Dirac Equation in Reissner-Nordström Metric

Transcription

1 Advanced Studies in Theoetical Physics Vol. 11, 017, no. 7, HIKARI Ltd, Diac Equation in Reissne-Nodstöm Metic Antonio Zecca 1 Dipatimento di Fisica dell Univesita, Via Celoia, 16 - Milano, Italy GNFM - Guppo Nazionale pe la Fisica matematica - Milano, Italy Copyight c 017 Antonio Zecca. This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact The Diac equation of a chaged paticle is studied in the Reissne- Nodstöm space-time by means of the Newman Penose fomalism based on a well known null tetad fame. The electostatic inteaction of the chage of the paticle with the chage associated to the Reissne-Nodstöm metic is taken into acount. The time and angula dependence ae sepaated and integated. The sepaated adial dependence is finally educed to the study of a single Fuchs type diffeential equation. The equation is integated by seies nea the egula singulaities and by asymptotic expansion fo lage distances. A definition of scala poduct is tested on the solutions of the Diac equation that have been found and the esults compaed with those elative to a diffeent definition of poduct. Keywods: Reissne-Nodstöm metic; Diac equation; Solution 1 Intoduction It is well known that the Reissne-Nodstöm (R-N metic is a spheically symmetic solution of the Einstein equation coupled to Maxwell equations [4]. A peculia chaacteistic of that solution is that the time component of the vecto potential is the only one component that does not vanish. It has the fom Q /, whee Q is an integation constant that can be intepeted as a cental chage. The esulting R-N space-time epesents an intemediate situation between the Schwazschild and the Ke one [4]. It funishes an inteesting context whee to study spin field equations. The solutions of field equation could give infomations about the physical effect of gavitation on 1 Retied fom: Physics Depatment, Milano Univesity - Italy.

2 98 Antonio Zecca mico systems [] and on Hawking evapoation of black holes [5]. The knowledge of the solutions could give also the possibility of defining the nomal modes in view of a quantization of the field. Accodingly the Diac equation is one of the most attactive case to study. The Diac equation has been discussed in diffeent ways in the R-N metic (e. g., [13, 11, 8, 6] and Refeences theein, often in the line of Chandasekha sepaation of Diac equation in Ke metic [4]. The object of the pesent pape is to popose once again a solution of the equation. The poblem has been aleady consideed by the autho fo neutal paticle [17]. In case the paticle has chage e 0 the inteaction with the chage Q of the R-N geomety has to be taken into account. Though the pape it is then assumed an electomagnetic inteaction of the fom e 0 Q / fo evey value of the adial coodinate. The equation is fomulated in the two spino fom adopted in [4], extended to include a potential tem (cf [15]. To solve the equation, the Newman Penose fomalim [1] is employed. It is based on the null tetad fame epoted in [4]. The Diac equation is solved by vaiable sepaation. The time and angula dependence of the spino wave function ae obtained by explicit integation. As to the adial dependence one is finally left with an odinay diffeential equation of Fuchs type. The integation of the equation is pefomed nea the egula singulaities by convegent seies expansions and at = by asymptotic seies expansion. A covaiant scala poduct of the solutions of the Diac equation is defined by means of the consevation of a cuent-like spino. The convegence of the scala poduct is tested on the solutions that have been detemined. The esults ae compaed with simila esults existing in the liteatue and based on a diffeent notion of poduct of states. It emains open the poblem of the detemination of the global solution of the educed adial equation. Spin 1/ field equation The object is to study the Diac equation in the context of the Reissne- Nodstöm space-time of line element ds = f( dt f( d [dθ + dϕ sin θ] (1 f = M + Q = ( ( 1 > 0 (,1 = M ± M Q ; > 1. The two spino fom of the Diac equation fo paticle of mass m 0 and chage e o can be witten as the natual genealization

3 Diac equation in Reissne-Nodstöm space-time 99 [15] of the one adopted in [4] (see also [7]: ( AA + iv AA P A + iµ QA = 0 (3 ( AA i V AA Q A + iµ PA = 0 (4 By using the Newman Penose fomalism [1] the equation can be witten in tems of diectional deivatives and spin coefficients. In a nomalized spino basis it eads (D + ɛ ρ + iv 00 P 1 (δ + π α + iv 10 P 0 + iµ Q0 = 0 (5 (δ + β τ + iv 01 P 1 ( + µ γ + iv 11 P 0 + iµ Q1 = 0 (6 (D + ɛ ρ i V 00 Q 1 (δ + π α + i V 10 Q 0 + iµ P0 = 0 (7 (δ + β τ + i V 01 Q 1 ( + µ γ + i V 11 Q 0 + iµ P1 = 0 (8 µ = m 0 is the mass of the paticles of the field AA is the covaiant spinoial deivative and V AA the inteacting potential of the chage e o with the chage Q of the black hole. The object is to solve as fa as possible the equation (1 in the R-N metic whose line element is of the fom The assumed null tetad fame {l i, n i, m i, m i } elative to the metic (1 is the one consideed in [4] with coesponding diectional deivatives and non zeo spin coefficients D = l i i = f( t +, = n i i = 1 t f( (9 δ = m i i = 1 ( θ + i csc θ ϕ, δ = δ (10 ρ = 1, γ = µ + M µ = f( 3, cot θ β = α = (11 = M Q 3 (1 By assuming the electomagnetic potential fou vecto V µ (e o Q /, 0, 0, 0, the coesponding spino vesion is given by V AA = σaa i V i = σaa 0 V 0 o [ ] V AA = 1 0 f( 1 V 0 = e [ 1 ] 0Q f( (13 3 whee the σaa i s ae the spin matices in the given null tetad fame (9, (10. [If Q = 0 the scheme educes to the one elative to the Schwazschild metic]. By using (9-(13 in (5-(8 and by the futhe assumption P A ( H 1 (, ts 1 (θ, H (, ts (θ e imϕ (14 Q A ( H 1 (, ts (θ, H (, ts 1 (θ e imϕ (15

4 300 Antonio Zecca (fo convenience it is assumed m = 0, ±1, ±,... the angula dependence can be sepaated with sepaation constant λ. The esulting angula equations can be solved unde the bounday condition S i 0 = S i (π = 0, i = 1, (e.g., [16, 9]. Thee esults that λ = (l + 1/, l = 0, 1,,.. and that the S i S ilm (θ, (i = 1, ae essentially given by Jacobi polynomials [1]. Moeove the functions A ilm = S ilm (θe imϕ, can be nomalized so that: (A jlm, A jl m L (Ω = δ ll δ mm, j = 1,. One is then left with (, t equations (D + iv 00 H 1 = ( iµ λ H, (16 ( γ + iv 11 H = ( iµ + λ H1 (17 By the substitution H i (, t H i ( exp(iσt, i = 1, the time dependence factos out in (11 and one obtains, by using also (9, (1, (13, the adial equations ( σ H 1 + i f + e oq ( M Q H + i σ if f H 1 = ( iµ e oq λ H = f H (18 ( iµ + λ H1 (19 The eqs. (18, (19 could be easily disentangled by substitution, but it seems bette to peliminay pefom anothe eduction. By setting (cf [17] H j = F j ( e i [σ f( + eoq ]d, j = 1, (0 the equations fo the F j s ae F = F 1 + F 1 = F j ( e iσ iσ iσ iµ λ ( b b a i e oq iσ i eoq (1 ( F 1 (µ + λ ( 1 ( F 1 = 0 (3 b = 1 iσ 1, b 1 = 1 iσ 1 1, a = λ/(iµ (4 One has then that F i F iσl (, H i H iσl (, i = 1,. The final equation (3 is of Fuchs class with egula singulaities at =, 1, a, 0 with exponents (0, 1 b, (0, 1 + b 1, (0,, (0, 1 + i eoq espectively, while = is a singulaity that is not egula.

5 Diac equation in Reissne-Nodstöm space-time Local solutions nea singulaities It seems a poblem to find global solutions of eq. (3. It is possible howeve to give local esults by applying the theoy of Fuchs equations. In case of finite singulaities ξ = 0, 1, two integals of eq. (3, nea the egula singulaity = ξ, ae given by F (j 1 = ( ξ α j 0 c (j n ( ξ n, j = 1, (5 whee the α j s ae the solutions of the index equation elative to ξ. The c (j n s (c 0 0, n 1 ae detemined by the ecuence elation obtained by inseting (3 into the eq. (1. The seies has a non tivial adius of convegence [14]. F (1 1, F ( 1 ae linealy independent except when α α 1 is an intege, a case that does not happen hee because a is not eal while does. Fo what concens the singulaity at =, note that the equation (3 can be put, fo lage, into the fom (e.g.,[3] F 1 + (p 0 + p 1 + p +...F 1 + (q 0 + q 1 + q +.. F 1 = 0 (6 p 0 = iσ, p 1 = ie 0 Q 4σM,... (7 q 0 = µ, q 1 = 4µ M,... (8 The solutions can then be asymptotically epesented by F 1 = α e χ 0 c n n (9 χ ± = iσ ± (µ σ 1/ (30 α ± = i(σm + e 0 Q 1/ ± M(µ σ (µ σ 1/ (31 Once conditions (30, (31 ae satisfied the c n s ae detemined by the ecuence elation obtained by inseting (9 into (6. One has now F ± 1 F ± α ± e χ ± (3 e ±(µ σ 1/ ±M(µ σ (µ σ 1/, σ < µ (33 [ e i(σ±(σ µ 1/ i σm+ e 0 Q M(µ σ 1/ (σ µ ], 1/ σ > µ (34 F 1 ± ( (35 Theefoe, if F 1, F ae solutions of (3, ( in (0,, the asymptotic be-

6 30 Antonio Zecca havio of F 1, F nea the egula singulaities ae of the fom F 1 F 0 H 0 + K 0 1+ieoQ (36 0 L 0 (37 F 1 1 H1 + K 1 ( 1 1/ iσ 1 /( 1 F 1 L1 + M 1 ( 1 1/ iσ /( 1 F 1 H + K ( 1/+iσ 1 /( 1 F F 1 F L + M ( 1/+iσ /( 1 (38 (39 (40 (41 H + F H F1 (4 K + F K F1 (43 with H j, K j, L j, j = 0, 1, ; M l, l = 1, and H ±, K ± ae constant that in geneal may depend on σ. 4 Poduct of solutions Fom the solutions ψ (Pi A, Q B i, i (σlm; ψ (Pi A, QB i, i (σ l m of the Diac equation one can define [10] the poduct (ψ, ψ = d 3 x g 1 nξ σ ξ AB J AB (ψ, ψ (44 = I I = 1/ ddω [ (P 0 0 i P 1/ f( 1/ [ d I i + Q0 0 i Q i f( H H 1/ f(1/ H H ] + f(1/ (Pi 1 P i 1 + Q1 1 i Q i (45 ] δ ll δ mm (46 (In the last integal it has also been set H i H iσl (, H i H iσ l ( i = 1,. The spino J AB (ψ, ψ = Pi A P B i + QA i QB i is divegence fee: AA J AA = α J α = 0 as a consequence of the Diac equation (3, (4 (J AA (ψ, ψ is the conseved cuent; n α (, 0, 0, 0 is a futue diected unit vecto, σ ξ f( AA is the ξ-genealized Pauli matix elative to the null tetad of (9, (10 so that σaa t = 1/ diag{, 1/} (cf (13. In paticula one has then f ( (ψ, ψ = 1/ d f( F 1 + f(1/ F 1/ (47 One can now test the convegence of the integal (47. If F 1, F ae known global solution, on account of the behavios (36-(43, the integal esults locally convegent in = 0, 1,, and in = if it happens that K + = H + = 0 and σ < µ. Hence it is convegent in I (0, +. On the contay it

7 Diac equation in Reissne-Nodstöm space-time 303 esults divegent in I if it happens that σ > µ. To see moe explicitly inside the situation one should have explicit global solutions of (3. In the massless case µ = 0 global solutions can be given, but they have a complex fom [10]. Developments in the mentioned diection ae investigated in [6] whee a diffeent definition of nomalization integal is employed. One can see that that definition coincides with (47 afte a multiplication of the expession in backets by /f(. If such modified expession of (47 is consideed, one can easily see, by using (36-(43, that thee is convegence in = 0, divegence in = 1, = and convegence-divegence in = in ageement with [6]. 5 Remaks and comments In the pevious Sections the poblem of the solution of Diac equation in R-N space-time has been educed to the solution of a adial second ode diffeential equation of Fuchs type. Even if global solutions of the equation have not yet explicitly given, thei local behavio has been analyzed. This gives indications on the convegence of a possible scala poduct of Diac states. The detemination of global solutions of the adial equation would be of inteest to establish the existence o non existence of nomal modes and hence of possible elementay quantization of the Diac field. Also the poblem of the existence of a discete spectum of values of σ, hee intepeted as the enegy of the system, is of inteest in that connection. The discussion of such poblem depends on the definition of scala poduct of solutions. Accoding to the esults of the pevious Section, thee ae indications that the expession (47 (that is deived fom Gauss theoem, is bounded fo a continuous set of values of σ. Even if futhe estictions on the possible values of σ could deive by selecting othogonal solutions, that popety seems unusual. On the othe hand, if one adopts the nomalization integal of [6] thee ae not bounded esults (in case of non extemal R-N metic at all. So anothe poblem is what is the best scala poduct to associate with solutions. Refeences [1] W. Abamovitz and I.E. Stegun, Handbook of Mathematical Functions, Dove Publication, New Yok, [] D. R. Bill and J. A. Wheele, Inteaction of Neutinos and Gavitational Fields, Rev. Mod. Phys., 9 (1957, [3] P. Caldiola, R. Cielli, G. M. Pospei, Intoduzione alla Fisica Teoica, UTET, Toino, 198.

8 304 Antonio Zecca [4] S. Chandasekha, The Mathematical Theoy of Black Holes, Oxfod Univesity Pess, New Yok, [5] T. Damou and R. Ruffini, Black-hole evapoation in the Klein-Saute- Heisenbeg-Eule fomalism, Physical Review D, 14 (1976, [6] V. I. Dokuchaev and Yu. N. Eoshenko, Stationay Solutions of the Diac Equation in the Gavitational Field of a Chaged Black Hole, Joun. Exp. Theo. Phys., 117 (013, [7] R. Illge, Massive fields of abitay spin in cuved space-times, Comm. Math. Phys., 158 (1993, [8] Y. Lyu and S.-F. Sun, Scatteing of Diac Waves of Reissne-Nodstöm Black Holes, Int. Jou. Theo. Phys., 5 (013, [9] E. Montaldi and A. Zecca, Neutino Wave Equation in the Robetson- Walke Geomety, Int. Joun. Theo. Phys., 33 (1994, [10] E. Montaldi and A. Zecca, Second Quantization of the Diac Field: Nomal Modes in the Robetson-Walke Space-Time, Int. Joun. Theo. Phys., 37 (1998, [11] B. Mukhopadhyay, Behavio of a spin-1/ paticle aound a chaged black hole, Class. Quantum Gav., 17 (000, [1] E. Newman, R. Penose, An Appoach to Gavitational Radiation by a Method of Spin Coefficients, Jou. Math. Phys., 3 (196, no. 3, [13] C. L. Pekeis and K. Fankowski, Solution of Diac s equation in Reissne- Nodstöm geomety, Poc. Nat. Acad. Sci. USA, 83 (1986, [14] F. G. Ticomi, Istituzioni di Analisi Supeioe, CEDAM, Padova, [15] A. Zecca, Diac Equation with Electomagnetic Potential: Discete Spectum of the Hydogen Atom in the Robetson Walke Space-Time, Int. Jou. Theo. Phys., 38 (1999,

9 Diac equation in Reissne-Nodstöm space-time 305 [16] A. Zecca, Vaiable sepaation and solutions of massive field equations of abitay spin in Robetson-Walke space-time, Adv. Stud. Theo. Phys., 3 (009, [17] A. Zecca, Diac neutal paticle in Reissne-Nodstöm space time: popeties of the solutions, Adv. Stud. Theo. Phys., 11 (017, Received: May 8, 017; Published: May 3, 017