GRAVITATIONAL FARADAY EFFECT INDUCED BY HIGH-POWER LASERS

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∗[email protected]

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

597

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615

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 .

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615

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)

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615


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

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615


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)

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615


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)

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

March 10, 2005 13:37 WSPC/146-MPLA 01615


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).

Mod

. Phy

s. L

ett.

A 2

005.

20:5

97-6

03. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by M

ASS

AC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

on

05/0

6/13

. For

per

sona

l use

onl

y.

Documents

GRAVITATIONAL FARADAY EFFECT INDUCED BY HIGH-POWER LASERS