Late-time expansion in the semiclassical theory

of the Hawking radiation

Pietro Menotti

Dipartimento di Fisica della Universita di Pisaand INFN Sezione di Pisa

The present contribution is concerned with the computation of the back-reaction effects on the Hawking radiation, in the semiclassical approxi-mation.

1) Hawking radiation (Hawking: Comm.Math.Phys. 43 (1975); E 46(1976))

F (ω)dω =1

2πΓ(ω)

1

e8π Mω

l2P − 1

dω (1)

Γ(ω) is the gray-body factor and l2P = G~/c3, lP = 1.62 10−35m. Tem-perature

T =l2P c4

8πGM=

~c3

8πGM

2) External field approach

Quantize a (massless) scalar field in the background gravitational fieldgenerated by the black-hole (known procedure due to the existence of atime like Killing vector). Definition of the physical vacuum which turns

out not to be the “Killing” or Boulware vacuum and thus a stationaryobserver at space infinity detects a radiation.

φ =

∫ ∞

0

dω√2ω

(uω(r)e−iωta(ω) + u∗ω(r)eiωta+(ω)) =

=

∫ ∞

0

dk√2k

(vk(r, t)b(k) + v∗k(r, t)b+(k)).

Unruh vacuum isb(k)|0u〉 = 0

where b(k) are the annihilation operators for certain modes which areregular at the horizon. Under such condition the falling detector doesnot experience excitations. Knowledge of u and v allows to computethe Bogoliubov transformation

a(ω) =

∫ ∞

0

dk(αωkb(k) + βωkb+(k))

from which one can compute the average value of the number operator

on the Unruh (true) vacuum

N(ω) = 〈0u|a+(ω)a(ω)|0u〉 =

∫ ∞

0

dk|βωk|2

Fredenhagen and Haag (Comm.Math.Phys. 127 (1990) and correctionin R. Haag: Local quantum physics, II edition).

Work outside the horizon. Again scalar field (simpler if massless).Construct the local observable

Q+Q

where

Q =

∫ √−gφ(x)h(x)d4x

h test function with 4-dimensional compact support. QT is Q withthe support translated by a large time T . They compute such a meanvalue by piling up (propagating back) the field on a space like surfaceτ = const. The result is that

〈0|Q+T QT |0〉

is expressed (rigorously for free fields) in terms of the Wightman (notFeynman) function

W (x, x′) = 〈0|φ(x)φ(x′)|0〉

τ=0

Q

Q

r = const.

Kruskal−Szekeres coordinates

T

computed on τ = const, thus at very short distances and very near thefuture event horizon.

W (x, x′) =1

4π

1

σε+ lower terms

in Minkowski would be

W (x, x′) =1

4π

1

(x − x′)2 − (x0 − x′0 − iε)2

Takes the place of the definition of the vacuum.

Independent of T (time): The Hawking radiation is a permanent phe-nomenon.

3) Semiclassical treatment

Introduced by Kraus and Wilczek (Nucl.Phys. B433 (1995)).Work out a dynamical model which takes into account conservation ofenergy.The model (4-dimensions): a spherical shell of matter (massive or mass-less) moving in the gravitational field generated by the black-hole andby the shell itself (only s-waves).Compute the modes (regular and singular) semiclassically i.e.

eiS/~

where S is the classical action of the shell moving in the gravitationalfield. Use such modes to compute the Bogoliubov coefficients.

The classical problem is exactly soluble even if non trivial.

(W. Fischler, D. Morgan J. Polchinski Phys.Rev. D 42(1990) , P.Kraus,F.Wilczek, Nucl.Phys. B433 (1995) , J. Friedman, J. Louko, S. Winters-Hilt,Phys. Rev. D 56 (1998), F. Fiamberti P.M. Nucl.Phys. B 794(2008); P.M. Class.Q.Gravity 27 (2010))

Choice of gauge and the conjugate momen-tum

In the general expression of the metric (ADM)

ds2 = −N2dt2 + L2(dr + Nrdt)2 + R2dΩ2 (2)

all quantities N, L, Nr, R are supposed functions only of the radial vari-able r and t thus realizing spherical symmetry.Painleve-Gullstrand metric characterized by setting L = 1. Such ametric has the advantage of being non singular at the horizon. Afterfixing L = 1 one has still a gauge choice on R. In presence of a shell ofmatter one cannot choose R = r. One has several choices(F.Fiamberti, P.M. Nucl.Phys. B 794 (2008); P.M. Class.Q.Gravity 27(2010))

We will use the “outer gauge” which is defined by R = r for r ≥ rwhere r denotes the shell position. At r = r, R is continuous as all theother functions appearing in (2), but its derivative is discontinuous.

r r

outer gauge

R

After solving the constraints one reaches the reduced action i.e. a formin which only the coordinate r of the shell and a conjugate momentumappears, in addition to the boundary terms.The reduced action in the outer gauge is given by ( F. Fiamberti P.M.Nucl.Phys. B 794 (2008); P.M. Class.Q.Gravity 27 (2010))***

S =

∫ tf

ti

(

pc˙r − M(t)

∫ r(t)

ri

∂F

∂Mdr − HN(re) + MN(ri)

)

dt (3)

With M as a datum and the normalization N(re) = 1 is equivalent to

S =

∫ tf

ti

(

pc˙r − H)dt. (4)

In the massless case (massive far more complicated)

pc =√

2Mr −√

2Hr − r log

√r −

√2H√

r −√

2M. (5)

H as implicit function (invertible) of r and t.One has to keep in mind that pc is not the kinetic momentum of the shellbut the conjugate momentum with respect to r of the whole system.

The action for the modes regular at the hori-zon

In the following we use the symbol r for r.At the semiclassical level the modes which are invariant under theKilling vector ∂

∂t are simply given by

eiS/l2P (6)

with l2P = G~ the square of the Planck length and

S =

∫ r1

pcdr − Ht + const. (7)

Such modes have the feature of being singular at the horizon; this isimmediately seen from the expression of pc eq.(5) which diverges at r =2H. The vacuum given by aω|0〉 = 0, being aω the destruction operatorrelative to the described modes gives rise to a singular description atthe horizon, while a free falling observer should not experiment anysingularity. They describe your detector.

Construction of regular (Unruh) modes

i) at time 0 the conjugate momentum is a given value k;ii) at time t the shell position r is a given value r1.

Structure already given by Kraus and Wilczek.

With the two conditions pc(0) = k and r(t) = r1 the action is

S(r1, t, k) = kr0(r1, t, k) +

∫ t

0

(

pcr − H(r(t′), pc(t′)))

dt′ =

= kr0(r1, t, k) +

∫ t

0

pcrdt′ − H[r1, t, k]t. (8)

The last equality is due to the fact that H along the motion is a constantdespite H depends on the boundary conditions as explicitely written.

S(r, 0, k) = kr

r0 denotes the value of r at time 0; also such a quantity depends onthe imposed boundary conditions. Taking into account that r and pc

depend both on the final time t and on the running time t′, and denotingwith a dot the derivative with respect to t′ one has

∂S

∂r1= k

∂r0

∂r1+

∫ t

0

(

pc∂r

∂r1+ pc

∂r

∂r1

)

dt′ = pc. (9)

Similarly

∂S

∂t= k

∂r0

∂t+ (pcr − H)|t +

∫ t

0

(

pc∂r

∂t+ pc

∂r

∂t

)

dt′ = −H. (10)

The action (8) has to be computed on the solution of the equation ofmotion, satisfying the described boundary conditions.

The equations of motion

In the outer gauge, that we adopt here, the equation of motion for rhas the form

dr

dt= 1 −

√

2H

r(11)

while dpc/dt can be obtained substituting r(t) in eq.(5). Eq.(11) canbe integrated in the form

t = 4H log

√r1 −

√2H

√r0 −

√2H

+ r1 − r0 + 2√

2Hr1 − 2√

2Hr0. (12)

The boundary condition at t = 0 gives

0 < k =√

2Mr0 −√

2Hr0 − r0 log

√r0 −

√2H

√r0 −

√2M

(13)

where 2M < 2H < r0 < r1; eq.(19) together with eq.(18) should deter-mine completely the motion.Given t, k and r1: determine r0(k, t) and H(k, t).

The occurrence of caustics (Lagrangean sin-gularities)

(P.M. arXiv:1107.3312 [hep-th])

It is very easy to show that the standard variational problem in whichr is fixed to r0 at time 0 and to r1 at time t presents no caustics.To investigate the occurrence of caustics we shall compute the derivativeof t with respect to r0 under the constraint of constant k.

(

∂t

∂r0

)

k

=

(

∂t

∂r0

)

H

+

(

∂t

∂H

)

r0

(

∂H

∂r0

)

k

= − 1

1 −√a[1 − I1 I2] (14)

with a = 2Hr0

.

I1 = (1 −√

a)

∫

√a

q

2Mr0

y2dy

(1 − y)2, I2 = (1 −

√a)

∫

√a

q

2Hr1

dz

z2(1 − z)2.

(15)One proves that for any given k (Unruh frequency), for large enough r1

the derivative (14), when r0 moves from r1 to 2M changes sign, thusvanishing at at least one intermediate point.This implies the occurrence of Lagrangean singularities (caustics).(Arnold: Mathematical methods of classical mechanics)

r1

t

r02 M

However analytically one can prove: for r1 < 10M no caustics; (numer-ically for r1 < 24M).

Also: Given a k for sufficiently large t no caustics arises, for any r1.

Kraus and Wilczek proposed to perform the time Fourier analysis ata point r1 not too far from the horizon, the reason being that thereone should expect the semiclassical approximation to be reliable. Weshowed above that for r1 < rc there are no ambiguities in the definitionof the action and in addition it is well known the time Fourier transformgives results independent of r1; thus we shall work with r1 < rc.

The late-time expansion

(Kraus-Wilczek)Expand the modes at late timesCompute the Bogoliubov coefficient by means of time Fourier transform.

F (ω)dω =dω

2π

1

e8π Mω

l2P

(1− ωM

) − 1=

dω

2π

1

e8π ω

l2P

(M−ω) − 1(16)

Repeated by Kraus and Keski-Vakkuri (Nucl.Phys. B491 (997)) with adifferent method to find

F (ω)dω =dω

2π

1

e8π Mω

l2P

(1− ω2M

) − 1. (17)

Take up again the late time expansion

(P.M. Phys.Rev. D85, 084005 (2012); arXiv:1107.3312 [hep-th])

t = 4H log

√r1 −

√2H

√r0 −

√2H

+ r1 − r0 + 2√

2Hr1 − 2√

2Hr0. (18)

The boundary condition at t = 0 gives

0 < k =√

2Mr0 −√

2Hr0 − r0 log

√r0 −

√2H

√r0 −

√2M

(19)

Introduce now an implicit time variable

T = exp(− t

4H)

which due to the bounds on H, for t → +∞ tends to 0. It is possibleto show that H(t) is a decreasing function of t; thus the above relationcan be inverted t = t(T ) and we can write

h ≡√

2H =√

2M + g(T ) ≡ m + g(T ) (20)

One can prove the following properties of g(T )

g′(T ) > 0, g(0) = 0, g(1) =√

2H1−m, g(T ) = c0HT +c1

HT 2+ · · ·(21)

where H1 is the solution of pc(r0 = r1) = k. One can solve eq.(20) byiteration starting with h0 =

√2M and by induction one proves that

hn+1 > hn (22)

and as h is bounded by√

2H1 the series converges for all 0 < t < ∞.

r

2 M

t0

1

2 H(t)

r (t)0

oo

Figure 2: Time development of 2H(t) and r0(t)

We give in Fig.2 a qualitative graph of the behavior in time of 2H(t)and r0(t).We have for the first two terms of such an iteration procedure

H(t) = M +√

2Mc0Hτ +

t

2M(c0

Hτ)2, τ = e−t

4M (23)

sufficient to give the O(ω/M) corrections to the Hawking distribution.Due to eq.(10) the time dependence of the mode which is regular at thehorizon, for fixed r1 is (recalling ∂S/∂t = −H )

S = f(r1) −∫ t

H(t′)dt′ = const − Mt + 4M√

2Mτ1 + tτ21 (24)

i.e. for the semiclassical mode we have

eiS/l2P = ei[q(r1)−Mt+4M√

2Mτ1+tτ2

1]/l2P (25)

where l2P = G~ is the square of the Planck length and

τ1 ≡ c0Hτ = c0

He−t

4M

The saddle point approximation

The Bogoliubov coefficients αωk and βωk are given by

αωk = c(r1)

∫

dt ei(S+Mt+ωt)/l2P , βωk = c(r1)

∫

dt ei(S+Mt−ωt)/l2P .

(26)The above integrals will be computed using the saddle point method

where l2P plays the role of asymptotic parameter. From what we derivedin the previous section, the exponent appearing in the integrands, mul-tiplied by −il2P apart from q(r1) which is constant in time and commonto both coefficients, are respectively

2m3τ1 + tτ21 + m2(s + 1)τ2

1 ± ωt with τ1 = c0Hτ (27)

where we used the notation of eq.(??), For the αωk case (i.e. uppersign) the saddle point is given by the value of time t which satisfies

0 = −H(t) + M + ω = −mτ1 −t

m2τ21 − sτ2

1 + ω (28)

which being ω > 0 has solution for real t and thus at a real value of theexponent in eq.(26). On the contrary for the βωk case (lower sign), thesaddle point equation

0 = −H(t) + M − ω = −mτ1 −t

m2τ21 − sτ2

1 − ω (29)

has solution for complex t. At such a value of time the exponent (27)(lower sign) equals

B = −2m2ω − t(τ21 + ω) − (s − 1)m2τ2

1 . (30)

The solution of eq.(29) to second order in ω, which is the order we areinterested in, is given by

τ1 = − ω

m

(

1 − 2ω

m2log(− ω

c0Hm

) +sω

m2

)

. (31)

From eq.(30) we see that to find the imaginary part of such exponentto order ω2 we simply need the imaginary part of t to first order in ω.Using (31) we have

Im t = −2πm2(1 − 2ω

m2). (32)

Substituting into eq.(30) we find

Im B = 2πm2ω(1 − ω

m2) = 4πMω(1 − ω

2M) (33)

which according to (26) has to be divided by l2P . Thus we have

|βωk|2|αωk|2

= e−8π Mω

l2P

(1− ω2M

)(34)

which is independent of k.

We see from eq.(23) that for t → +∞, H(t) tends to M and thus thetime Fourier transform of the exponential of the action (25) which refersto the whole system has a singularity at the frequency M .

F (ω)dω =1

2π

∑

k

|βωk|2dω

Using the property of the Bogoliubov coefficients∑

k

(αωkα∗ω′k − βωkβ∗

ω′k) = δω,ω′ (35)

F (ω)dω =1

2π

∑

k |βωk|2∑

k′(|αωk′ |2 − |βωk′ |2)dω

one reaches for the flux of the Hawking radiation

F (ω)dω =1

2π

1

e8π Mω

l2P

(1− ω2M

) − 1dω . (36)

This completes the explicit derivation of the ω2 correction to the Hawk-ing formula from the time Fourier transform of the semiclassical modes.

An alternative way to derive the above result was given by Keski-Vakkuri and Kraus (Nucl.Phys. B491 (1997), arXiv:hep-th/9610045)where it is proven that for the βωk coefficient the imaginary part of theaction at the saddle point (29) is given by

Im

∫ r1

r0

pcdr = Im

∫ 2M

2H

pcdr = π1

2((2M)2 − (2H)2) = 4πMω(1 − ω

2M)

(37)which is equivalent to eq.(33). The importance of equation (37) is toshow directly how the “tunneling” is due only to the imaginary part ofthe “space part” of the action.

With regard to the validity of the expansion we see from the saddlepoint value (28,29)

A(ek

2M − 1)√2ME

e−t

4M ≈ ω

2M(38)

that the series if effectively an expansion in ω/M and thus expectedto hold for ω/M << 1. From eq.(38) we see that for a given ω, largevalues of the wave number k contribute at times t which grow like 2k.The typical ω for the radiation emitted by a black hole of mass M isaccording to eq.(34)

ω ≈ l2P8πM

(39)

and thus the approximation expected to be reliable at the typical fre-quency (39) or below for l2P /8πM2 << 1 i.e. for black holes of massof a few Planck masses or of higher mass. Higher order in the saddlepoint gives corrections O(l2P /M2).

The relation by Keski-Vakkuri and Kraus

2Im

∫ r1

r0

pcdr = Im

∫ 2M

2H

pcdr = π1

2((2M)2−(2H)2) = 8πMω(1− ω

2M)

(40)Gives rise to the tunneling picture similar to ordinary Q.M. (withoutback reaction corrections; geodesic motion of a massless particle in theexternal gravitational field) (many papers).

Conclusions

1. There are rather clean derivations of the Hawking spectrum in theexternal field treatment but the external field treatment violates energyconservation.

2. The semiclassical approach takes into account the energy conserva-tions and allows to compute through the convergent late-time expansionthe back reaction effects to order ω/M .

3. The late time expansion gives agrees with the result of Keski-Vakkuriand Kraus.

4. The derivation gives the back reaction effects and it is independentof the tunneling method: in the tunneling interpretation only the imag-

inary part of the “space part” of the action is relevant not the imaginarypart of the whole action.

Appendix

Objections to the simple formula

1) The above is not invariant under canonical transformations: oneshould use

1

2Im

∮

pcdr

but then factor 1/2 (Chowdhury)

2) In the name of covariance one should take into account also the“time-part” of the action (Akhmedova ...), but then the action

∫ ro

ri

(pcdr − Hdt)

becomes a constant which vanishes inside and thus is zero (Pizzi)

3) One should split the two contributions and take different prescrip-tions then

∫ ro

ri

(pdr − Hdt)

becomes the double as wanted (Akhmedov...) cured by taking 1/2 ∗∮

(Singleton...)

4) One should take different prescription in treating the two contribu-tion equivalent to neglect the time contribution (Zerbini, Vanzo ... )but not going to the 1/2 ∗

∮

otherwise one obtains 1/2.

The mode analysis is completely independent of the tunneling picture.

Units and order of magnitudes

Time, mass, energy, momentum, measured as lengths. Wave length

ω ≈ l2P8πM

λ

2π≈ 4πrS

Evaporation time

t =5120πG2M3

~c4(41)

lP = 1.62 10−35mFor the sun t = 5 1074s;for M = 200 000kg, t = 0.67s and T = 50 TeV ;for the Planck mass (= 1.21 1019 GeV/c2 = 2.1710−8kg) t = 8.6 10−40 s.

The Unruh modes

ds2 = −16M2e−r2

4M2 dUdV + r2dΩ2 (42)

On H− given by V = 0 ξ = ∂U is a Killing vector.

Lξgµν = 0. (43)

U = −4Me−u

4M V = 4Mev

4M (44)

Null coordinates

u = r + 2M(log r/2M − 1) − t v = r + 2M(log r/2M − 1) + t (45)

φ = eikU (46)

Shell dynamics

We summarize here the essential formulas of the shell dynamics. Onestarts from the usual Hilbert-Einstein action to which the shell actionis added

S =1

16πG

∫

R√−g d4x + Sshell. (47)

We shall in the following use c = G = 1 which simply means that massesacquire the dimension of length i.e. they are measured by the relatedSchwarzschild radius divided by 2. As usual in gravity it is better towork on a bounded region of space-time. Employing the general spher-ically symmetric metric (2) the action can be rewritten in Hamiltonianform as

S =

∫ tf

ti

dt

∫ re

ri

dr(πLL + πRR − NHt − NrHr) +

∫ tf

ti

dt (−NrπLL +NRR′

L)

∣

∣

∣

∣

re

ri

+

∫ tf

ti

dt p ˙r (48)

where r denotes the radial coordinate of the shell. The constraints are

given byHr = πRR′ − π′

LL − p δ(r − r), (49)

Ht =RR′′

L+

R′2

2L+

Lπ2L

2R2−RR′L′

L2− πLπR

R−L

2+√

p2L−2 + m2 δ(r− r).

(50)The Painleve-Gullstrand gauge is defined by L ≡ 1. There is still onegauge freedom in the choice of R(r). In virtue of the constraints R′(r)has to be discontinuous at r = r. Here we will adopt the “outer gauge”[8] defined by R(r) = r for r ≥ r i.e. in the massless case

R(r) = r +p

rg(r − r) (51)

with g smooth function of support [−l, 0], g(0) = 0 and g′(0−) = 1.Other gauges could well be used The constraints can be solved and theaction in the outer gauge takes the form

S =

∫ tf

ti

(

pc˙r − M(t)

∫ r(t)

ri

∂F

∂Mdr − HN(re) + MN(ri)

)

dt (52)

where F is the generating function

F = RW + RR′(L − B) (53)

with

W =

√

R′2 − 1 +2MR

, L = log(R′−W ), B =

√

2MR

+log

(

1−√

2MR

)

.

(54)The general expression of the conjugate momentum pc is [8]

pc = R(∆L − ∆B) (55)

where ∆ represents the discontinuity of the related quantities across theshell position r. Contrary to p, pc is a gauge invariant quantity withinthe Painleve class of gauges. Its expression for the case of a masslessshell is given by eq.(5). Normalizing the lapse function N , which isconstant for r > r, as N(re) = 1 we have from the expression (5) of pc

and action (4) the equation of motion

∂H

∂pc= 1 −

√

2H

r= ˙r (56)

c0H = (ek/m2 − 1)

A

exp(−A(A+4m)4m2

(57)

A =√

r1 − m (58)

References

[1] P. Kraus, F. Wilczek, Nucl.Phys. B433 (1995) 403, e-Print:arXiv:gr-qc/9408003

[2] P. Kraus, F. Wilczek, Nucl.Phys. B437 (1995) 231, e-Print:arXiv:hep-th/9411219

[3] S. Hawking, Comm.Math.Phys. 43 (1975) 199; Erratum-ibid.46(1976) 206

[4] W. G. Unruh, Phys.Rev. D14 (1976) 870

[5] K. Fredenhagen, R. Haag, Commun.Math.Phys. 127 (1990) 273

[6] W. Fischler, D. Morgan, J. Polchinski, Phys.Rev. D41 (1990) 2638;Phys.Rev. D42 (1990) 4042

[7] J. L. Friedman , J. Louko S. Winters-Hilt, Phys.Rev. D56 (1997)7674, e-Print: arXiv:gr-qc/9706051

[8] F. Fiamberti, P. Menotti, Nucl.Phys. B794 (2008) 512, e-Print:arXiv:0708.2868 [hep-th]

[9] P. Menotti, Class.Quant.Grav. 27:135008 (2010), e-Print:arXiv:0911.4358 [hep-th]

[10] P. Menotti, J.Phys.Conf.Ser. 222:012051 (2010), e-Print:arXiv:0912.3873 [gr-qc]

[11] E. Keski-Vakkuri, P. Kraus, Nucl.Phys. B491 (1997) 249, e-Print:arXiv:hep-th/9610045

[12] M. K. Parikh, F. Wilczek, Phys.Rev.Lett. 85 (2000) 5042, e-Print:hep-th/9907001

[13] K. Srinivasan, T. Padmanabhan, Phys.Rev. D60 (1999) 24007, e-Print: gr-qc/9812028

[14] B. Chowdhury, Pramana 70 (2008) 593; ibid 70 (2008) 3, e-Print:arXiv:hep-th/0605197; P. Mitra, Phys.Lett. B648 (2007) 240, e-Print: arXiv:hep-th/0611265

[15] M. Angheben, M. Nadalini, L. Vanzo, S. Zerbini, JHEP 0505:014(2005), e-Print: arXiv:hep-th/0503081; A.J.M. Medved, C.Vagenas, Mod.Phys.Lett. A20 (2005) 2449, e-Print: arXiv:gr-qc/0504113

[16] E. T. Akhmedov, V. Akhmedova, D. Singleton, Phys.Lett. B642(2006) 124, e-Print: arXiv:hep-th/0608098

[17] V. Akhmedova, T. Pilling, A. de Gill, D. Singleton, Phys.Lett.B666 (2008) 269, e-Print: arXiv:0804.2289 [hep-th]

[18] S. A. Hayward , R. Di Criscienzo, L. Vanzo, M. Nadalini,S. Zerbini, Class.Quant.Grav. 26:062001 (2009), e-Print:arXiv:0806.0014 [gr-qc]

[19] R. Di Criscienzo, S. A. Hayward, M. Nadalini, L. Vanzo, S. Zerbini,e-Print: arXiv:0906.1725 [gr-qc]

[20] S.A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini,AIP Conf.Proc.1122:145 (2009), e-Print: arXiv:0812.2534 [gr-qc];e-Print: arXiv:0909.2956 [gr-qc]

[21] M. Pizzi, e-Print: arXiv:0904.4572 [gr-qc]; e-Print:arXiv:0907.2020 [gr-qc]; e-Print: arXiv:0909.3800 [gr-qc]

[22] L. Vanzo, G. Acquaviva, R. Di Criscienzo, e-Print:arXiv:1106.4153 [gr-qc]

[23] V. I. Arnold, “Mathematical methods of classical mechanics”Springer, Berlin (1978)

[24] R. C. Gunning and H. Rossi “Analytic functions of several complexvariables” Prentice-Hall Inc., Englewood Cliffs, N.J. (1965)

[25] K. Deimling “Non linear functional analysys” Springer-VerlagBerlin Heidelberg New York Tokyo (1985)

[26] A. Erdelyi, “Asymptotic expansions” Dover Publications, Inc. NewYork (1956)

[27] R. Kerner and R.B. Mann, Phys. Rev. D 73 (2006) 104010, e-Print:arXiv:gr-qc/0603019