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Electrodynamics in a rotating frame of reference
Volume 139, number 3,4
PHYSICS LETTERS A
31 July 1989
ELECTRODYNAMICS IN A ROTATING FRAME OF REFERENCE Bahram MASHHOON Department of Physics andAstronomy, University ofMissouri—Columbia, Columbia, MO 65211, USA Received 10 May 1989; accepted for publication 5 June 1989 Communicatedby J.P. Vigier
The propagation of electromagnetic waves in a uniformly rotating frame of reference is studied. The existence of a general coupling between the helicity of the radiation and the rotation of the frame is pointed out. The standard relativistic expressions for the Doppler principle and aberration are generalized to include the new coupling. The consequences of these results for the aberration of polarized light and for interferometry in a rotating system are briefly discussed.
1. Introduction The laws of physics ordinarily involve quantities that are defined with respect to inertial observers. To generalize these laws to connect measurements performed by observers in arbitrary motion, certain hypotheses are necessary to relate the measurements of an accelerated observer to those of an ideal inertial observer. The distinction between these observers may be illustrated by considering the path of an accelerated observer in Cartesian space. The motion in the neighborhood of a point on this curve at time can be approximated to first order as linear motion along the tangent vector v( t) at that point. To second order, the motion is essentially along an osculating circle if the tangential acceleration can be neglected. In fact, this Frenet approximation to the curve implies that to third order the curve in general twists out of the plane of the osculating circle. The curvature and torsion of the path measure the rate of rotation of the curve away from the velocity vector and the rate of twisting of the curve out of the osculating plane, respectively. For the interpretation of the result of an experiment, the requisite order of approximation should presumably depend on the intrinsic scale of the phenomenon under observation. The linear approximation would thus be valid for pointwise measurements. It follows from this line of thought that for phenomena involving no extension in spacetime (i.e., coincidences), the results ofmea-
surements performed by the accelerated observer at time t are identical with those of a hypothetical inertial observer that is instantaneously comoving with the noninertial observer at that time. This idea is the basis for the hypothesis oflocality which provides the foundation for the standard description of physics in accelerated systems and gravitational fields. This standard theory therefore provides a proper and consistent description of systems involving classical point particles and electromagnetic rays whose interactions, by definition, do not involve any extension in space and time. There exists, however, a natural extension of this theory to wave phenomena as follows: The hypothesis of locality can be employed to define the electromagnetic field quantities as measured by the accelerated observer. The (nonlocal) Fourier analysis ofthe field is then used to define the frequency and propagation vector of radiation in a covariant manner. This extended hypothesis of locality has been critically analyzed in a series of recent papers (cf. ref. [1] and the references cited therein). The purpose ofthe present study is to point out further consequences of this extension of the hypothesis of locality to wave phenomena. In particular, novel expressions for the Doppler principle and aberration are presented in the eikonal approximation, namely, w’ =y(a—cfik) —yRQ,
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Volume 139, number 3,4
PHYSICS LETTERS A
k’ =k+ (y— 1) (gk)ft—
~-
ywfi+
-~-
y(R~Q)fl, (2)
where the new terms represent the coupling of the helicity of the radiation with the rotation of the frame. This coupling is expected to have observable consequences for the aberration ofpolarized light and for interferometry in a rotating frame. Imagine electromagnetic waves of frequency w and wave vector k incident upon a uniformly rotating observer with velocity v(t)=cfl(t). The hypothesis of locality implies that the frequency & and wave vector ~ measured by the rotating observer are &=y(w—v~k)
(3)
,
~=k+(y—l)(ft.k)g—
!ywp,
31 July 1989
pothesis ofinlocality the wave vector k’ would related to 4~ the eikonal approximation as k’be k+ Q, where Q should be due to the influence of helicity—rotation coupling on wave propagation. This physical characterization of Q together with the requirements of invariance under parity and time reversal implies that Q must be proportional to (H Q)/i. It will be shown in section 2 that the proportionality factor is y/c as in eq. (2). Eqs. (1) and (2) thus contain a generalization of the standard formulas (3) and (4) beyond the eikonal limit. In section 2, these new relations for the Doppler effect and aberration are derived by a different approach. Their observational consequences are explored in sections 3 and 4.
(4) 2. Helicity—rotation coupling
as a consequence of the invariance of the phase of the ray. Here y= (1 fl) —1/2, and formulas (3) and (4) are the standard expressions for the Doppler effect and aberration of light as employed, for instance, in astronomy to take due account of the motion of the earth. On the other hand, the extended hypothesis of locality implies that the rotating observer would measure a spectrum of frequencies w’ given by [2] —
w’
=
y ( w MQ) —
,
(5)
where M 0, ±1, ±2 is the angular momentum parameter along the rotation axis and Q is the constant frequency of rotation of the observer. Eq. (5) reduces to eq. (3) in the eikonal limit. This transition may be seen from the following intuitive considerations: For a photon of energy E= hw, eq. (5) implies that E’ y(E—J•Q), where J is the angular momentum of the photon. In the eikonal approximarion J~xxhk+h1’~where Ris a unit vector given by /1= ±ck/w depending on whether the ray is right (upper sign; positive helicity) or left (lower sign; negative helicity) circularly polarized. Thus eq. (5) may be expressed in the eikonal approximation as w’ th—yfl~Q, where in the expression for &, v=Qxx(t) and x(t) is the position vector ofthe rotaring observer. It follows that the helicity—rotation coupling is due to the wave nature of light. Similarly, one might expect that according to the extended hy104
There exists a general coupling between the circular polarization of light and the rotation of the frame of the observer. To derive this coupling from the extended hypothesis of locality, the superposilion of wave amplitudes as measured by the rotating observer must be studied. The eikonal approximation is obtained from the constructive interference of such waves in the high frequency regime. It proves interesting, however, to follow a more intuitive procedure. To this end, imagine three classes of observers in Minkowski spacetime. Inertial observers ~ are static and refer their observations to a fixed set of Cartesian axes. Observers C* are also static, but they are not inertial since they refer their measurements to spatial axes that rotate uniformly with frequency Q with respect to Cartesian axes. Observers C’ are the usual rotating observers: they move with velocity v=QXx while their reference axes rotate with respect to Cartesian axes at the same uniform rate as for observers C*. Let observers C* measure the frequency and wave vector of a monochromatic wave definite helicity. It will be shown that in the eikonal approximation the measurements of C* and C are related by w*=w_R.Q,
k*=k,
(6)
where the frequency of a wave with definite helicity is given by the rate of rotation of electromagnetic fields along the direction of propagation. Thus, eq.
Volume 139, number 3,4
PHYSICS LETTERSA
(6) expresses the addition law of angular frequencies. To see this in detail, consider circularly polarized electromagnetic waves represented by the (electric or magnetic) field F, (7)
F=Re[a(i±ij)e’°],
where a is the wave amplitude, 0= —wt+kx is the phase, and 1~J, and /~=ck/wform an orthonormal triad that can be expressed in terms of Cartesian axes as 1= —cos 01 +sin 013, 1=12, and k= (sin 01~+ cos Ox3). Here 0 is a polar angle. Ac—
cording to observers C*, however, the properties of the wave are measured with respect to spatial axes 17, i = 1, 2, 3, which rotate about the £~direction and coincide with the Cartesian axes at t=0. The electromagnetic tensor is thus given by the projection of the Faraday tensor onto the natural orthonormal tetrad frame carried by observers C*. It follows that the (electric or magnetic) field simply transforms as a vector, F1=Re[a(—cosOcosQt±isinQt)e°], 0] , F~=Re[a(cos 0 sin Qt±icos Qt)e’ f 3=Re(asin0e°).
(8) (9) (10)
Let the observations begin at t=0. Observers C* measure the polarization ofthe wave with respect to the orthonormal triad ((*,ft, ~(*)given by (11)
i”= —(—cosOlT+ sin0cosQt1~), ~
j
1
.
.
,.
.
(sin 0 cos Qt xi + cos 0 x~)sin 0 sin Qt + ~
=
—
pX2
12
‘
k’= —sin 0 cos Qt1~+sin 0 sin Qt1~ —
cos
,
~13 .
.
pe~=cosQtpcos0sinQt.
so that attention is focused on waves with high frequencies w >> Q and periods that fall within the interval, then p~1 and e ~ Qt cos 0 to first order in Qt. Hence in the eikonal approximation (w>>Q), w*~w±Qcos 9 and k*=k in agreement with eq. (6). This result may be thought of as the electromagnetic analog of the phenomenon associated with the Foucault pendulum. The next step in this analysis involves the relationship between observers C* and C’. At any point t << Q’
in spacetime, these observers are related by a boost. This must be a Lorentz boost, however, even for very small /3. Otherwise, the phenomenon ofaberration of light cannot be explained. That is, the electromagnetic theory of light in combination with a Galilean transformation can only give the (first-order) Doppler effect. In fact, this absence of aberration led to the introduction of Lorentz transformations (to first order in/3) in 1895. This theory of Lorentz provided a satisfactory explanation for all electromagnetic phenomena in moving systems to first order in /3. The . generalization of this theory to incorporate FitzGerald contraction led naturally to theLorentz— Lorentz transformations [3]. Thus relating observers C’ to C” by a Lorentz boost fi, one finds w’ =y(w*_c/3.k*) ,
(16)
-.
-
k=k*+(y_l)(fi.k*)fi__yw*fi,
(17)
which amount to the same equations as (1) and (2) once eq. (6) is taken into account. The inverse re. lationship can be simply obtained from eqs. (16), (17) and (6) by letting fi-+—fl and noting k’Q =k•Q. The dispersion relation in the rotating frame is then a consequence of the inverse relations once
where p is a modulus defined by •
31 July 1989
(14)
Note that i~~=s,and P” andj” reduce to tandj respectively, at t = 0. It follows that F”=Re[a(i~”±ij~”)e~~. (15) Thus according to observers ~ the wave is circularly polarized as well. If the measurement ofwave properties is performed over a short time interval
w=ckisimposed. The new formulas for the Doppler effect and aberration illustrate the idea that wave optics is associated with absolute motion [4]. In the geometric optics limit, the relationship is relative, i.e., the inverse transformation is obtained from fl—’ —fi. However, when X 0 the absolute character of rotation is associated with the wave aspect. Relativity ofmotion is recovered in the X—~0limit. The standard relativistic expression for Doppler effect, eq. (3), implies that & can never become zero or negative. This important feature disappears when .
.
.
.
.
.
105
Volume 139, number 3,4
PHYSICS LETTERS A
helicity—rotation coupling is taken into account. This can be simply illustrated in the case of transverse Doppler effect when the incident radiation propagates along the axis of rotation. In this case, the approximate expression given by eq. (1) is in fact exact [21. The helicity—rotation coupling, which is a consequence of the extended hypothesis of locality, thus leads to the possibility of existence of rotating observers that can stay at rest with respect to an electromagnetic wave.
31 July 1989
larized radiation cannot propagate in the same direction except in a plane perpendicular to the axis of rotation. This implies, in general, a certain displacement of an astronomical source when observed with radiation of different helicities. Let k±and k’÷ be the wave vectors for circularly polarized radiation of definite frequency (w±=w, w’±=w’ ), then eqs. (1) and (2) imply that wfl.(k±—ki= —(k÷+k_Y~Q,
(19)
k’~—k~=k~—k+(l/y—l) (k±—k_)ft. (20) 3. Aberration of polarized radiation In the standard theory of aberration of starlight, the wavelength of the radiation does not enter the analysis. It is a purely geometric theory; i.e., the wave theory gives a result that is identical to one obtained from the law of transformation of velocity of a partide when it is replaced by a ray of light [5]. The same result can also be obtained from eq. (2) when averaged for unpolarized radiation. The situation is different, however, for polarized radiation due to the coupling of helicity with rotation. If the polarization of incident radiation is taken into account, then the aberration angle changes by a term of order /3Q/w. Let ,~and ~j’ be the angles formed by k and k’, respectively, with the instantaneous direction of yelocity v of the observer. The aberration angle, ij’ ~, can be calculated from y tan =sin ii/(cos n—fl) in the standard analysis. Taking the polarization of the radiation into account, the first-order contribution of helicity—rotation coupling to the aberration angle is given by —
,~‘
ô(~
~ sin ~ ~ (18) /3 cos For /3 ~z 1, the aberration angle for unpolarized light is /3 sin ,j’ and the magnitude of its variation due to polarization is generally smaller by Q/w. This variation appears to be too small to be significant for astronomical observations at present. In the standard theory of aberration and Doppler effect, the propagation of the wave does not depend on the state of polarization of the radiation. However, eq. (1) for the Doppler effect has an interesting consequence if the incident and observed frequencies are fixed, namely, right and left circularly po~).
—
—
106
For a minute angular displacement, it is useful to assume that ck±=ck±~wA,where A is orthogonal to k. To simplify the analysis, terms that are second order in A or /3 are neglected. This is reasonable for earth-based observations of astronomical sources. Hence it follows from eqs. (19) and (20) that ~wA. Here k’ and w2fl.4= —2ck~Qand ck’±=ck’ ± w’ satisfy the standard aberration and Doppler formulas for unpolarized radiation in terms of k and w. Using the fixed orthonormal triad introduced in the previous section, the displacement A can be written as A=pI+aJ where p cos Osin q+qcos ~=2Xr cos 0. Here ç~is the azimuthal angle of the rotating observer, r is its radius, and X is the reduced wavelength of the radiation. The quantities p and q represent the coordinates of an ellipse with semimajor axis 2 X /r and semiminor axis 2 X r ‘cos 0. The angular displacement A follows this ellipse in the opposite sense as the rotation of the observer. If the motion of the observer is caused by the proper rotation of the earth, the displacement due to the polarization of the radiation is inversely proportional to the cosine of the geographical latitude in contrast to the diurnal aberration which is directly proportional to it. Consider, for instance, radiation of frequency v = 0.33 GHz incident on the earth, then 2 X R~ lO_2 arcsecond. The angular resolution for earth-based observations at this frequency is 25 milliarcsec [6]. A threefold improvement in resolution is envisioned for the planned orbiting Quasat antenna [6]. At northern latitudes, for example, the angular displacement would be within the resolving power of the proposed Quasat antenna. It should be pointed out, however, that polarization-dependent displacements could also be produced by magnetic fields [7]. A detailed investigation is clearly neces/
Volume 139, number 3,4
PHYSICS LETTERS A
sary to determine if a rotational displacement can indeed be observed for polarized radiation received from a given astronomical source. Apart from these astronomical consequences of eqs. (1) and (2), it is interesting to explore possible laboratory tests. The next section is therefore devoted to interferometry with circularly polarized beams in a rotating system. The well-known Sagnac effect can be though of as a coupling between the orbital angular momentum of the radiation and the rotation of the frame [81. The coupling ofhelicity with rotation implies the existence of a new effect in interferometry with polarized radiation.
4. Interferometry in a rotating system The phase shift due to the rotation of an interferometer was first observed by Sagnac. The Sagnac shift, which is independent of the polarization of the radiation, can be given an invariant geometric interpretation in the eikonal limit. This effect has been extensively studied [9], and it has found wide-ranging applications in rotation sensing devices [10,11]. It follows from eqs. (1) and (2) that besides the Sagnac shift there is an additional phase shift in the rotating frame due to the coupling of the helicity of the radiation with rotation. Consider a rotating interferometer as depicted in fig. 1. Incident photon beams from a source S are coherently split along identical semicircular paths of n
31 July 1989
radius R and recombined at the detector D. The radiation is assumed to be right (left) circularly polarized along the counterclockwise (clockwise) path. The plane ofthe interferometer rotates about the SD direction with uniform angular frequency Q. The interferencephaseshiftisgivenby~Ø’=Ø~ where 0’ =
J~~’
—o~,
dt’ +k’ ~dx’) .
(21)
S
(o~,)
Here 0~ is the difference between the phase of the wave in the rotating frame at the detector at t’D and the phase at the source at t~,t’D —t~= itR/c, along the counterclockwise (clockwise) path. The axis of rotation lies in the plane of the interferometer; therefore, the Sagnac phase shift vanishes for the configuration of fig. 1. To determine the phase shift öØ’, the dispersion relation in the rotating frame must be calculated. In the eikonal approximation, this relation may be obtained from eqs. (1) and (2). The result to first order in RQ/c is w’ = ck’ ii’ Q. The spacetime for the rotating observer is stationary, hence electromagnetic waves of a definite frequency w’ may be considered. Thus, for w~=w’, c(k’÷ k~)=(k~ +L )~Q.It then follows from eq. (21) that ~O’ = 4RQ/c due to the helicity—rotation coupling. It is interesting to compare the dispersion relation in this case with an analogous result for the propagation of electromagnetic waves in the field of a rotating mass, where the helicity of the radiation couples to the dragging frequency ofthe local inertial frames [7]. More generally, let the rotation axis lie in the plane formed by the normal to the plane of the interfero—
.
—
the sum of contributions due to the Sagnac effect and the helicity—rotation coupling. It is therefore given by
LCP
RCP HWP
s
Fig. 1. Positive-helicity radiation is coherently split in a rotating interferometer. The helicity of the radiation along the clockwise path is inverted — e.g., by a half-wave plate (HWP) — immediately after splitting and again before recombination to allow interference to take place. The interference phase shift is then measured at the detector D in the rotating frame.
meter mathe SD direction. \RQ Then the phase shift 2(ltRsand +2cosa )—, (22)is where a is the angle jc that specifies the direction of rotation above the plane of the interferometer. The component of angular velocity normal to the plane of the interferometer, Q sin a, is responsible for the Sagnac effect, whereas the component along the SD direction, Q cos a, is responsible for the helicity—rotation coupling phase shift. The spin effect is generally smaller than the Sagnac effect by the ratio of 107
Volume 139, number 3,4
PHYSICS LETTERSA
the wavelength of the radiation to the perimeter of the interferometer. For an interferometer of radius R = 60 cm rotating at 102 rounds per second as in fig. 1, the helicity—rotation coupling phase shift amounts to 1 arcsecond. To distinguish the spin effect experimentally from the (usually much larger) Sagnac effect, it is important to note that the novel phase shift is independent of the wavelength and changes sign whenever the helicity of the incident beam is reversed. References [1] B. Mashhoon,Phys. Rev. Lett. 61(1988)2639.
108
31 July 1989
[2] B. Mashhoon, Found. Phys. 16 (1986) 619. [31HA. Lorentz, The theory of electrons (Dover, New York. 1952). [41B. Mashhoon, Phys. Lett. A 126 (1988) 393. 151 E.T. Whittaker, A history of the theories of aether and electricity, Vol. 1 (Nelson, London, 1951) ch. 4. [6] B. Dennjson and R.S. Booth, Mon. Not. R. Astron. Soc. 224 (1987) 927. [71B. Mashhoon, Phys. Rev. D 11(1975) 2679. [8] C.V. Heer, Phys. Rev. 134 (1964) A799. [91E.J. Post, Rev. Mod. Phys. 39 (1967) 475, [10] S. Ezekiel and H.J. Arditty, eds., Fiber-optic rotation sensors (Springer, Berlin, 1982). [ll]M. Faucheux, D. Fayoux and J.J. Roland, 1. Opt. (Paris) 19 (1988) 101.
PHYSICS LETTERS A
31 July 1989
ELECTRODYNAMICS IN A ROTATING FRAME OF REFERENCE Bahram MASHHOON Department of Physics andAstronomy, University ofMissouri—Columbia, Columbia, MO 65211, USA Received 10 May 1989; accepted for publication 5 June 1989 Communicatedby J.P. Vigier
The propagation of electromagnetic waves in a uniformly rotating frame of reference is studied. The existence of a general coupling between the helicity of the radiation and the rotation of the frame is pointed out. The standard relativistic expressions for the Doppler principle and aberration are generalized to include the new coupling. The consequences of these results for the aberration of polarized light and for interferometry in a rotating system are briefly discussed.
1. Introduction The laws of physics ordinarily involve quantities that are defined with respect to inertial observers. To generalize these laws to connect measurements performed by observers in arbitrary motion, certain hypotheses are necessary to relate the measurements of an accelerated observer to those of an ideal inertial observer. The distinction between these observers may be illustrated by considering the path of an accelerated observer in Cartesian space. The motion in the neighborhood of a point on this curve at time can be approximated to first order as linear motion along the tangent vector v( t) at that point. To second order, the motion is essentially along an osculating circle if the tangential acceleration can be neglected. In fact, this Frenet approximation to the curve implies that to third order the curve in general twists out of the plane of the osculating circle. The curvature and torsion of the path measure the rate of rotation of the curve away from the velocity vector and the rate of twisting of the curve out of the osculating plane, respectively. For the interpretation of the result of an experiment, the requisite order of approximation should presumably depend on the intrinsic scale of the phenomenon under observation. The linear approximation would thus be valid for pointwise measurements. It follows from this line of thought that for phenomena involving no extension in spacetime (i.e., coincidences), the results ofmea-
surements performed by the accelerated observer at time t are identical with those of a hypothetical inertial observer that is instantaneously comoving with the noninertial observer at that time. This idea is the basis for the hypothesis oflocality which provides the foundation for the standard description of physics in accelerated systems and gravitational fields. This standard theory therefore provides a proper and consistent description of systems involving classical point particles and electromagnetic rays whose interactions, by definition, do not involve any extension in space and time. There exists, however, a natural extension of this theory to wave phenomena as follows: The hypothesis of locality can be employed to define the electromagnetic field quantities as measured by the accelerated observer. The (nonlocal) Fourier analysis ofthe field is then used to define the frequency and propagation vector of radiation in a covariant manner. This extended hypothesis of locality has been critically analyzed in a series of recent papers (cf. ref. [1] and the references cited therein). The purpose ofthe present study is to point out further consequences of this extension of the hypothesis of locality to wave phenomena. In particular, novel expressions for the Doppler principle and aberration are presented in the eikonal approximation, namely, w’ =y(a—cfik) —yRQ,
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1) 103
Volume 139, number 3,4
PHYSICS LETTERS A
k’ =k+ (y— 1) (gk)ft—
~-
ywfi+
-~-
y(R~Q)fl, (2)
where the new terms represent the coupling of the helicity of the radiation with the rotation of the frame. This coupling is expected to have observable consequences for the aberration ofpolarized light and for interferometry in a rotating frame. Imagine electromagnetic waves of frequency w and wave vector k incident upon a uniformly rotating observer with velocity v(t)=cfl(t). The hypothesis of locality implies that the frequency & and wave vector ~ measured by the rotating observer are &=y(w—v~k)
(3)
,
~=k+(y—l)(ft.k)g—
!ywp,
31 July 1989
pothesis ofinlocality the wave vector k’ would related to 4~ the eikonal approximation as k’be k+ Q, where Q should be due to the influence of helicity—rotation coupling on wave propagation. This physical characterization of Q together with the requirements of invariance under parity and time reversal implies that Q must be proportional to (H Q)/i. It will be shown in section 2 that the proportionality factor is y/c as in eq. (2). Eqs. (1) and (2) thus contain a generalization of the standard formulas (3) and (4) beyond the eikonal limit. In section 2, these new relations for the Doppler effect and aberration are derived by a different approach. Their observational consequences are explored in sections 3 and 4.
(4) 2. Helicity—rotation coupling
as a consequence of the invariance of the phase of the ray. Here y= (1 fl) —1/2, and formulas (3) and (4) are the standard expressions for the Doppler effect and aberration of light as employed, for instance, in astronomy to take due account of the motion of the earth. On the other hand, the extended hypothesis of locality implies that the rotating observer would measure a spectrum of frequencies w’ given by [2] —
w’
=
y ( w MQ) —
,
(5)
where M 0, ±1, ±2 is the angular momentum parameter along the rotation axis and Q is the constant frequency of rotation of the observer. Eq. (5) reduces to eq. (3) in the eikonal limit. This transition may be seen from the following intuitive considerations: For a photon of energy E= hw, eq. (5) implies that E’ y(E—J•Q), where J is the angular momentum of the photon. In the eikonal approximarion J~xxhk+h1’~where Ris a unit vector given by /1= ±ck/w depending on whether the ray is right (upper sign; positive helicity) or left (lower sign; negative helicity) circularly polarized. Thus eq. (5) may be expressed in the eikonal approximation as w’ th—yfl~Q, where in the expression for &, v=Qxx(t) and x(t) is the position vector ofthe rotaring observer. It follows that the helicity—rotation coupling is due to the wave nature of light. Similarly, one might expect that according to the extended hy104
There exists a general coupling between the circular polarization of light and the rotation of the frame of the observer. To derive this coupling from the extended hypothesis of locality, the superposilion of wave amplitudes as measured by the rotating observer must be studied. The eikonal approximation is obtained from the constructive interference of such waves in the high frequency regime. It proves interesting, however, to follow a more intuitive procedure. To this end, imagine three classes of observers in Minkowski spacetime. Inertial observers ~ are static and refer their observations to a fixed set of Cartesian axes. Observers C* are also static, but they are not inertial since they refer their measurements to spatial axes that rotate uniformly with frequency Q with respect to Cartesian axes. Observers C’ are the usual rotating observers: they move with velocity v=QXx while their reference axes rotate with respect to Cartesian axes at the same uniform rate as for observers C*. Let observers C* measure the frequency and wave vector of a monochromatic wave definite helicity. It will be shown that in the eikonal approximation the measurements of C* and C are related by w*=w_R.Q,
k*=k,
(6)
where the frequency of a wave with definite helicity is given by the rate of rotation of electromagnetic fields along the direction of propagation. Thus, eq.
Volume 139, number 3,4
PHYSICS LETTERSA
(6) expresses the addition law of angular frequencies. To see this in detail, consider circularly polarized electromagnetic waves represented by the (electric or magnetic) field F, (7)
F=Re[a(i±ij)e’°],
where a is the wave amplitude, 0= —wt+kx is the phase, and 1~J, and /~=ck/wform an orthonormal triad that can be expressed in terms of Cartesian axes as 1= —cos 01 +sin 013, 1=12, and k= (sin 01~+ cos Ox3). Here 0 is a polar angle. Ac—
cording to observers C*, however, the properties of the wave are measured with respect to spatial axes 17, i = 1, 2, 3, which rotate about the £~direction and coincide with the Cartesian axes at t=0. The electromagnetic tensor is thus given by the projection of the Faraday tensor onto the natural orthonormal tetrad frame carried by observers C*. It follows that the (electric or magnetic) field simply transforms as a vector, F1=Re[a(—cosOcosQt±isinQt)e°], 0] , F~=Re[a(cos 0 sin Qt±icos Qt)e’ f 3=Re(asin0e°).
(8) (9) (10)
Let the observations begin at t=0. Observers C* measure the polarization ofthe wave with respect to the orthonormal triad ((*,ft, ~(*)given by (11)
i”= —(—cosOlT+ sin0cosQt1~), ~
j
1
.
.
,.
.
(sin 0 cos Qt xi + cos 0 x~)sin 0 sin Qt + ~
=
—
pX2
12
‘
k’= —sin 0 cos Qt1~+sin 0 sin Qt1~ —
cos
,
~13 .
.
pe~=cosQtpcos0sinQt.
so that attention is focused on waves with high frequencies w >> Q and periods that fall within the interval, then p~1 and e ~ Qt cos 0 to first order in Qt. Hence in the eikonal approximation (w>>Q), w*~w±Qcos 9 and k*=k in agreement with eq. (6). This result may be thought of as the electromagnetic analog of the phenomenon associated with the Foucault pendulum. The next step in this analysis involves the relationship between observers C* and C’. At any point t << Q’
in spacetime, these observers are related by a boost. This must be a Lorentz boost, however, even for very small /3. Otherwise, the phenomenon ofaberration of light cannot be explained. That is, the electromagnetic theory of light in combination with a Galilean transformation can only give the (first-order) Doppler effect. In fact, this absence of aberration led to the introduction of Lorentz transformations (to first order in/3) in 1895. This theory of Lorentz provided a satisfactory explanation for all electromagnetic phenomena in moving systems to first order in /3. The . generalization of this theory to incorporate FitzGerald contraction led naturally to theLorentz— Lorentz transformations [3]. Thus relating observers C’ to C” by a Lorentz boost fi, one finds w’ =y(w*_c/3.k*) ,
(16)
-.
-
k=k*+(y_l)(fi.k*)fi__yw*fi,
(17)
which amount to the same equations as (1) and (2) once eq. (6) is taken into account. The inverse re. lationship can be simply obtained from eqs. (16), (17) and (6) by letting fi-+—fl and noting k’Q =k•Q. The dispersion relation in the rotating frame is then a consequence of the inverse relations once
where p is a modulus defined by •
31 July 1989
(14)
Note that i~~=s,and P” andj” reduce to tandj respectively, at t = 0. It follows that F”=Re[a(i~”±ij~”)e~~. (15) Thus according to observers ~ the wave is circularly polarized as well. If the measurement ofwave properties is performed over a short time interval
w=ckisimposed. The new formulas for the Doppler effect and aberration illustrate the idea that wave optics is associated with absolute motion [4]. In the geometric optics limit, the relationship is relative, i.e., the inverse transformation is obtained from fl—’ —fi. However, when X 0 the absolute character of rotation is associated with the wave aspect. Relativity ofmotion is recovered in the X—~0limit. The standard relativistic expression for Doppler effect, eq. (3), implies that & can never become zero or negative. This important feature disappears when .
.
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helicity—rotation coupling is taken into account. This can be simply illustrated in the case of transverse Doppler effect when the incident radiation propagates along the axis of rotation. In this case, the approximate expression given by eq. (1) is in fact exact [21. The helicity—rotation coupling, which is a consequence of the extended hypothesis of locality, thus leads to the possibility of existence of rotating observers that can stay at rest with respect to an electromagnetic wave.
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larized radiation cannot propagate in the same direction except in a plane perpendicular to the axis of rotation. This implies, in general, a certain displacement of an astronomical source when observed with radiation of different helicities. Let k±and k’÷ be the wave vectors for circularly polarized radiation of definite frequency (w±=w, w’±=w’ ), then eqs. (1) and (2) imply that wfl.(k±—ki= —(k÷+k_Y~Q,
(19)
k’~—k~=k~—k+(l/y—l) (k±—k_)ft. (20) 3. Aberration of polarized radiation In the standard theory of aberration of starlight, the wavelength of the radiation does not enter the analysis. It is a purely geometric theory; i.e., the wave theory gives a result that is identical to one obtained from the law of transformation of velocity of a partide when it is replaced by a ray of light [5]. The same result can also be obtained from eq. (2) when averaged for unpolarized radiation. The situation is different, however, for polarized radiation due to the coupling of helicity with rotation. If the polarization of incident radiation is taken into account, then the aberration angle changes by a term of order /3Q/w. Let ,~and ~j’ be the angles formed by k and k’, respectively, with the instantaneous direction of yelocity v of the observer. The aberration angle, ij’ ~, can be calculated from y tan =sin ii/(cos n—fl) in the standard analysis. Taking the polarization of the radiation into account, the first-order contribution of helicity—rotation coupling to the aberration angle is given by —
,~‘
ô(~
~ sin ~ ~ (18) /3 cos For /3 ~z 1, the aberration angle for unpolarized light is /3 sin ,j’ and the magnitude of its variation due to polarization is generally smaller by Q/w. This variation appears to be too small to be significant for astronomical observations at present. In the standard theory of aberration and Doppler effect, the propagation of the wave does not depend on the state of polarization of the radiation. However, eq. (1) for the Doppler effect has an interesting consequence if the incident and observed frequencies are fixed, namely, right and left circularly po~).
—
—
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For a minute angular displacement, it is useful to assume that ck±=ck±~wA,where A is orthogonal to k. To simplify the analysis, terms that are second order in A or /3 are neglected. This is reasonable for earth-based observations of astronomical sources. Hence it follows from eqs. (19) and (20) that ~wA. Here k’ and w2fl.4= —2ck~Qand ck’±=ck’ ± w’ satisfy the standard aberration and Doppler formulas for unpolarized radiation in terms of k and w. Using the fixed orthonormal triad introduced in the previous section, the displacement A can be written as A=pI+aJ where p cos Osin q+qcos ~=2Xr cos 0. Here ç~is the azimuthal angle of the rotating observer, r is its radius, and X is the reduced wavelength of the radiation. The quantities p and q represent the coordinates of an ellipse with semimajor axis 2 X /r and semiminor axis 2 X r ‘cos 0. The angular displacement A follows this ellipse in the opposite sense as the rotation of the observer. If the motion of the observer is caused by the proper rotation of the earth, the displacement due to the polarization of the radiation is inversely proportional to the cosine of the geographical latitude in contrast to the diurnal aberration which is directly proportional to it. Consider, for instance, radiation of frequency v = 0.33 GHz incident on the earth, then 2 X R~ lO_2 arcsecond. The angular resolution for earth-based observations at this frequency is 25 milliarcsec [6]. A threefold improvement in resolution is envisioned for the planned orbiting Quasat antenna [6]. At northern latitudes, for example, the angular displacement would be within the resolving power of the proposed Quasat antenna. It should be pointed out, however, that polarization-dependent displacements could also be produced by magnetic fields [7]. A detailed investigation is clearly neces/
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sary to determine if a rotational displacement can indeed be observed for polarized radiation received from a given astronomical source. Apart from these astronomical consequences of eqs. (1) and (2), it is interesting to explore possible laboratory tests. The next section is therefore devoted to interferometry with circularly polarized beams in a rotating system. The well-known Sagnac effect can be though of as a coupling between the orbital angular momentum of the radiation and the rotation of the frame [81. The coupling ofhelicity with rotation implies the existence of a new effect in interferometry with polarized radiation.
4. Interferometry in a rotating system The phase shift due to the rotation of an interferometer was first observed by Sagnac. The Sagnac shift, which is independent of the polarization of the radiation, can be given an invariant geometric interpretation in the eikonal limit. This effect has been extensively studied [9], and it has found wide-ranging applications in rotation sensing devices [10,11]. It follows from eqs. (1) and (2) that besides the Sagnac shift there is an additional phase shift in the rotating frame due to the coupling of the helicity of the radiation with rotation. Consider a rotating interferometer as depicted in fig. 1. Incident photon beams from a source S are coherently split along identical semicircular paths of n
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radius R and recombined at the detector D. The radiation is assumed to be right (left) circularly polarized along the counterclockwise (clockwise) path. The plane ofthe interferometer rotates about the SD direction with uniform angular frequency Q. The interferencephaseshiftisgivenby~Ø’=Ø~ where 0’ =
J~~’
—o~,
dt’ +k’ ~dx’) .
(21)
S
(o~,)
Here 0~ is the difference between the phase of the wave in the rotating frame at the detector at t’D and the phase at the source at t~,t’D —t~= itR/c, along the counterclockwise (clockwise) path. The axis of rotation lies in the plane of the interferometer; therefore, the Sagnac phase shift vanishes for the configuration of fig. 1. To determine the phase shift öØ’, the dispersion relation in the rotating frame must be calculated. In the eikonal approximation, this relation may be obtained from eqs. (1) and (2). The result to first order in RQ/c is w’ = ck’ ii’ Q. The spacetime for the rotating observer is stationary, hence electromagnetic waves of a definite frequency w’ may be considered. Thus, for w~=w’, c(k’÷ k~)=(k~ +L )~Q.It then follows from eq. (21) that ~O’ = 4RQ/c due to the helicity—rotation coupling. It is interesting to compare the dispersion relation in this case with an analogous result for the propagation of electromagnetic waves in the field of a rotating mass, where the helicity of the radiation couples to the dragging frequency ofthe local inertial frames [7]. More generally, let the rotation axis lie in the plane formed by the normal to the plane of the interfero—
.
—
the sum of contributions due to the Sagnac effect and the helicity—rotation coupling. It is therefore given by
LCP
RCP HWP
s
Fig. 1. Positive-helicity radiation is coherently split in a rotating interferometer. The helicity of the radiation along the clockwise path is inverted — e.g., by a half-wave plate (HWP) — immediately after splitting and again before recombination to allow interference to take place. The interference phase shift is then measured at the detector D in the rotating frame.
meter mathe SD direction. \RQ Then the phase shift 2(ltRsand +2cosa )—, (22)is where a is the angle jc that specifies the direction of rotation above the plane of the interferometer. The component of angular velocity normal to the plane of the interferometer, Q sin a, is responsible for the Sagnac effect, whereas the component along the SD direction, Q cos a, is responsible for the helicity—rotation coupling phase shift. The spin effect is generally smaller than the Sagnac effect by the ratio of 107
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the wavelength of the radiation to the perimeter of the interferometer. For an interferometer of radius R = 60 cm rotating at 102 rounds per second as in fig. 1, the helicity—rotation coupling phase shift amounts to 1 arcsecond. To distinguish the spin effect experimentally from the (usually much larger) Sagnac effect, it is important to note that the novel phase shift is independent of the wavelength and changes sign whenever the helicity of the incident beam is reversed. References [1] B. Mashhoon,Phys. Rev. Lett. 61(1988)2639.
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[2] B. Mashhoon, Found. Phys. 16 (1986) 619. [31HA. Lorentz, The theory of electrons (Dover, New York. 1952). [41B. Mashhoon, Phys. Lett. A 126 (1988) 393. 151 E.T. Whittaker, A history of the theories of aether and electricity, Vol. 1 (Nelson, London, 1951) ch. 4. [6] B. Dennjson and R.S. Booth, Mon. Not. R. Astron. Soc. 224 (1987) 927. [71B. Mashhoon, Phys. Rev. D 11(1975) 2679. [8] C.V. Heer, Phys. Rev. 134 (1964) A799. [91E.J. Post, Rev. Mod. Phys. 39 (1967) 475, [10] S. Ezekiel and H.J. Arditty, eds., Fiber-optic rotation sensors (Springer, Berlin, 1982). [ll]M. Faucheux, D. Fayoux and J.J. Roland, 1. Opt. (Paris) 19 (1988) 101.