Global and local perspectives of gravitationally assisted negative-phase-velocity propagation of electromagnetic waves in vacuum

Transcription

1 Physis Letters A 336 (2005) wwwelsevierom/loate/pla Global and loal perspetives of gravitationally assisted negative-phase-veloity propagation of eletromagneti waves in vauum Akhlesh Lakhtakia a,1, Tom G Makay b,, Sandi Setiawan b a CATMAS Computational & Theoretial Materials Sienes Group, Department of Engineering Siene and Mehanis, Pennsylvania State University, University Park, PA , USA b Shool of Mathematis, University of Edinburgh, Edinburgh EH9 3JZ, UK Reeived 30 September 2004; reeived in revised form 6 January 2005; aepted 7 January 2005 Available online 18 January 2005 Communiated by PR Holland Abstrat Consistently with the Einstein equivalene priniple and using an eletromagneti formulation first suggested by Tamm, we show that a loal observer annot observe negative-phase-veloity (NPV) propagation of eletromagneti waves in vauum, whereas a global observer an appreiate that phenomenon Using the speifi example of the Kerr metri, we also demonstrate the possibility of NPV propagation within the ergosphere of a rotating blak hole 2005 Elsevier BV All rights reserved PACS: 0420Cv; 0350De Keywords: General theory of relativity; Negative phase veloity; Speial theory of relativity; Kerr metri 1 Introdution Reently, we have shown that eletromagneti plane waves an propagate in lassial vauum in suh a way that the phase veloity vetor has a negative projetion on the time-averaged Poynting vetor, provided that the vauum is nontrivially affeted by a gravitational field [1] Our approah is based upon Tamm s eletromagneti formulation [2], involving a gravitomagneti metri The negative-phase-veloity (NPV) propagation harateristi * Corresponding author addresses: akhlesh@psuedu (A Lakhtakia), tmakay@edauk (TG Makay), ssetiawan@edauk (S Setiawan) 1 Also affiliated with Department of Physis, Imperial College, London SW7 2BZ, UK /$ see front matter 2005 Elsevier BV All rights reserved doi:101016/jphysleta

2 90 A Lakhtakia et al / Physis Letters A 336 (2005) depends on the diretion of the propagation wavevetor but not on the frequeny NPV propagation harateristis in ertain homogeneous material mediums lead to the phenomenon of negative refration [3 5], whih suggests the possibility of gravitationally assisted negative refration of eletromagneti waves by vauum, with attendant sientifi impliations [1,6] Researhers have studied eletromagneti wave propagation in terrestrial environments for several enturies, and our planet is ertainly affeted by the solar gravitational field Yet, NPV propagation in lassial vauum has never been reported in the terrestrial ontext Although the solar gravitational field evidently does not satisfy the onditions for NPV propagation that were reported elsewhere [1], evidene aumulated from the Pioneer 10 mission has prompted suggestions that new physis may be found even at the outer reahes of our solar system [7] These and other onsiderations beg the question: are global and loal perspetives of gravitationally assisted NPV propagation different? In this ommuniation, we answer that question in the affirmative 2 Maxwell equations in universal spaetime In the absene of harges and urrents, eletromagneti fields obey the ovariant Maxwell equations 2 f αβ;ν + f βν;α + f να;β = 0, h αβ ;β = 0, where f αβ and h αβ are, respetively, the ovariant and the ontravariant eletromagneti field tensors whereas the subsript ; ν indiates the ovariant derivative with respet to the νth spaetime oordinate The spaetime denoted by the vetor x α ontaining the normalized time oordinate x 0 = t (with as the maximum speed of light in the absene of the gravitational field) and the spae oordinates x 1, x 2,andx 3 is Riemannian, with the metri g αβ being a funtion of spaetime and arrying the signature (+,,, ) [8] It is ommonplaeto follow up on a suggestion of Tamm [2,9 11] and hangethe formof (1) for appliation to eletromagneti fields in vauum The Maxwell equations (1) may be expressed in nonovariant form as (1) f αβ,ν + f βν,α + f να,β = 0, [ ( g) 1/2 h αβ ],β = 0, (2) wherein g = det[g αβ ] and the subsript,ν denotes ordinary differentiation with respet to the νth spaetime oordinate We note that generalizing the Maxwell equations from nonovariant to ovariant formulations is not totally unambiguous [12,13] In the absene of experimental evidene to eliminate this ambiguity, we adopt the standard generalization (1) Let us introdue the eletromagneti field vetors E l, B l, D l and H l via the standard deompositions } E l = f l0, B l = (1/2)ε lmn f mn (3) D l = ( g) 1/2 h l0, H l = (1/2)ε lmn ( g) 1/2 h mn with ε lmn being the three-dimensional Levi-Civita symbol Thereby the nonovariant Maxwell equations (2) assume the familiar form } B l,l = 0, B l,0 + ε lmn E m,n = 0 (4) D l,l = 0, D l,0 + ε lmn H m,n = 0 In vauum, the omponents of the eletromagneti field tensors are onneted by the onstitutive relations h αβ = g αµ g βν f µν, f αβ = g αµ g βν h µν (5) 2 Roman indexes take the values 1, 2 and 3; Greek indexes take the values 0, 1, 2 and 3; summation is implied over any repeated index; and Gaussian units are used

3 A Lakhtakia et al / Physis Letters A 336 (2005) These onstitutive relations of vauum an be stated for the eletromagneti field vetors as } D l = γ lm E m + ε lmn Γ m H n, B l = γ lm H m ε lmn Γ m E n where 1/2 glm γ lm = ( g), g 00 Γ m = g 0m (8) g 00 Eqs (1) and (5) employ urved spaetime So do (4) and (6), but the differene is that they look like the familiar eletromagneti equations in flat spaetime applied to an instantaneously reating, bianisotropi medium Tehniques ommonly employed to handle eletromagneti problems in the absene of gravitational fields should therefore be useful for solving (4) and (6) Before proeeding further with our analysis, let us therefore reast the url equations in (4) and (6) using the onventional 3-vetors and 3 3 dyadis as E(t, r ) + t B(t, r ) = 0 (9) H (t, r ) t D(t, r ) = 0 and D(t, r ) = ɛ 0 γ (t, r ) E(t, r ) 1 Γ (t, r ) H (t, r ) B(t, r ) = µ 0 γ (t, r ) H (t, r ) + 1 Γ (t, r ) E(t, r ), wherein γ (t, r ) is the dyadi-equivalent of γ lm, Γ (t, r ) is the vetor-equivalent of Γ m, the salar onstants ɛ 0 and µ 0 denote the permittivity and permeability of vauum in the absene of a gravitational field, and SI units are adopted (6) (7) (10) 3 Global perspetive Suppose that we wish to solve these equations in a ertain region X of spaetime, subjet to speifi boundary onditions A fairly standard proedure would be to partition X into subregions X (n = 1, 2, 3,), in eah of whih we would replae the nonuniform metri g αβ by the uniform metri g αβ Correspondingly, the nonuniform vetor with omponents Γ l would be replaed by the uniform vetor with omponents l in eah subregion X The spaetime x α would still remain urved After solving (4) and (6) in eah subregion, we ould stith bak the subregional solutions into the regional solution This pieewise uniform approximation tehnique is very ommon for solving differential equations with nonhomogeneous oeffiients [14] Aordingly, let us examine the eletromagneti fields in the nth subregion A three-dimensional Fourier transform of the eletromagneti field vetors an be taken, with the wavevetor k denoting the Fourier variable orresponding to r; thus, E(t, r) = 1 H (t, r) = 1 E(ω/, k) exp [ i(k r ωt) ] dωdk 1 dk 2, H(ω/, k) exp [ i(k r ωt) ] dωdk 1 dk 2 (11)

4 92 A Lakhtakia et al / Physis Letters A 336 (2005) where i = 1andω is the usual temporal frequeny The wavevetor omponent k 3 is determined by first substituting (11) in (9), and then formulating a 4 4 matrix ordinary differential equation whih is then solved as an eigenvalue problem [15] As our fous lies here on propagating plane waves, evanesent (nonuniform) solutions are eliminated from further onsideration by enforing the restrition k 3 R We remark that E(ω/, k) and H(ω/, k) are the omplexvalued phasors of the eletri and magneti fields, respetively, assoiated with a plane wave, for whih a timeaveraged Poynting vetor P(ω/, k) t an be derived When the projetion of k on P(ω/, k) t is negative, ie, k P(ω/, k) t < 0, we say that the phase veloity is negative The ruial quantity k P(ω/, k) t is derived as follows [1]: Combining the Fourier representations (11) with the onstitutive relations (10) and the Maxwell url postulates (9),we get where p E(ω/, k) = ωµ 0 γ H(ω/, k), p H(ω/, k) = ωɛ 0 γ E(ω/, k), p = k ω Γ (12) (13) (14) The use of (12) to eliminate H(ω/, k) from (13) provides, after some manipulation, the eigenvetor equation {[( ) ω 2 γ } p γ p] I + p p γ E(ω/, k) = 0 for representing E(ω/, k), and the orresponding dispersion relation [ ( ) ω 2 p γ p γ ] 2 = 0 (16) Herein, I is the identity dyadi, and γ is the determinant of γ Considering (15) in light of (16), we see that all E(ω/, k) eigenvetors solutions must satisfy the relation (15) p γ E(ω/, k) = 0 (17) Let us introdue the eigenvalues γ 1,2,3 and orresponding eigenvetors ṽ 1,2,3 of γ For the purposes of illustration, suppose we onsider planewave propagation along the diretion parallel to ṽ 3, while the vetor Γ lies at an angle θ in the plane of the eigenvetors ṽ 1 and ṽ 3,asper 1 = sin θ 2 = 0 (18) 3 = os θ After exploiting the orthogonality ondition (17), the general solution to (15) may be expressed as the sum of two independent modes as E(ω/, k) = A a (ω/, k)e a (ω/, k) + A b (ω/, k)e b (ω/, k), where e a (ω/, k) = ṽ 2 ( e b (ω/, k) = γ 3 k ω ) os θ ṽ 1 + ω γ 1 Γ sin θ ṽ 3 (19) (20)

5 A Lakhtakia et al / Physis Letters A 336 (2005) The omplex-valued salars A a,b are unknown amplitude funtions that an be determined from initial and boundary onditions The orresponding general solution for H(ω/, k) follows straightforwardly by ombining (19) with the Maxwell url postulates (9) as with H(ω/, k) = A a (ω/, k)h a (ω/, k) + A b (ω/, k)h b (ω/, k) h a (ω/, k) = 1 [( ω γ 1 ωµ 0 h b (ω/, k) = 1 ωµ 0 γ 1 ) os θ k ṽ 1 ω ] sin θ ṽ 3 [( k ω 2 os θ) γ 3 + ( ) ω 2 ] sin θ γ 1 The dispersion equation (16) reduesto a k -quadratiexpression whih yields the two wavenumbers k = ω ( os θ ± γ 1 γ 2 ) γ 1 2 sin 2 θ γ 3 ṽ 2 (21) (22) (23) Finally, ombining (19), (21) with (23),wefindthat k P(ω/, k) [( t = 1 ) γ γ 2µ 1 γ 0 γ sin 2 θ 3 γ 3 ± Γ os θ γ 1 γ 2 γ 1 γ 3 Γ 2 sin 2 θ ] ( Aa (ω/, k) 2 + γ 3 ω 2 γ Ab (ω/, k) 2 ) (24) Let us probe the physial signifiane of (24) To do so, we fous upon a speifi gravitomagneti metri, namely the Kerr metri We begin by noting that for a rotating blak hole of geometri mass m rbh and angular veloity a rbh in onventional units [16], the Kerr metri yields and γ = ( R 4 + a 2 rbh z 2 ) 2 ( R 4 + a 2 rbh z 2 2m rbh R 3 ) 2 γ 1 = R 4 + a 2 z rbh 2 γ 2 = R 4 + arbh 2 z 2 2m rbh R 3 γ 3 = 1, (25) (26) where x, ỹ and z are the values of the three spae oordinates at some representative point in the subregion X The quantity R is defined impliitly via R 2 = x 2 + ỹ 2 + z [ 2 arbh 2 1 ( z R ) 2 ] Sine both γ > 0and γ 3 > 0, we see from (24) that NPV propagation in X arises provided that the inequality γ Γ γ Γ < 0 (28) is satisfied (27)

6 94 A Lakhtakia et al / Physis Letters A 336 (2005) It is illuminating to reast (28) in terms of the metri omponents as per g 00 g 00 < 0 (29) For the Kerr metri we have [16] g 00 g 00 = R 4 + a 2 rbh z 2 + 2m rbh R 3 R 4 + a 2 rbh z 2 2m rbh R 3 (30) Within the ergosphere whih lies between the outer event horizon and the stationary limit surfae we have ( ) z 2 m rbh + m 2 rbh a2 rbh < R <m rbh + m 2 rbh a2 rbh, (31) R and m 2 rbh >a2 rbh Therefore, inside the ergosphere R 4 + arbh 2 z } 2 2m rbh R 3 < 0, R >0 (32) and the NPV ondition (29) is evidently satisfied for the hosen wave-propagation ase Thus, when the traversal of an eletromagneti signal is traed from point P (p) X to point Q (q) X, with p q, the possibility of NPV propagation in some subregions of X annot be ruled out a priori 4 Loal perspetive The previous two setions are from a global perspetive Let us now attend to the loal perspetive of an observer loated at some point This observer is onstrained to formulate an eletromagneti theory valid only in some small neighborhood of, and this theory must emerge from observations made only in that neighborhood The gravitational field in this neighborhood an be held to be virtually uniform, the metri from the global perspetive being g αβ However, given the admissibility of the uniform-gravity theory for all loal observers in this neighborhood, we show here that the loal observer would end up formulating a loal metri whih is the same as in the speial theory of relativity By virtue of the Einstein equivalene priniple [17], g αβ is onstrained to be suh that there exists a matrix Λ α β yielding η µν = Λ α µ g αβ Λ β ν, where η µν = diag[1, 1, 1, 1] is the Lorentzian spaetime metri [8] The matrix Λ α β an be used to onstrut a flat spaetime x α for all x α in the speifi neighborhood of as per Λ α β = x α x β (34) dx 0 dx 1 dx 2 dx 3 = det [ Λ α β] dx 0 dx 1 dx 2 dx 3 We must note that, as Λ α µ depends upon g αβ and therefore on x α, a global oordinate transformation whih simultaneously transforms g αβ into η µν for all x α X annot be realized unless gravity is totally ignored [8] In the loal oordinate system x α, the spaetime metri in the speifi neighborhood is given as η αβ Repeating the steps outlined in Setion 2 with η αβ substituting for g αβ, we find that the loal onstitutive relations for the (33)

7 A Lakhtakia et al / Physis Letters A 336 (2005) neighborhood redue to the trivial form D l = } E l B l = H l (35) with respet to x α NPV propagation is not possible for the medium haraterized by the onstitutive relations (35) [18] 5 Conluding remarks Thus, we have shown that gravitationally assisted negative-phase-veloity propagation in vauum an be appreiated only from a global perspetive based on urved spaetime; whereas a loal observer, onstrained to a flat spaetime, must onlude NPV propagation is impossible in vauum The diversity of the loal and the global perspetives is in full aord with the Einstein equivalene priniple, and an be explained as follows: the subregional metri (m) g αβ appears from a suitable subregional averaging of g αβ Whereas g αβ must satisfy the ditates of the Einstein equivalene priniple at every x α X, there is no reason for (m) g αβ to satisfy the same ditates at even one x α (m) X The feasibility of NPV propagation within the ergosphere of a rotating blak hole has been demonstrated in this ommuniation A detailed numerial study is alled for in order to further explore the irumstanes and impliations of suh type of propagation, whih we hope to report in due ourse of time Aknowledgements We gratefully aknowledge disussions with Dr Martin W MCall of Imperial College, London We thank an anonymous referee for helpful suggestions, espeially relating to the Kerr metri SS aknowledges EPSRC for support under grant GR/S60631/01 Referenes [1] A Lakhtakia, TG Makay, J Phys A: Math Gen 37 (2004) L505; A Lakhtakia, TG Makay, J Phys A: Math Gen 37 (2004) 12093, Corrigendum [2] IE Tamm, Zhurnal Russkogo Fiziko-Khimiheskogo Obshhestva, Otdel Fiziheskii (J Russ Phys-Chem So, Phys Setion) 56 (1924) 248 [3] A Lakhtakia, MW MCall, WS Weiglhofer, in: WS Weiglhofer, A Lakhtakia (Eds), Introdution to Complex Mediums for Optis and Eletromagnetis, SPIE, Bellingham, WA, 2003, pp [4] JB Pendry, DR Smith, Phys Today 57 (2004) 37 (June issue) [5] DR Smith, Phys World 17 (2004) 23 (May issue) [6] TG Makay, A Lakhtakia, Curr Si 86 (2004) 1593 [7] JD Anderson, PA Laing, EL Lau, AS Liu, MM Nieto, SG Turyshev, Phys Rev D 65 (2004) [8] BF Shutz, A First Course in General Relativity, Cambridge Univ Press, Cambridge, UK, 1985, Chapter 6 [9] D Bini, C Cherubini, B Mashhoon, Class Quantum Grav 21 (2004) 3893 [10] GV Skrotskii, Sov Phys Dokl 2 (1957) 226 [11] J Plebanski, Phys Rev 118 (1960) 1396 [12] S Weinberg, Gravitation and Cosmology, Wiley, New York, 1972, Chapter 7 [13] AP Lightman, WH Press, RH Prie, SA Teukolsky, Problem Book in Relativity and Gravitation, Prineton Univ Press, Prineton, 1975, Chapter 14 [14] JD Hoffman, Numerial Methods for Engineers and Sientists, MGraw Hill, New York, 1992

8 96 A Lakhtakia et al / Physis Letters A 336 (2005) [15] A Lakhtakia, Optik 90 (1992) 184 [16] R d Inverno, Introduing Einstein s Relativity, Claredon Press, Oxford, 1992, Chapter 19 [17] HA Lorentz, et al, The Priniple of Relativity, Dover, New York, 1952, p 120 [18] TG Makay, A Lakhtakia, J Phys A: Math Gen 37 (2004) 5697