- Email: [email protected]
Black hole as a model of computation
Results in Physics 13 (2019) 102188
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Black hole as a model of computation
T
G.R. Andrews III Independent Researcher, 8805 Jenny Ln Fredericksburg, VA 22407, United States
A R T I C LE I N FO
A B S T R A C T
Keywords: Black hole computation Kerr/CFT correspondence Holographic principle Information theory Gamma-ray spectroscopy Shannon entropy
This paper focuses on an alternative, more physically realistic model of computation than Etesi and Németi’s relativistic computer in a Malament-Hogarth spacetime (2002) that uses the black hole itself combined with an external observer equipped with a source and some method of measurement of gamma-rays, as opposed to sending a classical computer into a black hole and exploiting the properties of the spacetime to achieve hypercomputation. The source of output, Hawking radiation, is considered along with the constraints imposed by the holographic principle which limit the number of degrees of freedom in the system and consequently the maximum usable information. The Bekenstein-Hawking entropy is converted from the traditional form in terms of the horizon area to that of the Shannon entropy, establishing an analogy between the physical and computational perspectives of the system. Next examples are considered to establish the approximate order of the necessary excitation energy and the resulting gamma-ray interactions which form the input from the observer. Finally, the Turing completeness of the language for this model is considered through a simulation of the Turing machine. The goal is to introduce a model of computation that can later be used to study the relationship between computability and physical systems.
Introduction
Geometry of holographic duals
In contrast to past research on computation in black holes which focused on the idea of sending a computer into the black hole and exploiting its relativistic properties (as in the case of [7], here it is proposed that the black hole itself be used as a model of computation, motivated by the unique properties arising from holographic duality and the greater feasibility of such a system in the real world. This is achieved by using the energy states of molecules behind the event horizon as the states of the computer, with the movement of the external observer measuring the system corresponding to the shifting of the computer’s tape and the change in states combined with the resulting Hawking radiation corresponding to the output. A stream of photons at the appropriate energy levels is used as input to change the states of the computer. In this way a correspondence is formed between the physical and computational models of the system which may then contribute to a proof of its Turing completeness. In the following sections the methods for essential components of the computer will be developed which will then contribute to the proof of the Turing completeness of this model of computation. The goal is to introduce a model of computation which can be used to investigate the effects of physical constraints imposed by the holographic principle and the Kerr/CFT correspondence on computation and vice versa.
The AdS/CFT correspondence allows for a dual description of an anti-de Sitter space and a conformal field theory of one less dimension, one of the most well known of which is the correspondence between the AdS5 × S5 space and the D = 4, Ɲ = 4 supersymmetric SU(N) YangMills theory. For a realistic model of a black hole we need to find the holographic dual to a Kerr solution, since we assume that the black hole is in an equilibrium state and has a nondegenerate event horizon [16]. This duality is called the Kerr/CFT correspondence and the associated conformal field theory is a chiral half of a two-dimensional CFT [9]. The symmetry of the two-dimensional CFT is described by two copies of the Virasoro algebra with generators that satisfy the below commutation relations, where m, n ∈ Z and c denotes the central charge.
[Lm , Ln] = (m − n) Lm + n +
c (m3 − m) δm + n,0 12
[c, Ln] = 0 We only consider one copy corresponding to a chiral half of the CFT. Representations of the Virasoro algebra arise from the highest weight state |h > = ϕ(0)|0 > which satisfies L0|h > = h|h > , where ϕ is a primary field and h denotes the conformal weight [8].
E-mail address: [email protected]. https://doi.org/10.1016/j.rinp.2019.102188 Received 19 February 2019; Accepted 9 March 2019 Available online 14 March 2019 2211-3797/ © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Results in Physics 13 (2019) 102188
G.R. Andrews
Black hole output
Hawking entropy to the form of a Shannon entropy. This analogy arises from the Bekenstein-Hawking entropy’s dependence on the horizon area [17]. The entanglement (or von Neumann) entropy may be written in similar terms to the Bekenstein-Hawking entropy, as shown below in the case of a ℝ1,d conformal field theory [17].
The fact that particles cannot escape black holes through classical means (arising from the second law of black hole thermodynamics [1] poses a challenge to obtaining an observable output, however Hawking radiation circumvents this problem through quantum tunneling. Section 2.1 considers Hawking radiation, a process described by the tunneling of particles out of the potential well behind the black hole horizon [10,14]. A relationship between the energy of the tunneling shell and the emission rate from the black hole is also established. Section 2.2 considers the constraints imposed by the holographic principle on information density in the black hole.
SBH =
Ah 4GN
SA = −trA ρA logρA =
A γA 4GN(d + 2)
ρA = trB |Ψ > < Ψ|
Hawking radiation
This entanglement entropy itself may be written in the form of the Shannon entropy [3].
Hawking radiation provides a source of information via tunneling that avoids the problem caused by the second law of black hole thermodynamics. Hawking presents such an argument for the emission of particles (1975). Furthermore, Parikh and Wilczek [14] derive this phenomenon directly from the description of it as a tunneling process. The exponential part of the emission rate Γ from Hawking radiation is related to the mass of the black hole and the energy of the particle that will potentially escape:
ρA =
∑ λj |j > < j| → SA = − ∑ λjlogλj j
j
The next step is to relate the Bekenstein-Hawking entropy directly to the Shannon entropy, which intuitively follows from the previously shown correspondence involving the entanglement entropy and is shown below. Let Ah denote the area of the horizon which corresponds to the area of the holographic screen. Let Ω and β denote the inside and outside of the black hole respectively. Then we change the form of SBH to that of the entanglement entropy, writing the density matrix μβ in terms of an orthonormal basis |Ψi > .
ω
Γ e−8πω (M − 2 ) Parikh & Wilczek [14] from this relation by introducing proportionality constant α we may obtain a graph relating Γ and ω which is shown in Fig. 1. According to Fig. 1 Γ may only be made large if ω is either small or large relative to the mass M of the black hole (close to 0 in the former case and 2 M in the latter case). Since M < 0 is ignored we must excite the particles to be emitted past ω = 2 M (where Γ will blow up) in order to achieve arbitrarily large values of Γ (which must be large enough in order that output from the black hole may be detected by distant observers in a reasonable amount of time). The mass of the black hole may be calculated through its relation to the Hawking temperature and proper distance from the black hole in Rindler coordinates [4]. This will involve a measurement of the temperature at a sufficiently far distance from the black hole since it varies inversely with the distance.
SBH =
Ah → SBH = −trβ μβ logμβ 4GN
μβ = trΩ |Ψ > < Ψ| Now we may write SBH in the form of the Shannon entropy by writing μβ in terms of its eigenvectors |Ψj > .
μβ =
∑ μ βj |Ψj > < Ψj| j
⎛ ∑ μ βj |Ψj > < Ψj| ∑ logμ βi |Ψi > < Ψi|⎞⎟ ⎜ i ⎝ j ⎠ |Ψj > < Ψj | ∑ logμ βi |Ψi > < Ψi| Ψk >
SBH = −trβ (μβ logμβ ) = −trβ = − ∑ < Ψk | ∑ μ βj k
The Bekenstein-Hawking entropy as a Shannon entropy
j
i
= − ∑ δkj μ βj δji logμ βi δik = − ∑ μ βj logμ βj
To develop the correspondence between the physical and computational interpretations of the model, we will convert the Bekenstein-
k , j, i
j
All of this work merely serves to show the similarity between the Bekenstein-Hawking entropy and the holographic entanglement entropy and its resulting potential application to our computer. For this reason we are interested in whether the Bekenstein-Hawking entropy may be physically interpreted as a holographic entanglement entropy. Solodukhin considers this question at length, posing several problems which must be solved in order to make such a connection (2011): (1) that the UV divergence of the entanglement entropy may render it inconsistent with the Bekenstein-Hawking entropy which is finite; (2) that the entanglement entropy depends on the number of types of fields unlike the Bekenstein-Hawking entropy; and (3) that the different behavior of non-minimally coupled fields such as gauge bosons and gravitons with respect to the entanglement entropy may affect the possibility of the interpretation of the Bekenstein-Hawking entropy as an entanglement entropy. Examples where the two entropies are equal include a model with minimally coupled fields [18], a 3-brane with the Z2 symmetry group in an AdS spacetime [11,18], and the Kaluza-Klein model [18]. We will only consider the first example here for the sake of simplicity, clarity, and brevity. In a minimally coupled field theory with N0 scalars and N1/ 2 Dirac fermions, the induced gravitational action reads
Fig. 1. Emission rate as a function of escaping particle energy. 2
Results in Physics 13 (2019) 102188
G.R. Andrews
Wind = −
1 16πGind
∫E R
High energy excitations
g d 4x
In order to establish the practicality of excitations on the order of 2 M we must estimate the required energy levels for a realistic black hole. For example, we may choose its mass to equal to that of earth.
with
1 N = , N = N0 + 2N1/2 Gind 12π ∊2
M = 5.972 × 1024kg
where N denotes the total number of field species in the model. We can show that
Sent =
Then the required energy ω to achieve the emission rate Γ = α may be approximated.
N 1 A (Σ) = A (Σ) = SBH , 48π ∊2 4Gind
ω = 2M = 1.194 × 1023cm−1 Converting to the more familiar energy units, we have
noting that the renormalization statement assures that Gind and Sent are inversely proportional. This example circumvents (1) and (2), however the absence of any non-minimally coupled fields leaves (3) unanswered. The general solution to calculating the entanglement entropy is believed by some to be solvable with a string theoretic approach [18], which Susskind and Uglum proposed [19]. A direct calculation of the entanglement entropy would allow for a complete answer to the comparison of the Bekenstein-Hawking and holographic entanglement entropies. The complexity of such an approach, however, has rendered progress towards such a solution slow. This is one topic of interest for future research in the development of this model of computation.
E = 1.480 × 1019eV , which corresponds to the energy of gamma-rays. In general the energy required to achieve ω = 2 M necessitates the use of gamma-rays for realistic black holes (those that can be detected using current methods). Low energy excitations In order to maximize the emission rate ω must be minimized to the smallest energy which the contents of the black hole will absorb. Continuing with our example and letting Γ = α/2, we obtain ω.
Arguments from the holographic principle
ω=M−
We now turn our attention to the holographic principle, which relates the world volume to the boundary of the system and from which arises the AdS/CFT correspondence, starting with the Bekenstein bound in order to obtain a limit on the usable information for a given volume. The Bekenstein bound limits the maximum amount of entropy in the system which is related to the number of quantum states in that system [5]. It too may be considered an analogue to some relation involving information entropy like in the case of the Bekenstein-Hawking entropy. The equation for this bound is given below.
4π 2M 2 − π ln2 ≈ 0cm−1 2π
Since the difference caused by the π ln 2 term is small when M is large (requiring extreme precision when calculating the solution), it is advisable to find a bound for ω. A well known inequality gives us
4π 2M 2 − π ln2 > 2π
4π 2M 2 − 2π
π ln2 2π
From this inequality we have a bound for ω:
ω<
2πkER S≤ hc
π ln2 = 0.235m−1 = 2.35 × 10−3cm−1 2π
E < 2.91 × 10−7eV By considering this analogy and the relation between the number of degrees of freedom in the given volume and the horizon area we may conclude that the holographic principle limits the usable information for a given volume by limiting its number of degrees of freedom, i.e. the number of quantum states in the system. This intuitively follows from the analogous limitations on information density imposed by the number of individual states of a computer (which for the purposes of this discussion is considered to be a Turing machine). The corresponding bound from an information theoretic perspective is given below, where E is the energy and I is the amount of information enclosed by a sphere with radius R, which Bekenstein proposed [20,2].
I≤
This corresponds to the Very High Frequency (VHF) range. The calculation shows that a high-energy excitation is preferable due to the higher attainable emission rate. Gamma ray interactions Gamma rays with energies of more than 1 TeV, such as was considered in the example in Section 4.1, mostly interact with infrared photons with a wavelength greater than 1 μm [12]. For a typical high energy excitation (> 10 MeV) caused by gamma-rays such as the example in Section 4.1, pair production gamma-ray interactions are dominant [15]. In such an interaction an electron-positron pair is created provided that a sufficient energy 2m0c2 (the mass of the electronpositron pair) is supplied by the gamma-ray (Rittersdorf). When a positron annihilates an electron upon contact, two 0.511 MeV photons are emitted (Rittersdorf). Secondary electron escape and Bremsstrahlung escape may be disregarded due to the localization of particles caused by the black hole’s gravitational pull. This leaves only tunneling via Hawking radiation as a viable method of escape. For our example black hole the 0.511 MeV photons do not have sufficient energy for the “critical point” Γ = α to be achieved. Thus for a black hole with the mass of earth the emission rate for the photons created via annihilation in pair production gamma-ray interactions is small. Future research could examine the process of pair production near the horizon of a Kerr black hole taking into account the Kerr/CFT correspondence and the associated string theory.
2πER M ⎞⎟ ⎛ R ⎞ bits = 2.57 × 10 45 ⎛⎜ hc ln2 1 kilogram ⎝ ⎠ ⎝ 1meter ⎠
Thus we have a bound for the maximum usable information for a topologically spherical black hole which depends on the mass and radius. Further work will involve describing Hawking radiation for a Kerr solution through the Kerr/CFT correspondence near the horizon and an associated string theory and finding a stronger information bound. Black hole input In order to excite the particles inside the black hole and thus change their states we will consider irradiating them with photons. Fig. 1 showed that the emission rate Γ peaks for values of ω larger than 2 M and close to 0, which sections 3.1 and 3.2 consider respectively. 3
Results in Physics 13 (2019) 102188
G.R. Andrews
Turing Completeness of Language for Black Hole
where the propagator − − 1 < X (z , z ) X (w, w ) > = − log|z − w| 2
Since as we have seen the black hole possesses inherent properties which are analogues to components of a Turing machine, the most obvious and direct way to prove its Turing completeness is through a simulation of the Turing machine, showing that corresponding parts perform equivalent operations. The contents of the black hole itself serve as the tape and may be scaled to an arbitrarily large size by increasing the mass of the black hole; therefore the requirement for an indefinitely extensible tape is satisfied. In this model the observer serves as a head which can move in order to shift the tape and read the output from the machine via Hawking radiation. The states assigned to each position on the tape are encoded in the energy levels for each unit of matter. The external observer is limited in the number of measurable states by the choice of equipment (since the energy levels associated with all types of particles are quantized and there is a finite amount of energy in the universe). Finally, there exists a set of instructions which form a Turing-complete language from these physical components: the state of a position on the tape may be changed by irradiating that particle with light, the head may be moved, and information may be read by the head from the Hawking radiation. We will now define the corresponding Turing machine in more formal terms to present directions for future research in more complete physical descriptions of each of the components. This Turing machine is represented by
with z = ½(σ1 + iσ0) and 2idz^dz ̅ = dσ1^dσ0. X(z, z)̅ may be written in terms of left and right movers (thus separating holomorphic and antiholomorphic dependence): −
X (z , z ) =
− 1 (xL (z ) + xR (z )) 2
where the corresponding propagators are −
−
−
−
< xL (z ) xL (w ) > = −log(z − w ), < xR (z ) xR (w ) > = −log(z − w ). This quantization procedure is used for the case of the free boson and is described far more completely in texts on conformal field theory such as [8]. To differentiate between different input symbols in Σ we use the energy of the photon from the Schrodinger equation absorbed by the black hole. Tape alphabet Γ There are massive particles behind the black hole horizon which form the tape. The tape alphabet Γ is composed of the set of possible energy eigenstates for the corresponding unit quantum systems (i.e. the smallest unit which is equivalent to a single location on the tape) which are ultimately constrained by the Planck dimensions. The simplest such unit is a free fermion, which will be considered briefly here. For a free massless fermion, after preserving only the left moving (or holomorphic) half (since the CFT2 being considered is a chiral half), the action becomes
M = (Q, Σ, Γ, δ , q0 , ⧠, F ) where Q is the set of internal states, Σ is the input alphabet, Γ is the tape alphabet, δ is the transition function, □ is the blank symbol, q0 is the initial state, and F is the set of final states. This is the universally accepted definition of a Turing machine which may be found in elementary textbooks on theoretical computer science (such as [13]. The sets of states may be defined on the boundary two dimensional conformal field theory in terms of the possible energy eigenvalues of the involved particles (using the Schrodinger equation). The description of interacting fields in the resulting interactions on the CFT2 (an abbreviation for a two dimensional conformal field theory) falls outside the scope of the paper, however we will consider the case of free bosons and fermions which together compose the idealized states in Q, Σ, and Γ. In what follows the resource used is a set of lecture notes presented by Ginsparg [8].
1 8π
S=
−
∫ ψ∂ψ.
The field ψ satisfies the short distance singularity
ψ (z ) ψ (w ) = −
1 z−w −
and the operator ∂ is related to the Feynman’s Dirac operator in the following way.
∂x − i∂y ⎞ ⎛ 0 ∂ ⎞ 0 D = σx ∂x + σy ∂y = ⎛⎜ ⎟ ⎟ ⎜− ∂ + ∂ i 0 ⎠ ⎝∂ 0⎠ y ⎝ x A more realistic field may be specified by considering the case of a free massive Dirac fermion whose Euclidean action in d dimensions is given by
Set of internal states Q The internal state of the black hole may be represented by the set of quasi-primary fields of the CFT2 which may be used to generate the secondary fields (consisting of the derivatives of all orders of the primary fields ϕi which are referred to as descendant fields). Thus we may represent the set of internal states Q as the set of all possible configurations of the set of quasi-primary fields which represent their respective conformal families.
−
S [ψ , ψ] =
−
∫ ddxψ (γμ ∂μ + m) ψ
where the γμ∂μ term is the Dirac operator alternatively represented in Feynman slash notation [6]. The energy eigenvalues of the free fermion form elements of the set Γ as in Section 5.2.
Q = {(ϕ1, ⋯, ϕn )1, ⋯, (ϕ1, ⋯, ϕn )i} Conclusion Input alphabet Σ
We have shown that black holes combined with an external beam of gamma-rays form a viable model of computation; the Turing completeness of its associated language arises intuitively from direct analogues between the Turing machine and the physical device. Although it is currently not feasible to test such a model of computation in the real world, its unconventionality provides insights into new ways of thinking about leveraging nature for computational purposes as well as analyzing physical processes from a computational perspective (since physical events themselves may be regarded as instances of transition functions) and the absence of a classical computer behind the horizon
For the input we consider a free (non-interacting) boson on a CFT2. With Euclidean space and time coordinates σ1 and σ0 we compactify the space coordinate by σ1 = σ1 + 2π which defines a cylinder. This defines the geometry for the radial quantization procedure to take place as described in [8]. The string theory normalization for the action is given by −
S=
∫ L = 21π ∫ ∂X ∂X 4
Results in Physics 13 (2019) 102188
G.R. Andrews
renders the model more physically realistic due to the inevitable crushing of any such computer by the tidal forces of the black hole. Future work should look for extensions to the Turing completeness of this model, i.e. the theoretical potential for hypercomputation as has been examined for relativistic computation in Malament-Hogarth spacetimes [7], as well as further details in the feasibility of methods for observer-provided gamma-ray beams and measurement of the gammarays emitted from the black hole. Additionally, the theoretical description of pair production gamma-ray interactions and Hawking radiation in the context of the Kerr/CFT correspondence and string theory should be formulated in future research and stronger information bounds must be found to better define the limitations of this model of computation.
quantum entanglement. Cambridge University Press; 2006. [4] Bigatti D, Susskind L. TASI lectures on the holographic principle. Strings, Branes, and Gravity. 2001. p. 883–933. [5] Bousso R. The holographic principle. Rev Mod Phys 2002;74(3):825. [6] Chandrasekharan S, Wiese U-J. An introduction to chiral symmetry on the lattice. Prog Part Nucl Phys 2004;53(1):373–418. [7] Etesi G, Németi I. Non-Turing computations via Malament-Hogarth space-times. Int J Theor Phys 2002;41:341–70. [8] Ginsparg P. Applied conformal field theory, 1988. arXiv preprint hep-th/9108028. [9] Guica M, Hartman T, Song W, Strominger A. The kerr/cft correspondence. Phys Rev D 2009;80(12):124008. [10] Hawking SW. Particle creation by black holes. Commun Math Phys 1975;43(3):199–220. [11] Hawking SW, Maldacena J, Strominger A. DeSitter entropy, quantum entanglement and AdS/CFT. J High Energy Phys 2001;2001(05):001. [12] Kneiske TM, Bretz T, Mannheim K, Hartmann DH. Implications of cosmological gamma-ray absorption – II. Modification of gamma-ray spectra. A & A 2004;413(3):807–15. [13] Linz P. An introduction to formal languages and automata. 5th ed. Jones & Bartlett Learning; 2012. [14] Parikh MK, Wilczek F. Hawking radiation as tunneling. Phys Rev Lett 2001;85(24):5042. [15] Rittersdorf I. Gamma ray spectroscopy. Nucl Eng Radiol Sci 2007:18–20. [16] Robinson DC. Uniqueness of the Kerr black hole. Phys Rev Lett 1975;34(14):905. [17] Ryu S, Takayanagi T. Holographic derivation of entanglement entropy from AdS/ CFT. Phys Rev Lett 2008;96(18):181602. [18] Solodukhin SN. Entanglement entropy of black holes. Living Rev Relativ 2011;14(1):8. [19] Susskind L, Uglum J. Black hole entropy in canonical quantum gravity and superstring theory. Phys Rev D 1994;50(4):2700. [20] Tipler RJ. Feynman-Weinberg quantum gravity and the extended standard model as a theory of everything; 2007. arXiv preprint arXiv:0704.3276.
Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Bardeen JM, Carter B, Hawking SW. The four laws of black hole mechanics. Commun Math Phys 1973;31(2):161–70. [2] Bekenstein JD. Black holes and information theory. Contemp Phys 2003;45(1):31–43. [3] Bengtsson I, Zyczkowski K. Geometry of quantum states: an introduction to
5
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Black hole as a model of computation
T
G.R. Andrews III Independent Researcher, 8805 Jenny Ln Fredericksburg, VA 22407, United States
A R T I C LE I N FO
A B S T R A C T
Keywords: Black hole computation Kerr/CFT correspondence Holographic principle Information theory Gamma-ray spectroscopy Shannon entropy
This paper focuses on an alternative, more physically realistic model of computation than Etesi and Németi’s relativistic computer in a Malament-Hogarth spacetime (2002) that uses the black hole itself combined with an external observer equipped with a source and some method of measurement of gamma-rays, as opposed to sending a classical computer into a black hole and exploiting the properties of the spacetime to achieve hypercomputation. The source of output, Hawking radiation, is considered along with the constraints imposed by the holographic principle which limit the number of degrees of freedom in the system and consequently the maximum usable information. The Bekenstein-Hawking entropy is converted from the traditional form in terms of the horizon area to that of the Shannon entropy, establishing an analogy between the physical and computational perspectives of the system. Next examples are considered to establish the approximate order of the necessary excitation energy and the resulting gamma-ray interactions which form the input from the observer. Finally, the Turing completeness of the language for this model is considered through a simulation of the Turing machine. The goal is to introduce a model of computation that can later be used to study the relationship between computability and physical systems.
Introduction
Geometry of holographic duals
In contrast to past research on computation in black holes which focused on the idea of sending a computer into the black hole and exploiting its relativistic properties (as in the case of [7], here it is proposed that the black hole itself be used as a model of computation, motivated by the unique properties arising from holographic duality and the greater feasibility of such a system in the real world. This is achieved by using the energy states of molecules behind the event horizon as the states of the computer, with the movement of the external observer measuring the system corresponding to the shifting of the computer’s tape and the change in states combined with the resulting Hawking radiation corresponding to the output. A stream of photons at the appropriate energy levels is used as input to change the states of the computer. In this way a correspondence is formed between the physical and computational models of the system which may then contribute to a proof of its Turing completeness. In the following sections the methods for essential components of the computer will be developed which will then contribute to the proof of the Turing completeness of this model of computation. The goal is to introduce a model of computation which can be used to investigate the effects of physical constraints imposed by the holographic principle and the Kerr/CFT correspondence on computation and vice versa.
The AdS/CFT correspondence allows for a dual description of an anti-de Sitter space and a conformal field theory of one less dimension, one of the most well known of which is the correspondence between the AdS5 × S5 space and the D = 4, Ɲ = 4 supersymmetric SU(N) YangMills theory. For a realistic model of a black hole we need to find the holographic dual to a Kerr solution, since we assume that the black hole is in an equilibrium state and has a nondegenerate event horizon [16]. This duality is called the Kerr/CFT correspondence and the associated conformal field theory is a chiral half of a two-dimensional CFT [9]. The symmetry of the two-dimensional CFT is described by two copies of the Virasoro algebra with generators that satisfy the below commutation relations, where m, n ∈ Z and c denotes the central charge.
[Lm , Ln] = (m − n) Lm + n +
c (m3 − m) δm + n,0 12
[c, Ln] = 0 We only consider one copy corresponding to a chiral half of the CFT. Representations of the Virasoro algebra arise from the highest weight state |h > = ϕ(0)|0 > which satisfies L0|h > = h|h > , where ϕ is a primary field and h denotes the conformal weight [8].
E-mail address: [email protected]. https://doi.org/10.1016/j.rinp.2019.102188 Received 19 February 2019; Accepted 9 March 2019 Available online 14 March 2019 2211-3797/ © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Results in Physics 13 (2019) 102188
G.R. Andrews
Black hole output
Hawking entropy to the form of a Shannon entropy. This analogy arises from the Bekenstein-Hawking entropy’s dependence on the horizon area [17]. The entanglement (or von Neumann) entropy may be written in similar terms to the Bekenstein-Hawking entropy, as shown below in the case of a ℝ1,d conformal field theory [17].
The fact that particles cannot escape black holes through classical means (arising from the second law of black hole thermodynamics [1] poses a challenge to obtaining an observable output, however Hawking radiation circumvents this problem through quantum tunneling. Section 2.1 considers Hawking radiation, a process described by the tunneling of particles out of the potential well behind the black hole horizon [10,14]. A relationship between the energy of the tunneling shell and the emission rate from the black hole is also established. Section 2.2 considers the constraints imposed by the holographic principle on information density in the black hole.
SBH =
Ah 4GN
SA = −trA ρA logρA =
A γA 4GN(d + 2)
ρA = trB |Ψ > < Ψ|
Hawking radiation
This entanglement entropy itself may be written in the form of the Shannon entropy [3].
Hawking radiation provides a source of information via tunneling that avoids the problem caused by the second law of black hole thermodynamics. Hawking presents such an argument for the emission of particles (1975). Furthermore, Parikh and Wilczek [14] derive this phenomenon directly from the description of it as a tunneling process. The exponential part of the emission rate Γ from Hawking radiation is related to the mass of the black hole and the energy of the particle that will potentially escape:
ρA =
∑ λj |j > < j| → SA = − ∑ λjlogλj j
j
The next step is to relate the Bekenstein-Hawking entropy directly to the Shannon entropy, which intuitively follows from the previously shown correspondence involving the entanglement entropy and is shown below. Let Ah denote the area of the horizon which corresponds to the area of the holographic screen. Let Ω and β denote the inside and outside of the black hole respectively. Then we change the form of SBH to that of the entanglement entropy, writing the density matrix μβ in terms of an orthonormal basis |Ψi > .
ω
Γ e−8πω (M − 2 ) Parikh & Wilczek [14] from this relation by introducing proportionality constant α we may obtain a graph relating Γ and ω which is shown in Fig. 1. According to Fig. 1 Γ may only be made large if ω is either small or large relative to the mass M of the black hole (close to 0 in the former case and 2 M in the latter case). Since M < 0 is ignored we must excite the particles to be emitted past ω = 2 M (where Γ will blow up) in order to achieve arbitrarily large values of Γ (which must be large enough in order that output from the black hole may be detected by distant observers in a reasonable amount of time). The mass of the black hole may be calculated through its relation to the Hawking temperature and proper distance from the black hole in Rindler coordinates [4]. This will involve a measurement of the temperature at a sufficiently far distance from the black hole since it varies inversely with the distance.
SBH =
Ah → SBH = −trβ μβ logμβ 4GN
μβ = trΩ |Ψ > < Ψ| Now we may write SBH in the form of the Shannon entropy by writing μβ in terms of its eigenvectors |Ψj > .
μβ =
∑ μ βj |Ψj > < Ψj| j
⎛ ∑ μ βj |Ψj > < Ψj| ∑ logμ βi |Ψi > < Ψi|⎞⎟ ⎜ i ⎝ j ⎠ |Ψj > < Ψj | ∑ logμ βi |Ψi > < Ψi| Ψk >
SBH = −trβ (μβ logμβ ) = −trβ = − ∑ < Ψk | ∑ μ βj k
The Bekenstein-Hawking entropy as a Shannon entropy
j
i
= − ∑ δkj μ βj δji logμ βi δik = − ∑ μ βj logμ βj
To develop the correspondence between the physical and computational interpretations of the model, we will convert the Bekenstein-
k , j, i
j
All of this work merely serves to show the similarity between the Bekenstein-Hawking entropy and the holographic entanglement entropy and its resulting potential application to our computer. For this reason we are interested in whether the Bekenstein-Hawking entropy may be physically interpreted as a holographic entanglement entropy. Solodukhin considers this question at length, posing several problems which must be solved in order to make such a connection (2011): (1) that the UV divergence of the entanglement entropy may render it inconsistent with the Bekenstein-Hawking entropy which is finite; (2) that the entanglement entropy depends on the number of types of fields unlike the Bekenstein-Hawking entropy; and (3) that the different behavior of non-minimally coupled fields such as gauge bosons and gravitons with respect to the entanglement entropy may affect the possibility of the interpretation of the Bekenstein-Hawking entropy as an entanglement entropy. Examples where the two entropies are equal include a model with minimally coupled fields [18], a 3-brane with the Z2 symmetry group in an AdS spacetime [11,18], and the Kaluza-Klein model [18]. We will only consider the first example here for the sake of simplicity, clarity, and brevity. In a minimally coupled field theory with N0 scalars and N1/ 2 Dirac fermions, the induced gravitational action reads
Fig. 1. Emission rate as a function of escaping particle energy. 2
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Wind = −
1 16πGind
∫E R
High energy excitations
g d 4x
In order to establish the practicality of excitations on the order of 2 M we must estimate the required energy levels for a realistic black hole. For example, we may choose its mass to equal to that of earth.
with
1 N = , N = N0 + 2N1/2 Gind 12π ∊2
M = 5.972 × 1024kg
where N denotes the total number of field species in the model. We can show that
Sent =
Then the required energy ω to achieve the emission rate Γ = α may be approximated.
N 1 A (Σ) = A (Σ) = SBH , 48π ∊2 4Gind
ω = 2M = 1.194 × 1023cm−1 Converting to the more familiar energy units, we have
noting that the renormalization statement assures that Gind and Sent are inversely proportional. This example circumvents (1) and (2), however the absence of any non-minimally coupled fields leaves (3) unanswered. The general solution to calculating the entanglement entropy is believed by some to be solvable with a string theoretic approach [18], which Susskind and Uglum proposed [19]. A direct calculation of the entanglement entropy would allow for a complete answer to the comparison of the Bekenstein-Hawking and holographic entanglement entropies. The complexity of such an approach, however, has rendered progress towards such a solution slow. This is one topic of interest for future research in the development of this model of computation.
E = 1.480 × 1019eV , which corresponds to the energy of gamma-rays. In general the energy required to achieve ω = 2 M necessitates the use of gamma-rays for realistic black holes (those that can be detected using current methods). Low energy excitations In order to maximize the emission rate ω must be minimized to the smallest energy which the contents of the black hole will absorb. Continuing with our example and letting Γ = α/2, we obtain ω.
Arguments from the holographic principle
ω=M−
We now turn our attention to the holographic principle, which relates the world volume to the boundary of the system and from which arises the AdS/CFT correspondence, starting with the Bekenstein bound in order to obtain a limit on the usable information for a given volume. The Bekenstein bound limits the maximum amount of entropy in the system which is related to the number of quantum states in that system [5]. It too may be considered an analogue to some relation involving information entropy like in the case of the Bekenstein-Hawking entropy. The equation for this bound is given below.
4π 2M 2 − π ln2 ≈ 0cm−1 2π
Since the difference caused by the π ln 2 term is small when M is large (requiring extreme precision when calculating the solution), it is advisable to find a bound for ω. A well known inequality gives us
4π 2M 2 − π ln2 > 2π
4π 2M 2 − 2π
π ln2 2π
From this inequality we have a bound for ω:
ω<
2πkER S≤ hc
π ln2 = 0.235m−1 = 2.35 × 10−3cm−1 2π
E < 2.91 × 10−7eV By considering this analogy and the relation between the number of degrees of freedom in the given volume and the horizon area we may conclude that the holographic principle limits the usable information for a given volume by limiting its number of degrees of freedom, i.e. the number of quantum states in the system. This intuitively follows from the analogous limitations on information density imposed by the number of individual states of a computer (which for the purposes of this discussion is considered to be a Turing machine). The corresponding bound from an information theoretic perspective is given below, where E is the energy and I is the amount of information enclosed by a sphere with radius R, which Bekenstein proposed [20,2].
I≤
This corresponds to the Very High Frequency (VHF) range. The calculation shows that a high-energy excitation is preferable due to the higher attainable emission rate. Gamma ray interactions Gamma rays with energies of more than 1 TeV, such as was considered in the example in Section 4.1, mostly interact with infrared photons with a wavelength greater than 1 μm [12]. For a typical high energy excitation (> 10 MeV) caused by gamma-rays such as the example in Section 4.1, pair production gamma-ray interactions are dominant [15]. In such an interaction an electron-positron pair is created provided that a sufficient energy 2m0c2 (the mass of the electronpositron pair) is supplied by the gamma-ray (Rittersdorf). When a positron annihilates an electron upon contact, two 0.511 MeV photons are emitted (Rittersdorf). Secondary electron escape and Bremsstrahlung escape may be disregarded due to the localization of particles caused by the black hole’s gravitational pull. This leaves only tunneling via Hawking radiation as a viable method of escape. For our example black hole the 0.511 MeV photons do not have sufficient energy for the “critical point” Γ = α to be achieved. Thus for a black hole with the mass of earth the emission rate for the photons created via annihilation in pair production gamma-ray interactions is small. Future research could examine the process of pair production near the horizon of a Kerr black hole taking into account the Kerr/CFT correspondence and the associated string theory.
2πER M ⎞⎟ ⎛ R ⎞ bits = 2.57 × 10 45 ⎛⎜ hc ln2 1 kilogram ⎝ ⎠ ⎝ 1meter ⎠
Thus we have a bound for the maximum usable information for a topologically spherical black hole which depends on the mass and radius. Further work will involve describing Hawking radiation for a Kerr solution through the Kerr/CFT correspondence near the horizon and an associated string theory and finding a stronger information bound. Black hole input In order to excite the particles inside the black hole and thus change their states we will consider irradiating them with photons. Fig. 1 showed that the emission rate Γ peaks for values of ω larger than 2 M and close to 0, which sections 3.1 and 3.2 consider respectively. 3
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Turing Completeness of Language for Black Hole
where the propagator − − 1 < X (z , z ) X (w, w ) > = − log|z − w| 2
Since as we have seen the black hole possesses inherent properties which are analogues to components of a Turing machine, the most obvious and direct way to prove its Turing completeness is through a simulation of the Turing machine, showing that corresponding parts perform equivalent operations. The contents of the black hole itself serve as the tape and may be scaled to an arbitrarily large size by increasing the mass of the black hole; therefore the requirement for an indefinitely extensible tape is satisfied. In this model the observer serves as a head which can move in order to shift the tape and read the output from the machine via Hawking radiation. The states assigned to each position on the tape are encoded in the energy levels for each unit of matter. The external observer is limited in the number of measurable states by the choice of equipment (since the energy levels associated with all types of particles are quantized and there is a finite amount of energy in the universe). Finally, there exists a set of instructions which form a Turing-complete language from these physical components: the state of a position on the tape may be changed by irradiating that particle with light, the head may be moved, and information may be read by the head from the Hawking radiation. We will now define the corresponding Turing machine in more formal terms to present directions for future research in more complete physical descriptions of each of the components. This Turing machine is represented by
with z = ½(σ1 + iσ0) and 2idz^dz ̅ = dσ1^dσ0. X(z, z)̅ may be written in terms of left and right movers (thus separating holomorphic and antiholomorphic dependence): −
X (z , z ) =
− 1 (xL (z ) + xR (z )) 2
where the corresponding propagators are −
−
−
−
< xL (z ) xL (w ) > = −log(z − w ), < xR (z ) xR (w ) > = −log(z − w ). This quantization procedure is used for the case of the free boson and is described far more completely in texts on conformal field theory such as [8]. To differentiate between different input symbols in Σ we use the energy of the photon from the Schrodinger equation absorbed by the black hole. Tape alphabet Γ There are massive particles behind the black hole horizon which form the tape. The tape alphabet Γ is composed of the set of possible energy eigenstates for the corresponding unit quantum systems (i.e. the smallest unit which is equivalent to a single location on the tape) which are ultimately constrained by the Planck dimensions. The simplest such unit is a free fermion, which will be considered briefly here. For a free massless fermion, after preserving only the left moving (or holomorphic) half (since the CFT2 being considered is a chiral half), the action becomes
M = (Q, Σ, Γ, δ , q0 , ⧠, F ) where Q is the set of internal states, Σ is the input alphabet, Γ is the tape alphabet, δ is the transition function, □ is the blank symbol, q0 is the initial state, and F is the set of final states. This is the universally accepted definition of a Turing machine which may be found in elementary textbooks on theoretical computer science (such as [13]. The sets of states may be defined on the boundary two dimensional conformal field theory in terms of the possible energy eigenvalues of the involved particles (using the Schrodinger equation). The description of interacting fields in the resulting interactions on the CFT2 (an abbreviation for a two dimensional conformal field theory) falls outside the scope of the paper, however we will consider the case of free bosons and fermions which together compose the idealized states in Q, Σ, and Γ. In what follows the resource used is a set of lecture notes presented by Ginsparg [8].
1 8π
S=
−
∫ ψ∂ψ.
The field ψ satisfies the short distance singularity
ψ (z ) ψ (w ) = −
1 z−w −
and the operator ∂ is related to the Feynman’s Dirac operator in the following way.
∂x − i∂y ⎞ ⎛ 0 ∂ ⎞ 0 D = σx ∂x + σy ∂y = ⎛⎜ ⎟ ⎟ ⎜− ∂ + ∂ i 0 ⎠ ⎝∂ 0⎠ y ⎝ x A more realistic field may be specified by considering the case of a free massive Dirac fermion whose Euclidean action in d dimensions is given by
Set of internal states Q The internal state of the black hole may be represented by the set of quasi-primary fields of the CFT2 which may be used to generate the secondary fields (consisting of the derivatives of all orders of the primary fields ϕi which are referred to as descendant fields). Thus we may represent the set of internal states Q as the set of all possible configurations of the set of quasi-primary fields which represent their respective conformal families.
−
S [ψ , ψ] =
−
∫ ddxψ (γμ ∂μ + m) ψ
where the γμ∂μ term is the Dirac operator alternatively represented in Feynman slash notation [6]. The energy eigenvalues of the free fermion form elements of the set Γ as in Section 5.2.
Q = {(ϕ1, ⋯, ϕn )1, ⋯, (ϕ1, ⋯, ϕn )i} Conclusion Input alphabet Σ
We have shown that black holes combined with an external beam of gamma-rays form a viable model of computation; the Turing completeness of its associated language arises intuitively from direct analogues between the Turing machine and the physical device. Although it is currently not feasible to test such a model of computation in the real world, its unconventionality provides insights into new ways of thinking about leveraging nature for computational purposes as well as analyzing physical processes from a computational perspective (since physical events themselves may be regarded as instances of transition functions) and the absence of a classical computer behind the horizon
For the input we consider a free (non-interacting) boson on a CFT2. With Euclidean space and time coordinates σ1 and σ0 we compactify the space coordinate by σ1 = σ1 + 2π which defines a cylinder. This defines the geometry for the radial quantization procedure to take place as described in [8]. The string theory normalization for the action is given by −
S=
∫ L = 21π ∫ ∂X ∂X 4
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renders the model more physically realistic due to the inevitable crushing of any such computer by the tidal forces of the black hole. Future work should look for extensions to the Turing completeness of this model, i.e. the theoretical potential for hypercomputation as has been examined for relativistic computation in Malament-Hogarth spacetimes [7], as well as further details in the feasibility of methods for observer-provided gamma-ray beams and measurement of the gammarays emitted from the black hole. Additionally, the theoretical description of pair production gamma-ray interactions and Hawking radiation in the context of the Kerr/CFT correspondence and string theory should be formulated in future research and stronger information bounds must be found to better define the limitations of this model of computation.
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Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Bardeen JM, Carter B, Hawking SW. The four laws of black hole mechanics. Commun Math Phys 1973;31(2):161–70. [2] Bekenstein JD. Black holes and information theory. Contemp Phys 2003;45(1):31–43. [3] Bengtsson I, Zyczkowski K. Geometry of quantum states: an introduction to
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