A matrix solution to Maxwell's equations in 2 + 1 dimensional curved space: Two examples
A matrix solution to Maxwell’s equations in 2 + 1 dimensional curved space: Two examples S.G.Kamath* [0000-0002-1148-3787]
Department of Mathematics, Indian Institute of Technology Tirupati Renigunta Road, Tirupati 517506,India *e-mail:[email protected]
Abstract : In an effort that puts together a paper by Plebanski[1] with a matrix approach to the solution of Maxwell’s equations in flat space by Moses[2], Maxwell’s equations in 2 + 1 dimensional curved space are solved in two separate cases of the metric g given by: a. Banados, Teitelboim and Zanelli[3] and b. Deser, Jackiw and ‘tHooft[4] and Clement[5] to obtain the respective time – independent solutions; an extension to time – dependent solutions with the same point of view is also briefly indicated. Keywords: 2 + 1 dimensional curved space, solution of Maxwell’s equations, partial differential equations in curved spaces. PACS: ,ln yxrr is an obvious solution to the Laplace equation , with the general solution easily worked out as cossin, m mmmmmm mdmcrbrar (1) While the simplicity of the Coulomb potential is lost in (1), there have been several recent calculations [6,7,8,9]on the correction to the hydrogen spectrum from the inclusion of the Schwarzschild metric as a perturbation for example. In this context a complementary investigation that is wanting is the quantum mechanics of an hydrogen ion subject to an electrostatic potential , r without the inconvenience of eq. (1) and the matrix solution to the Maxwell’s equations in flat space by Moses that yields both the static and time – dependent solutions as an integral given by eqs.(3.9) and (3.14) therein is perhaps an answer. Going further, the restriction to flat space in Moses[2] motivates this report to build on Ref.2 and solve Maxwell’s equations in curved space; the answer thus obtained would be the counterpart of the flat space results mentioned above. In this effort it pays to link the twin ideas of Plebanski[1] and Moses[2] and as a first we solve Maxwell’s equations in 2 + 1 dimensional curved space in this paper; the calculation is lengthy and a reader–friendly approach seems in order. With the equations formally given in 3 + 1 dimensions as FFAAFFggF FjcFgg ** (2) it helps to elaborate eq.(2) in 2 + 1 dimensions as the four equations involving partial derivatives namely, EEBjcHD jcHDjcD ii (2a) with[1] ,,, FBFEFgHFgD aaaa (3) The last entry in eq.(2a) is the Bianchi identity, and for the examples we shall consider here g is a constant with g . ith the pair of column vectors defined as TT jc ijc ijc iiDEiDEiHB (4) eqs.(2a) can now be written following Ref.2 as I (5) with I being the unit matrix 4 x 4 matrix and the i defined by (6) The operator has eigenvalues ipp , with pp , with the respective linearly independent set of orthonormal eigenvectors being TxpiTxpi TxpiTxpi pppeppppep ipppeppipipep (7) These eigenvectors satisfy the orthonormality and the completeness property respectively given by yxpypxyxyx abi p biaiijjp i )2(41 *)2( ,,, (8) The subscripts ba , in eq.(8) label the elements of the column vectors i with the operation of complex(hermitian) conjugation shown as * ib i . A diligent application of the work by Moses[2] now helps to determine the matrix elements in the column vector T iDEiDEiHB (9) in terms of those in T jc ijc ijc i (10) rom the expansion pi p iii p ii pdtpgtph (11) With p hiphiphphp (12) the time – independent solutions to (5) can now be obtained in two steps using eqs.(8),(12) through (13) One first gets ,,, igphigphgphgph (14) and the use of eqs.(14) and then eqs.(8) enables the rewrite p z ziipx (15) with the i and their hermitian conjugates being functions of zx , respectively. With eqs.(7), eq.(15) reads as zippipp ippipp ippipp ippipppex p z zxpi
21 21 12 123 (16) Or, p z Tzxpi jpjpipjippjippjpjpippec ix The last term will be zero by current conservation and the momentum integration then leads to z Tyx zxjzxjzx zxjcix (17) quating this to T iDEiDEiHB it is easy to infer that log RxgradiDiE (18) when
Rzcj reflecting an unit charge at Rz . By extension this implies that z zx zxjciiHB (18a) The relation between EB , and DH , now follows from FggFFggF bccaba , (19) and eqs.(3).In detail, one has FggggFggggFggggF FggggFggggFggggF aaaaaaa leading to,
FiggggFggggFggggiFFiHB
FggggFggggFiggggiFFiDE (20)
FggggFiggggFggggiFFiDE
Writing eqs.(20) as
FFFggggigggggggg ggggggggigggg igggggggggggg iDEiDEiHB T (21) one can now determine through matrix inversion the column vector on the right hand side of (21) as eqs.(18) and (18a) define the left hand side of eq.(21) for an unit charge at Rz . his will be done first for the g given in Cartesian coordinates by: 1. The black – hole metric of M.Banados, C.Teitelboim and J.Zanelli[3]: ,4112 112 22 yxrrJlrMf fr fxyfrxyrJx frxyfr fyxrJy rJxrJylrMg (22a) eq.(22a) being obtained from rJlrMfNJNrdtNdrdrfdtNds (22b) In this case eq.(21) reads as
211 211 122
FFFfrJyifrMfyfxrfrMffrxy frJxfrMffrxyifrMfxfyr iffrJyfrJxFFFL iDEiDEiHB
T T (23) The matrix L is invertible as fflrMifiL (24) and acyacxlrMgic rf xkr yfidrfkrfdxyacy rfkrfdxyrf ykr xfidacxLL
22 2222 222222 222222 22 221 (25) with lrMikfigfidficfr Ja (26) Eq.(23) thus yields
22 1122 2222 222222 222222 22 22 22111210201 iDE iDE iHBacyacxlrMgic rf xkr yfidrfkrfdxyacy rfkrfdxyrf ykr xfidacxL iDEiDEiHBLFFF TT (27) With eqs.(18) and (18a) defining the column vector on the right hand side of (27) it is easy to determine the required answer for the metric given by (22a). A similar effort informs the second example below: 2. The metric of S.Deser, R.Jackiw and G. ’tHooft[4] and G.Clement[5] is obtained from
JJGkkJdrdrdcdtds ,8,2, as: yx xyx xyyx x yx xyyx yyx y yx xyx yg (28) being the gravitational constant and J the spin of the massless particle, eq.(28) being labeled as the rotating solution of Ref.4 by Clement[5]. In this case one first gets
10 01 1
FFFyx yi yx xi yxiyx yyx xFFFLiDE iDE iHB (29) with iryyxiiL (30) and
22 1122 222 2210201
112 21 21det1 iDE iDE iHBGiAii Cyx yiiAGGi AGCyx xiiAiLFFF (31) where
22 22222 ,, yxCyx yGyx xA (32) Eqs.(31) and (32) together are the counterpart of (27) for this metric and this completes the derivation of the time – independent solutions to eqs.(2). From eq.(5) the time – dependent solutions can also be determined as above from p i p ii ghiphhiphhphhphI (33) and the four counterparts of eqs.(14) ,,, giphthgiphthgphthgphth (34) Eqs.(34) have the respective solutions spgedstphspgedstph spgedstphspgedstph tsipstip tspstp ,,,,, ,,,,, (35) Eq.(15) now writes as zedsedsedseds htx p z tsipstiptspstpi p ii , (36) with ii , being functions of zx , respectively. For the integration over sp , , the first term in (36) yields: aJaJaJ aJaJaJaJ aJaJaJaJK zKepdpdszeppippip pipppp pipppppeds stpp stpzxpi
011 1202 1220 0221 22221 121212. sincos0 sin2cos212sin210 cos2sin212cos210 0000 4000 00002 (37) with
22 11 tan, zx zxzxpa in (37). From the integrals: baba abaaaxJxedx bab aaxJxedxbab baxJxedx bx bxbx (38) nd the labels zxJzxLzxK zxRzx tGtHzxzxt (39) one gets after the integration in (37) the result RGyGx GyHLJH GxJHHK
11 11 sinhsinh0 sinh2cos2sin0 sinh2sin2cos0 000041 (40) For the remaining terms in (36) one obtains successively: 2 nd term: aJaJaJ aJaJaJaJ aJaJaJaJL zLepdpdszeppippip pipppp pipppppeds tspp tspzxpi
011 1202 1220 0221 22221 121212. sincos0 sin2cos212sin210 cos2sin212cos210 0000 4000 00002 (41) The integration in eq.(41) yields RGyGx GyELJE GxJEEK
11 11 sinhsinh0 sinh2cos2sin0 sinh2sin2cos0 000041 (42) with tEzx as opposed to tHzx in eq.(42).Thus the sum of eqs.(40) and (42) works to RGyGx GyTLJT GxJTTK
11 11 sinhsinh0 sinh2cos2sin0 sinh2sin2cos0 000021 (43) with
Tzx .Continuing, one gets for the 3 rd term: aJaJaJaiJ aJaJaJaiJ aiJaiJaJJ zJepdpdszeppppp ppppp ppppppeds tsipttsipp t zxpi (44) and the integration in Eq.(43) leads to
11 11
SiKiJSzx txi iJSSiLzx tyi zx txizx tyiiW (45) with ~11,~,~ zxWitSzxtzx (46) 4 th term: Using ~ itTzx below one obtains aJaJaJaiJ aJaJaJaiJ aiJaiJaJL zLepdpdszeppppp ppppp ppppppeds tsipttsipp t zxpi (47) On completing the integration eq.(47) yields
11 11
TiKiJTzx txi iJTTiLzx tyi zx txizx tyiiW (48) and the sum of eqs.(45) and (48) works to
22 22 zxtzxt zxtzxt (49) Finally, the sum of eqs.(43) and (49) yields
RGyGx GyELJE GxJEEK
11 11 sinhsinh0 sinh2cos2sin0 sinh2sin2cos0 000021 (50) with tEzx .Eq.(36) thus becomes zRGyGx GyELJE GxJEEKtx z
11 11 sinhsinh0 sinh2cos2sin0 sinh2sin2cos0 000021, (51) with
222 222122 sincos,112, ,cossin,11,, zxLzxJtEzx zxKzxRzx tGzxt (52) and is the counterpart of eq.(14) for the static case. rom eq.(51) it is clear that iHB will be zero unlike DiE and this is therefore one departure from eqs.(17) and (17a); eq.(21) also reminds us that the column vector T FFF is got from the column vector T iDEiDEiHB by inversion and as eqs.(23) and (30) show, one should expect despite iHB being zero, non – trivial answers for the electric and magnetic fields given by T FFF ; this feature merits a separate discussion and will be taken up elsewhere. In conclusion, one has determined the solutions to Maxwell’s equations in curved space in a form that does not have the infirmity of eq.(1) and can be used to meet the objectives stated in the introduction to this paper; admittedly the presentation has been belaboured given the steps involved. A preliminary version of this report was presented as ‘Three partial differential equations in curved space and their respective solutions’ at QTS11,Centre de Recherches Mathematiques,Universite de Montreal, Canada and will appear as part of the QTS Proceedings.
References Jerzy Plebanski, Phys.Rev. , 1396 (1960). 2.
H. E. Moses, Phys.Rev. ,1670 (1959). 3.
M. Banados, C. Teitelboim and J. Zanelli, Phys.Rev.Lett. , 1849 (1992). 4. S. Deser, R. Jackiw and G. ‘tHooft, Ann.Phys. ,220 (1984). 5. G. Clement, Int.J.Theor.Phys. , 267 (1985). 6. Zhao Zhen-Hua, Liu Yu-Xiao and Li Xi-Guo, Phys.Rev.D ,064016(2007). 7. L. Parker, and L.O. Pimentel, Phys.Rev.D ,3180 (1982). 8. E. Fischbach, B. S. Freeman, and Chang Wen-Kwei, Phys.Rev.D ,2157(1981). 9..A. S. Lemos, G. C. Luna, E. Maciel and F. Dahia,Class.Quant.Grav.36