The Dirac equation in Rindler space: A pedagogical introduction

Transcription

1 The Drc equton n Rndler spce: A pedgogcl ntroducton Dvd McMhon Snd Ntonl Lbortores Albuquerque, NM Eml: dmmcmh@snd.gov Pul M. Alsng Deprtment of Physcs nd Astronomy, Unversty of New Meco, Albuquerque, NM lsng@hpc.unm.edu Pedro Embd Deprtment of Mthemtcs nd Sttstcs, Unversty of New Meco, Albuquerque, NM December 8, 5 Abstrct A pedgogcl ntroducton to the Drc equton for mssve prtcles n Rndler spce s presented. The spn connecton coeffcents re eplctly derved usng technques from generl reltvty. We then pply the Lgrnge-Green dentty to gretly smplfy clculton of the nner products needed to normlze the sttes. Fnlly, the Bogolubov coeffcents reltng the Rndler nd Mnkowsk modes re derved n n ntutve mnner. These dervtons re useful for students nterested n lernng bout quntum feld theory n curved spce-tme. 1 Introducton One of the most drmtc results n the erly ttempts to mrry quntum feld theory to generl reltvty ws the dscovery by Hwkng tht blck holes rdte 1. Ths nterestng result ws soon followed by the dscovery by Unruh tht unformly ccelerted detector would be mmersed n bth of therml rdton wth temperture T = ħ /π, where s the ccelerton of the observer ( k = c= 1. Moreover, n ccelerted observer n flt spce-tme hs hs or her own event horzon. The result of Unruh cn be encpsulted by syng ech observer hs hs or her own vcuum. These celebrted results, together wth bt of smplcty tht flt spce-tme brngs, hve mde Rndler spce very nterestng ren wthn whch to study quntum feld theory. Whle most tretments hve focused on the sclr feld, one effort mny yers go brefly showed tht unformly ccelerted observer would see flu of mssve Drc prtcles 3. Due to the recent nterest n quntum nformton theory n reltvstc contet, n prtculr the consderton of teleportton wth n ccelerted observer 4, n ths prmrly pedgogcl

2 pper we recll the mn results ssocted wth Drc prtcles n Rndler spce. Our motvton s two fold. Frst, ths pper s desgned to ssst students nterested n studyng quntum feld theory n curved spce-tme. In ddton, we m to put the necessry mchnery on the tble so tht lter work cn consder quntum nformton theory wth ccelerted observers usng complete reltvstc frmework. The outlne of ths pper s s follows. After brefly descrbng Rndler coordntes nd the form of the Drc equton n curved spce-tme, we derve the spn connecton coeffcents by consderng the geodesc equton. In the net secton, we ddress the crucl problem of normlzng the sttes, whch re gven n terms of Hnkel functons. Ths s non-trvl problem. Ths pper ddresses tht fct by pplyng the Lgrnge-Green dentty to gretly smplfy the clculton. To our knowledge, ths pproch hs not been used n ths cse. Fnlly, fter normlzng the sttes, we show how to wrte the Mnkowsk wve functons n terms of the Rndler wve functons nd vce vers. Ths llows us to conclude the pper by dervng the Bogolubov coeffcents. Rndler Coordntes In ths pper we consder the fmlr cse of unformly ccelerted observer n Mnkowsk spce-tme. As shown n Fg. 1, we consder two-dmensonl Mnkowsk spce wth coordntes tz, dvded nto four regons. These re the rght Rndler wedge (regon I, the left Rndler ( wedge (regon II nd the future (F nd pst (P. Settng ħ = c = 1, we defne Rndler vu, whch cn be relted to Mnkowsk coordntes n the followng wy coordntes ( t = usnhv z = ucoshv n regons I, II v t z u z z t ( ( 1 = tnh / = sgn t = ucoshv z = usnhv n regons F, P v z t u t t z ( ( 1 = tnh / = sgn (1 Snce the clcultons descrbed n ths pper wll be smlr for ll four regons, we focus on regon I, whch we denote s Rndler spce. In ths regon, strtng wth the two-dmensonl Mnkowsk lne element ds = dt dz nd usng (1, t s esy to show tht the lne element tkes the form ds = udv du ( Note tht v s tmelke coordnte nd u s spcelke coordnte-n regon I. Ths wll chnge n the other regons. Usng ths lne element we cn red off the components of the metrc tensor. These cn be rrnged n mtr s follows u g µν = 1 (3

3 The components of g µν cn be found esly by nvertng (3. We fnd tht 1/ u g µν = 1 (4 In the net secton, we proceed to derve the form of the Drc equton n Rndler spce. Fg. 1. The dvson of Mnkowsk spce nto four regons, the rght Rndler wedge (I, the left Rndler wedge (II, future (F, nd pst (P. The hyperbol s the worldlne of n ccelerted observer. 3 The Drc equton n Rndler spce Gven generl curvlner coordntes wth metrc g µν, the Drc equton s wrtten s 5 γ µ µ +Γµ m ψ = (5 where the Γ µ re known s the spn connecton coeffcents. These re 4 4 mtrces tht cn be wrtten n terms of the Gmm mtrces n the followng wy ν 1 γ Γ = γ +Γ µ 4 ν µ ν λµ γ λ (6 Mny reders wll recognze the ν Γ λµ s the fmed Chrstoffel symbols fmlr from generl µ reltvty. Note tht the γ shown here re not the usul Drc mtrces from flt spce, nsted they depend on the metrc n the followng wy

4 γ γ + γ γ = g (7 µ ν ν µ µν We cn use ths equton to relte these gmm mtrces to the usul flt spce vrety. If we denote the flt spce gmm mtrces by together wth (7 nd (3, we mmedtely deduce tht Usng g µν to rse ndces t cn be shown tht µ γ, usng the fct tht ( ( 3 γ =+ 1 nd ( γ = 1 γ = g = u γ, γ = uγ (8 ( γ = g = 1 = γ, γ = γ ( γ = γ, γ = γ (1 u Our net tsk on the rod to computng the spn connecton coeffcents s to turn to the problem of fndng the form of the Chrstoffel symbols. To do ths we begn wth bref study of the geodesc equton. Generl reltvty tells us tht the pths of prtcles movng wthout the nfluence of ny eternl forces re determned by the followng equton µ ν λ ν +Γ = (11 where the dervtves ndcted by dot re tken wth respect to n ffne prmeter. The eplct form of the Chrstoffel symbols nd therefore of the spn connecton coeffcents cn be determned n frly pnless fshon by pplyng Lgrngn procedure 6. We strt by defnng quntty whch we lbel s K In the cse of Rndler coordntes n regon I, we hve 7 λµ 1 µ ν K = g µν (1 1 K = uv u (13 Cllng the ffne prmeter τ, we cn use K to rewrte the geodesc equton, nd then compre t wth (11 to smply red off the Chrstoffel symbols. We hve K d K µ µ = dτ (14 Frst we consder the v coordnte. Settng µ = vwe fnd

5 K d K =, = v dτ v (15 Usng the defnton of K gven n (13 ths leds to the followng equton d K = uuv + uv dτ v (16 Rerrngng terms nd settng equl to zero by vrtue of (15 gves v + uv = (17 u Comprson wth the geodesc equton (11 llows us to red off the vlues of the followng Chrstoffel symbols v v Γ uv =Γ vu = (18 u A smlr procedure usng µ = ushows tht the only other non-zero Chrstoffel symbol s gven by Γ = u (19 u vv Now tht we hve eplct epressons for the Chrstoffel symbols, we cn wrte down the spn connecton coeffcents usng (6. We wll clculte one emple eplctly. The equton for Γ s gven by v 1 γ 1 v Γ v = γ + γ Γ λvγ 4 v 4 λ ( We re followng the summton conventon, so the nde λ s to be summed over ll coordntes. Wrtng ths result n terms of the flt spce gmm mtrces nd eplctly wrtng out the sum, we hve Γ = uγ Γ γ + γ Γ γ 4 4 u = ( γ γ + γ γ = γ γ v 3 u v uv 3 vv (1

6 A smlr procedure cn be used to show tht Γ =. We now hve everythng we need to wrte down the full Drc equton. Usng (1 together wth (1 n (5 we fnd tht u µ = γ +Γ m µ µ ψ ψ ψ ψ = + u v u u ( 3 3 γ γ γ γ γ ψ mψ ( γ Usng ( =+ 1 nd rerrngng terms gves ψ ψ = u + + m v u 3 3 γ γ γ γ ψ γψ (3 Now we defne 3 γ γ = α 3 nd denote γ by β to wrte the Drc equton s ψ 1 = u α3 + + βm ψ v u u (4 In ths pper we use the representton where α =, β = (5 Our net tsk s to wrte down the solutons to (4 nd normlze the sttes. 3 Normlzton wth the Lgrnge-Green Identty The ultmte gol of the present eercse s to relte the complete Mnkowsk wve functon to the Rndler wve functons n ll four regons. In order to do ths properly we wll need to normlze the sttes. Ths turns out to be frly mthemtclly dffcult nd my even pper bt mysterous to most reders. However, s we ll see below, ths process s gretly smplfed by pplyng the Lgrnge-Green dentty. The frst step s to look t (4 by thnkng of the Drc equton s dfferentl opertor. We wll cll tht opertor L nd note by lookng t the rght hnd sde of the Drc equton (4, we cn wrte L s 1 L = u α3 + + βm u u (6

7 Of course we cn vew the Drc equton s n opertor nd look t the left sde of (4 s well. In tht cse we cn wrte v Sttonry sttes wll be of the form ψ e ψ ( u mmedtely yelds L = (7 v =, n whch cse the pplcton of (7 Lψ = ψ (8 To proceed further we wll need the eplct form of the sttes. In ths pper we consder the spnup cse whereψ s four component column vector gven by 3 ψ ( uv, = e v φ φ ( mu φ ( mu + ( mu + φ ( mu + (9 The φ ± ( mu stsfy the followng dfferentl equton d d ± ± u u φ ( mu = mu φ ( mu (3 du du Solutons of (3 re gven n terms of Hnkel functons of the frst knd In regon I, Drc stte L [ nd φ s gven by (1 ( mu H ( mu φ ± ± 1/ (,, = (31 4 ψ C. Therefore the nner product between two sttes ψ ψφ ψφdu = (3 In order to normlze the sttes ψ s gven n (9 we wll need to compute the nner product ψ ψ = ψψ du (33

8 We cn clculte ths ntegrl by pplyng the Lgrnge-Green dentty (descrbed n Append B. We begn by emnng the djont of the dfferentl opertor gven n (6. In prtculr, we wsh to show tht ( Lψ φ = ( uψαφ + ψ ( Lφ u 3 (34 Ths relton corresponds to tht descrbed by (9 n the ppend. In due course we wll see the vlue of ths relton by notng tht when consderng nner products of sttes, the presence of the term ( uψαφ 3 wll llow us to wrte down the results of the relevnt ntegrls lmost by u nspecton. Let s consder the epresson ( Lψ φ n detl. Usng (4 nd (6, we see tht ths s ψ α3 ψ α3 3 3 α u + ψ + βmψ φ = α u φ+ ψ φ + ( βmψ φ u u u u (35 We pproch the gol of rewrtng ths epresson so tht L s ppled to φ by consderng ech pece on the rght hnd sde of (35 n turn. Begnnng wth the frst term, we hve ψ ψ ψ α u φ u α φ α uφ u = = u u ( (36 However, notce tht ( uφ ψ φ ψ ( ψ α3uφ = ( α3uφ + ψ αφ 3 + ψ α3u = ( α3uφ + ψ α3 (37 u u u u u And so we cn wrte ( uφ ψ α3u φ ( ψ α3uφ ψ α3 u = u u (38 Turnng to the second term n (35 we hve 3 α3 α ψ φ = ψ φ (39 Fnlly, for the lst term we fnd

9 ( m β ψ φ = ψ mβφ (4 Puttng these results together gves φ u u α3 + ψ φ+ ψ mβφ φ α3 = ( ψα3uφ + ψ α3u φ+ mβφ u u ( Lψ φ = ( ψα uφ ψ α u ψ ( αφ (41 However, lookng bck t the defnton of L, we hve φ α3 α3u φ+ mβφ= Lφ u (4 Therefore we hve shown tht ( Lψ φ = ( uψαφ + ψ ( Lφ u 3 (43 Notce tht usng the result of the Lgrnge-Green dentty s ppled to the Drc equton, we hve demonstrted tht b b b ( Lψ φdu = ψ ( uα3 φ + ψ ( Lφ du (44 We ll see very shortly tht t s now possble to compute the normlzton ntegrl gven n (33 usng (44. Now we set ψ ψ nd φ ψ nd recll (8, where we found tht Lψ = ψ. Then the left hnd sde of (44 becomes Now, mkng the substtutons ψ of (44 gves us ( becomes b ( Lψ ψ du = ( ψ ψ du= ψψ du (45 ψ nd L du = du φ ψ n the lst term of the rght-hnd sde ψ ψ ψ ψ. Puttng our results together (44

10 b b b ψψ du = ψ( uα3 ψ + ψψ du (46 Movng both ntegrls to the left hnd sde nd dvdng through by gves the result we seek u ψ ψ = ψψ du = ψαψ 3 (47 u To proceed we need to wrte down n eplct epresson for ψαψ 3. Recllng the form of the spn-up sttes gven n (9, fter some lgebr we fnd tht u * * u + + ψαψ 3 = ( φ φ ( φ φ u (1 (1 (1 (1 = H 1/( mu H 1/( mu H + 1/( mu H + 1/ ( mu (48 Now we proceed to evlute ths epresson t the lmts gven n (47. Ths wll be smplfed somewht due to the symptotc form of the Hnkel functons wth lrge rgument. For lrge z, 1 Hν ( z cn be wrtten s 8 1 νπ π Hν ( z ep z π z 4 (49 The presence of the term becomes π z ensures tht s z, these terms wll vnsh. Therefore (47 u u ψψ du = ψαψ 3 = lm ψαψ 3 (5 u Snce both terms n (48 re smlr, let s just focus on one of them. We consder u (1 (1 lm H + 1/ mu H + 1/ mu u ( ( (51 Hnkel functons cn be wrtten n terms of the modfed Bessel functons of the second knd n the followng wy

11 π νπ/ (1 π / Kν ( z = e Hν ( ze (5 Ths s convenent becuse for smll z the symptotc form of the K ( z K ν ( z ν 1 Γ Invertng (5 nd usng (53 we cn wrte (51 s ( ν ν s gven by (53 ν z 1/ 1/ u + 1/ π / Γ + 1/ + 1/ π/ Γ + 1/ lm e e u + 1/ + 1/ π π ( ( ( ( mu ( + / ( ( ( ( ( ( mu π 4 e 1 = lm Γ( + 1/ Γ u mπ m u ( + 1/ We cn wrte ths result n more convenent form by notcng tht (54 ( ( = ep ln = ep ( ln u u u (55 Usng Euler s dentty nd rerrngng terms, (54 becomes 4 π lm e u m ( cos ( ln( 1/ u ( + / π m ( ( u sn ln 1/ + Γ( + 1/ Γ ( + 1/} (56 In Append A we wll show tht

12 1 cos( ln lm u = u 1 sn( ln u lm = πδ u ( (57 Usng these lmts (56 smplfes to u (1 (1 4 lm H + 1/ mu H + 1/ mu e u mπ 1/ 1/ δ (58 π ( ( = Γ( + Γ ( + ( To obtn the fnl result, note tht the Gmm functons stsfy ( 1/ ( 1/ ( 1/ π Γ + Γ + = Γ + = (59 cosh π Ths mens tht we cn rewrte (58 n the form π u (1 (1 4e lm H + 1/ mu H + 1/ mu δ u mcoshπ ( ( = ( (6 The mnus sgn n (5 cncels the one here. Moreover, we obtn the ect sme result modulo the mnus sgn for the other term n (48. Addng these results together we fnd tht the normlzton s gven by π 8e ψ ψ = ψψ du = δ ( (61 mcoshπ 3 Epressng Mnkowsk wve functons n terms of Rndler wvefunctons tz plne. Therefore n order to epress t n terms of Rndler wve functons, we need to know the form of the Rndler sttes n ll four regons. We now descrbe the sttes n regons II, F, nd P nd then pece them together to wrte down the Mnkowsk wve functon. The Mnkowsk wve functon s defned over the entre (,

13 In order to dstngush the sttes of regons I, II, F, nd P we wll dopt the notton ψ I, II, F, nd P ψ ψ ψ respectvely. Furthermore, t s necessry to dstngush between postve nd negtve frequency modes n the future nd pst regons. These cn be denoted by F( + F( P( + P( ψ, ψ, ψ, nd ψ. Fnlly, we lbel the Mnkowsk wve functons by ψ ± where ( ± denotes prtcle/nt-prtcle mode respectvely. The dervton nd normlzton of the Rndler sttes n the other regons s smlr to tht used n the prevous secton. So we smply stte the results, contnung to emne the spn-up cse. In the future regon, the Drc equton ssumes dfferent form becuse u becomes tmelke coordnte nd v becomes spcelke coordnte. The spn-up stte n the future regon s gven by 3 ψ Φ Φ + F( + v = e + Φ +Φ (6 In ths cse the components of the wve functon re gven by Hnkel functons of the second knd The negtve frequency stte s found to be ( ± 1/ ( mu ± Φ = H (63 ψ φ φ + F( v = e + φ + φ (64 (1 where φ ± = H ± 1/( mu. Usng the procedure outlned n secton, the student cn verfy tht the normlzton of the sttes n the future regon s gven by kπ F( l F( k 16e ψ ψ = δ ( δkl ( kl, =± 1 (65 m The sttes n regons P nd II re relted to the sttes n regons F nd I n the followng wy ( ± ( ( ( ( ( ψ tz, = αψ t, z, ψ tz, = αψ t, z P F II I 3 3 (66 As n sde, note tht strctly spekng t s necessry to wrte down the wve functons so tht they re only defned n the pproprte regon. Ths cn be done usng the Hevsde step

14 functon. For our purposes, t won t be necessry to worry bout ths eplctly. We smply note tht ech Rndler wve functon s defned n ts gven regon nd s zero elsewhere. Now let s turn to the problem of epressng the Mnkowsk sttes n terms of the Rndler sttes. In prevous work 3 comple rgument nvolvng source terms on the lght cone (whch s bound to confuse the student s used to suggest the form of the Mnkowsk wve functon. We nsted tke more heurstc pproch bsed on smple mthemtcl rguments. Observng tht the Mnkowsk modes must cover the entre ( tz, plne, we construct them by ptchng together the Rndler wve functons ψ, ψ ( ±, ψ ( ±, ndψ contnuty t the boundres of ech regon. I F P II. Then we demnd To ptch together the Rndler wve functons, let s just sum them up wth rbtrry constnts. A postve frequency Mnkowsk mode cn then be wrtten n terms of the Rndler modes n the followng wy I F( P( II ( ψ = N ψ + ψ + ψ + ψ (67 where N s normlzton constnt. A negtve frequency Mnkowsk mode s defned smlrly I ( ( ( F P II ψ = M bψ + bψ + bψ + bψ (68 In our smple pproch, to determne the vlues of the s we cn compre the wve functons n ech regon long the boundres where contnuty n Mnkowsk spce requres they mtch up. For emple, consder the Rndler wve functons n regons I nd II, whch must mtch up t u =. We cn fnd the constnts by tkng the lmt n ech regon s u. Ths cn be demonstrted eplctly by consderng the upper component of ech spnor. ( In the followng we use ψ I U nd ψ IIU ( to denote the upper component of the spnors n regons I nd II respectvely. In the cse of regon I, the reder cn refer to (9 to recll tht the upper component of the spnor s gven by ( ( IU ( (1 (1 ψ + = H 1/ mu + H 1/ mu (upper component (69 Usng (5 we cn wrte ths n terms of the modfed Bessel functons of the second knd, gvng IU ( + ψ = 1/ + + 1/ ( 1/ π / ( 1/ π/ ( ( e K mu e K mu π (7 Recllng the behvor of the Bessel functons for smll rgument, for smll u ths becomes ψ ( ( ( ( Γ 1/ Γ + 1/ π mu mu 3/ 1/ IU ( π/ π/4 π/4 e e + e 1/ + 1/ (71

15 To determne the form of the wve functon n regon II for smll u, we pply (66 together wth the orgnl defnton of the coordntes s gven n (1. The u,v coordntes n ths regon re gven by ( v z t u z z t = rctnh /, = sgn( (7 From ths we see tht settng z zt, tleves v unchnged whle u u. We fnd tht the upper component of the spnor n regon II s u s gven by ψ ( ( ( ( 3/ 1/ IIU ( π/ π/4 Γ 1/ Γ + 1/ e e + 1/ + 1/ π mu mu (73 Now, for the (+ frequency mode, we hve 1 = e π. Therefore we cn proceed s follows ψ ( ( ( ( Γ 1/ Γ + 1/ π mu mu (74 ( ( π ( mu ( mu 3/ 1/ IIU ( π/ π/4 + 1/ 1/ e e ( 1 + ( 1 1/ + 1/ 3/ 1/ 3 π/ π /4 Γ 1/ Γ + 1/ = e e + 1/ + 1/ Comprson wth the epresson we obtned for the wve functon n regon I (71 s u shows tht the followng reltonshp must hold = (75 II I e π ψ ψ A smlr procedure cn be used to compre the sttes n the future nd pst regons to tht n regon I. We fnd tht ψ = e e ψ, ψ = e e ψ (76 F( + π / /4 I P( + π / /4 I Lookng bck t the epresson used for the Mnkowsk wve functon (67 nd tkng 1 to be unty, (75 nd (76 tell us tht the postve frequency Mnkowsk wve functon should be wrtten s ( I π / π /4 F ( π / π /4 P ( π II ψ = N ψ e e ψ + e e ψ e ψ (77 The normlzton constnt cn be determned by requrng tht ( ψ ψ = δ (78 + +

16 Usng (61 nd (65 together wth (66 nd (77 nd (78, we fnd tht N = m/48 (79 A smlr procedure cn be used to show tht the normlzton constnt for the negtve frequency modes s the sme nd tht the Mnkowsk cn be wrtten s ( π I π / π /4 F( π / π /4 P( II ψ = m/48 e ψ + e e ψ + e e ψ + ψ (8 4 Clcultng the Bogolubov coeffcents In ths secton we wll determne the Bogolubov coeffcents whch llow us to wrte the Rndler wve functon n terms of postve nd negtve frequency Mnkowsk modes. These were orgnlly stted by Soffel 3, nd cn be obtned by smple lgebr. The key concept n ths dervton s tht the Rndler observer n regon I s cuslly dsconnected from regon II nd vce vers. If we focus on n observer n regon I, ths mens tht we seek combnton of ψ + nd ψ II tht reflects ths fct by elmntng ψ. Let s denote the wve functon s seen by ths Rndler observer n regon I by R ψ lner combnton of ψ + nd ψ we seek s ( RI ( π + π/ π/ + π / I. The ψ = e ψ + ψ = e e ψ + e ψ (81 It s necessry to normlze ths stte. Usng (78 we fnd tht ( + ( + ( + R I R I π / π / + + π / ψ ψ = e e ψ ψ + e ψ ψ ( π δ( π = e cosh (8 Addng one fnl lyer of notton, we denote the properly normlzed Rndler wve functon RI ( by ψ. Usng (8 ths s wrtten s ψ = RI ( R I π 1 e cosh ( π 1 = e e + e π e cosh ( π e = + cosh ( π cosh( π + ( ψ ψ π/ π/ π / π/ π / + ψ ψ = α ψ + β ψ + ψ e (83

17 The Bogolubov coeffcents re gven by α π/ π / e e =, β = (84 cosh ( π cosh( π A smlr procedure cn be ppled to the stte n the left wedge, by notng tht the observer n regon II s cuslly dsconnected from regon I. Ths s good eercse for the student. 5 Concluson Ths pper hs provded pedgogcl ntroducton to workng wth mssve Drc prtcles n Rndler spce. In ddton, we hve ppled the Lgrnge-Green dentty whch gretly smplfes the problem of normlzng the sttes, obtnng the ect result found n the lterture. To the knowledge of the uthor ths pproch hs not be used before. We concluded by pplyng smple mthemtcl rguments to epress the Mnkowsk wve functons n terms of the Rndler sttes. The mchnery descrbed n ths pper cn be ppled to further reserch. In prtculr, t cn be used to gve full reltvstc tretment to problems n quntum nformton theory tht nvolve mssve Drc prtcles nd n ccelerted observer. We ntend to pursue ths lne of reserch n future pper. The vlue of dong so hs been demonstrted by recent nterest epressed n the propertes of entngled prtcles n grvttonl feld 9. Append A: Mthemtcl Prelmnres We now prove three crucl propostons tht were used to clculte the normlzton of the Rndler sttes. In the followng, we wll consder functons φ ( on the rel numbers tht re nfntely dfferentble nd tht vnsh outsde of some regon. These functons re clled test functons re denoted by wrtng φ ( C ( R. We begn by consderng the well-known Remnn-Lebesgue lemm, whch bsclly sttes tht the Fourer coeffcents f ˆk of perodc ntegrble functon vnsh s k. Proposton lmsnk = n the dstrbuton sense. k Proof Let φ ( ( R wth support contned n [ RR, ] C. Then

18 cosk R R cos k sn kφ( d = φ( φ ( d k R R k 1 = k R R cos kφ ( d R R R Now, snce cos φ ( cos φ ( φ ( k d k d d R R nd 1 s k R k, we conclude tht sn φ ( sense. k d s k, tht s, lmsnk = n the dstrbuton k Wth ths result n hnd, we cn show tht lm πδ( k Proposton sn k lm = πδ( n the dstrbuton sense. k Proof Let φ ( ( R wth support contned n [ RR, ] C ( ( + ( φ φ ψ where ψ ( sn k =, whch we do n the net proof.. Usng Tylor epnson t follows tht s contnuous, nfntely dfferentble functon tht my not hve compct support. Usng ths epnson we hve snk R snk R sn k φ( d = φ( d = φ( + ψ( d R R R snk R = φ( d + sn k ψ( d R R From the Remnn-Lebesgue lemm, we know tht the second term vnshes s k. Therefore, we only need consder the frst term, but t s well known tht R snk kr sn y d= dy π s k R kr y sn k lm φ( d πφ( πδ ( φ ( d k = =. Therefore we fnd tht

19 sn k = n the dstrbuton sense. Tht s, lm πδ( k cos k The fnl result we need s to show tht vnshes s k. The proof s smlr to the one we just dd wth some mnor modfctons. Proposton cos k lm PV.. = k n the dstrbuton sense, where P.V. s prncpl vlue. Proof Let φ ( C ( R wth support contned n [ RR, ] n Tylor φ( φ( + ψ ( where ψ ( tht my not hve compct support. Now, by the defnton of cosk R cosk PV.. φ( d= PV.. φ( d R. Followng the lst emple, we epnd s contnuous, nfntely dfferentble functon cos k PV.. we hve cosk cosk = lm φ( d lm φ( ψ( d + = ε ε< < R + + ε ε< < R cos k = lmφ( d lm ( cos k ψ( + + d ε ε< < R + ε ε< < R cos k cos k Snce s n odd functon, d =. On the other hnd, the ntegrnd ε < < R cos k ψ s regulr t =, hence ( ( R ( ψ ( = ( ψ ( lm cosk d cosk d + ε ε< < R R However, f we let k nd use the Remnn-Lebesgue lemm, we conclude tht cosk R lm PV.. φ( d lm ( cosk ψ ( d k = k = R Tht s, cos k lm PV.. = n the dstrbuton sense. k

20 The results n derved n ths ppend led to the two results used drectly n the pper, tht s 1 cos( ln lm u = u 1 sn( ln lm u = πδ u ( Append B: The Lgrnge-Green Identty The dscusson n ths secton s from Stckgold ( From elementry dfferentl equtons we recll tht Green s formul cn be used to epress n ntegrl n terms of ts boundry condtons by usng ntegrton by prts. Our gol s to eplot ths procedure so tht we cn evlute n nner product by nspecton. whch re contnuous twce dfferentble functons of rel vrble tht we re denotng by. For the most generl cse, we cn wrte such n opertor s Consder second order dfferentl opertor L wth coeffcents ( d d L = ( + 1( + o ( (85 d d Net, consder the followng ntegrl b b dg dg f ( Lg( d= f ( ( + f ( 1( + f ( o ( g( d d d (86 where f ( nd ( g re contnuous, twce dfferentble functons of rel vrble. Integrton by prts cn trnsform ths epresson nto b b ( ( = (, + ( ( b f Lg d J f g g L f d (87 Ths s Green s formul. J s known s the conjunct of the functons f nd g nd gven (85 cn be wrtten s df dg d J ( f, g = g f + 1 fg d d d (88 L s the djont of L whch n generl s found to be

21 d d d d d 1 L = + 1 o d d + + d d d (89 Tkng the dervtve of (87 wth respect to b nd then settng b = gves us Lgrnge s dentty glf dj = + flg (9 d Tken together (87 nd (9 re sometmes known s the Lgrnge-Green dentty. When dfferentl opertor shows up n n nner product we cn use ths result to trnsform the ntegrl nto one nvolvng totl dervtve,.e. dj ( glf flg d= d= J ( f, g (91 d References 1. S. W. Hwkng, Nture 48, 3 ( W. G. Unruh, Phys. Rev. D14, 87 ( M. Soffel, B. Muller, nd W. Grener, Phys. Rev. D, 1935 ( P. M. Alsng nd G. J. Mlburn, Phys.Rev.Lett. 91, 1844 (3. 5. N. D. Brrell nd P. C. W. Dves, Quntum Felds n Curved Spce (Cmbrdge Unversty Press, N. Y., R. D Inverno, Introducng Ensten s Reltvty (Oford Unversty Press, D. McMhon, Reltvty Demystfed (McGrw-Hll, N.N. Lebedev, Specl Functons nd Ther Applctons (Dover, H. Tershm nd M. Ued, Phys. Rev. A 69, 3113 (4. 1. I. Stkgold, Green s Functons nd Boundry Vlue Problems (Wley, 1998.