The Unruh thermal spectrum through scalar and fermion tunneling
TThe Unruh thermal spectrum through scalar and fermiontunneling
Debraj Roy ∗ S.N. Bose National Centre for Basic Sciences,Block–JD, Sector III, Salt Lake, Kolkata-700098, India.
Abstract
The thermal spectrum seen by accelerated observers in Minkowski space vacuum, the Unruh effect, isderived within the tunneling mechanism. This is a new result in this mechanism and it completes the treat-ment of Unruh effect via tunneling. Both Bose-Einstein and Fermi-Dirac spectrum is derived by consideringtunneling of scalar and spin half particles respectively, across the accelerated Rindler horizon. Full solutionsof massless Klein-Gordon and Dirac equations in the Rindler metric are employed to achieve this, insteadof approximate solutions.
Linearly accelerated observers should detect a ther-mal background with black-body spectrum in placewhere an inertial observer detects no particles. Thisresult is known as the Fulling-Davies-Unruh effect[1] . Several analysis of non-inertial observers havebeen done through a quantized field theory construc-tion in the coordinates adapted to an accelerated ob-server – Rindler coordinates. Now while analyzingthe closely related Hawking effect [3] for black-holes,an easier and more conceptually transparent single-particle analysis, the tunneling mechanism , was de-veloped [4, 5]. This considers a virtual pair of par-ticles formed just inside the horizon, one ‘ingoing’towards the black-hole i.e. away from horizon, whilethe other ‘outgoing’ one traveling towards the hori-zon. Now, as against classical general relativity, thisoutgoing particle is taken to quantum-mechanicallytunnel outside, with a small probability.The tunneling analysis has also been done in theUnruh effect [4, 6, 7, 8] where the Rindler wedges II and I (see Fig. 1) act as Black-hole interior and exte-rior regions, with the accelerated horizon playing therole of the black-hole horizon. However the tunnelingmechanism could only derive the Unruh (or Hawking)temperature and not explicitly the spectrum. Re-cently this drawback was removed [9] through a quan-tum statistical analysis of a system of particles that are tunneling across the horizon, leading to a clearderivation of the Hawking black-body spectrum, incase of a spherically symmetric black-hole. Other ap-plications of this method may be found in [10] wherethe Kerr-Newman black hole was considerd, and in[11] where black hole solutions in Lovelock gravity isdiscussed.In this paper, I calculate the thermal spectrum inUnruh effect, within the tunneling formalism. Themethodology follows [9], but with a modification inthe calculation of tunneling modes. Instead of theusually applied WKB approximations, I use full so-lutions of the Klein Gordon and Dirac equations in(3 + 1)-D Rindler spacetime, as the tunneling modes.This is possible as the flat Rindler metric is inherentlysimpler than, say, the Schwarzschild metric where fullsolution of the Klein Gordon equation is not known.Thus, this article provides a new and conceptuallyappealing derivation of the Unruh effect. The path of constant, linear acceleration (say α ) inMinkowski spacetime, is described by the hyperbola X − T = 1 α (1) ∗ E-mail: [email protected] for a recent review and extensive references, see [2] a r X i v : . [ h e p - t h ] O c t igure 1: The Rindler wedges shown on the Minkowskiplane. ( X , T ) and ( x, t ) are Minkowski and Rindler coor-dinates. The infinities lie outside the diagram, towards thedirections shown. Tunneling occurs from wedge II to I . where T & X are Minkowski coordinates, with X -axis being the direction of acceleration. If I con-sider a range (0 < α < ∞ ) of accelerated observers,the entire ‘wedges’ I and IV can be covered with theresultant hyperbolae. These wedges are known as‘ Rindler wedges ’. A new coordinate system – theRindler coordinates – can now be set up taking thesetimelike hyperbolae and corresponding spacelike lines(Fig. 1) as the coordinate lines. They are related tothe Minkowski coordinates, in wedges I and IV (indi-cated as subscripts), through the following relations: T = F ( x I ) sinh( at I ) X = F ( x I ) cosh( at I ) Y = y I , Z = z I (2)where F ( x ) = 1 /α and a is a constant. TheMinkowski line element ds = − dT + dX + dY + dZ can now be written as ds = − a F ( x ) dt + F (cid:48) ( x ) dx + dy + dz . (3)This is a generalized form of the Rindler met-ric and for different choices of the function F ( x ), wecan get all the different forms of Rindler metric seenin literature. Though this metric covers the entireMinkowski plane, the coordinates t and x change inthe different wedges. The Rindler horizon occurs at F ( x ) = 0. The two wedges I and II on two sides,act like the black-hole exterior and interior regionsrespectively, as seen in the case of Schwarzschild so-lution.Changing to a tortoise coordinate x (cid:63) appropriate for the Rindler metric (3) through dx (cid:63) = F (cid:48) ( x ) aF ( x ) dxx (cid:63) = 1 a ln F ( x ) (4)the metric (3) becomes ds = ( a e ax(cid:63) ) (cid:0) − dt + dx (cid:63) (cid:1) + dy + dz . (5)Returning to equation (1), though the hyperbola de-scribes a real accelerating particle only in wedge I ,the hyperbolae can be analytically extended to thewedges II & III as X − T = − α . The transforma-tion relating the Rindler coordinate to the Minkowskicoordinate then becomes T = F ( x II ) cosh( at II ) X = F ( x II ) sinh( at II ) Y = y II , Z = z II . (6)Now the maximal extension of the Rindler coordinatesystem ( t, x, . . . ) is the Minkowski ( T, X, . . . ) whichis defined everywhere throughout the four Rindlerwedges, i.e. the entire Minkowski plane, irrespec-tive of accelerated horizons which the Rindler ob-server encounters. The Rindler coordinates howeverundergo a finite shift through the horizon, as can eas-ily be seen through a comparison of (2) and (6). Arelation between the t − x (cid:63) coordinate pair outside(Region I ) and inside (Region II ) can then be writ-ten as t II = t I − iπ ax (cid:63) II = x (cid:63) I + iπ a . (7)The y and z coordinates on the other hand, remainunchanged across the horizon. It is to be noted thatthe transformation t II = t I + iπ a ; x (cid:63) II = x (cid:63) I − iπ a alsosuffice in relating the coordinates II and I . Howeverthis second pair leads to some problems in taking theclassical limit of the tunneling probability, and so isnot considered at the accelerated horizon, as will bediscussed later. Such transforms were reported ear-lier in [12, 13]. The massless Klein Gordon equation (cid:3)
Φ = 0, writ-ten in the Rindler metric (5), reads e − ax (cid:63) a (cid:2) − ∂ t Φ + ∂ x (cid:63) Φ (cid:3) + ∂ y Φ + ∂ z Φ = 0 . (8)2ince the metric is independent of the coordinates t, y & z , I take an ansatz for Φ asΦ( t, x (cid:63) , y, z ) = φ ( x (cid:63) ) e − i (cid:126) (Ω t + k y y + k z z ) , (9)where Ω is a constant. This Ω is related to thelocally observed energy ω at some Rindler space-time point x (cid:63) through a red-shift [14, 15] relation E V = E V = Ω connecting the observed ener-gies E & E at two different points in a gravitat-ing system at equilibrium. The result ensures thatthough the observed energies E and the Tolman red-shift factor V vary locally as functions of the coordi-nates (here x (cid:63) ), their product Ω is a constant. In agravitational system in equilibrium, this condition ofthe constancy of Ω characterizes the equilibrium, justas is done by temperature in a laboratory thermo-dynamic system at thermal equilibrium [14]. Here,this locally observed energy E is the energy ω ofthe tunneling-particle and the quantity V becomes (cid:112) | g | . Thus I haveΩ = ω ae ax (cid:63) = a ωα (10)where appropriate definitions F ( x (cid:63) ) = 1 /α = e ax (cid:63) ( see metrics φ ( x (cid:63) ) φ (cid:48)(cid:48) ( x (cid:63) ) + (cid:18) Ω (cid:126) − a e ax(cid:63) k ⊥ (cid:126) (cid:19) φ ( x (cid:63) ) = 0 , (11)with k ⊥ = (cid:113) k y + k z .Some observations can immediately be made fromequation (11). Near the horizon, as x (cid:63) → −∞ , theterm containing k ⊥ drops out and a simple harmonictype equation with plane wave solutions is obtained.Again at large spatial distances, x (cid:63) → ∞ , and nowthe term containing Ω becomes negligible. Thisleaves an equation with exponentially increasing anddecreasing solutions I (cid:16) k ⊥ (cid:126) e ax (cid:63) (cid:17) and K (cid:16) k ⊥ (cid:126) e ax (cid:63) (cid:17) ,where I and K represent the zero-th order mod-ified Bessel functions of first and second types re-spectively. Thus throwing away the I solutions, wehave an exponentially vanishing solution at infinity,in K . Similar conclusions have also been reached atby Boulware [16].The solution of the full equation (11) that is welldefined through the horizon is, φ ( x (cid:63) ) = A − e π Ω2 a (cid:126) Γ (cid:18) − i Ω a (cid:126) (cid:19) I − i Ω a (cid:126) (cid:18) k ⊥ (cid:126) e ax (cid:63) (cid:19) + A + e − π Ω2 a (cid:126) Γ (cid:18) i Ω a (cid:126) (cid:19) I i Ω a (cid:126) (cid:18) k ⊥ (cid:126) e ax (cid:63) (cid:19) (12) where A ∓ are arbitrary integration constants. Forsmall arguments, the appropriate expansion of themodified Bessel function is I ν ( z ) (cid:39) ( z/ ν Γ(1+ ν ) . Thisholds if k ⊥ (cid:28) Ω and also especially near the hori-zon where x (cid:63) → −∞ . Therefore (12) simplifies to φ ( x (cid:63) ) (cid:39) A ∓ e ± π Ω2 a (cid:126) (cid:18) k ⊥ (cid:126) (cid:19) ∓ i Ω a (cid:126) e ∓ i (cid:126) Ω x (cid:63) . The total wave function Φ( t, x (cid:63) , y, z ) near the horizonis thenΦ( t, x (cid:63) , y, z ) = B in e − i (cid:126) [Ω( t + x (cid:63) )+ k y y + k z z ] + B out e − i (cid:126) [Ω( t − x (cid:63) )+ k y y + k z z ] , (13)with all the constants clubbed together within B in / out . The subscript ‘ in ’ here stands for the ingoingmode which travels toward the accelerated horizonat x (cid:63) = −∞ , while the subscript ‘ out ’ stands for theoutgoing mode traveling away from horizon, i.e. to-wards x (cid:63) = ∞ . particles Spinors are introduced on a general curved space-time with metric g , by going to a local Lorentz framewith metric η , at each spacetime point [17]. Thisprocedure is best done by constructing tetrad fieldswhich map curved spacetime tensors to local Lorentzframe and vice-versa. For a Rindler metric in thetortoise-like coordinate system (5), the tetrad field V aµ is defined through the relation g µν = V aµ V bν η ab where latin ( a, b, . . . ) and greek ( µ, ν, . . . ) letters runover local Lorentz and curved space indices respec-tively. The explicit choice of the tetrad field V aµ adopted here is V aµ = diag( ae ax (cid:63) , ae ax (cid:63) , ,
1) (14)and the metric signature, both global and local, iskept same as ( − , + , + , +).The massless Dirac equation is written as [18, 2][ γ a V µa ( ∂ µ + Γ µ )] Ψ = 0 (15)where γ a are the Dirac matrices obeying the usualalgebra (cid:2) γ a , γ b (cid:3) = 2 η ab and Γ µ are connection coeffi-cients given by Γ µ = 12 Σ ab V νa V bν ; µ Σ ab = 14 (cid:104) γ a , γ b (cid:105) . V bν ; µ = ∂ µ V bν − Γ αµν V bα , where Γ αµν is the Christoffel symbol.On using the properties of γ matrices and the diag-onal choice of the tetrad (14), the spin-connectionbecomes Γ µ = − Σ ab Γ λµν V νa V bλ . The Dirac equa-tion (15) then turns out to be (cid:2) (cid:0) ∂ t − a Σ (cid:1) − γ γ ∂ x (cid:63) − ae ax (cid:63) (cid:0) γ γ ∂ y + γ γ ∂ z (cid:1) (cid:3) Ψ = 0 . (16)The ansatz for the spinor Ψ is taken asΨ( t, x (cid:63) , y, z ) = ψ ( x (cid:63) ) e − i (cid:126) (Ω t + k y y + k z z ) ψ ( x (cid:63) ) = A ( x (cid:63) )0 B ( x (cid:63) )0 . (17)Upon using this ansatz, equation (16) can be castinto a Schr¨odinger like equationˆ H ψ ( x (cid:63) ) = Ω ψ ( x (cid:63) ) (18)where the Hamiltonian-like operator ˆ H isˆ H = i (cid:126) (cid:0) a Σ + γ γ ∂ x (cid:63) (cid:1) + ae ax (cid:63) (cid:0) k y γ γ + k z γ γ (cid:1) . (19)The above equation, on squaring, gives ˆ H ψ = Ω ψ .Now using the ansatz (17) and adopting the con-vention for the gamma matrices as γ = (cid:0) − i i (cid:1) , γ j = (cid:16) − iσ j iσ j (cid:17) (where j = 1 , , σ j are thePauli matrices), the following equation for the spinorcomponent functions is obtained (cid:7) (cid:48)(cid:48) + a (cid:7) (cid:48) + (cid:34) Ω (cid:126) − a e ax (cid:63) k ⊥ (cid:126) + a (cid:35) (cid:7) = 0 . (20)Here (cid:7) stands for the functions A ( x (cid:63) ) and B ( x (cid:63) ). Asin the case of scalar particles (11), a study of asymp-totic behaviour of this equation show that solutionsnear the horizon are oscillatory, and that near to in-finity are vanishing in nature. Solution for the fullequation (20) turns out to be (cid:7) = e − ax(cid:63) (cid:20) M ( (cid:7) ) − e π Ω2 a (cid:126) Γ (cid:18) − i Ω a (cid:126) (cid:19) I − i Ω a (cid:126) (cid:18) k ⊥ (cid:126) e ax (cid:63) (cid:19) + M ( (cid:7) )+ e − π Ω2 a (cid:126) Γ (cid:18) i Ω a (cid:126) (cid:19) I i Ω a (cid:126) (cid:18) k ⊥ (cid:126) e ax (cid:63) (cid:19) (cid:21) . (21)It is to be noted that both (21) and (12) are fullsolutions of the respective differential equations (20) and (11), without any approximations of parameters.This can be easily verified by substituting back thesesolutions in the original differential equations, to seethat they satisfy them without using any approxi-mations on the parameters. However, an approxi-mation is used to get a simpler solution from (21),by using the appropriate expansion of the modifiedBessel function for small arguments which is givenas I ν ( z ) (cid:39) ( z/ ν Γ(1+ ν ) . This holds if k ⊥ (cid:28) Ω and alsoespecially near the horizon where x (cid:63) → −∞ . So, forthe region near to the horizon, the total spinor Ψ canfinally be written asΨ( t, x (cid:63) , y, z ) = ξ in e − ax(cid:63) e − i (cid:126) [Ω( t + x (cid:63) )+ k y y + k z z ] + ξ out e − ax(cid:63) e − i (cid:126) [Ω( t − x (cid:63) )+ k y y + k z z ] , (22)with ξ in / out being constant spinors and the subscript‘ in ’ or ‘ out ’ standing for ‘ingoing’ or ‘outgoing’ modes. Uptil now, I had found single-particle wave-functionsfor bosons and fermions, by solving for the Klein-Gordon and Dirac equations in the Rindler coor-dinates. These solutions are valid in both Rindlerwedges I and II , but in coordinates ( t I , x I ) and( t II , x II ) respectively. Classically, both the ingoingand outgoing modes (say of a virtual pair instanta-neously produced) in wedge II are trapped, as noth-ing from inside can cross the horizon and come outto wedge I . However in the tunneling mechanism, anoutgoing particle can quantum-mechanically tunnelout across the horizon and into wedge I . This processoccurs with a probability given by the Maxwell term e − πω (cid:126) a , that appropriately goes to zero in the classical( (cid:126) →
0) limit. Now to find the energy distributionof a collection of such particles, I will (following [9])construct a suitable density matrix for both bosonsand fermions, and find out the average number ofparticles having some particular energy ω .Starting first with bosonic particles, the relationbetween inside and outside wave-functions is foundby using the connection between coordinates (7) in4quation (13) for the modes. B in e − i (cid:126) [Ω( t II + x (cid:63) II )+ k y y II + k z z II ] = B in e − i (cid:126) [Ω( t I + x (cid:63) I )+ k y y I + k z z I ] B out e − i (cid:126) [Ω( t II − x (cid:63) II )+ k y y II + k z z II ] = (cid:16) e − π Ω a (cid:126) (cid:17) B out e − i (cid:126) [Ω( t I − x (cid:63) I )+ k y y I + k z z I ] (23)Now, let there be ‘ n ’ pair of free particles (ingoingand outgoing) in wedge II . The total state of thissystem of particles, with each being described by thesector II modes in equation (13), is | χ B (cid:105) = N B ∞ (cid:88) n =0 | n in II (cid:105) ⊗ | n out II (cid:105) = N B ∞ (cid:88) n =0 (cid:16) e − nπ Ω a (cid:126) (cid:17) | n in I (cid:105) ⊗ | n out I (cid:105) (24)where N B is a normalization constant definedthrough (cid:104) χ B | χ B (cid:105) = 1. The sum over n runs from0 to ∞ here, in the case of bosons. But in case offermions, as will be used later, n is limited to 0 and1 by Pauli’s exclusion principle. The normalizationof | χ B (cid:105) leads to N B ∞ (cid:88) n,m =0 e − ( n + m ) π Ω a (cid:126) (cid:16) (cid:104) m out I | ⊗ (cid:104) m in I | (cid:17)(cid:16) | n in I (cid:105) ⊗ | n out I (cid:105) (cid:17) = 1 ⇒ N B = (cid:34) ∞ (cid:88) n =0 e − πn Ω a (cid:126) (cid:35) − , (25)and finally for bosons, we have N B = (cid:16) − e − π Ω a (cid:126) (cid:17) . (26)The density matrix operator for this system ofbosons is defined as usualˆ ρ B = (cid:16) − e − π Ω a (cid:126) (cid:17) ∞ (cid:88) n,m =0 e − ( n + m ) π Ω a (cid:126) | n in I (cid:105) ⊗ | n out I (cid:105)(cid:104) m out I | ⊗ (cid:104) m in I | . (27)Since ingoing waves are trapped within the horizonand outgoing particles contribute to spectrum, wetrace out over the ingoing particles, to form the den-sity matrix for outgoing modes,ˆ ρ out B = (cid:16) − e − π Ω a (cid:126) (cid:17) ∞ (cid:88) n =0 e − πn Ω a (cid:126) | n out I (cid:105)(cid:104) n out I | . (28) The average number of outgoing particles is then cal-culated as (cid:104) ˆ n B (cid:105) = Tr out [ˆ n B ˆ ρ out B ] = (cid:16) − e − π Ω a (cid:126) (cid:17) ∞ (cid:88) n =0 ne − πn Ω a (cid:126) = 1 e π Ω a (cid:126) −
1= 1 e πω (cid:126) α − T U , the Unruh temperature[1], given as T U = (cid:126) α π (30)with α being the local acceleration.For fermions, the same method goes through stepby step. The connection between spinorial wave-functions in wedges II and I is obtained by using(7) in (22). ξ in e − i (cid:126) [Ω( t II + x (cid:63) II )+ k y y II + k z z II ] = ξ in e − i (cid:126) [Ω( t I + x (cid:63) I )+ k y y I + k z z I ] ξ out e − i (cid:126) [Ω( t II − x (cid:63) II )+ k y y II + k z z II ] = (cid:16) e − π Ω a (cid:126) (cid:17) ξ out e − i (cid:126) [Ω( t I − x (cid:63) I )+ k y y I + k z z I ] (31)The normalization of the total state ket for fermions | χ F (cid:105) = N F (cid:88) n =0 | n in II (cid:105) ⊗ | n out II (cid:105) = N F (cid:88) n =0 (cid:16) e − nπ Ω a (cid:126) (cid:17) | n in I (cid:105) ⊗ | n out I (cid:105) (32)is again done through (cid:104) χ F | χ F (cid:105) = 1. The sum overnumber of particles in a given state, in fermionic cal-culations, always run from 0 to 1, following Pauli’sexclusion principle. The normalization constant N F turns out to be N F = 1 (cid:113) e − π Ω a (cid:126) . (33)The density operator for fermions ˆ ρ F is defined as | χ F (cid:105)(cid:104) χ F | . Using the outgoing fermionic density oper-ator ˆ ρ out F the spectrum is calculated as average number5f outgoing particles (cid:104) ˆ n F (cid:105) = Tr out [ˆ n F ˆ ρ out F ] = 11 + e − π Ω a (cid:126) (cid:88) n =0 ne − πn Ω a (cid:126) = 1 e π Ω ah + 1= 1 e πω (cid:126) α + 1 (34)where equation (10) was used in the last step. This isthe Fermi-Dirac distribution, at the Unruh temper-ature T U defined in (30), and constitutes the Unruheffect for accelerated fermions [19]. In this paper, I calculated the thermal spectrum forUnruh effect within the tunneling mechanism. TheUnruh temperature was identified via a comparisonbetween the calculated thermal distribution and thestandard form of Bose or Fermi distributions.However, the temperature can also be calculateddirectly via tunneling. Say, when an observer inRindler wedge I observes an outgoing particle, com-ing from within wedge II , he will see the wave func-tion from inside wedge II with a factor as shownin equation (23) for scalar particles, and in (31)for fermions. The ingoing wave, however, does notchange by such a factor between the two wedges.To calculate the temperature, we can now use theprinciple of detailed balance, P out P in = e − ωT , where P out/in = | wave-function | is the outgoing/ingoingprobability, ω is the observed energy at that pointand T is the temperature. This gives the Unruh tem-perature T U = (cid:126) α π after suitably using the red-shiftdefinition of Ω given in (10) at the observer’s point.For a more detailed discussion on the connection be-tween the earlier approaches to tunneling and the oneused in this paper, the reader is directed to [9, 13].Another point to note concerns an apparent am-biguity in the sign of the factors chosen to con-nect the coordinates between wedges I and II inequation (7). It can be verified that the relations t II = t I + iπ a ; x (cid:63) II = x (cid:63) I − iπ a , which have a change insign between the t and x (cid:63) term signs, also connect thetransformations (2) and (6). However this set is notemployed as it produces an exponential factor with awrong exponent sign (cid:16) e π Ω a (cid:126) (cid:17) in the relations betweenwave-functions in wedges I and II (23 & 31). Withthis alternate sign, the probability of outgoing parti-cles to tunnel out across the horizon, diverges at the classical limit of (cid:126) →
0. This is clearly unacceptable.So this obvious criterion of conformity with classicallimit is used to remove the above said ambiguity ofsigns.
Acknowledgments:
I thank Prof. R. Banerjee forsuggesting the problem and for constant encourage-ment throughout. I also thank Mr. B.R. Majhi andMr. S. Kulkarni for discussions.
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