Boughn, Stephen P.

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Haverford College Haverford Scholarship Faculty Publications Astronomy 1975 Electromagnetic radiation induced by a gravitational wave Stephen P. Boughn Haverford College, sboughn@haverford.edu Follow this and additional works at: https://scholarship.haverford.edu/astronomy_facpubs Repository Citation "Electromagnetic Radiation Induced by a Gravitational Wave", Phys. Rev. D 11, 248 (1975). This Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. PHYSICAL REVIE% D VOLUME 11, NUMBER 2 15 JANUARY 1975Electromagnetic radiation induced by a gravitational waveStephen BoughnDepartment of Physics, Stanford University, Stanford, California 94305(Received 8 July 1974)The electromagnetic radiation induced by a gravitational wave incident on a point charge anda static point dipole is calculated using a mul. ipole expansion of the fields. The calculationis easily extended to the case of a relativistic source. The presence of this radiation doesnot violate the principle of equivalence. Estimates -are made of the electromagnetic radiationinduced by a plane gravitational wave incident on a plasma. For hot plasmas the radiated powercan be considerably enhanced; however, it is still rather small and the frequency of the radia-tion would probably be too low for this process to be of any astrophysical significance.INTRODUCTIONSince Einstein first introduced the principle ofequivalence, the question of whether or not acharge radiates when falling freely in a gravita-tional field has generated much discussion. Onlyrecently have explicit calculations been carriedout within the framework of Einstein-Maxwelltheory which bear upon this problem. ' ' Thesecalculations involved the motions of particles nearSchwarzschild and Reissner -Nordstrom blackholes. In this paper I will investigate the radia-tion from a point charge and static point dipole in-duced by a plane, polarized gravitational wave.The results are interesting examples of how grav-ity can induce electromagnetic radiation; however,it is doubtful that these processes are of any astro-physical significance.The mathematical procedure is straightforward.The generalized Maxwell's equations will besolved in a space-time that has been slightlycurved by the gravity wave. For a polarized grav-itational wave propagating in the x, direction, thedeviation of the metric tensor g&, from the Min-kowskian form g» is most simply expressed inthe special "transverse-traceless" (TT) coordinatesystem':0 0 0 0,.(„„,) 0 0 0 0b'av= gPv —'gPv =he' ""~ 0 0 1 00 0 0 -1The radiation is assumed to be weak, i.e. , h «1.In the presence of the gravity wave Maxwell'sequations must be expressed in their covariantform, '-A 'Bg+A BAH—-4'where A is the electromagnetic vector potential,is the 4-current density, A is the Ricci tensor,and the semicolon denotes covariant differentiation.If we assume that the region of space-time ofinterest has negligible stress-energy, then Ein-stein's equations tell us that 8~~=0.COORDINATE CHOICEBefore solving Maxwell's equations one mustdecide upon an appropriate coordinate system. Al-though formally the results are independent of thechoice, one should pick a frame in which the inter-pretation of these results is made simple. In gen-eral an appropriate frame is one which is closeenough to a Lorentz frame that gravitational cor-rections to the behavior of measuring instrumentsare negligible compared to the measurement ofinterest. Only in such a coordinate system willelectric and magnetic field vectors have the samephysical significance they have in special relativ-ity. From this standpoint the transverse-tracelessgauge is acceptable since it deviates from a Lo-rentz frame by an amount of order It, &&1 at largedistances where the electromagnetic field is tobe sampled. Corrections due to gravitational ef-fects in measuring instruments will then be oforder h and can be neglected.The specification of the source J is also coordi-nate -dependent. The generally covariant expres-sion for the 4-current of a source consisting ofpoint charges is'~"l, &)-z "f~, t)g v&lx. -x)( d, )where g= —detg 8. The trajectories of noninteract-ing point particles in the presence of gravity aregiven by the geodesic equationx i„ dx dxdT '- dT d7 'In the TT gauge of Eq. (1) I',",=0, and the solutionto this equation for particles initially at rest isdx /dT=dx"/dt=(1, 0, 0, 0). Also, g '= I+termsof order h' a, nd higher. Equation (3), therefore,E LE CT ROMAGNE TIC RADIATION INDUC ED 8 Y A. . .reduces to the classical formJ"(x) =p q„&(x„—x)5, +terms of order h'.Ji ' s (k&y -~ t) 'vQrh"e'4m BX2(8)(4)The above analysis neglected any interaction be-tween the particles. If a dipole consists of twooppositely charged particles separated by a dis-tance d and held apart by a spring, then in generalthe gravitational wave will excite oscillations ofthe system. The resulting time-dependent dipolemoment will serve as a source for radiation in theusual way. This radiation is well understood clas-sically and will not be considered in this paper.Therefore, we treat the source as consisting ofpoint charges sufficiently free so that Eq. (4) holds.SOLVING MAXWELL'S EQUATIONSIf we expand the covariant derivatives in termsof the metric tensor in Eq. (1), Maxwell's equa-tions becomeUA„—(gas/rt~)k 8yA„S-qM(&~it aAg +27„aAs 8+I'as A„s) =-47'J„,(5)J& ~ ei, (k~&- ~t)47t ~x3The fields from A.' fall off as x ' or faster, ' conse-quently, any radiation will be associated with theperturbation 4 . In effect, the gravitational waveinduces a 4-current density J' in a flat space-timebackground, and it is this current which acts as asource for electromagnetic radiation. One cancheck that J~ satisfies the important requirementthat it be conserved, i.e. , J' = 0.Although J' can be expressed analytically, it isnot a simple source and the only practical approachto the problem is to expand the radiation field inmultipoles. The power radiated by a source $(x, t)= e '"' J(x) can be expressed by the sum'&= (8»') 'P [IA, (&, m) I'+ IA, (t, m) I']SmA„and A~ are the coefficients of the magnetic andelectric multipole fields and are given by the ex-pressionswhere terms of order h' and higher have been neg-lected (note I"„~SO-h), I' ~8 are the usual Riemann-Christoffel symbols, and the comma denotesordinary differentiation. Since h «1 we expect Ato be close to the flat-space-time solution A'„whereAa(l, m) = 4miA~(t, m) = —4j, (ur)X+ (Vx J)d'x,j,(kr )X,* ~ (~ x g x J)d 'x, (10)A = -4' (6) where j& are spherical Bessel functions and X,are vector spher ical har monic s.For both of the sources treated in this paper A~is of the form Ao =($, 0, 0, 0). Consider a pertur-bation ~~ to this solution. Substituting A~-A.~+ eA„ into Eg. (5), using Eq. (6), and droppingterms of order A, we obtainRADIATION FROM A POINT CHARGEThe scalar potential for a point charge q isq/r (the-minus sign is a. result of using thecovariant form of the vector potential); therefore,from Eq. (8) the induced current density becomes=-4mJ',where & has been set equal to 1 for convenience,the connection coefficients I' ~8 are expressed interms of h, and we have introduced the 4-vectorzxJ' = 0, ', u qhe' k"~ ', ,'cuqhe'""~ -LX0 4~~3 4++3(11)If one substitutes Eq. (11) into Eq. (10) and usingthe expansionJ' =01&0..—~)4~ ~x' ~x'2 3e'""' = P [4m(2 l' + 1)]' 'i' Y, „j, (kr),=0the integrals can be evaluated and one obtains arather lengthy expression for Az(l, m):250 STE PHEN BOUGHN3 i/2 m 2~+1A~ = i&u'qh(2('/2)' '(-1) l(l+1)&QI' (2l'+l)I2mW3( )'(2 )-( ) (2+I+2)(I'+I)F( )F'( )+ i [(1+m)(f —m+1)]' ' +[(&—m)(&+I +1)]. l'l1 l' I 1 l 1000 0 —m+1 1 0 -~ —1 —1(, (I' —I+2) (I —F+2)—v3 [(&+m)(l —m+1)]' 2 .+[(l —m)(l+rn+1)]l' l 2x (I'+I+2)(l'+l)F( )F )where!t' f,(m, m, m,is the Wigner 3-j coefficient and I'(n) is the gammafunction. The magnetic multipole coefficientsA„(l, m) all vanish. From the above expressionone can see that the only nonvanishing As(l, m)are l ~ 2 and m=+2. This is not too surprisingconsidering the spin-2 nature of the graviton. Fig-ure 1 is a plot of the multipole coefficients and thepower radiated in each multipole,P, = (s~~')-'g [l&„(l,~) I'+ l&~(f, I) I'1proximately a factor of q'/~' less than the crosssection for scattering electromagnetic radiation(Thomson scattering) and approximately q'/m'times greater than the cross section for scatteringgravitational radiation. ' If the particle is an elec-tron, q'/m' = 5 &&10" and o =10 "cm'.The results presented above are readily gener-alized to the case where the source is a highlyrelativistic, y»1, point charge. The metric ten-sor of the gravity wave in the rest frame of a par-ticle moving with velocity v = —vx, takes exactlythe same form as in Eq. (1), except that the fre-quency is Doppler -shifted, ~' = 2y~. Therefore,the power radiated in the rest frame of the sourceisFor large f, P, ~ 1/l so the sum P"P, diverges.Therefore, an infinite plane wave incident on apoint charge induces an infinite amount of electro-magnetic radiation. The divergence is avoided ifeither the charge is screened at some distance dor the gravity wave is in the form of a wave packetof dimension d. In each case the sum will beeffectively terminated at l = kd. Physically this isbecause the large angular momenta associated'h high multipoles arise from radiation incidentwith large angular momenta, i.e. , large impactparameters. An approximate expression for theradiation emitted with the sum terminated at multi-pole number l isO. I3~w O.OI—IOEJ'Ol3EIleP= [0.35+ kin(L/10)](d'q%' for f~ 10. (12)The cross section o for the production of electro-magnetic radiation is obtained by dividing E(l. (12)by the flux of incident gravitational radiation, ' += (1/32')(o%':0.001 IIOMULTIPOLE NUMBER, LO. IIOOo = P/F = [35+42(in(l/10)]q'. (13)For a particle of mass ~ and charge q, o is ap-FIG. 1. Power P& and coefficients Az(l, m) of themultipole components of the induced radiation field ofa point charge.ELECT ROMAGNETIC RADIATION INDUC ED B Y A. . .P = [1.4+-', In(f/10)]~'y'q'h'. (14)RADIATION FROM A POINT DIPOLENow consider a static point dipole which has theusual flat-space-time scalar potential Q = -(p x)/r'.To simplify the calculation I have assumed p = t)xy.Az(l, m) and A„(l, m) are calculated using the sameprocedure as above. In this case, however, thedipole field falls off rapidly enough that for largel, P, ~1/P and the infinite sum converges. Thefirst term is again quadrupolar,Pl = 2 5.6 x 10-4p '~'h ' . (15)For a dipole formed by two charges +q and -qseparated by a distance d, this power is roughlya factor (hd)' less than that of a single charge q.It is interesting to compare this case with theradiation implied by changes in the "proper" lengthd of a finite dipole. Both the expression for properlength' and the equation of geodesic deviation tellus that particles lying on the x, axis are not accel-erated with respect to one another and from thispoint of view should not radiate, contrary to theabove calculation. The point is that radiation isa global, not local, phenomenon, and cannot bedetermined by investigating the relative accelera-tions of charged particles in a local inertial frame.Recall that in this analysis all charges are as-sumed to be noninteracting. Suppose we attachthe point charges to the ends of a stiff spring (i.e. ,a spring with a resonant frequency larger than thefrequency of the gravitational wave) of length d. Itis a well-known result" that in a TT frame thespring responds to a gravitational plane wave withan oscillation of amplitude -kd if the spring istransverse to the direction of propagation. Thus,the dipole moment has a time-dependent componentdp/dt —vhqd. From classical electrodynamics thepower radiated from an oscillating dipole iss ~5p~'tu'-p'&u'h'. This is of the same order givenWe recall from relativistic kinematics that in theoriginal frame of the observer most of this radia-tion will be emitted within an angle 6=1/y of theforward direction -x, and with a frequency betweeny~' and 2yo'. From the transformation propertiesof the fields the intensity of the radiation is in-creased by y' for radiation in the forward direction.The total power emitted in the observer's frameis then of the same order as that in the particle'srest frame. Hence, by Eq. (14) the cross sectiono for a relativistic particle is up by y' from thatfor the nonrelativistic case. In the forward direc-tion the intensity is increased by a factor of y4 andat a frequency of about y' times the frequency ofthe gravity wave.in Eq. (15). Consequently, this type of radiationmust not be neglected for tightly bound dipoleslying perpendicular to the propagation vector ofthe gravity wave.ASTROPHYSICAL SIGNIFICANCEThe incredibly small cross sections togetherwith the over-all electrical neutrality of matterindicate that the processes described in this paperare of little astrophysical significance. Still, letus estimate the radiation produced by a plane waveincident on a plasma of large dimension. For ourpurposes a simple model of a plasma is a distribu-tion of randomly oriented dipoles of magnitude p= qn ' ', where n is the number density of chargedparticles. This model is the easiest way to in-corporate the constraint of zero net charge of theplasma. Because of random orientation the radiat-ed power yer unit volume will be roughly n timesthe power emitted by a single diyole,dPyy dipole~yzP . np ~~/ = q2~ ggj 3Within a tenuous plasma, electromagnetic radia-tion with a frequency less than the plasma frequen-cy, &v~=4& qn' m' (where m is the electron mass),is exponentially damped with a penetration depth of5=1/u&~. " Consequently, we expect only the outerlayer 5 of the plasma to emit the gravitationallyinduced radiation. Therefore, the induced lux ison the order of—~- ~'q~W-~«m~2dPinduced yyThe ratio of this to the incident flux +&& of gravitywaves isff = E;„g„„~/Fp~ qQJ n mIf we choose the typical values of m =electronmass, q = electron charge, ~ = 10' sec ', andn =10"/cm', then B-10 ". If the plasma is veryhot the electrons may be relativistic and conse-quently emit a rather broad band spectrum of radia-tion while the heavier positive cha, rges will emitat the primary frequency. Then the two compo-nents of the plasma act independently and one wouldexpect coherent radiation to occur for regions ofdimension A. 5 on the outer la.yer of the plasma,(where A, denotes the radiation wavelength). Thepower emitted by such a region is, from Eq. (10),P- (n5/e')'v'q'h',which results in a flux of roughly &-mnh'. Theratio of this to the flux of gravity waves isR-mn/v'. For the values of m, n, and tu chosenabove A-10 '. This ratio is still too small to252 STEPHEN BOUGHNbe interesting, furthermore, the expected fre-quency of gravity waves is so low that the electro-magnetic radiation produced would have troublepropagating through interstellar space and certain-ly would never penetrate the ionosphere of theearth. Highly rela, tivistic particles, y»1, asdemonstrated, would raise the frequency of theinduced radiation by a factor -y' and the radiatedflux by -z'; however, it is difficult to imagine asource which mould produce a sufficient density ofsuch particles to give this process astrophysicalsignificance.DISCUSSIONThough apparently of little astrophysical signifi-cance, the processes treated in this paper areinteresting examples of the interaction of gravita-tional and electromagnetic waves. Strictly speak-ing, a gravity wave cannot interact with a, pointparticle. A point charge, however, has a non-vanishing quadrupole moment due to the distribu-tion of field energy about the charge. It is there-fore able to absorb energy from the gravitationalwave. The subsequent reradiation of this energyas electromagnetic waves was the subject treatedin this paper. Since the energy quadrupole mo-ment of a point charge diverges we might expectan infinite amount of energy to be absorbed fromthe gravity wave. This agrees with the divergentcross section for electromagnetic emission cal-culated above.Momentum as well as energy is transferred inthis process. Although it would take a detailedcalculation to determine the momentum carriedamay by the particle, it is quite possible that it iscomparable to that of the radiation. If this is thecase the particle will experience a radiation reac-tion force on the order of h q2u . By attaching anaccelerometer to the charge one could in principle de-tect this force. The magnitude of the response de-pends only on the force -k'q'~', which is defined at apoint, and the response time is a property of theaccelerometer alone. Consequently, this measure-ment might be construed as a violation of theequivalence principle as it is usually stated, i.e. ,"no local measurement can detect the curvatureof space -time. " This statement is rather vaguein that no specification of what constitutes a localmeasurement is given. Possibly one could pre-scribe conditions of locality which the above mea-surement fails to meet. A much more satisfac-tory resolution is to restate the equivalenceprinciple in the only may general relativistsever use it, namely, "all the fundamental lawsof physics reduce to their special-relativisticform at the origin of a local inertial frame, i.e. ,a frame such that I'" z-—0 at the spatial origin. "The phenomena dealt with in this paper are notlaws of physics but rather global solutions to thelaws of physics and depend on their structure ina large region of space-time. The presence of a,radiation reaction force in no way affects the localstructure of physical laws and does not constitutea violation of the equivalence principle.ACKNOWLEDGMENTSI would like to acknowledge helpful discussionswith Sandy Fetter and Bob Wagoner. This workwas supported in part by the National ScienceFoundation under Grant No. GP 38610.~R. Ruffini, Phys. Lett. 418, 334 (1972).2R. Ruffini, J. Tiomno, and C. U. Uishveshwara, NuovoCimento Lett. 3, 211 (1972).D. M. Chitre, R. H. Price, and V. D. Sandberg, Phys.Rev. Lett. 31, 1018 (1973).4M. Johnston, R. Ruffini, and F. Zerilli, Phys. Rev. Lett.31, 1317 (1973).C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973), Chap. 35.6Reference 5, p. 389. In this equation and in what followswe use units where G = c=l.J. D. Jackson, C/assica/ Electrodynamics (Wiley,New York, 1962), Chap. 16.P. C. Peters, Phys. Rev. D 9, 2207 (1974).L. D. Landau and Z. M. Lifshitz, The Classica/ Theoryof Fields (Addison-Wesley, Reading, Mass. , 1962),Chap. 10.Reference 5, Chap. 35.~~Jackson, Ref. 7, p. 227.

Electromagnetic radiation induced by a gravitational wave

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Electromagnetic radiation induced by a gravitational wave

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