Global and Local Horizon Quantum Mechanics
aa r X i v : . [ g r- q c ] J a n Global and Local Horizon QuantumMechanics
Roberto Casadio ab ∗ , Andrea Giugno c † and Andrea Giusti ab ‡ a Dipartimento di Fisica e Astronomia, Universit`a di Bolognavia Irnerio 46, I-40126 Bologna, Italy b I.N.F.N., Sezione di Bologna, IS - FLAGvia B. Pichat 6/2, I-40127 Bologna, Italy c Arnold Sommerfeld Center, Ludwig-Maximilians-Universit¨atTheresienstraße 37, 80333 M¨unchen, Germany
September 16, 2018
Abstract
Horizons are classical causal structures that arise in systems with sharply definedenergy and corresponding gravitational radius. A global gravitational radius oper-ator can be introduced for a static and spherically symmetric quantum mechanicalmatter state by lifting the classical “Hamiltonian” constraint that relates the gravita-tional radius to the ADM mass, thus giving rise to a “horizon wave-function”. Thisminisuperspace-like formalism is shown here to be able to consistently describe also thelocal gravitational radius related to the Misner-Sharp mass function of the quantumsource, provided its energy spectrum is determined by spatially localised modes.
A black hole can be viewed as a gravitationally bound state confining all possible signalswithin its horizon. According to General Relativity, such extreme causal structures occurin very compact gravitating systems, and understanding black holes from the perspective of ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] . The peculiarity of this approach is that it aims at de-scribing quantum mechanically the existence of trapping surfaces (which reduce to horizonsin the static case) from the quantum state of the source that produces the black hole. TheHQM can therefore be viewed as complementary to approaches which consider the horizonand black holes as purely gravitational (or metric) objects, in that the gravitational degreesof freedom are taken “on-shell” with respect to a suitable constraint with the matter state.The HQM can then be naturally applied to systems which do not turn out to be black holes,that is, to matter systems with a negligible probability of being black holes, or to objects“on the verge” of being black holes [15].In more details, our construction is based on the classical key concept of the gravitationalradius of a static and spherically symmetric self-gravitating source, for which this quantitydetermines the existence of horizons. We recall that we can always write a spherically For an attempt at including time evolution, see Refs. [13, 14]. g µν asd s = g ij d x i d x j + r ( x i ) (cid:0) d θ + sin θ d φ (cid:1) , (1.1)where and x i = ( x , x ) are coordinates on surfaces of constant θ and φ . A horizon then existswhere the expansion of null geodesics vanishes, g ij ∇ i r ∇ j r = 0, ∇ i r being perpendicular tosurfaces of constant area A = 4 π r . If we set x = t and x = r , and denote the staticmatter density as ρ = ρ ( r ), Einstein equations tell us that g rr = 1 − r H ( r ) /r , where r H ( r ) = 2 ℓ p m ( r ) m p (1.2)is the gravitational radius determined by the Misner-Sharp mass function m ( r ) = 4 π Z r ρ (¯ r ) ¯ r d¯ r . (1.3)A horizon then exists where g rr = 0, or where the gravitational radius satisfies r H ( r ) = r , (1.4)for r >
0. In the vacuum outside the region where the source is located, the Misner-Sharpmass approaches the Arnowitt-Deser-Misner (ADM) mass of the source,lim r →∞ m ( r ) = M , (1.5)and the gravitational radius likewise becomes the Schwarzschild radius R H = 2 ℓ p Mm p . (1.6)If the source is described by quantum physics, the quantities that define the Misner-Sharp mass m (and ADM mass M ) should become quantum variables and one expectsthe gravitational radius will undergo the same fate. The HQM was precisely proposed [11]in order to describe the “fuzzy” Schwarzschild (or gravitational) radius of a localised (butlikewise fuzzy) quantum source. It is important to emphasise once more that the HQMdiffers from most previous attempts in which the gravitational degrees of freedom of thehorizon, or of the black hole metric, are quantised independently of the state of the source.In our case, the gravitational radius is instead quantised along with the matter source thatproduces it, somewhat more in line with the highly non-linear general relativistic descriptionof the gravitational interaction in the strong regime.Clearly, the HQM becomes particularly relevant for sources of the Planck size [15, 18],for which quantum effects may not be neglected. The Heisenberg principle of quantum Let us remark this is the frame in which the Tolman-Oppenheimer-Volkoff equation is usually derived [16]. We shall use units with c = 1, and the Newton constant G = ℓ p /m p , where ℓ p and m p are the Plancklength and mass, respectively, and ~ = ℓ p m p . λ M ≃ ℓ p m p /M , and R H only makes sense if R H & λ M ⇔ M & m p . (1.7)In fact, this is the argument that grants the Planck mass (and Planck length) a remarkablerole in the search for a quantum theory of gravity [19]. It is comforting that the HQMpredicts a particle is very likely a black hole only if (1.7) holds [12, 20, 21]. Remarkably,it also predicts that a truly macroscopic black hole cannot be produced by a very localisedsource [13, 12], but could be associated with an extended source [22, 23, 24].In the next Section, we shall first review the general foundations of the HQM, andthen show that the same physical states that describe the global horizon also describe localhorizons, provided the spectral decomposition involves spatially localised energy eigenmodes.The consequences of the discrete spectrum for a GUP and Hawking radiation will also bebriefly considered. We shall finally comment on our results in Section 3. The HQM emerges from relating the matter source to its gravitational radius at the quantumlevel [11, 13], and allows us to put on more quantitative grounds the condition (1.7) thatshould distinguish black holes from regular particles.Before reviewing and extending this formalism, let us clarify the underlying viewpointby noting that one could describe formally the state of a quantum system for which thereexist two relevant sets of variables, say X and Y , as | Ψ i = X α,β c αβ | X α , Y β i . (2.1)Without loss of generality, we can group terms in the above superposition as | Ψ i = X α,β ( A αβ | X α , Y β i + B αβ | X α i| Y β ; X α i + D αβ | Y β i| X α ; Y β i + C αβ | X α i| Y β i ) , (2.2)where ˆ X | X α i = X α | X α i and ˆ Y | Y β i = Y β | Y β i . (2.3)In the sum in Eq. (2.2), the first term has no specific features; the second (third) termis of the kind that admits the Born-Oppenheimer approximation with X (respectively Y )representing slow degrees of freedom compared to Y (respectively X ); finally, the fourthterm contains the contribution from the direct product of the two separate Hilbert spacesof the eigenstates (2.3). We can now view X as “matter” degrees of freedom, such as theusual standard model fields, and Y as “gravitational” degrees of freedom. Upon furtherassuming the state | Ψ i only contains P β | Y β i ∼ | Y s i which reproduces a (semi-)classical4onfiguration, the third term in Eq. (2.2) would reduce to the usual approach of QFT on agiven (curved) background [25], | Ψ i ∼ P α | Y s i | X α ; Y s i , in which only the matter fieldsretain their full quantum properties .One could also do without the (semi)classical approximation. We shall in fact see belowthat the states of relevance for the HQM are of the fourth kind in the sum (2.2), provided wesuitably reduce the matter degrees of freedom to X = H (the “matter energy”) and Y = R H (the gravitational radius). This is not very different from the usual quantum mechanicaltreatment of the hydrogen atom, in which one quantises the (reduced) electron’s position,whereas terms containing | Y s i replaced by an electron’s energy level yield the Lamb shiftdue to the QFT description of the Coulomb field. We only consider spherically symmetric sources which are both localised in space and atrest in the chosen reference frame. If the specific source is not at rest, one should thereforechange coordinates accordingly before applying the following analysis. We denote with α theset of (discrete or continuous) quantum numbers parametrising the spectral decompositionof the source, so that our matter state can be written as | ψ S i = X α C S ( E α ) | E α i , (2.4)where the sum formally represents the spectral decomposition in Hamiltonian eigenmodes,ˆ H = X α E α | E α ih E α | , (2.5)and H should be specified depending on the system we wish to consider. We can then replacethe r.h.s. of Eq. (1.5) defining the ADM mass with the expectation value of the Hamiltonian, M → h ψ S | ˆ H | ψ S i = h ψ S | X α E α | E α ih E α | ψ S i = X α | C S ( E α ) | E α , (2.6)which follows from the orthonormality of the energy eigenstates, h E α | E β i = δ αβ , (2.7)where δ is the Kronecker delta for a discrete energy spectrum and the Dirac delta for acontinuous spectrum . Let us then introduce the gravitational radius eigenstatesˆ R H | R H β i = R H β | R H β i . (2.8) Conversely, but perhaps of less interest, the second term would be useful in order to describe states inwhich matter can be approximated classically but gravity remains fully quantum. This point is purely technical in the global approach, but will become crucial in the local analysis.
5e can now show that a physical state for our system can be described by linear combinationslike the fourth term in Eq. (2.2), | Ψ i = X α,β C ( E α , R H β ) | E α i| R H β i . (2.9)In fact, the algebraic (Hamiltonian) constraint (1.6) now reads0 = (cid:18) ˆ H − m p ℓ p ˆ R H (cid:19) | Ψ i = X α,β (cid:18) E α − m p ℓ p R H β (cid:19) C ( E α , R H β ) | E α i| R H β i , (2.10)and is clearly solved by the combination in Eq. (2.9) with C ( E α , R H β ) = C ( E α , ℓ p E α /m p ) δ αβ . (2.11)By tracing out the gravitational radius part, we should recover the matter state, that is | ψ S i = X α C ( E α , ℓ p E α /m p ) | E α i , (2.12)which implies C ( E α , ℓ p E α /m p ) = C S ( E α ) . (2.13)Now, by integrating out the matter states, we will obtain | ψ H i = X α C S ( m p R H α / ℓ p ) | R H α i , (2.14)where m p R H α / ℓ p = E ( R H α ). We have thus recovered the HWF [11] ψ H ( R H α ) = h R H α | ψ H i = C S ( m p R H α / ℓ p ) , (2.15)where the values R H α form a discrete (continuous) spectrum if E α is discrete (continuous)in the quantum number α .If the spectral decomposition (2.4) is continuous, so will be the HWF, and we can write ψ H ( R H ) = N H C S ( m p R H / ℓ p ) , (2.16)with N − = 4 π R ∞ | C S ( m p R H / ℓ p ) | R d R H determined by the scalar product h ψ H | φ H i = 4 π Z ∞ ψ ∗ H ( R H ) φ H ( R H ) R d R H . (2.17)In this continuous case, the normalised wave-function (2.16) yields the probability density P H ( R H ) = 4 π R | ψ H ( R H ) | (2.18) Note the integration is formally extended from zero to infinity, although it will be naturally limited toa smaller range if the spectral decomposition of the source is limited above and/or below. R H associated with the particle in thequantum state | ψ S i . Moreover, we can define the conditional probability density that theparticle lies inside its own gravitational radius R H as P < ( r < R H ) = P S ( r < R H ) P H ( R H ) , (2.19)where P S ( r < R H ) = 4 π R R H | ψ S ( r ) | r d r is the usual probability that the particle is foundinside a sphere of radius r = R H . One can also view P < ( r < R H ) as the probability densitythat the sphere r = R H is a trapping surface. Finally, the probability that the particledescribed by the state | ψ S i is a black hole (regardless of the horizon size), will be obtainedby integrating (2.19) over all possible values of R H , namely P BH = Z ∞ P < ( r < R H ) d R H , (2.20)which will depend on the observables and parameters of the specific matter state. We have seen that the global gravitational radius can be described irrespectively of whetherthe spectral decomposition is discrete or continuous. In order to show that the same physicalquantum states (2.9) with coefficients given in Eq. (2.11) also allow for a local description ofthe gravitational radius, we shall instead need localised energy eigenmodes and correspond-ingly discrete energy quantum numbers.Instead of the ADM mass (1.5), we now start from Misner-Sharp mass at finite radius,and again assume the system is static, so that m = m ( r ). We first observe that the totalHamiltonian (2.5) associated with the ADM mass can also be written asˆ H = X α E α | E α ih E α | E α ih E α | = X α E α ∞ X r =0 | E α ih E α | r ih r | E α ih E α | = 4 π X α E α Z ∞ | ψ E α (¯ r ) | ¯ r d¯ r | E α ih E α | . (2.21)which follows from the discrete orthogonality condition (2.7) and the (continuous) decompo-sition of the identity P ∞ r =0 | r ih r | ≡ π R ∞ ¯ r d¯ r | r ih r | = ˆ I . We can analogously introducethe radius-dependent Hamiltonianˆ H ( r ) = X α E α r X r ′ =0 | E α ih E α | r ′ ih r ′ | E α ih E α | = X α E α P α ( r ) | E α ih E α | , (2.22)7here we defined P α ( r ) = 4 π Z r | ψ E α (¯ r ) | ¯ r d¯ r , (2.23)and note that lim r →∞ P α ( r ) = 1. This property only holds for square integrable (that is, localised ) energy eigenmodes, a restriction which was not necessary in the global case, sincethe norm of these modes never entered explicitly in that calculation. However, the condition h E α | E α i = 1 is now necessary in order to obtain (2.21) above, which means the localconstruction requires the existence of localised (bound) Hamiltonian eigenmodes.Assuming again the spectral decomposition (2.4), the Misner-Sharp mass function cannow be replaced by the radius-dependent quantity m ( r ) → h ψ S | ˆ H ( r ) | ψ S i = X α | C S ( E α ) | E α P α ( r ) . (2.24)The gravitational radius (1.2) analogously depends on r , and we introduce the local gravi-tational radius eigenstates, ˆ r H ( r ) | R H β i = r H β ( r ) | R H β i . (2.25)where it is again the operator that carries a radial label. If we now recall a physical statethat satisfies the global (Hamiltonian) constraint can be written as | Ψ i = X α C S ( E α ) | E α i| R H α i , (2.26)it immediately follows that it will also satisfy the local (Hamiltonian) constraint (1.2) for allvalues of r , that is 0 = (cid:20) ˆ H ( r ) − m p ℓ p ˆ R H ( r ) (cid:21) | Ψ i = X α (cid:20) E α ( r ) − m p ℓ p r H α ( r ) (cid:21) C S ( E α ) | E α i| R H α i , (2.27)provided the local eigenvalues r H α ( r ) = P α ( r ) R H α . (2.28)Since the spectral decomposition must now be discrete, so are the above eigenvalues, andone has h ψ H | ˆ r H ( r ) | ψ H i = X α | C S ( E α ) | P α ( r ) R H α = X α | ψ H ( R H α ) | P α ( r ) R H α , (2.29)8here ψ H is the (discrete) global HWF. Finally, the classical local condition (1.4) for theexistence of a (static) trapping surface at the radius r can now be directly replaced by h ψ H | ˆ r H ( r ) | ψ H i = r , (2.30)which therefore defines quantum local horizons.It is worth discussing further what would go wrong if the spectral decomposition (2.4)did not contain only localised energy eigenmodes but also, say spatially homogenous modes,like the plane waves. The function P α = P α ( r ) would increase monotonically with r withoutbounds, and any attempt at regularising it, for example by enclosing the system in a “box” ofsize R , would lead to expressions like P α ∼ r/R that explicitly depend on the (arbitrary) cut-off R , a clear sign of inconsistency. This is of course implicitly seen already from Eq. (2.21)which again only makes sense if h E α | E α i = 1, as we remarked above. In Ref. [12], a GUP was obtained by combining (linearly) the uncertainty in the source size h ∆ˆ r i encoded in the matter state (2.12) with the uncertainty in the horizon size h ∆ ˆ R i given by a (continuous) global HWF (2.16). In particular, for Gaussian matter states, ψ S ( r ) ≃ e − r ℓ , (2.31)where ℓ ≃ ℓ p m p /m is the Compton width of the source of mass m , the GUP was shown totake the form ∆ r ≃ ℓ p m p ∆ p + γ ℓ p ∆ pm p , (2.32)with γ an arbitrary coefficient, h ∆ˆ r i = 4 π Z ∞ | ψ S ( r ) | r d r − (cid:18) π Z ∞ | ψ S ( r ) | r d r (cid:19) ≃ ℓ , (2.33)and h ∆ ˆ R i = 4 π Z ∞ | ψ H ( R H ) | R d R H − (cid:18) π Z ∞ | ψ H ( R H ) | R d R H (cid:19) ≃ ℓ ℓ . (2.34)Finally, the global uncertainty in radial momentum is given by∆ p = 4 π Z ∞ | ψ S ( p ) | p d p − (cid:18) π Z ∞ | ψ S ( p ) | p d p (cid:19) ≃ m ℓ ℓ , (2.35)where ψ S ( p ) ≃ e − p m . (2.36)9s we have just shown, were one to employ the local description of Section 2.2, thespectrum of the source should be discrete, and consequently so would be the HWF. Inparticular, we can now define a local uncertainty in the source size h ∆ˆ r ( r ) i = 4 π Z r | ψ S (¯ r ) | ¯ r d¯ r − (cid:18) π Z r | ψ S (¯ r ) | ¯ r d¯ r (cid:19) . (2.37)Likewise, and replacing integrals with sums over the spectral index α , we can also definelocal uncertainties for the gravitational radius h ∆ˆ r ( r ) i = X α | C S ( E α ) | R H2 α P α ( r ) − X α | C S ( E α ) | R H α P α ( r ) ! , (2.38)and for the radial momentum∆ p ( r ) = X α | C S ( E α ) | p α P α ( r ) − X α | C S ( E α ) | p α P α ( r ) ! , (2.39)where p α = p ( E α ).It is interesting to note that, for spectral eigenmodes, the above sums would reduce tosingle terms and one finds h ∆ˆ r ( r ) i R H2 α = P α ( r ) [1 − P α ( r )] = ∆ p ( r ) p α . (2.40)By again combining linearly the size uncertainty (2.37) with the uncertainty in the gravi-tational radius (2.38), one obtains a local GUP of the form (2.32) at each (finite) value of r . Since P α ( r → ∞ ) = 1, one also finds h ∆ˆ r ( r → ∞ ) i = h ∆ ˆ R i = 0 = ∆ p ( r → ∞ ),in agreement with the fact that the global GUP reduces to the standard Heisenberg uncer-tainty relation for spectral eigenmodes. Of course, Gaussian states (2.31) may now not bein the discrete Hilbert space , or they could instead be energy eigenstates, depending onthe details of the system at hand. In any case, it is hard to conceive that macroscopic blackholes are simple spectral eigenmodes, and some form of global GUP should therefore apply.A more drastic consequence follows for the Hawking radiation, as one expects only quantacorresponding to transitions between states of the discrete spectrum be allowed. Moreover,this argument would further support the idea that macroscopic black holes cannot be spectraleigenmodes, as those would not support any Hawking emission. It is in fact tempting todraw a connection with corpuscular models of black holes [22], as they appear to be boundstates in the sense shown in Ref. [26], and do not suffer of the paradoxes related to thestandard description of the Hawking radiation. More technically, one can view 4 π h ∆ˆ r ( r ) i as the uncertainty in the area of a sphere of coordinateradius r . Of course, they might still be obtained as the limit of suitable series. Conclusions
We have analysed the quantum constraint that relates the gravitational radius of a sphericallysymmetric source to its spectral decomposition, and shown that the same quantum state canbe employed in order to describe both the global radius associated with the ADM mass andthe local radius associated with the Misner-Sharp mass function.A crucial difference that emerges between the local and global gravitational radius atthe quantum level is that the former requires the spectral decomposition is done in termsof localised energy eigenmodes, whereas the global radius can be defined in any case. Fromthe physical point of view, one can argue the global gravitational radius is an asymptoticproperty of a self-gravitating system and should therefore be rather insensitive to the detailsof its internal structure, whereas the local gravitational radius should be determined bythe precise internal structure of the source. It therefore appears consistent that the localgravitational radius can be defined only provided the inner structure of the source is properlycharacterised as well. Finally, the fact the spectral decomposition must be discrete does notconstitute a real limitation in most practical situations, since any realistic astrophysicalsources, like stars, should have very finely-spaced energy levels.So far we have only considered the formal extension of the HQM to the local gravitationalradius, and some general considerations regarding the GUP and Hawking radiation, but itwill next be important to apply this approach to specific models of self-gravitating objects,and also to compare our findings with specific proposals of quantum black holes [9, 15, 22,24, 27, 28].
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