Physics Letters B 670 (2009) 446–448

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Physics Letters B

www.elsevier.com/locate/physletb

Fermions tunneling from apparent horizon of FRW universe

Ran Li, Ji-Rong Ren ∗, Dun-Fu Shi

Institute of Theoretical Physics, Lanzhou University, Lanzhou, 730000 Gansu, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 September 2008Received in revised form 9 November 2008Accepted 11 November 2008Available online 18 November 2008Editor: M. Cvetic

PACS:04.62.+v04.70.Dy98.80.Jk

Keywords:FermionsTunnelingHawking radiationFRW universe

In the paper [R.-G. Cai, L.-M. Cao, Y.-P. Hu, arXiv: 0809.1554], the scalar particles’ Hawking radiationfrom the apparent horizon of Friedmann–Robertson–Walker (FRW) universe was investigated by usingthe tunneling formalism. They obtained the Hawking temperature associated with the apparent horizon,which was extensively applied in investigating the relationship between the first law of thermodynamicsand Friedmann equations. In this Letter, we calculate fermions’ Hawking radiation from the apparenthorizon of FRW universe via tunneling formalism. Applying WKB approximation to the general covariantDirac equation in FRW spacetime background, the radiation spectrum and Hawking temperature ofapparent horizon are correctly recovered, which supports the arguments presented in the paper[R.-G. Cai, L.-M. Cao, Y.-P. Hu, arXiv: 0809.1554].

Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.

In the recent paper [1], Cai et al. discussed the scalar parti-cles’ Hawking radiation from the apparent horizon of FRW uni-verse, which revealed that there is a Hawking temperature associ-ated with the apparent horizon. In this Letter, we will investigatefermions’ tunneling from the apparent horizon of FRW universe.The results in the present Letter together with that in [1] fill inthe gap existing in the literatures investigating the relationship be-tween the first law of thermodynamics and Friedmann equationsof FRW universe.

Ted Jacobson [2] has been able to derive Einstein equation fromthe proportionality of entropy and horizon area together with thefundamental Clausius relation δQ = T dS connecting heat, entropy,and temperature. The key idea is to demand that this relationholds for all the local Rindler causal horizons through each space-time point, with δQ and T interpreted as the energy flux andUnruh temperature seen by an accelerated observer just inside thehorizon. This perspective suggests that Einstein field equation canbe viewed as an equation of state of spacetime. The idea has alsoapplied to f (R) theory [3] and scalar–tensor theory [4], where thenon-equilibrium thermodynamic must be taken into account.

It has been proved that the above idea can also be applied inestablishing the relationship between the Friedmann equations andthe first law of thermodynamics in the framework of Friedmann–

* Corresponding author.E-mail address: renjr@lzu.edu.cn (J.-R. Ren).

0370-2693/$ – see front matter Crown Copyright © 2008 Published by Elsevier B.V. Alldoi:10.1016/j.physletb.2008.11.029

Robertson–Walker (FRW) universe. Assuming that the apparenthorizon of FRW universe has the temperature T = 1/2π r A and en-tropy S = A/4, where r A and A are the radius and the area ofthe apparent horizon, respectively, the Friedmann equations canbe derived from the Clausius relation [5] and the first Friedmannequation can be cast into the form of the unified first law [4,6].For some related investigations see also [7–15].

However, whether there is a Hawking temperature associatedwith the apparent horizon of FRW universe is still an open ques-tion to investigate. In a recent paper [1], the scalar particles’ Hawk-ing radiation from the apparent horizon of FRW universe was in-vestigated by using the tunneling formalism. They obtained theHawking temperature associated with the apparent horizon of FRWuniverse. They found that the Kodama observer inside the apparenthorizon do see a thermal spectrum with temperature T = 1/2π r A ,which is caused by particles tunneling from the outside apparenthorizon to the inside apparent horizon.

The semi-classical derivation of Hawking radiation as tunnel-ing process [16] was initially proposed by Parikh and Wilczek. Inrecent years, it has already attracted a lot of attention. In thismethod, the imaginary part of the action is calculated using thenull geodesic equation. Zhang and Zhao extended this method tothe charged Reissner–Nordström black hole [17] and the rotatingKerr–Newman black hole [18]. See also [19] for a different dis-cussion. M. Angheben et al. [20] also proposed a derivation ofHawking radiation by calculating the particles’ classical action fromthe Hamilton–Jacobi equation, which is an extension of the com-

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R. Li et al. / Physics Letters B 670 (2009) 446–448 447

plex path analysis of T. Padmanabhan et al. [21]. Very recently,a new calculation concerning fermions’ radiation from the station-ary spherical symmetric black hole was done by R. Kerner andR.B. Mann in [22]. This method has been generalized to the moregeneral and complicated spacetime background [23–29] and dy-namical black hole [30].

In this Letter, we will calculate the fermions’ Hawking radiationfrom the apparent horizon of FRW universe via tunneling formal-ism. Applying WKB approximation to the general covariant Diracequation in FRW spacetime background, we also obtain the Hawk-ing temperature of apparent horizon of FRW universe. Our resultsobtained by taking the fermion tunneling into account support thearguments presented in the previous paper [1].

For convenience, we firstly give some results related to the FRWuniverse. The FRW metric is given by

ds2 = −dt2 + a2(t)

(dr2

1 − kr2+ r2(dθ2 + sin2 θ dφ2)), (1)

where a(t) is the scale factor and k = 1, 0 and −1 represent theclosed, flat and open universe, respectively. Introducing r = ar, theFRW metric (1) can be rewritten as

ds2 = hab dxa dxb + r2 dΩ22 , (2)

where xa = (t, r), hab = diag(−1, a2

1−kr2 ) and dΩ22 represents the

line element of S2. The apparent horizon is defined by the equa-tion

hab∂ar∂br = 0, (3)

which gives us the location of apparent horizon explicitly as

r A = 1√H2 + k/a2

, (4)

with H = a/a being the Hubble parameter. For simplicity, we willuse the (t, r) coordinates, in which the FRW metric can be rewrit-ten as

ds2 = − 1 − r2/r2A

1 − kr2/a2dt2 − 2Hr

1 − kr2/a2dt dr

+ 1

1 − kr2/a2dr2 + r2 dΩ2

2 . (5)

The inverse of the metric of the (t, r) parts is given by

gtt = −1, gtr = −Hr, grr = 1 − r2/r2A . (6)

Now we calculate the fermions’ Hawking radiation from the ap-parent horizon of the FRW universe via the tunneling formalism.Let us start with the massless spinor field Ψ obeyed the generalcovariant Dirac equation

−ihγ aeμa ∇μΨ = 0, (7)

where ∇μ is the spinor covariant derivative defined by ∇μ = ∂μ +14 ωab

μ γ[aγb] , and ωabμ is the spin connection, which can be given in

terms of the tetrad eμa . The γ matrices are selected as

γ 0 =(

i 00 −i

),

γ 1 =(

0 σ 3

σ 3 0

),

γ 2 =(

0 σ 1

σ 1 0

),

γ 3 =(

0 σ 2

σ 2 0

),

where the matrices σ k (k = 1,2,3) are the Pauli matrices. Accord-ing to the line element (5) and the inverse metric given in (6), thetetrad field can be selected to be

eμ0 = (1, Hr,0,0),

eμ1 = (

0,

√1 − kr2/a2,0,0

),

eμ2 = (

0,0, r−1,0),

eμ3 = (

0,0,0, (r sin θ)−1).Without loss of generality, we employ the following ansatz forspinor field with the spin up

Ψ =⎛⎜⎝

A(t, r, θ,φ)

0B(t, r, θ,φ)

0

⎞⎟⎠exp

[i

hI(t, r, θ,φ)

]. (8)

It should be noted that the spin-down case is just analogous. In or-der to apply the WKB approximation, we can insert the ansatz (8)for the spin-up spinor field Ψ into the general covariant Diracequation (7). Dividing by the exponential term and neglecting theterms with h, one can arrive at the following four equations⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

i A(∂t I + Hr∂r I) + B√

1 − kr2/a2∂r I = 0,

B( 1r ∂θ I + i

r sin θ∂φ I) = 0,

A√

1 − kr2/a2∂r I − iB(∂t I + Hr∂r I) = 0,

A( 1r ∂θ I + i

r sin θ∂φ I) = 0.

(9)

Note that although A and B are not constant, their derivatives andthe spin connections are all of the factor h, so can be neglected tothe lowest order in WKB approximation.

To carry out the separation of variables for the above equations,we must analysis the symmetries of the metric (5). For the met-ric (5), the Kodama vector is given by [1]

K a =√

1 − kr/a2

(∂

∂t

)a

. (10)

The Kodama vector in dynamical spacetime is of the same signifi-cance with the Killing vector in static spacetime. It should be notedthat the Kodama vector is time-like, null and space-like as r < r A ,r = r A and r > r A , respectively. Using the Kodama vector, one candefine the energy measured by the Kodama observer

ω = −K a∂a I = −√

1 − kr/a2∂t I. (11)

Using the definition of energy, the classical action can be separatedas

I = −∫

ω√1 − kr/a2

dt + R(r) + P (θ,φ). (12)

Substituting the above ansatz into Eq. (9) yields⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

i A(− ω√1−kr/a2

+ Hr∂r I) + B√

1 − kr2/a2∂r I = 0,

B( 1r ∂θ P (θ,φ) + i

r sin θ∂φ P (θ,φ)) = 0,

A√

1 − kr2/a2∂r I − iB(− ω√1−kr/a2

+ Hr∂r I) = 0,

A( 1r ∂θ P (θ,φ) + i

r sin θ∂φ P (θ,φ)) = 0.

(13)

It is easy to see that P must be a complex function, which meansit will yield a contribution to the imaginary part of the classicalaction. The contribution of P to the tunneling rate is cancelledout when dividing the outgoing probability by the ingoing prob-ability because P is completely the same for both the outgoingand ingoing solutions. It is no need to solve the equations aboutthe complex function P . So it is essential to work out the imagi-nary part of R(r). From the first and third formulae of the above

448 R. Li et al. / Physics Letters B 670 (2009) 446–448

equation, there will be a non-trivial solution for A and B if andonly if the determinant of the coefficient matrix vanishes, whichresults

∂r I = ω

(r2/r2A − 1)

√1 − kr/a2

(Hr ±

√1 − kr2/a2

), (14)

where the +/− sign corresponds to the incoming/outgoing so-lutions, respectively. It should be noted that the imaginary partof R± can be calculated using the above equation. Integrat-ing the pole at the apparent horizon as in Refs. [31,32], wehave

Im R+ = Im∫

ω

(r2/r2A − 1)

√1 − kr/a2

(Hr +

√1 − kr2/a2

)dr

= πωr A,

Im R− = Im∫

ω

(r2/r2A − 1)

√1 − kr/a2

(Hr −

√1 − kr2/a2

)dr

= 0. (15)

In the WKB approximation, the tunneling probability is related tothe imaginary part of the action as

Γ = P in

Pout= exp[−2(Im R+ + Im P )]

exp[−2(Im R− + Im P )] = exp[−2πωr A]. (16)

Comparing the tunneling probability and the thermal spectrumΓ = exp[−ω/T ], Hawking temperature associated with the appar-ent horizon can be determined as

T = 1

2π r A. (17)

The Hawking temperature associated with the apparent horizonof FRW universe obtained by using the fermions tunneling methodtakes the same form as that obtained in [1]. It should be noted thatthe Kodama observer is inside the apparent horizon. The results in-dicate that the Kodama observer does see a thermal spectrum withtemperature T = 1/2π r A , which is caused by fermions tunnelingfrom the outside apparent horizon to the inside apparent horizon.The results in the present Letter together with that in [1] fill inthe gap existing in the literatures investigating the relationship be-tween the first law of thermodynamics and Friedmann equationsof FRW universe.

On the other hand, the higher terms about ω are neglectedin our derivation and the expression (16) for tunneling probabil-ity implies the pure thermal radiation. When energy conservationis taken into account, the emission spectrum is no longer purelythermal and contains the higher terms about ω.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China and Cuiying Project of Lanzhou University.

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