Alternative derivation of the correspondence between Rindler. Rua Pamplona S~ao Paulo, S~ao Paulo. Brazil. Abstract

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1 Alterntive derivtion of the correspondence between Rindler nd Minkowski prticles George E.A. Mtss Instituto de Fsic Teoric, Universidde Estdul Pulist Ru Pmplon S~o Pulo, S~o Pulo Brzil Abstrct We develop n lterntive derivtion of Unruh nd Wld's seminl result tht the bsorption of Rindler prticle by detector s described by uniformly ccelerted observers corresponds to the emission of Minkowski prticle s described by inertil observers. Actully, we present itinninverted version, nmely, tht the emission of Minkowski prticle corresponds in generl to either the emission or the bsorption of Rindler prticle v, 4.7.Dy Typeset using REVTEX

2 We present brute force, but strightforwrd method to reobtin Unruh nd Wld's seminl result [] tht the bsorption of Rindler prticle by detector s described by uniformly ccelerted observers corresponds to the emission of Minkowski prticle s described by inertil observers. Actully, we present itinninverted version, nmely, tht the emission of Minkowski prticle in the inertil vcuum will correspond in generl to therml emission or bsorption of Rindler prticle. In the prticulr cse where the source is uniformly ccelerted Unruh{DeWitt detector [2,3], the emission of Minkowski prticle will uniuely correspond to the bsorption of Rindler prticle. We believe tht our pproch my be prticulrly useful in understnding the behvior of relistic sources following rbitrry worldlines. We ssume nturl units (h = c = k B = ), nd n n- dimensionl Minkowski spcetime with signture (+ ::: ) for ske of generlity. In order to cpture only the essentil fetures of untum device without mking use of some prticulr detector, let us motivte the use of complex currents. The current describing the excittion of DeWitt{like detector [3] cn be dened s j() =hej^m()je i; () where ^m() is the monopole which represents the detector, is its proper time, nd E = E E is the energy gp of the detector. In the Heisenberg picture the monopole is time{ evolved s ^m() =e ih ^m()e ih, where H is the free Hmiltonin of the detector. Thus, current () is clerly complex since j() / e ie. This is the reson why we will consider here rbitrry complex currents. These currents should be interpreted s describing the trnsition of non{necessrily pointlike untum device following n rbitrry worldline. For ske of simplicity, we couple our complex current to rel mssless sclr eld through the interction opertor ^S = d n x jg(x)j j(x)^(x): (2) Concluded the preliminries, let us begin by computing the emission rte of Minkowski prticles in the inertil frme. The emission mplitude of Minkowski prticle s clculted in the inertil frme is 2

3 M A em (k; k) = M hkj^sji M ; (3) where k = jkj. Expnding the sclr eld ^(x) in terms of positive nd negtive energy modes with respect to inertil observers [4] ^(x )= d n k 2k(2) n (^ M k e ikx +H:c:); (4) we express from (2) the emission mplitude (3) s M A em (k; k) = h 2k(2) n i =2 d n xj(x)e ikx ; (5) where x (t; x; y 2in ). Thus, the emission probbility of Minkowski prticle s described by inertil observers is M P em = d n k j M Aem(k; k)j 2 ; (6) where M A em is given in (5). Next, we im to express the emission rte of Minkowski prticles (6) in terms of the emission nd bsorption of Rindler prticles. For this purpose, it is convenient tointro- duce Rindler coordintes (;;y 2in ). These coordintes re relted with Minkowski coordintes (t; x; y 2in )by t= e e sinh ; x = cosh : (7) Aworldline dened by ; y 2in = const describes n observer with constnt proper ccelertion e.we will denominte these observers Rindler observers. The nturl mnifold to describe Rindler observers is the Rindler wedge, i.e. the portion of the Minkowski spce dened by x>jtj. It is crucil for our purposes tht the current j(x ) be conned inside this wedge. The Rindler wedge is globlly hyperbolic spcetime in its own right, with Killing horizon t x = t ( = ) ssocited with the boost Killing t its boundry. Using (7), the Minkowski line element restricted to the Rindler wedge is ds 2 = e 2 (d 2 d 2 ) 3 nx i=2 (dy i ) 2 : (8)

4 Solving the mssless Klein-Gordon eution 2 = in the Rindler wedge, we obtin complete set of Klein-Gordon orthonormlized functions [5] " #! 2 sinh!= u!k? (x 2 )= K (2) n i!= e e iy i! ; (9) where j j= Pn i=2 (k y i) 2,! is the freuency of the Rindler mode, nd K i (x) is the McDonld function. Using (9), we express the sclr eld in the uniformly ccelerted frme in terms of positive nd negtive freuency modes with respect to the boost Killing ^(x )= d n 2 + o R d! n^!k? u!k? (x )+H:c: ; () where ^ R!k? is the nnihiltion opertor of Rindler prticles, nd obeys the usul commuttion reltion i R h^!k? ; ^Ry! k = (!! )( k??): () It is possible now tofourier nlyze the current j(x ) in terms of Rindler modes. We dene its Rindler{Fourier trnsform s ~ R (! ;!; ) d n x jg(x)jj(x )u!k? e i(!! ) : (2) This reltion cn be esily inverted j(x )= e d n 2 d!! d! ~ R (! ;!; )u!k? (x )e i(!! ) (3) by using the completeness reltion [6] + d!! sinh! K i!= e! K i!= e! = 2 2 ( ): (4) In order to relte the prticle emission nd bsorption rtes in both reference frmes, we substitute (3) nd (9) in (5). The y i nd integrls re esy to perform, while the nd integrls cn be solved by noting tht (use the chnge of vribles = e in conjunction with E of Ref. [7]) 4

5 + " # d e i(kt kxx! ) = 2 i!=2 k + kx e! =2 K i! = e! ; (5) nd by using the following orthonormlity reltion [8,9]: +!! 2 d K i!= e K i! = e = 2! sinh(!=) (!! ) (6) for!;! 2 R +. With this procedure we reduce the emission mplitude (3) to M A em (k; k) = + (4k) =2 d! sinh =2 (!=) +~ R (!;!; ) 8 < : ~ R(!;!; )! +i!=2! i!=2 e!=2 e!=2 9 = ; : (7) Our next tsk will be to interpret ~ R (!;!; ) in terms of the emission nd bsorption mplitudes of Rindler prticles R A em = R h! j ^Sji R ; R A bs = R hj ^Sj! i R (8) respectively, where ^S is given in (2). Using explicitly (2) nd () in (8) we obtin ~ R (!;!; )= R A em (!; ); ~ R (!;!; )= R A bs (!; ): (9) As conseuence, we cn express the Minkowski prticle emission mplitude (7) s M A em (k; k) = + (4k) =2 d! sinh =2 (!=) + R A bs (!; ) 8 < : R A em (!; )! +i!=2! i!=2 Now, it is useful to introduce the following representtion of the delt function 2 + dk x k e!=2 e!=2 9 = ; : (2) " k + kx # i(!!)=2 = (!! ); (2) where k = kx 2 + k2?. (This delt function representtion cn be cst in the more fmilir form R + dke ik(!! ) =2(!! ) fter the chnge of vribles k x! K = ln[(k + k x )=(k k x )].) Finlly, introducing (2) in (6), nd using (2) to perform the integrl in k x we re ble to express the totl emission rte of Minkowski prticles in its nl form s 5

6 M P em = R P em + R P bs ; (22) where nd R P em = R P bs = + d n 2 d!j R A em (!; )j + 2 e 2!= ; (23) d n 2 + d!j R A bs (!; )j 2 e 2!= : (24) The therml fctors which pper in (23) nd (24) re in greement with the fct tht the Minkowski vcuum corresponds to therml stte with respect to uniformly ccelerted observers [2]. The physicl content of (22) combined with (23) nd (24) cn be summrized s follows: The emission of Minkowski prticle in the vcuum with some xed trnsverse momentum s described by n inertil observer (6) will correspond either to the emission of Rindler prticle with the sme trnsverse momentum, or to the bsorption of Rindler prticle with trnsverse momentum described by uniformly ccelerted observers. from the Dvies-Unruh therml bth s Notice tht the conservtion of trnsverse momentum in both frmes ppers nturlly enclosed in this result. This is mndtory since the trnsverse momentum is invrint under boosts. In the prticulr cse where the source is uniformly ccelerted Unruh-DeWitt detector, R A em = implying tht in this cse the bsorption of Rindler prticle corresponds uniuely to the emission of Minkowski prticle. However, s seen bove, in more generl situtions where the detector is switched on/o [] or follows some rbitrry worldline, the excittion of the detector usully ssocited with the bsorption of Rindler prticle cn be lso ssocited with the emission of Rindler prticle. This is lso in greement with energy conservtion rguments. Acknowledgements I m relly indebted to Atsushi Higuchi for vrious enlightening discussions. I m lso very grteful to Ulrich Gerlch for criticlly reding previous version of this mnuscript. This work ws prtilly supported by Conselho Ncionl de Desenvolvimento Cientco e Tecnologico. 6

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