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Bekenstein–Hawking Entropy by energy quantization from Schwarzschild-de Sitter black hole
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Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole M. Atiqur Rahman, M. Jakir Hossain, M. Ilias Hossain PII: DOI: Reference:
S0927-6505(15)00095-X 10.1016/j.astropartphys.2015.05.005 ASTPHY 2052
To appear in:
Astroparticle Physics
Received date: Revised date: Accepted date:
26 March 2015 27 May 2015 31 May 2015
Please cite this article as: M. Atiqur Rahman, M. Jakir Hossain, M. Ilias Hossain, Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole, Astroparticle Physics (2015), doi: 10.1016/j.astropartphys.2015.05.005
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ACCEPTED MANUSCRIPT
Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole
Abstract
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M. Atiqur Rahman† Department of Applied Mathematics, Rajshahi University, Rajshahi - 6205, Bangladesh. M. Jakir Hossain,‡ and M. Ilias Hossain§ Department of Mathematics, Rajshahi University, Rajshahi - 6205, Bangladesh.
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The energy of a test particle orbiting a Schwarzschild-de Sitter (SdS) black hole is quantized with the help of the quantization of angular momentum. We have investigated the change of entropy between two nearby states using energy quantization. We have shown that the change of entropy as well as purely thermal emission rate is dependent of quantum number and approach to zero for large quantum number. Keywords: Energy Quantization, SdS black hole. E-mail: [email protected]†, mjakir− [email protected]‡, ilias− [email protected]§
Introduction
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and the gravitational acceleration, respectively. In this GEM, the source of magnetic field is consider Recently quantization of black holes is an important as the matter current density in accordance with issues for the researchers who deal with quantum the Biot-Savart law and is called the GSM magnetic gravity theory of physics [1]. For the quantization field which is a divergenceless quantity everywhere. of black holes there has been no satisfactory solution In this GEM analogy, Zee [16] has considered the exyet. Since the radius is the only one parameter for istence of a gravitipole following Dirac. Due to the the black holes having no charge and angular mohypothetical nature of the gravitipole one can splitmentum, for an observer it is not possible to observe ting the upper bound of energy without the quanwhat happen inside it from outside. One should tization effects on energy level splittings in atoms consider all the information are reserved on the surand molecules [16]. Also, by expanding the action face which is known as event horizon of black hole. of the test particle one can quantize the mass of the In 1973 Bekenstein [2] first considered the black text particle. hole horizon area as an adiabatic invariant quantity In the present work, we have used the method known as black hole entropy which is proportional of Simanek [19] to obtain the Hawking purely therto the area spectrum [3, 4, 5, 6, 7]. A wonderful mal emission rate as well as the Bekenstein-Hawking fact of black hole radiation [8, 9] have discovered by Entropy by quantizing the energy having the gravHawking in 1975 and several works have been done itational field of SdS black hole which is a nonto calculate this quantum effect [10]. Nowadays, the asymptotically flat and spherical symmetric linear radiation of black holes is called ‘Hawking radiadilaton black hole solution in 4-dimensions. We tion’. The entropic framework given in Ref. [11, 12] have quantized the energy of a test particle orbitindeed support new idea on quantum properties of ing SdS black hole from the quantization of angular gravity beyond classical physics. The quantization momentum. It is noted for the spherically symmetof gravity presented in this work, however, should ric black hole that the canonical formulation can be not be interpreted as only a support of quantization developed to study quantization by proposing a foof black hole as an entropic force [12, 13, 14], it also liation in the radial parameter because it is only a suggests a new way to unify gravity with quantum function of the Lagrangian coordinates. Although theory and therefore will lead to new understanding many works on Hawking radiation, entropic force, and perspectives on gravity of a black hole. higher dimensions have been done [20, 21, 22, 23, In 1931, it has been shown by Dirac that the ex24, 25, 26, 27, 28, 29], the quantization of a black istence of magnetic monopoles lead to quantization hole or it’s gravity is very limited. Up to date the of electric charge [15]. Similar to the Dirac theory problem of the quantization of the SdS black hole or Zee [16] has also presented a new gravitational anait’s gravity has not been studied, yet. So, we want log of Dirac quantization condition in 1985 which is to fill this gap in the literature. known as the theory of gravitoelectromagnetism(GEM) The remainder of the paper is arranged as fol[17, 18]. In this GEM analogy, the electric charge lows: Section 2 is devoted for calculating the Laand the electric field of Maxwell electromagnetic grangian and canonical momenta of a test particle theory play the role of the mass of the test particle of the schwarzschild-de sitter line element near
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the event horizon according to Simanek [19]. Effective potential for radial motion have been calculated in section 3. In section 4, we have quantized the circular orbit of the test particle moving around the black hole due to the fact that the azimuthal angle is a periodic function of time. The Hawking purely thermal emission rate has been derived by quantizing the energy of the SdS black hole gravity in section 5. Finally, we finish with the conclusions in Section 6.
2
Lagrangian and canonical momenta of test particle
2
ds
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In this section we have used the canonical formulation [30, 31, 32, 33] to quantize the SdS black hole gravity. Cavagli‘a et. al. [32, 33] have developed a canonical formulation deal with the spherically symmetric spacetime to quantize the Schwarzschild black hole by imposing a restriction in the radial parameter r due to the fact that the Lagrangian coordinates is only the functions of r. The solution of Einstein equations with a positive Λ(= 3/`2 ) term of Schwarzschild-de Sitter black hole can be configured as [34] −1 r2 2M r2 2M 2 2 dr2 + r2 (dθ2 + sin2 θdφ2 ), − 2 c dt + 1 − − 2 = − 1− r ` r `
(1)
rsds = 2M
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where M being the mass of the black hole, the coordinates are defined such that −∞ ≤ t ≤ ∞, r ≥ 0, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π. Note that this spacetime is not asymptotically flat. At large r, the metric (1) tends to the dS space limit. The SdS black hole horizons are described by the real positive roots of 1 `2 r (r − rsds )(r − rc )(r− − r) = 0 which is [34] 4M 2 1+ 2 +··· . `
(2)
The terms in the bracket is greater than one and indicate that SdS black hole radius is larger than Schwarzschild Black hole (rs = 2M ). For the simplicity we can rewrite the (1) of the following form −1 2M r3 r3 2M 2 2 = − 1− 1+ 1 + c dt + 1 − dr2 + r2 (dθ2 + sin2 θdφ2 ). (3) r 2M `2 r 2M `2
M
ds2
−1 4M 2 2M 4M 2 2M 1+ 2 c2 dt2 + 1 − 1+ 2 dr2 + r2 (dθ2 + sin2 θdφ2 ). = − 1− r ` r `
(4)
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ds2
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If we take the first approximation r0 = 2M then the above metric (3) can be written as
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Now, we consider a test particle of mass m orbiting along the circular geodesics in the equatorial plane around SdS black hole. Then according to the Ref. [35, 36] the metric given in Eq. (4) can be taken of the following form ds2
=
−1 2M 4M 2 2M 4M 2 − 1− 1+ 2 c2 dt2 + 1 − 1+ 2 dr2 + r2 dφ2 . r ` r `
(5)
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We assume that the black hole mass M is larger than the Planck mass so that the Compton radius, rc = h ¯ /M c is very smallest then SdS black hole rsds . In this situation the quantum fluctuations of the black hole disregards [37]. One can define the Lagrangian of the test particle in terms of the metric components gij as " # −1 m m 2M 4M 2 2M 4M 2 i j 2 ˙2 2 2 2 ˙2 2 ˙2 £ = − gij x˙ x˙ = − 1− 1+ 2 c t + 1− 1+ 2 r˙ + r sin θφ + r θ . 2 2 r ` r ` (6) Since the SdS spacetime is static and spherically symmetric, there exist two constants of motion for the test particles, associated with two Killing vectors as 1 ∂£ 2M 4M 2 ˙ = mc2 1 − 1+ 2 t, (7) E= 2 ∂ t˙ r ` L=
1 ∂£ ˙ = mr2 sin2 θφ, 2 ∂ φ˙
2
(8)
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where E and L denote corresponding specific energy and the specific angular momentum of the orbiting test particle, respectively. With the help of the canonical momenta defined by pα = ∂∂£ the other two xα ˙ components can be written as −1 2M 4M 2 ∂£ r, ˙ = −m 1 − 1+ 2 ∂ r˙ r `
(9)
∂£ ˙ = −mr2 θ. ∂ θ˙
(10)
pθ =
3
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pr =
Radial motion and Effective Potential
The radial motion of a geodesic can be written as
g 00 p20 + g rr p2r + g φφ p2φ + g θθ p2θ + m2 c2 = 0,
(11)
the four-vector (p0 = E/c, p) describe the magnitude of the energy-momentum. Inserting Eqs (7-10) we have [19]
2M 1− r
−1 −1 m2 4M 2 2M 4M 2 L2 2 1+ 2 +m 1− 1+ 2 r˙ 2 + 2 2 + 2 θ˙2 + m2 c2 = 0. (12) ` r ` r r sin θ
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E2 − 2 c
When θ˙2 = 0, and sin2 θ = 1 then the above Eq. (12) becomes −
E2 c2
1−
2M r
4M 2 `2
1+
−1
−1 2M 4M 2 L2 + m2 1 − 1+ 2 r˙ 2 + 2 + m2 c2 = 0. r ` r
(13)
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We introduce the energy and momentum of the test particle per unit rest mass as given in Ref. [19] of the form ˜ = E ,L ˜= L E (14) m m and using this into Eq. (13) we get " # −1 −1 ˜2 ˜2 E 4M 2 4M 2 2M 1 2M L 2 2 2 m c − 4 1− 1+ 2 1+ 2 (15) + 2 1− r˙ + 2 2 + 1 = 0. c r ` c r ` c r
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The radial motion of the test particle can be obtained from the above equation of the form ! ˜2 ˜2 r˙ 2 E 1 L 2M 4M 2 2 = 2− +c 1− 1+ 2 . 2 2c 2 r2 r `
(16)
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For the time like particle orbit here the velocity is described by two parameters the energy and the angular momentum. Note that here r˙ is different from and so-called the effective potential Vef f . Therefore, the effective potential for the radial motion can be written as ! ˜2 4M 2 1 L 2M 2 1 + Vef f = + c 1 − . (17) 2 r2 r `2 The radial acceleration of the test particle can be derive by taking the derivative of Eq. (17) with respect ∂Vef f to the proper time and taking ∂r = 0 the maximum potential can be obtained as c2 M r4
1+
4M 2 `2
(r2 −
c2 M
˜2 L 1+
4M 2 `2
r +
˜2 3L ) = 0. c2
(18)
Equation (18) is a quadratic equation and gives the two roots of the form ˜2 L R± = 2c2 M 1 +
˜2 L ± 4M 2 2c2 M 1 + `2 3
4M 2 `2
!2
21 2 ˜ 3L − 2 , c
(19)
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where R is the radius of the circular orbit. We can write the Eq. (19) in an alternative form as 2 12 4M 2 2 2 12c M 1 + `2 ˜2 L 1 ± 1 − R± = 2 . 4M 2 2 ˜ 2c M 1 + L
(20)
`2
˜ 2 ≥ 12c2 M 2 1 + It is clear from the Eq. (20) that R± is real only when L
4M 2 `2
2
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. For smallest stable 2 ˜ 2 = 12c2 M 2 1 + 4M2 2 . orbit, the square root on the right hand side of Eq. (20) vanishes, hence we have L ` 2 2 4M 2 4M 2 2 2 2 2 2 2 ˜ ˜ We also correspond the conditions L ≥ 12c M 1 + `2 and L >> 12c M 1 + `2 for large and largest stable circular orbits, respectively.
Quantization of Circular Orbit
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According to the Wilson [38] and Sommerfeld [39] openion angular momentum can be quantized as a periodic function of time and help to quantize energy because it is closely related to the quantize angular momentum of the orbiting test particle. For the periodic motion we can quantized angular momentum Jφ with the help of canonical momentum L conjugate to the angular variable of the form Z 2π Jφ = Ldφ = nh. (21) 0
Since L is a constant of motion, Eq. (14) gives the quantization condition for the angular momentum of the form ˜ 0 = n0 /m¯ L h.
so that
M
˜ = n¯h, L = mL
(22)
As mention in the previous section, we have for smallest circular orbits
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2 2 ˜ 2 = 12c2 M 2 1 + 4M L , `2
(23)
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which can be written with the help of Eq. (22) interms of n0 as n20 =
12M 2 1 +
4M 2 `2
rc2
2
,
(24)
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h ¯ where rc is the Compton radius of the test particle defined by mc . The radius of the different stable circular orbit of the particle corresponds to n0 can be obtained by using Eq. (22) into Eq. (20) of the form 1
rc2 R± = n20 2M 1 +
12M 2 1 + 4M `2 1 + 1 − 2 n2 4M 2 r c 0 `2
2
2
2
.
(25)
Using Eq. (24) into Eq. (25), the radius of the first circular orbit denoted by R0 can be approximated as R0 ≈ n20
rc2 2M 1 + r2
4M 2 `2
,
(26)
in the limiting case when ` → ∞ this becomes R0 = n20 rcs and agree with the result given in Ref. [19], where rs = 2M is the Schwarzschild radius. The position of the next higher circular orbit R1 can be obtained from Eq. (25) by replacing n1 = n0 + 1 of the form 1 2
rc2 R1 = (n0 + 1)2 2M 1 +
12M 2 1 + 4M `2 1 + 1 − 2 4M 2 rc (n0 + 1)2 `2 4
2
2
.
(27)
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For simplicity, we consider 2M 1 + n20 [1 +
2 n0 ]
4M 2 `2
>> rc so that n0 >> 1. We therefore can be written (n0 + 1)2 ≈
and using this into Eq. (27) we get
R1 = n20 (1 +
2 rc2 ) n0 2M 1 +
4M 2 `2
2
12M 1 + 1 − 1 + rc2 n20 (1 +
2 12 .
4M 2 `2 2 2 n0 )
(28)
Therefore, Eq. (28) reduced to
r 2 2 R1 = R0 1 + 1+ . n0 n0
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The term in the first bracket can be approximated with the help of Eq. (24) of the form 2 21 r 4M 2 1/2 2 12M 1 + 2 ` 1 2 1 + 1 − = 1 + ≈ 1 + 1 − . (1 + 2/n0 )2 n0 rc2 n20 (1 + n20 )2
(29)
(30)
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In the similar fashion, the radius of the next higher circular orbit R2 of the particle can be calculated from Eq. (25) of the form q r 1 + n40 1 + n40 4 4 q . R2 = R0 1 + 1+ = R1 (31) n0 n0 1 + n20 1 + n20
(32)
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Finally, in general form the radius of the stable circular orbits of the particle can be written as q 2n+2 1 + 1 + 2n+2 n0 n0 q , Rn+1 = Rn 2n 2n 1 + n0 1 + n0
Energy quantization and Hawking Radiation
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5
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which means that gravity of the SdS black holes can be quantized in discrete states with radius Rn+1 . When n0 → ∞ we have Rn+1 = Rn . Therefore, we may conclude that for large quantum number two nearby states coincide.
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In this section, we have quantized the energy of the orbiting test particle with the help of angular momentum because the quantization of energy is closely related to the quantization of the angular momentum. Equation (16) gives the energy for zero velocity at r = R ! 4M 2 2M 1 + 2 ˜ 2 ` ˜ 2 = c2 L + c2 1 − . E (33) R2 R
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But we have from Eq. (18) at r = R
˜2 L = R2
Therefore, Eq. (33) can be rewritten as E2 ˜2 =E m2
=
=
c2 2R 2 2M 1+ 4M2 `
2M 1 +
4M 2 `2
. −3
(34)
+ c2 , R R − 3 2 M 1+ 4M2 ` 2 2M 1+ 4M2 ` 4M 2 1 − 2M 1 + `2 R , m2 c4 1 − 2 3M 1+ 4M2 R ` 1− R
c2 1 −
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c2
(35)
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and after some calculations gives 2 2M 1+ 4M2 ` 1− R E = mc2 1/2 . 1−
M 1+ 4M2
If we neglect the terms corresponding to 4th and the higher power of (M/`) of the SdS black hole radius given in Eq. (2) then we have 4M 2 . (45) rsds ≈ 2M 1 + 2 `
(36)
2
`
2R
Therefore, the Eq. (44) can be written for SdS black hole 2 mc2 rsds 1 1 δE = − . (46) 8rc2 n2 (n + 1)2
which gives with the help of Eq. (23) for the circular orbits corresponding to n0 >> 1 2 12M 1 + 4M `2 << 1. (38) R
It is clear that δE decreases with the increase of n. When n → ∞ we have δE → 0. Therefore, for a larger circular orbit the change of energy between two nearby states approaches to zero. In classical thermodynamics one can defined the change in the entropy for a given temperature as dS(E) = dE/T (E). Therefore, the Bekenstein-Hawking Entropy of SdS black hole can be written with the help of Eq. (46) of the form
For large values of n, the bracket can be replaced by 2/n3 so that
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Substituting Eq. (27) into Eq. (39) we have 2 4M 2 2 2 c M 1 + `2 E ≈ mc2 1 − . 2 ˜ 2L
(39)
(40)
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(42)
The energy corresponding to n+1 label can be written from Eq. (42) as 2 4M 2 2 M 1 + 2 ` (43) En+1 ≈ mc2 1 − . 2rc2 (n + 1)2
= ≈
δS
= =
δE TH 2 πmc2 rsds 1 1 − , 4κrc2 n2 (n + 1)2
Γ ∼ exp(−2Im(I))
Thus, from the quantized energy formula we get the energy difference between two nearby states of the form δE
(47)
(48)
κ where TH = 2π is Hawking temperature and κ is the surface gravity of SdS black hole. Thus, for larger circular orbit (large distance from black hole) the change of entropy between two nearby states approaches to zero and possesses low energy continuum behavior. Also, the temperature approaches to zero for this orbit. This means that between two nearby states by transition a black hole can absorb a particle with small energy or between far-away states a big one cab be absorbed. A mini black hole can only be emitted or absorbed particles with finite energy and definite quantum numbers like the atomic spectrum of quantum mechanism. According to the WKB approximation, the emission rate for an outgoing particle with positive energy E coming from inside Rin (M ) to outside Rout (M − E) of a circular orbit can be related to the imaginary part of the particle’s action Im(I) as
The quantized energy En for n energy label can be obtained as 2 4M 2 2 2 c M 1 + 2 ` En ≈ mc2 1 − (41) , 2 ˜ 2Ln which with the help of Eq. (23) becomes 2 4M 2 2 M 1 + 2 ` En ≈ mc2 1 − . 2n2 rc2
2 c4 m3 rsds 2 3 . 4¯h n
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δE ≈
M
Therefore, Eq. (36) can be approximated to 2 M 1 + 4M 2 ` . E = mc2 1 − 2R
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To simplify the above Eq. (36), we recall the Eq. (20) and setting the mechanical stability condition ˜ 2 = M GR. The second term in the parenthesis of L the Eq. (20) can be written as 2 2 4M 2 2M 1 + 12c2 M 2 1 + 4M 2 2 ` ` 2GM = 3c2 2 ˜2 c M GR L 2 12M 1 + 4M `2 = , (37) R
(49)
where Im(I) = − 12 [S(M − E) − S(M )] = − 12 δS. Therefore, for the SdS black hole the thermal emission rate and the entropy change satisfies 2 πmc2 rsds 1 1 − . (50) Γ ∼ exp(δS) = exp 4κrc2 n2 (n + 1)2
En+1 − En 2 mc2 1 4M 2 1 2 − M 1 + 2 (44) . 2rc2 n2 (n + 1)2 ` 6
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During the evaporation process the SdS black hole loses it’s mass M due to emitted energy and at the final state of the evaporation, energy (mass) goes to M κ = 2πR zero so that S → 0. Also, TH = 2π 2 → 0 n at M → 0. Therefore, we have S, TH (M → 0) → 0, which agree with the result obtained by Sakalli et al. [25]. When ` → ∞ the result reduced for Schwarzschild black hole.
[16] A. Zee; Phys. Rev. Lett. 55, 2379 (1985).
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[19] E. Simanek; arXiv:1209.3791v1 [gr-qc].
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H. Pasaoglu, I. Sakalli; Int. J. Theor. Phys. 48, 3517 (2009), arXiv:0910.1198. S.H. Mazharimousavi, I. Sakalli and M. Halilsoy; Phys. Lett. B 672, 177, arXiv:0902.0666. S.H. Mazharimousavi, M. Halilsoy, I. Sakalli and O. Gurtug; Class. Quantum Grav. 27, 105005 (2010), arXiv:0908.3113.
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[20] In summary, we have calculated the change of entropy for two nearby circular orbit around Schwarzschildde Sitter Black Hole by Energy Quantization pro[21] cess. The present work shows that the different energy labels of black hole in the nature can be performed in the same way as that for the electron oc[22] cupy different energy labels outside the atom like quantum theory which is interesting [40]. The entropic framework given in this work indeed support [23] the new perspectives on quantum properties of gravity given in Refs. [11, 12] beyond classical physics, however, suggests a new idea to unify gravity with [24] quantum theory.
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Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole M. Atiqur Rahman, M. Jakir Hossain, M. Ilias Hossain PII: DOI: Reference:
S0927-6505(15)00095-X 10.1016/j.astropartphys.2015.05.005 ASTPHY 2052
To appear in:
Astroparticle Physics
Received date: Revised date: Accepted date:
26 March 2015 27 May 2015 31 May 2015
Please cite this article as: M. Atiqur Rahman, M. Jakir Hossain, M. Ilias Hossain, Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole, Astroparticle Physics (2015), doi: 10.1016/j.astropartphys.2015.05.005
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Bekenstein-Hawking Entropy by Energy Quantization from Schwarzschild-de Sitter Black Hole
Abstract
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M. Atiqur Rahman† Department of Applied Mathematics, Rajshahi University, Rajshahi - 6205, Bangladesh. M. Jakir Hossain,‡ and M. Ilias Hossain§ Department of Mathematics, Rajshahi University, Rajshahi - 6205, Bangladesh.
1
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The energy of a test particle orbiting a Schwarzschild-de Sitter (SdS) black hole is quantized with the help of the quantization of angular momentum. We have investigated the change of entropy between two nearby states using energy quantization. We have shown that the change of entropy as well as purely thermal emission rate is dependent of quantum number and approach to zero for large quantum number. Keywords: Energy Quantization, SdS black hole. E-mail: [email protected]†, mjakir− [email protected]‡, ilias− [email protected]§
Introduction
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and the gravitational acceleration, respectively. In this GEM, the source of magnetic field is consider Recently quantization of black holes is an important as the matter current density in accordance with issues for the researchers who deal with quantum the Biot-Savart law and is called the GSM magnetic gravity theory of physics [1]. For the quantization field which is a divergenceless quantity everywhere. of black holes there has been no satisfactory solution In this GEM analogy, Zee [16] has considered the exyet. Since the radius is the only one parameter for istence of a gravitipole following Dirac. Due to the the black holes having no charge and angular mohypothetical nature of the gravitipole one can splitmentum, for an observer it is not possible to observe ting the upper bound of energy without the quanwhat happen inside it from outside. One should tization effects on energy level splittings in atoms consider all the information are reserved on the surand molecules [16]. Also, by expanding the action face which is known as event horizon of black hole. of the test particle one can quantize the mass of the In 1973 Bekenstein [2] first considered the black text particle. hole horizon area as an adiabatic invariant quantity In the present work, we have used the method known as black hole entropy which is proportional of Simanek [19] to obtain the Hawking purely therto the area spectrum [3, 4, 5, 6, 7]. A wonderful mal emission rate as well as the Bekenstein-Hawking fact of black hole radiation [8, 9] have discovered by Entropy by quantizing the energy having the gravHawking in 1975 and several works have been done itational field of SdS black hole which is a nonto calculate this quantum effect [10]. Nowadays, the asymptotically flat and spherical symmetric linear radiation of black holes is called ‘Hawking radiadilaton black hole solution in 4-dimensions. We tion’. The entropic framework given in Ref. [11, 12] have quantized the energy of a test particle orbitindeed support new idea on quantum properties of ing SdS black hole from the quantization of angular gravity beyond classical physics. The quantization momentum. It is noted for the spherically symmetof gravity presented in this work, however, should ric black hole that the canonical formulation can be not be interpreted as only a support of quantization developed to study quantization by proposing a foof black hole as an entropic force [12, 13, 14], it also liation in the radial parameter because it is only a suggests a new way to unify gravity with quantum function of the Lagrangian coordinates. Although theory and therefore will lead to new understanding many works on Hawking radiation, entropic force, and perspectives on gravity of a black hole. higher dimensions have been done [20, 21, 22, 23, In 1931, it has been shown by Dirac that the ex24, 25, 26, 27, 28, 29], the quantization of a black istence of magnetic monopoles lead to quantization hole or it’s gravity is very limited. Up to date the of electric charge [15]. Similar to the Dirac theory problem of the quantization of the SdS black hole or Zee [16] has also presented a new gravitational anait’s gravity has not been studied, yet. So, we want log of Dirac quantization condition in 1985 which is to fill this gap in the literature. known as the theory of gravitoelectromagnetism(GEM) The remainder of the paper is arranged as fol[17, 18]. In this GEM analogy, the electric charge lows: Section 2 is devoted for calculating the Laand the electric field of Maxwell electromagnetic grangian and canonical momenta of a test particle theory play the role of the mass of the test particle of the schwarzschild-de sitter line element near
1
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the event horizon according to Simanek [19]. Effective potential for radial motion have been calculated in section 3. In section 4, we have quantized the circular orbit of the test particle moving around the black hole due to the fact that the azimuthal angle is a periodic function of time. The Hawking purely thermal emission rate has been derived by quantizing the energy of the SdS black hole gravity in section 5. Finally, we finish with the conclusions in Section 6.
2
Lagrangian and canonical momenta of test particle
2
ds
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In this section we have used the canonical formulation [30, 31, 32, 33] to quantize the SdS black hole gravity. Cavagli‘a et. al. [32, 33] have developed a canonical formulation deal with the spherically symmetric spacetime to quantize the Schwarzschild black hole by imposing a restriction in the radial parameter r due to the fact that the Lagrangian coordinates is only the functions of r. The solution of Einstein equations with a positive Λ(= 3/`2 ) term of Schwarzschild-de Sitter black hole can be configured as [34] −1 r2 2M r2 2M 2 2 dr2 + r2 (dθ2 + sin2 θdφ2 ), − 2 c dt + 1 − − 2 = − 1− r ` r `
(1)
rsds = 2M
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where M being the mass of the black hole, the coordinates are defined such that −∞ ≤ t ≤ ∞, r ≥ 0, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π. Note that this spacetime is not asymptotically flat. At large r, the metric (1) tends to the dS space limit. The SdS black hole horizons are described by the real positive roots of 1 `2 r (r − rsds )(r − rc )(r− − r) = 0 which is [34] 4M 2 1+ 2 +··· . `
(2)
The terms in the bracket is greater than one and indicate that SdS black hole radius is larger than Schwarzschild Black hole (rs = 2M ). For the simplicity we can rewrite the (1) of the following form −1 2M r3 r3 2M 2 2 = − 1− 1+ 1 + c dt + 1 − dr2 + r2 (dθ2 + sin2 θdφ2 ). (3) r 2M `2 r 2M `2
M
ds2
−1 4M 2 2M 4M 2 2M 1+ 2 c2 dt2 + 1 − 1+ 2 dr2 + r2 (dθ2 + sin2 θdφ2 ). = − 1− r ` r `
(4)
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ds2
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If we take the first approximation r0 = 2M then the above metric (3) can be written as
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Now, we consider a test particle of mass m orbiting along the circular geodesics in the equatorial plane around SdS black hole. Then according to the Ref. [35, 36] the metric given in Eq. (4) can be taken of the following form ds2
=
−1 2M 4M 2 2M 4M 2 − 1− 1+ 2 c2 dt2 + 1 − 1+ 2 dr2 + r2 dφ2 . r ` r `
(5)
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We assume that the black hole mass M is larger than the Planck mass so that the Compton radius, rc = h ¯ /M c is very smallest then SdS black hole rsds . In this situation the quantum fluctuations of the black hole disregards [37]. One can define the Lagrangian of the test particle in terms of the metric components gij as " # −1 m m 2M 4M 2 2M 4M 2 i j 2 ˙2 2 2 2 ˙2 2 ˙2 £ = − gij x˙ x˙ = − 1− 1+ 2 c t + 1− 1+ 2 r˙ + r sin θφ + r θ . 2 2 r ` r ` (6) Since the SdS spacetime is static and spherically symmetric, there exist two constants of motion for the test particles, associated with two Killing vectors as 1 ∂£ 2M 4M 2 ˙ = mc2 1 − 1+ 2 t, (7) E= 2 ∂ t˙ r ` L=
1 ∂£ ˙ = mr2 sin2 θφ, 2 ∂ φ˙
2
(8)
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where E and L denote corresponding specific energy and the specific angular momentum of the orbiting test particle, respectively. With the help of the canonical momenta defined by pα = ∂∂£ the other two xα ˙ components can be written as −1 2M 4M 2 ∂£ r, ˙ = −m 1 − 1+ 2 ∂ r˙ r `
(9)
∂£ ˙ = −mr2 θ. ∂ θ˙
(10)
pθ =
3
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pr =
Radial motion and Effective Potential
The radial motion of a geodesic can be written as
g 00 p20 + g rr p2r + g φφ p2φ + g θθ p2θ + m2 c2 = 0,
(11)
the four-vector (p0 = E/c, p) describe the magnitude of the energy-momentum. Inserting Eqs (7-10) we have [19]
2M 1− r
−1 −1 m2 4M 2 2M 4M 2 L2 2 1+ 2 +m 1− 1+ 2 r˙ 2 + 2 2 + 2 θ˙2 + m2 c2 = 0. (12) ` r ` r r sin θ
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E2 − 2 c
When θ˙2 = 0, and sin2 θ = 1 then the above Eq. (12) becomes −
E2 c2
1−
2M r
4M 2 `2
1+
−1
−1 2M 4M 2 L2 + m2 1 − 1+ 2 r˙ 2 + 2 + m2 c2 = 0. r ` r
(13)
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M
We introduce the energy and momentum of the test particle per unit rest mass as given in Ref. [19] of the form ˜ = E ,L ˜= L E (14) m m and using this into Eq. (13) we get " # −1 −1 ˜2 ˜2 E 4M 2 4M 2 2M 1 2M L 2 2 2 m c − 4 1− 1+ 2 1+ 2 (15) + 2 1− r˙ + 2 2 + 1 = 0. c r ` c r ` c r
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The radial motion of the test particle can be obtained from the above equation of the form ! ˜2 ˜2 r˙ 2 E 1 L 2M 4M 2 2 = 2− +c 1− 1+ 2 . 2 2c 2 r2 r `
(16)
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For the time like particle orbit here the velocity is described by two parameters the energy and the angular momentum. Note that here r˙ is different from and so-called the effective potential Vef f . Therefore, the effective potential for the radial motion can be written as ! ˜2 4M 2 1 L 2M 2 1 + Vef f = + c 1 − . (17) 2 r2 r `2 The radial acceleration of the test particle can be derive by taking the derivative of Eq. (17) with respect ∂Vef f to the proper time and taking ∂r = 0 the maximum potential can be obtained as c2 M r4
1+
4M 2 `2
(r2 −
c2 M
˜2 L 1+
4M 2 `2
r +
˜2 3L ) = 0. c2
(18)
Equation (18) is a quadratic equation and gives the two roots of the form ˜2 L R± = 2c2 M 1 +
˜2 L ± 4M 2 2c2 M 1 + `2 3
4M 2 `2
!2
21 2 ˜ 3L − 2 , c
(19)
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where R is the radius of the circular orbit. We can write the Eq. (19) in an alternative form as 2 12 4M 2 2 2 12c M 1 + `2 ˜2 L 1 ± 1 − R± = 2 . 4M 2 2 ˜ 2c M 1 + L
(20)
`2
˜ 2 ≥ 12c2 M 2 1 + It is clear from the Eq. (20) that R± is real only when L
4M 2 `2
2
4
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. For smallest stable 2 ˜ 2 = 12c2 M 2 1 + 4M2 2 . orbit, the square root on the right hand side of Eq. (20) vanishes, hence we have L ` 2 2 4M 2 4M 2 2 2 2 2 2 2 ˜ ˜ We also correspond the conditions L ≥ 12c M 1 + `2 and L >> 12c M 1 + `2 for large and largest stable circular orbits, respectively.
Quantization of Circular Orbit
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According to the Wilson [38] and Sommerfeld [39] openion angular momentum can be quantized as a periodic function of time and help to quantize energy because it is closely related to the quantize angular momentum of the orbiting test particle. For the periodic motion we can quantized angular momentum Jφ with the help of canonical momentum L conjugate to the angular variable of the form Z 2π Jφ = Ldφ = nh. (21) 0
Since L is a constant of motion, Eq. (14) gives the quantization condition for the angular momentum of the form ˜ 0 = n0 /m¯ L h.
so that
M
˜ = n¯h, L = mL
(22)
As mention in the previous section, we have for smallest circular orbits
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2 2 ˜ 2 = 12c2 M 2 1 + 4M L , `2
(23)
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which can be written with the help of Eq. (22) interms of n0 as n20 =
12M 2 1 +
4M 2 `2
rc2
2
,
(24)
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h ¯ where rc is the Compton radius of the test particle defined by mc . The radius of the different stable circular orbit of the particle corresponds to n0 can be obtained by using Eq. (22) into Eq. (20) of the form 1
rc2 R± = n20 2M 1 +
12M 2 1 + 4M `2 1 + 1 − 2 n2 4M 2 r c 0 `2
2
2
2
.
(25)
Using Eq. (24) into Eq. (25), the radius of the first circular orbit denoted by R0 can be approximated as R0 ≈ n20
rc2 2M 1 + r2
4M 2 `2
,
(26)
in the limiting case when ` → ∞ this becomes R0 = n20 rcs and agree with the result given in Ref. [19], where rs = 2M is the Schwarzschild radius. The position of the next higher circular orbit R1 can be obtained from Eq. (25) by replacing n1 = n0 + 1 of the form 1 2
rc2 R1 = (n0 + 1)2 2M 1 +
12M 2 1 + 4M `2 1 + 1 − 2 4M 2 rc (n0 + 1)2 `2 4
2
2
.
(27)
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For simplicity, we consider 2M 1 + n20 [1 +
2 n0 ]
4M 2 `2
>> rc so that n0 >> 1. We therefore can be written (n0 + 1)2 ≈
and using this into Eq. (27) we get
R1 = n20 (1 +
2 rc2 ) n0 2M 1 +
4M 2 `2
2
12M 1 + 1 − 1 + rc2 n20 (1 +
2 12 .
4M 2 `2 2 2 n0 )
(28)
Therefore, Eq. (28) reduced to
r 2 2 R1 = R0 1 + 1+ . n0 n0
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The term in the first bracket can be approximated with the help of Eq. (24) of the form 2 21 r 4M 2 1/2 2 12M 1 + 2 ` 1 2 1 + 1 − = 1 + ≈ 1 + 1 − . (1 + 2/n0 )2 n0 rc2 n20 (1 + n20 )2
(29)
(30)
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In the similar fashion, the radius of the next higher circular orbit R2 of the particle can be calculated from Eq. (25) of the form q r 1 + n40 1 + n40 4 4 q . R2 = R0 1 + 1+ = R1 (31) n0 n0 1 + n20 1 + n20
(32)
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Finally, in general form the radius of the stable circular orbits of the particle can be written as q 2n+2 1 + 1 + 2n+2 n0 n0 q , Rn+1 = Rn 2n 2n 1 + n0 1 + n0
Energy quantization and Hawking Radiation
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5
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which means that gravity of the SdS black holes can be quantized in discrete states with radius Rn+1 . When n0 → ∞ we have Rn+1 = Rn . Therefore, we may conclude that for large quantum number two nearby states coincide.
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In this section, we have quantized the energy of the orbiting test particle with the help of angular momentum because the quantization of energy is closely related to the quantization of the angular momentum. Equation (16) gives the energy for zero velocity at r = R ! 4M 2 2M 1 + 2 ˜ 2 ` ˜ 2 = c2 L + c2 1 − . E (33) R2 R
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But we have from Eq. (18) at r = R
˜2 L = R2
Therefore, Eq. (33) can be rewritten as E2 ˜2 =E m2
=
=
c2 2R 2 2M 1+ 4M2 `
2M 1 +
4M 2 `2
. −3
(34)
+ c2 , R R − 3 2 M 1+ 4M2 ` 2 2M 1+ 4M2 ` 4M 2 1 − 2M 1 + `2 R , m2 c4 1 − 2 3M 1+ 4M2 R ` 1− R
c2 1 −
5
c2
(35)
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and after some calculations gives 2 2M 1+ 4M2 ` 1− R E = mc2 1/2 . 1−
M 1+ 4M2
If we neglect the terms corresponding to 4th and the higher power of (M/`) of the SdS black hole radius given in Eq. (2) then we have 4M 2 . (45) rsds ≈ 2M 1 + 2 `
(36)
2
`
2R
Therefore, the Eq. (44) can be written for SdS black hole 2 mc2 rsds 1 1 δE = − . (46) 8rc2 n2 (n + 1)2
which gives with the help of Eq. (23) for the circular orbits corresponding to n0 >> 1 2 12M 1 + 4M `2 << 1. (38) R
It is clear that δE decreases with the increase of n. When n → ∞ we have δE → 0. Therefore, for a larger circular orbit the change of energy between two nearby states approaches to zero. In classical thermodynamics one can defined the change in the entropy for a given temperature as dS(E) = dE/T (E). Therefore, the Bekenstein-Hawking Entropy of SdS black hole can be written with the help of Eq. (46) of the form
For large values of n, the bracket can be replaced by 2/n3 so that
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Substituting Eq. (27) into Eq. (39) we have 2 4M 2 2 2 c M 1 + `2 E ≈ mc2 1 − . 2 ˜ 2L
(39)
(40)
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(42)
The energy corresponding to n+1 label can be written from Eq. (42) as 2 4M 2 2 M 1 + 2 ` (43) En+1 ≈ mc2 1 − . 2rc2 (n + 1)2
= ≈
δS
= =
δE TH 2 πmc2 rsds 1 1 − , 4κrc2 n2 (n + 1)2
Γ ∼ exp(−2Im(I))
Thus, from the quantized energy formula we get the energy difference between two nearby states of the form δE
(47)
(48)
κ where TH = 2π is Hawking temperature and κ is the surface gravity of SdS black hole. Thus, for larger circular orbit (large distance from black hole) the change of entropy between two nearby states approaches to zero and possesses low energy continuum behavior. Also, the temperature approaches to zero for this orbit. This means that between two nearby states by transition a black hole can absorb a particle with small energy or between far-away states a big one cab be absorbed. A mini black hole can only be emitted or absorbed particles with finite energy and definite quantum numbers like the atomic spectrum of quantum mechanism. According to the WKB approximation, the emission rate for an outgoing particle with positive energy E coming from inside Rin (M ) to outside Rout (M − E) of a circular orbit can be related to the imaginary part of the particle’s action Im(I) as
The quantized energy En for n energy label can be obtained as 2 4M 2 2 2 c M 1 + 2 ` En ≈ mc2 1 − (41) , 2 ˜ 2Ln which with the help of Eq. (23) becomes 2 4M 2 2 M 1 + 2 ` En ≈ mc2 1 − . 2n2 rc2
2 c4 m3 rsds 2 3 . 4¯h n
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δE ≈
M
Therefore, Eq. (36) can be approximated to 2 M 1 + 4M 2 ` . E = mc2 1 − 2R
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To simplify the above Eq. (36), we recall the Eq. (20) and setting the mechanical stability condition ˜ 2 = M GR. The second term in the parenthesis of L the Eq. (20) can be written as 2 2 4M 2 2M 1 + 12c2 M 2 1 + 4M 2 2 ` ` 2GM = 3c2 2 ˜2 c M GR L 2 12M 1 + 4M `2 = , (37) R
(49)
where Im(I) = − 12 [S(M − E) − S(M )] = − 12 δS. Therefore, for the SdS black hole the thermal emission rate and the entropy change satisfies 2 πmc2 rsds 1 1 − . (50) Γ ∼ exp(δS) = exp 4κrc2 n2 (n + 1)2
En+1 − En 2 mc2 1 4M 2 1 2 − M 1 + 2 (44) . 2rc2 n2 (n + 1)2 ` 6
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During the evaporation process the SdS black hole loses it’s mass M due to emitted energy and at the final state of the evaporation, energy (mass) goes to M κ = 2πR zero so that S → 0. Also, TH = 2π 2 → 0 n at M → 0. Therefore, we have S, TH (M → 0) → 0, which agree with the result obtained by Sakalli et al. [25]. When ` → ∞ the result reduced for Schwarzschild black hole.
[16] A. Zee; Phys. Rev. Lett. 55, 2379 (1985).
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[19] E. Simanek; arXiv:1209.3791v1 [gr-qc].
[17] I. Ciufolini, A.J. Wheeler; Gravitation and Inertia, (Princeton University Press, Princeton, 1995).
Concluding Remarks
H. Pasaoglu, I. Sakalli; Int. J. Theor. Phys. 48, 3517 (2009), arXiv:0910.1198. S.H. Mazharimousavi, I. Sakalli and M. Halilsoy; Phys. Lett. B 672, 177, arXiv:0902.0666. S.H. Mazharimousavi, M. Halilsoy, I. Sakalli and O. Gurtug; Class. Quantum Grav. 27, 105005 (2010), arXiv:0908.3113.
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[20] In summary, we have calculated the change of entropy for two nearby circular orbit around Schwarzschildde Sitter Black Hole by Energy Quantization pro[21] cess. The present work shows that the different energy labels of black hole in the nature can be performed in the same way as that for the electron oc[22] cupy different energy labels outside the atom like quantum theory which is interesting [40]. The entropic framework given in this work indeed support [23] the new perspectives on quantum properties of gravity given in Refs. [11, 12] beyond classical physics, however, suggests a new idea to unify gravity with [24] quantum theory.
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[18] B. Mashhoon; arXiv:gr-qc/0311030 (3rd chapter of The Measurement of Gravitomagnetism: A Challenging Enter- prise, edited by L. Iorio (Nova Science, New York, 2007), pp. 29-39)
M
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