Dirac Particles Emission from Reissner-Nordstrom-Vaidya Black Hole

Transcription

1 Journal o Physics: Conerence Series PAPER OPEN ACCESS Dirac Paricles Emission rom Reissner-Nordsrom-Vaidya Black Hole To cie his aricle: Yuan Tiandho and Triyana 016 J. Phys.: Con. Ser View he aricle online or updaes and enhancemens. This conen was downloaded rom IP address on 1/01/018 a 3:06

2 6h Asian Physics Symposium Journal o Physics: Conerence Series 739 (016) doi: / /739/1/01146 Dirac Paricles Emission rom Reissner-Nordsrom-Vaidya Black Hole Yuan Tiandho and Triyana* Theoreical High Energy Physics and Insrumenaion Division, Faculy o Mahemaics and Naural Sciences, Insiu Teknologi Bandung, Jl. Ganesha No. 10, Bandung 4013, Indonesia *Corresponding auhor: riyana@i.ib.ac.id Absrac. Using Hamilon-Jacobi mehod, we sudy he Dirac paricles emission rom Reissner-Nordsrom-Vaidya (RNV) black hole. The Dirac paricles are described by Dirac equaion in curved spaceime and emission process is deined as unneling eec. The probabiliy o Dirac paricles emission is relaed o he Hawking emperaure and we obain ha his emperaure is equal o emperaure ha derived hrough spinless paricles emission. Furhermore, we also show ha he mass o Dirac paricles does no aec o he Hawking emperaure. 1. Inroducion Using quanum mechanics heory, Hawking proposed ha a black hole has a emperaure, meaning ha he black hole can emi paricles no as saed in he classical heory [1-3]. In he irs explanaion o he black hole emperaure, Hawking considered he Schwarzschild black hole. Is Hawking emperaure is [3], c TH (1) 8 km where k is he Bolzmann consan and M is he black hole mass. I urns ou ha he emperaure o black holes is inversely proporional o is mass. Calculaion o he Hawking emperaure a dieren ypes o black holes wih various mehods is a ho opic in a recen decade [4-6]. In his work, we chose he Reissner-Nordsrom-Vaidya (RNV) black hole. Compared wih he Schwarzschild black hole, he Vaidya black hole is more realisic because is mass depends on space and ime [7-8]. Previously, one o us has sudied he Hawking emperaure o Vaidya black hole [9] and RNV black hole [10] or spinless and massless paricles emission. Thereore, in his paper we would like o exend his sudy or RNV black hole wih Dirac paricles (spin ½ paricles) emission. In his work we use semi classical Hamilon-Jacobi mehod or complex pah mehod [11] o calculae he Hawking emperaure. In his mehod, he wave uncion which is deined as uncion o he acion is subsiued ino he Dirac equaion wih he RNV spaceime as background. This deailed discussion is in Secion. Through he Hamilon-Jacobi mehod we can obain he acion or calculae he probabiliy o Dirac paricles emission. By using he balanced principle, we will know ha he probabiliy corresspond o he Hawking emperaure. Furhermore, in his work we also invesigaed Conen rom his work may be used under he erms o he Creaive Commons Aribuion 3.0 licence. Any urher disribuion o his work mus mainain aribuion o he auhor(s) and he ile o he work, journal ciaion and DOI. Published under licence by Ld 1

3 6h Asian Physics Symposium Journal o Physics: Conerence Series 739 (016) doi: / /739/1/01146 he inluence o paricles mass ha are emied o he emperaure. Behaviour o he massive unneling paricles is showed in Secion 3. In he las secion we give a conclusion o our work.. Dirac equaion in RNV spaceime The line inerval o RNV spaceime is deined by, ds d ddr r d sin d () where 1 M p r Q r. In his paper M and Q correspond o mass and charge o RNV black hole, υ is Eddingon ime coordinae, and p is arbirary uncion o mass and charge, p (M, Q). By using Eddingon coordinae ransormaion he above meric can be wrien as, ds d 1 dr r d sin d (3) From ha meric, i is clear ha he RNV black hole has wo even horizons, r M p M p Q (4) where plus (minus) sign correspond o ouer (inner) even horizon and single singulariy is reached or neural black hole. However, his orm o meric does no give inormaion on he velociy o massive 1 paricle. Accordingly, we use Painleve coordinaes by ransorm, dr and he meric in eq. (3) can read as. 1 ds d 1 drd dr r d (5) Dirac paricles is described by he Dirac equaion equaion in curved spaceime, m i D 0 (6) 1 ab where m is mass o emission paricles. The covarian derivaive is given by D ab, where 4 1 ab ab correspond o commuaor o Minkowskian spaceime gamma marices ab a, b and is a spin connecion. We use he la spaceime gamma marices as, i ,,, i (7) a a The la gamma marices and he curved gamma marices are relaed by e a and hose are speciied by deiniion, g I. There are several dieren expression o gamma marices and in his work we choose he represenaion or Dirac marices o be [1], ,, (8) , 1 r 0 rsin 0 k where are Pauli marices. Spinor wave uncion ψ has wo spin saes: spin-up and spin-down. Because RNV black hole is spherically symmeric so he Hawking radiaion depends on r and only. Thus, he uncions or spin-up and spin-down paricles respecively saisy,

4 6h Asian Physics Symposium Journal o Physics: Conerence Series 739 (016) doi: / /739/1/01146 a, r a, r i 0 i exp S, r exp S, r b, r b, r 0 (9) 0 c, r c, r i i exp S, r exp S, r d, r 0 d, r (10) where S and S are acion o emission paricles or spin-up paricles and spin-down paricles. However, we only analyse he spin-up case since he spin-down case is jus analogous. By subsiuing eq. (9) ino eq. (6) and recalling ha he Planck consan is very small we have, 1 1 S b rs ma 0 (11) 1 1 S a rs mb 0 Through he Hamilon-Jacobi mehod, he acion can be expressed in wo pars: he ime par which has he orm o E and he par ha relaes o radial expressed as R (r), The erm 0 ' 0 ' ', S E d R r (1) E d ' is a generalizaion o E because energy can vary in ime. Subsiuing eq. (1) ino eq. (11) we obain, 1 1 b E R r, rrr, ma a E Rr, rrr, mb 0 The wo equaions above have wo posibble soluions o R, 1 1 a 0 rrr E R r, 1 1 b 0 rrr E R r, (13) (14). 3. Behaviour o he massive Dirac paricles The equaion o moion beween massive paricles and massless paricles are dieren. When we consider unneling o massless paricles we may using radial null geodesic mehod bu or massive paricles he mehod is no valid. Since he world line o massive paricles is no ligh-like. By using 3

5 6h Asian Physics Symposium Journal o Physics: Conerence Series 739 (016) doi: / /739/1/01146 Landau heory o he coordinae clock synchronizaion and he deiniion o phase velociy o de Broglie wave, Wen can obain he velociy o massive paricles [13], 1 g00 r (15) g01 Subsiuing g 00 and g 01 (eq. (5)) ino eq. (15), we obain, 1 r (16) 1 Accordingly, soluion o wo equaion in eq. (14) can be wrien as, dr R R (17) dr r r where rcan be obained rom eq. (16). Near he even horizon, he meric coeicien can be expressed by Taylor series. Since we only need heir approximaion values or shor disances rom even horizon, we can apply he Taylor expansion a a ixed ime,, ', r r r r O r r (18) Considering a slowly varying R, he uncion R in he near even horizon may be obained by inegraing eq. (17) wih respec o r. Noice ha R r has a pole a horizon bu R r does no have a pole and well deined limi a he even horizon. Thus we may conclude ha he soluion o R - uncion is zero and R + uncion is, Finally, he complee expression o acion is, R 1 1 E ie dr (19) ' r r ' ie ' '; ' ' (0) ' r S E d S E d 0 0 In unneling process, he energy o an emied paricle is less han barrier poenial. Accordingly, he paricle s momenum and he acion uncion are imaginary. Thus Ed can be wrien asi Im Ed. Probabiliies o ingoing and ougoing paricles respecively are, Pin exp Im E ' d ' (1) E Pou exp Im E ' d ' ' r, I all ingoing paricles absorbed by black hole or Pin 1, he probabiliy o ougoing paricle is, 4 E Pou exp () ' r, The unneling probabiliy can be expressed in a Bolzmann acor and is energy Pou exp E. Thus he Hawking emperaure due o Dirac massive paricles emission rom he RNV black hole is, TH ' r, (3) 4 k Recalling he explici orm o meric coeicien he Hawking emperaure or he RNV black hole becomes, M ' p' M p QQ' Q TH 3 (4) 4 k r r r 4

6 6h Asian Physics Symposium Journal o Physics: Conerence Series 739 (016) doi: / /739/1/01146 The plus (minus) sign correspond o he emperaur in ouer (inner) even horizon. The above expression is exacly he same wih ha in [10]. The above expression also does no conain paricle mass. I can be concluded ha he Hawking emperaure does no depend on paricle spin and paricle mass. I is clear ha or Q = 0, p = 0 and M is a consan, he Hawking emperaure above corresponds ha or he Schwarzschild black hole. Conclusions We have successully exended our consideraion abou Hawking emperaure o RNV black hole by using a unneling mehod or ermion paricles. The analysis has showed ha he Hawking emperaure in his work has same value as ha or he RNV black hole when analysed hrough spinless paricles emission. In addiion, he mass o emied paricles also does no aec o he emperaure. The black hole emperaure is inversely proporional o is mass or direcly proporional o derivaive o he meric coeicien. Acknowledgemens This work was suppored by Hibah Desenralisasi DIKTI. Reerences [1] S. W. Hawking, Naure, 48, 30 (1974). [] S. W. Hawking, Commun. Mah, Phys, 43, 199 (1975). [3] M. K. Parikh and F. Wilczek, Physical Review Leers, 85, 504 (000). [4] L. Kai and Y. Zheng, Chinese Physics B, 18, 154 (009). [5] H. Ding, Fron Phys, 6, 106 (011). [6] H. Gohar, American Journal o Space Science, 1, 94 (013). [7] P. C. Vaidya, Proceeding o he Indian Academy o Sciences, 33, 64 (1951). [8] S. Zhou and W. Liu, Modern Physics Leers, 4, 099 (009). [9] H. M. Siahaan and Triyana, In. J. Mod. Phys, 5, 145 (010). [10] Triyana and A. N. Bowaire, In. Mah Fund. Sci, 45, 114 (013). [11] K. Srinivasan and T. Padmanabhan, Physical Review D, 60, (1999). [1] R. Kerner and R. B. Mann, Fermions Tunneling rom Black Holes, arxiv: [13] H. Y. Wen, Chinese Physics, 16, 93 (007). 5