOU and H. Q. LU
Department of Physics, Shanghai University, Shanghai, 200436, P. R. China∗pyji@staff.shu.edu.cn
Received 18 October 2004
Gravitational field produced by high-power laser is calculated according to the linearizedEinstein field equation in weak field approximation. Gravitational Faraday effect of elec-tromagnetic wave propagating in the above gravitational field is studied and the rotationangle of polarization plane of electromagnetic wave is derived. The result is discussedand estimated in the condition of present experimental facility.
Keywords: Curved spacetime; Faraday effect.
PACS Nos.: 04.40.-b, 04.80.Cc
1. Introduction
It is well known that the plane of polarization of a light ray undergoes a rotation
as it propagates through a plasma in the presence of magnetic field, which is called
electro-magnetic Faraday effect. Gravitational Faraday effect is a gravitational ana-
log of the usual Faraday effect due to the structure of the curved spacetime. The
effect was first discussed by Skrotskii1 and many authors have examined this grav-
itational effect in different gravitational backgrounds.2–4
Relativistic gravitational effects mostly concern about physical phenomena in
large scale spacetime. With the rapid development of high-intensity laser technology
in recent years, the intensity of laser pulse coming up to the order of magnitude
above 1020 W/cm2, it brings hope to detect gravitational effects in the laboratory.
Studying general relativistic effects in the gravitational field produced by high-
power lasers goes back to 1979, when Scully considered the deflection and the
phase shifting of a probe pulse in the above gravitational field.5 We also examined
the energy shifting of hydrogen atoms6 in the curved spacetime induced by high-
intensity lasers. In this paper we will investigate gravitational Faraday effect for
∗Corresponding author
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598 P. Y. Ji, H. Zhou & H. Q. Lu
an electromagnetic probe pulse propagating in the gravitational field induced by
high-intensity lasers.
This paper is as follows. In Sec. 2 the gravitational field produced by a high-
power laser is calculated using the linearized Einstein field equation in the weak
field approximation. In Sec. 3, Skrotskii effect in the above-mentioned gravitational
field is explored and the rotation angle is derived via applying the method developed
by Kopeikin and Mashhoon.2 Finally, the result obtained in Sec. 2 is discussed and
estimated numerically.
2. The Gravitation Field Produced by High-Power Laser
The gravitational field caused by the energy–momentum of high-power lasers be-
longs to weak field. The metric can be expressed as
gµν(t, r) = ηµν + hµν(t, r) , (1)
where ηµν is Minkowski metric of a flat spacetime given as
ηµν =
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
. (2)
Here we choose the natural unit system with c = 1. hµν denotes the perturbation
describing a curved spacetime, which can be calculated by linearized Einstein field
equation
�hµν = −16πG
(
Tµν −1
2ηµνT
)
, (3)
where � = ∂µ∂µ denotes D’Alembert operator, G is gravitational constant, T =
T αα and the energy–momentum tensor Tµν is defined as
Tµν = FµλFνλ−
1
4ηµνF ρσFρσ . (4)
The covariant electromagnetic field tensor Fµν is given by
Fαβ =
0 −E1 −E2 −E3
E1 0 B3 −B2
E2 −B3 0 B1
E3 B2 −B1 0
. (5)
Its contravariant tensor and mixed tensor can be expressed as
F µν = ηµρηνσFρσ ,
Fµλ = ηλσFµσ .
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Gravitational Faraday Effect Induced by High-Power Lasers 599
We assume that a pulse is propagating in +z direction with velocity of v and its
electric field and magnetic field can be shown as
Ex = ε(r, t) sin(ωt − kz) , (6)
By = vε(r, t) sin(ωt − kz) , (7)
Bz = (1 − v2)1/2ε(r, t) cos(ωt − kz) , (8)
where ε(r, t) denotes the envelope of pulse given by
ε2(r, t) = E20A[θ(v(t + T ) − z) − θ(vt − z)]δ(x)δ(y) , (9)
where E0 is the amplitude of pulse, A denotes the cross-section of laser pulse and
T represents the duration of laser. The step function, θ(z), is written as
θ(z) =
{
1 , z > 0 ,
0 , z < 0 .
Inserting Eqs. (6)–(8) into Eq. (4), we obtain the nonzero components of energy–
momentum tensors as follows:
T00 =1
2ε2(r, t)[(1 − v2) + 2v2 sin2(ωt − kz)] , (10)
T02 =1
2ε2(r, t)(1 − v2)1/2 sin 2(ωt − kz) , (11)
T03 = −1
2ε2(r, t)v sin 2(ωt − kz) , (12)
T11 =1
2ε2(r, t)[(1 − v2) − 2(1 − v2) sin2(ωt − kz)] , (13)
T22 =1
2ε2(r, t)(1 − v2) , (14)
T23 = −1
2ε2(r, t)v(1 − v2)1/2 sin 2(ωt − kz) , (15)
T33 =1
2ε2(r, t)[−(1 − v2) + 2 sin2(ωt − kz)] . (16)
From the expression of Tµν we note that the perturbation hµν(r, t) can be expressed
as hµν(r, t) = h(z − vt, y, z), and then the D’Alembert operator can be written as
−∂2
∂t2+
∂2
∂x2+
∂2
∂y2+
∂2
∂z2=
∂2
∂x2+
∂2
∂y2+ (1 − v2)
∂2
∂z2. (17)
Assuming
x = x , (18)
y = y , (19)
z = (1 − v2)1/2z , (20)
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600 P. Y. Ji, H. Zhou & H. Q. Lu
one can recast Eq. (17) in the form
∂2
∂x2+
∂2
∂y2+
∂2
∂z2= ∇
2 . (21)
The energy–momentum tensor defined by Eq. (4) has the property that T = 0.
Hence the Einstein field equation (3) is simplified as:
∇2hµν = −16πGTµν . (22)
Inserting (10)–(16) into Eq. (22), we can get the nonzero components of hµν as
follows:
h00 = [2(1 − v2) + 4v2 sin2(−kvt′)]h(r, t) , (23)
h02 = 2(1 − v2)1/2 sin(−2kvt′)h(r, t) , (24)
h03 = −2v sin(−2kvt′)h(r, t) , (25)
h11 = [2(1 − v2) − 4(1 − v2) sin2(−kvt′)]h(r, t) , (26)
h22 = 2(1 − v2)h(r, t) , (27)
h23 = −2v(1− v2)1/2 sin(−2kvt′)h(r, t) , (28)
h33 = [−2(1− v2) + 4 sin2(−kvt′)]h(r, t) , (29)
where
h(r, t) =GE0
2AvT
[(1 − v2)(x2 + y2) + (z − vt)2]1/2. (30)
hµν is a symmetric tensor, that is hµν = hνµ, so hµν has ten nonzero components.
In the above calculation of hµν , we obtain the value of integration approximately
by using the integration mean value theorem, where t′ is a small quantity and its
range is 0 < t′ < T and the approximation condition, vT/[(1 − v2)(x2 + y2) +
(z − vt)2]1/2 � 1, for the ultrashort pulse.
3. Faraday Effect in the Gravitational Field Produced by
High-Power Laser
We assume that a probe pulse is propagating in the +z direction following the
high-power laser. According to general relativity the light rays parallel transport
along the null geodesics in the curved spacetime.7 The motion equation for the
electromagnetic field of the probe in the curved space is represented by
F αβ;β = 0 , (31)
where “;” is the covariant derivative. Under the approximation of geometrical optics,
the solution of Maxwell equation can be expressed as
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Gravitational Faraday Effect Induced by High-Power Lasers 601
F αβ = Fαβ exp(ıϕ) , (32)
where Fαβ is a slowly variation function while ϕ is a quickly variation phase function
depending on the position of spacetime. Inserting the approximate solution Eq. (32)
into Eq. (31), one can obtain the geodesic equation
deα(µ)
dλ+ Γα
βγlβeγ(µ) = 0 , (33)
where polarization vector fields eα(β) define a local frame and they are parallel trans-
ported along the null geodesic. We set the following initial condition:
eα(1)(0) = (0, a1, a2, a3) , eα
(2)(0) = (0, b1, b2, b3) ,
eα(3)(0) = (0, k1, k2, k3) ,
(34)
where the three-dimensional spatial vectors a = (a1, a2, a3) and b = (b1, b2, b3)
construct a plane of polarization of light and the vector k defines the propaga-
tion direction of light ray. Inserting the metric (1) into Eq. (33), the propagation
equation of the polarization vectors eα(β) can be written as
d
dτ
(
eα(µ) +
1
2hα
βeβ(µ)
)
=1
2ηαν(∂νhγβ − ∂γhνβ)kβeγ
(µ) , (35)
where we replace the parameter τ with λ and assume dxα/dτ = kα +O(h). Accord-
ing to the initial condition Eq. (34), we have e0(i) = O(h) at any time. Since we are
only interested in the solution about the polarization vector, Eq. (35) is simplified
as
d
dτ
(
ei(n) +
1
2hije
j(n)
)
= εlijεlpq∂p
(
1
2hqβkβ
)
ej(n) , n = 1, 2 , (36)
where we define the quantity Ai as
Ai = −εijl∂j
(
1
2hlβkβ
)
. (37)
For the convenience of analysis we express Ai with two parts which are parallel and
perpendicular, respectively, to the unit vector k, i.e.
Ai = (k · A)ki + pijA
j , (38)
where pij = δi
j − kikj is the operator of projection onto the plane orthogonal to the
vector k. Using Eqs. (37) and (38), Eq. (36) is rewritten as
d
dτ
(
ei(n) +
1
2hije
j(n)
)
= −(k · A)εijlklej
(n) − εijlplqA
qej(n) , n = 1, 2 . (39)
Integrating the above equation from 0 to τ with respect to τ , we obtain
ei(1) = ai
−1
2hija
j +
(∫ τ
0
k · A dτ
)
bi− εijlP
lq
(∫ τ
0
Aq dτ
)
aj (40)
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602 P. Y. Ji, H. Zhou & H. Q. Lu
and
ei(2) = bi
−1
2hijb
j−
(∫ τ
0
k · A dτ
)
ai− εijlP
lq
(∫ τ
0
Aq dτ
)
bj , (41)
where we take into account the initial condition Eq. (34) and the equalities
εijlklaj = −bi, εijlk
lbj = ai. It is revealed from the above solutions that the
polarization vectors variate along the null geodesic due to the perturbational terms
hij of the curved spacetime. The first and third terms in the solutions just show
that the polarization vector undergoes rotation, i.e. the plane of polarization ro-
tates with the angle assumed as φ in this paper, while the fourth terms of (40) and
(41) indicate that polarization vectors rotate toward the wave vector k. One can
derive the variation of the rotation angle experienced by the polarization vectors
approximately from Eqs. (40) and (41)
dφ
dτ=
∫ τ
0
k · A dτ . (42)
Inserting (37) into (42), we get
dφ
dτ= −
1
2kαkiεijl∂jhlα . (43)
The parameter τ can be replaced with time t in the laboratory frame. Then the
wave vector k, the derivative of xα with respect to t, has the value as k0 = 1,
k1 = 0, k2 = 0, k3 = v. Therefore the above equation becomes
dφ
dt= −
1
2k3(∂xh2αkα
− ∂yh1αkα) . (44)
Inserting (24) and (28) into (44), we get
dφ
dt= GE2
0AvTv(1 − v2)5/2x sin(−2kvt′)
[(1 − v2)(x2 + y2) + (z − vt)2]3/2. (45)
As the probe pulse always keeps the same step with the high-intensity laser, just
like riding on the high-intensity laser by the probe, the term z − vt in the above
equation can be seen as a constant independent of time. Integrating Eq. (45) from
0 to t, we obtain the expression for the rotation angle of the polarization plane of
the probe pulse
φ =GE2
0AvTv(1 − v2)5/2x sin(−2kvt′)
[(1 − v2)(x2 + y2) + (z − vt)2]3/2t . (46)
4. Numerical Estimate and Discussion
Because the probe pulse propagates in the same step as the high-power laser, we can
manage to control the experiment to get the maximum effect, i.e. we can assume
sin(−2kvt′) = 1 theoretically. Choosing the case of z = vt, expression (46) is
rewritten as
φ ∼GE2
0AvTv(1 − v2)x
(x2 + y2)3/2
t , (47)
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Gravitational Faraday Effect Induced by High-Power Lasers 603
here x2 + y2 can be seen as the effective cross-section of the pulse. If x ∼ y ∼ r,
where r is the radius of the effective cross-section, the expression of φ becomes
φ ∼GE2
0AvTv(1 − v2)
r2t . (48)
From the above result, we can see that only when the velocity of pulse is less than
the speed of light, can the effect occur which agrees with the study of Scully.5
E20AvT can be seen as the energy of the pulse. Setting E2
0AvT ∼ 108 J, r ∼ 10−3 m
and v = 0.9, the order of the rotating angle φ is approximately estimated as
φ ∼ 10−23t . (49)
The above result indicates that the pulses must propagate a very long distance in
order to get a measurable effect in the laboratory. It is difficult to travel such long
distance in the present experimental condition. But if the pulses can be restricted
in a ring waveguide, the gravitational effect can be amplified by prolonging the
propagating time t of the pulses. With the development of laser technique and the
diagnose technology, we have the reason to believe that the realization of gravita-
tional Faraday effect in the laboratory will be workable.
Acknowledgments
This work was partly supported by the Science Technology Developing Foundation
of Shanghai, China under Contract No. 011911029 and Shanghai Leading Academic
Discipline Program.
References
1. G. V. Skrotskii, Dokl. Akad. Nauk SSSR 114, 73 (1957) (in Russian).2. S. Kopeikin and B. Mashhoon, Phys. Rev. D65, 064025 (2002).3. M. Nouri-Zonoz, Phys. Rev. D60, 024013 (1999).4. A. B. Balakin and J. P. S. Lemos, Class. Quantum Grav. 19, 4897 (2002).5. M. O. Scully, Phys. Rev. D19, 3582 (1978).6. P. Y. Ji, S.-T. Zhu and W.-D. Shen, Int. J. Theor. Phys. 37, 1779 (1998).7. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman and Co.,
1973).
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