Apeion, Vol. 9, No., Apil 00 6 Sufces of Constnt Retded Distnce nd Rdition Coodintes J. H. Cltenco, R. Lines y M., J. López-Bonill Sección de Estudios de Posgdo e Investigción Escuel Supeio de Ingenieí Mecánic y Eléctic Instituto Politécnico Ncionl Edif. Z, Acc. 3, 3e piso, Col. Lindvist CP07738, México, DF E-mil:lopezl@hotmil.com hclte@my.esimez.ipn.mx We constuct the element of volume vecto coesponding to sufce of constnt etded distnce ound of n ity timelike cuve; the method employed is sed in the dition coodintes of Floides-McCe-Synge fo Riemnnin - spces. Ou esults hve inteest in the study of the electomgnetic Liénd-Wiechet field in cuved spcetimes. Keywods: dition coodintes, Riemnnin -spces, univese function.. Intoduction In this wok the element of volume vecto d is clculted fo sufce with constnt etded distnce, which is constucted ound 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 63 of the tectoy of n electic chge with ity motion in Riemnnin spce. This is geneliztion tht ws done y Synge [] in specil eltivity. The employed method is suggested y the dition coodintes y intoduced in [,3] fo the study of gvittionl dition; hee they e used in electomgnetic dition nd they e vey well dpted fo this pupose ecuse with such coodintes the cuved spce ehves like flt spce in some spects. In othe wods, the use of y implies tht wht ws lened in Minkowski spce cn e tnslted ntully to Riemnn s spces. Ou expession fo d coincides with Villoel s esults otined in [] y mens of the pocedue tht DeWitt-Behme [5] use when constucting sufce with constnt instntneous distnce [6,7]. Howeve, we think tht ou method is moe simple nd poweful ecuse it tuns immedite the esults on dition tensos deduced in [8]. We shll use the Univese Function of Ruse [9], which llows hving covint expnsions in cuved spce. This function emined fogotten duing long time nd its pesent elevnce my e seen in [, 5, 8, 0-7].. Rdition Coodintes We ssume the Einstein convention fo the ddition of epeted indices (,, 3, nd,, fo geeks nd ltin indices, espectively) η =,,, - nd tht the metic loclly tkes the digonl fom ( ) ( ) t ny event. In ode to constuct the dition coodintes y of [] we need timelike cuve C (which in this cse will e the electon tectoy) with n othonoml tetd on it. ( ) e ( ) = η ( ), ( ) e e e = g, () 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 6 dx whee e( ) = υ = is the unity tngent vecto to C, nd x is ds i totlly ity coodinte system with ds = gidx dx. The pimed indices lel points on C. Now let us see how x genetes new coodintes. Fo evey P we constuct the pst sheet of its null cone which intesects to C in P (etded point ssocited to P). We pmetize the null geodesic P P in the fom x ( u) with u = u0 t P nd u = u > u0 t P with dx V = s its tngent vecto stisfying V V = 0. The ssigned du dition coodintes to P e given y: whee y [ ] ( ) + sυ e = () denote the covint deivtive of []: ( 0 ) V The expession () is equivlent to: = u u, = 0. (3) y e ( ) =, y = υ + s, () which implies tht in dition coodintes the cuve C educes to y = 0, y = s. If we intoduce the nottion: K = w = υ, (5) then () dopts the fom of the eltion (9.3) of [] fo flt spce: y ( ) y K e = =, y y = w + s. (6) = In this sense the cuved spce ehves like Minkowski spce, which is vey useful. 00 C. Roy Keys Inc.
( ) ( ) Apeion, Vol. 9, No., Apil 00 65 At P the metic tenso cn e witten in tems of the tetd s g = e e υ υ, then fom (3) nd (5): thus y y ( g + υ ) w = K υ K i K = ( ), (7) = y e + wυ, (8) theefoe ( y ) y ehves like null vecto ecuse ( y y )( y y ) = 0. Ou expessions (7) nd (8) coincide with (9.) nd (9.5) of []. Following the coesponding pocedue in flt spce we intoduce new system of coodintes: z = y, z = y y y = s, (9) tht is, z emins constnt on the null cone with vetex t P It is cle tht the Jcoin of the tnsfomtion: y z is equl to one, theefoe: z y J = J (0) x x Let us clculte (0). If we use tht = ( u u0 ) V x = s = w υ, υ deivtives: nd, then fom (6) nd (9) we otin the ptil 00 C. Roy Keys Inc.
thus i i = = w z J x i e Apeion, Vol. 9, No., Apil 00 66 ( ) ( ) ( ) i = w ikm + w det i υ e + d e ds ( ) ( ) ( 3) t ( ) e e e 3 k m t whee we hve employed the popety i, ), () m = m nd the ikm ntisymmety of the Levi-Civit symol. Fom (3) it is evident tht cn e expessed in tems of the tetd s ( ) ( ) = e + we, then () implies the finl fom: such tht z J x = g ( ) = det ( ), ( ) = det i ( ) ( ) ( ) ( ) ( P), (3) g P g g P g. () D= det, = g P g P D With (3) it is ppent the emk of [5] p. 3 nd [] p. 5: the geodesics emeging fom P egin thei intesection when = 0, ising the so-clled custic sufce. Theefoe we shll ccept tht P is ne to P. The eltions (9) nd (3) pemit to conside the volume element of the cuved spce- time, in fct: 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 67 x 3 d x = J d z = g ( P) dsd z, (5) z ut fom (6), (7) nd (9) it is cle tht z z = w, thus z cn e seen s 3- vecto t P of mgnitude w nd spheicl coodintes ( ) θ, ϕ with espect to the tid e, then: 3 d z = w dwdγ, dγ = sin θdθdϕ (6) eing d γ the element of solid ngle in the est fme of the chge. In this wy (5) tuns out to e: ( P) w dsdwdγ d x = g, (7) which togethe with (3) epesent the geneliztion to Riemnnin spces fom the following esults (9.5) nd (9.) of Synge [] (who uses imginy coodintes) fo Minkowski spce: z J =, d x = w dsdwdγ (8) x In the next section we will pply (7) to the pticul cse of the sufce w = constnt, which is impotnt when studying the electomgnetic dition. 3. Sufce of constnt etded distnce We conside the 3-spce w = constnt, thus the covint deivtive is othogonl to tht sufce. Then it is evident tht its vecto w ; volume element is given y: ; ; ; d = w w w d, (9) 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 68 eing d the 3-element of volume. But when uilding the shell fomed y w, w + dw nd the null cones t two points on C, we get fo its -volume d ; x = w w dwd nd fte compison with ; ; (7) it implies tht w w d g ( P) w dsdγ dopts the fom d, On the othe hnd, fom (5): with the nottion: ˆ =, thus (9) ( P) w w dsdγ ; w = g. (0) ( χ + W ) = ˆ ; w () = i υ, χ = υ υ, () d W = υ = K, ds whee is the cceletion of the chge. Sustituting () in (0) we find the esult (3.35) of Villoel []: d = g ( P) w[ wˆ ( χ + W ) ] dsdγ, (3) which is the geneliztion to cuved spces of (0.6) of Synge []. The deduction of (3) ws simple thnks to the dition coodintes tht oiginted (7). Nevetheless, this is not the end of the usefulness of z ; in ou opinion, its tue impotnce lies on the nlogies tht we cn estlish with the Minkowski spce, which will e seen moe clely in the next section. 00 C. Roy Keys Inc.
. Rdition tensos Apeion, Vol. 9, No., Apil 00 69 In the flt spce we hve the following ditive pt of the Mxwell tenso coesponding to the Liénd-Wiechet etded field = : ( ) T R s ( w W ) K K s = q w () which is dition tenso ecuse it stisfies: T = 0, T = 0. (5) K s R s, s R s The continuity eqution (5) is consequence of: ( wkk s) ( wwkk s), s 6, s = 0 which in tun e pticul cses of the identity; n m, s [ ( ) w W K K ] = 0 s = 0, (6) f, n + m, (7) f eing n ity function of. It is quite ntul to sk ouselves if () cn e extended to the cuved spce. The nswe is ffimtive unde the two pesciptions: Identify K with, see (5). Multiply () y ( P) fcto ( P) g due to the fct tht d x contins the g with espect to the coesponding expession fo the flt spce, see (7). Thus: T R s = q g ( P) w ( w W ) s, (8) 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 70 stisfies (5) with covint deivtive, nd it is immedite the geneliztion of (6): ; s s = g ( P) w 0. (9) ; s 6 g ( P) w W s = 0 Moeove, fom (7) nd (8) we hve the eltion: T R d x = q w ( w W ) ds dwdγ, (30) which is impotnt when pefoming some integtion ound the wold line of q. We notice tht (8) nd (9) coespond to the esults (.8),.,(.3) of Villoel [8], ut in ou focusing they emeged ntully though the coespondence with the Minkowski spce. Refeences [] J. L. Synge, Point-pticles nd enegy tensos in specil eltivity, Ann. Mt. Pu Appl., 8 (970), 33. [] P. S. Floides, J. McCe nd J. L. Synge, Rdition coodintes in genel eltivity, Poc. Roy. Soc. Lond., A9 (966),. [3] P. S. Floides nd J. L. Synge, Coodinte conditions in Riemnnin spce fo coodintes sed on suspce, Poc. Roy. Soc. Lond., A33 (97),. [] D. Villoel, Lmo fomul in cuved spces, Phys. Rev., D (975), 733. [5] B. S. DeWitt nd R. W. Behme, Rdition dmping in gvittionl field, Ann. of Phys., 9 (960), 0. [6] P. A. M. Dic, Clssicl theoy of diting electons, Poc. Roy. Soc. Lond., A67 (938), 8. [7] J. López-Bonill, Electodynmics of clssicl chged pticles, Rev. Colom. Fis., 7 (985),. 00 C. Roy Keys Inc.
Apeion, Vol. 9, No., Apil 00 7 [8] D. Villoel, Rdition fom electon in cuved spces, Phys. Rev., D (975), 383. [9] H. S. Ruse, Tylo s theoem in the tenso clculus, Poc. Lond. Mth. Soc., 3 (93), 87. [0] J. L. Synge, Opticl osevtions in genel eltivity, Rend. Sem. Mt. Fis. Milno, 30 (960),. [] J. L. Synge, Reltivity sed on chonomety, in Recent developments in genel eltivity, Pegmo n Pess, Oxfod (96),. [] B. S. DeWitt, Quntum theoy of gvity. III. Applictions of the covint theoy, Phys. Rev., 6 (967), 39. [3] J. D. Lthop, Covint desciption of motion in genel eltivity, Ann. of Phys., 79 (973), 580. [] J. D. Lthop, On Synge s covint consevtion lws fo genel eltivity, Ann. of Phys., 95 (975), 508. [5] J. L. Synge, Reltivity: the genel theoy, Noth Hollnd, Amstedm (976). [6] H. A. Buchdhl nd N. P. Wne, On the wold function of the Schwzschild field, Gen. Rel. Gv., 0 (979), 9. [7] J. López-Bonill, J. Moles nd M. Rosles, The Loentz-Dic eqution in cuved spces, Pmn J. Phys., 3 (99), 73. 00 C. Roy Keys Inc.