On Black Hole Schrodinger Equation And" Gravitational Fine Structure

Home , Fine structure

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M 2 V (r)= E , (1) − r 1 ∂2X 2 ∂X + + VX = EX. (2) − 2M ∂r2 r ∂r E ME Here E = 2 is the BH total energy and ME is the BH effective mass introduced by the− Author in a series of previous works, see [20, 21] and references within. ME is the average of the initial and final masses in a BH quantum transition. It represents the BH mass during the BH expansion (contraction), due to an absorption (emission) of a particle, see [20, 21] for details. Its rigorous definition is [20, 21] ω ME M , (3) ≡ ± 2 where ω represents the mass-energy of the absorbed (emitted) particle. Thus, on one hand, the introduction of the BH effective mass in the BH dynamical equations is very intuitive. On the other hand, such an introduction is rigor- ously justified through Hawking periodicity argument [26], see again [20, 21] for details. Thus, at the quantum level the SBH is interpreted as a two-particle system where the two components strongly interact with each other through the quantum gravitational interaction of Eq. (1) [2]. The two-particle system BH seems to be nonsingular from the quantum point of view and is analogous to the hydrogen atom because it consists of a “nucleus” and an “electron” [2]. Thus, the Schrodinger equation for the SBH of Eq. (2) is formally identical to the traditional Schrodinger equation of the s states (l = 0) of the hydrogen atom with the traditional Coulombian potential [27, 28]

e2 V (r)= . (4) − r One observes that, on one hand, Eq. (2) seems simpler than the corresponding Schrodinger equation for the hydrogen atom because there is no angular dependence; on the other hand it seems also more complicated, because, diﬀer- ently from the Schrodinger equation for the hydrogen atom, where the electron’s charge is constant, in Eq. (2) the BH mass variates, and this is exactly the rea- son because the BH eﬀective mass, that is indeed a dynamical quantity, has

2 been introduced, see [2, 20, 21] for details. One also recalls that e2 = α is the e2 fine structure constant, which, in standard units combines the constants 4πǫ0 from electromagnetism, ~ from quantum mechanics, and the speed of light c 1 from relativity, into the dimensionless irrational number α 137,036 , which is one of the most important numbers in Nature. Thus, from the≃ present approach one argues that the gravitational analogous of the fine structure constant is not a constant. It is a dynamical quantity instead. In fact, if one labels the “gravitational fine structure constant” as αG, by confronting Eqs. (1) and (4) one gets 2 M α = E , (5) G m p in standard units, where mp is the Planck mass. It will be indeed shown that αG has a spectrum of values which coincides with the set of natural numbers N 1 2 . In analogous way, the Rydberg constant, R∞ = 2 meα , is defined in terms of the electron mass me and on the fine structure constant. In the current gravitational approach the effective mass and the “charge” are the same. Hence, the quantum gravitational analogous of the Rydberg constant is

5 ME (R∞)G = 5 . (6) 2mplp in standard units, where lp is the Planck length. Again, it will be shown that (R∞)G is not a constant but a dynamical quantity having a particular spectrum of values. By introducing the variable y

y(r) rX (7) ≡ Eq. (2) becomes 1 d2 M 2 + E + E y =0. (8) − 2M dr2 r E Setting y′ d and using E = ME , Eq. (8) can be rewritten as ≡ dr − 2

′′ 2M y + M 2 E 1 y =0. (9) E r − As it is E < 0, the asymptotic form of the solution, which is regular at the origin, is a linear combination of exponentials exp(MEr) , exp( MEr). If one wants this solution to be an acceptable eigensolution, the coeﬃcient− in front of exp(MEr) must vanish. This happens only for certains discrete values of E. Such values will be the energies of the discrete spectrum of the SBH and the corresponding wave function represents one of the possible SBH bound states. If one makes the change of variable

x =2MEr (10)

3 Eq. (9) results equivalent to

d2 M 2 1 + E y =0, (11) dx2 x − 4 and y is the solution which goes as x at the origin. For x very large, it increases exponentially, except for certain particular values of the SBH eﬀective mass x where it behaves as exp 2 . One wants to determine such special values and their corresponding eigenfunctions.− One starts to perform the change of function x y = x exp z (x) , (12) − 2 which changes Eq. (11) to

d2 d + (2 x) 1 M 2 z =0. (13) dx2 − dx − − E This last equation is a Laplace-like equation. Within a constant, one finds only a solution which is positive at the origin. All the other solutions have a singularity in x−1. One can show that this solution is the confluent hypergeometric series ∞ Γ 1+ i M 2 1 xi Z = − E . (14) Γ (1 M 2 ) (1 + i)! i! i=1 E X − In fact, one expands the solution of Eq. (13) in Mc Laurin series at the origin as 2 i z =1+ α1x + α2x + ...αix + ... (15) By inserting Eq. (15) in Eq. (13) one writes the LHS in terms of a power series of x. All the coefficients of this expansion must be null. Thus,

2 2α1 =1 M − E 2 2 3α2 = 2 M α1 ∗ − E (16) ....

2 i (1 + i) αi = i M αi−1. − E Then, one gets

2 2 2 i ME i 1 ME 1 ME 1 αi = − − − ... − , (17) (1 + i) (1 + i 1) 1+1 i! − which is exactly the coeﬃcient of xi in the expansion (15). From a mathematical point of view the series (14) is inﬁnite and behaves as exp x 2 x(1+ME )

4 for large x. Consequently, y behaves in the asymptotic region as

x exp 2 2 xME and cannot be, in general, an eigensolution. On the other hand, for particular 2 values of ME all the coeﬃcients will vanish from a certain order on. In that case, the series (14) reduces to a polynomial. The requested condition is

1 M 2 0, with 1 M 2 N, (18) − E ≤ − E ∈ which implies the quantization condition ′ ′ M 2 = n = n +1, withn =0, 1, 2, 3, .... + . (19) E ∞ The hypergeometric series becomes a polynomial of degree n′ and y behaves in the asymptotic region as xn x . exp 2 Then, the regular solution of the SBH Schrodinger equation becomes an acceptable eigensolution. The quantization condition (19) permits to obtain the spectrum of the eﬀective mass as

(ME) = √n, withn =1, 2, 3, .... + . (20) n ∞ The quantization condition (19) gives also the spectrum of the “gravitational fine structure constant” that one writes as 2 (ME)n (αG) = = n, withn =1, 2, 3, .... + . (21) n m ∞ p Thus, the gravitational analogous of the fine structure constant is not a constant. It is a dynamical quantity instead, which has the spectrum of values (21) which coincides with the set of non-zero natural numbers N 0 . In the same way, one finds the spectrum of the “gravitational Rydberg constan−{ } t” as

5 5 2 (ME)n n [(R∞)G]n = 5 = , withn =1, 2, 3, .... + . (22) 2mplp 2 ∞

Now, from the quantum point of view, one wants to obtain the mass eigenvalues as being absorptions starting from the BH formation, that is from the BH having null mass, where with “the BH having null mass” one means the situation of the gravitational collapse before the formation of the ﬁrst event horizon. This implies that one must replace M 0 and ω M in Eqs. (3) and must take the sign + (absorptions) in the same→ equations.→ Thus, one gets M ME , rE 2ME = M. (23) ≡ 2 ≡

5 By combining Eqs. (20) and (23) one immediately gets the SBH mass spectrum as Mn =2√n. (24) Eq. (24) is consistent with the SBH mass spectrum found by Bekenstein in n 1974 [4]. Bekenstein indeed obtained En = 2 by using the Bohr-Sommerfeld quantization condition because he argued that the SBH behaves as an adiabatic invariant. It is also consistent with other BHp energy spectra in the literature, ME see for example [3, 6–9, 19]. The relation E = 2 permits also to ﬁnd the negative energies of the SBH bound states as −

n En = . (25) − 4 r The (unnormalized) wave function corresponding to each energy level is

n−1 1 2MEr i (n 1)! i X(r)= ( ) − (2MEr) . (26) r exp x − (n 1 i)! (i + 1)!i! " 2 i=0 ! # X − − One observes that the number of nodes of the wave function is exactly n 1 and that the energy spectrum (25) contains a denumerably inﬁnite of levels because− n can take all the inﬁnite values n N 0 . One can calculate the energy jump between two neighboring level∈ as −{ }

n n +1 1 E = En+1 En = = . (27) △ − 4 − 4 − n n+1 r r 4 4 + 4 q For large n one obtains p 1 E △ ≃−4√n when n + one gets E 0. Thus, one finds that the energy levels become more and→ more∞ closely spaced△ → and their difference tends to E =0 at the limit, at which point the continuous spectrum of the SBH begins.△ One also observes that a two particle Hamiltonian p2 M 2 H (~p, r)= E , (28) 2ME − r which governs the SBH quantum mechanics, must exist in correspondence of Eqs. (1) and (2). Therefore, the square of the wave function (26) must be interpreted as the probability density of a single particle in a finite volume. Thus, the integral over the entire volume must be normalized to unity as

2 dx3 X =1. (29) | | Z For stable particles, this normalization must remain the same at all times of the SBH evolution. This issue has deep implications for the BH information

6 paradox [29] because it guarantees preservation of quantum information. As the wave function (26) obeys the SBH Schrödinger equation (2), this is assured if and only if the Hamiltonian operator (28) is Hermitian [30]. In other words, the Hamiltonian operator (28) must satisfy for arbitrary wave functions y1 and y2 the equality [30]

3 ∗ 3 ∗ dx [HX2] X1 = dx X2 HX1. (30) Z Z One notes that both ~p and r are Hermitian operators. Thus, the Hamiltonian (28) will automatically be a Hermitian operator if it is a sum of a kinetic and a potential energy [30] H = T + V. (31) This is always the case for non-relativistic particles in Cartesian coordinates [30] and works also for SBHs. Thus, it has been shown that the SBH is a perfect quantum system, which obeys Schrodinger’s theory. In a certain sense it is a “gravitational hydrogen atom”. One also observes that studying the SBH in terms of a well deﬁned quantum mechanical system, having an ordered, discrete quantum spectrum, looks consistent with the unitarity of the underlying quantum gravity theory and with the idea that information should come out in BH evaporation.

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