What is the maximum rate at which entropy of a string can increase?
K. Ropotenko
备注：
11 pages, no figures; formulas (18), (20) are corrected (the quantum constant is added), a point concerning a relation between the Hawking and Hagedorn temperatures is corrected, conclusions unchanged; accepted by Physical Review D for publication
According to Susskind, a string falling toward a black hole spreads exponentially over the stretched horizon due to repulsive interactions of the string bits. In this paper such a string is modeled as a self-avoiding walk and the string entropy is found. It is shown that the rate at which information/entropy contained in the string spreads is the maximum rate allowed by quantum theory. The maximum rate at which the black hole entropy can increase when a string falls into a black hole is also discussed.
The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string.
The string-black hole correspondence is considered in the context of the correspondence principle proposed recently by Horowitz and Polchinski. We demonstrate that the entropy of string states and the entropy of a Schwarzschild black hole can be matched including the subleading terms which depend on mass logarithmically. We argue the necessity to include the string interaction (with coupling $g$) in the consideration and propose the $g^2$-dependent modification of the string entropy. The matching of it with the entropy of Schwarzschild black hole is analyzed.
Suggested correspondence between a black hole and a highly excited elementary string is explored. Black hole entropy is calculated by computing the density of states for an open excited string. We identify the square root of oscillator number of the excited string with Rindler energy of black hole to obtain an entropy formula which, not only agrees at the leading order with the Bekenstein-Hawking entropy, but also reproduces the logarithmic correction obtained for black hole entropy in the quantum geometry framework. This provides an additional supporting evidence for correspondence between bl…
We provide formulas for the entropy of free-string states depending on their mass, charges and size, both in bosonic and superstring theory (IIA or IIB). We properly define these quantities in full-fledged string theory. We then investigate the corrections to the entropy due to self-interactions of the string for states with fixed mass, charge and size, both for BPS and non-BPS configurations. Again, the analysis is performed using string theory techniques.
For extremal black holes, the thermodynamic entropy is not proportional to the area. The general form allowed by thermodynamics is worked out for three classes of extremal black hole solutions of string theory and shown to be consistent with the entropy calculated from the density of elementary string states. On the other hand, the entanglement entropy does not in general agree with these results.
I show that gravitational entropy can be ascribed to spacetimes containing Misner strings (the gravitational analogues of Dirac strings), even in the absence of any other event horizon (or bolt) structures. This result follows from an extension of proposals for evaluating the stress-energy of a gravitational system which are motivated by the AdS/CFT correspondence.
We study the entanglement entropy for open bosonic strings on multiple $Dp$-branes by using the covariant open string field theory. Choosing one of the spatial coordinates which are tangential to the hyperplane on which $Dp$-branes are located, we divide the hyperplane into two halves. By using the string wavefunction in the Fock space representation, we evaluate the entanglement entropy. The entanglement entropy is found to be proportional to the area of $(p-1)$-dimensional boundary of the bipartite hyperplanes and divergent in the ultraviolet (UV) region as well as in the infrared (IR) regio…
We compute the entropy of extremal black strings in three dimensions, using Strominger's approach to relate the Anti-de-Sitter near-horizon geometry and the conformal field theory at the asymptotic infinity of this geometry. The result is identical to the geometric Bekenstein-Hawking entropy. We further discuss an embedding of three-dimensional black strings in $N=1 D=10$ supergravity and demonstrate that the extremal strings preserve 1/4 of supersymmetries.
It is observed that the three-dimensional BTZ black hole is a supersymmetric solution of the low-energy field equations of heterotic string theory compactified on an Einstein space. The solution involves a non-zero dilaton and NS-NS H-field. The entropy of the extreme black hole can then be computed using string theory and the asymptotic properties of anti-de Sitter space, without recourse to a D-brane analysis. This provides an explicit example of a black hole whose entropy can be computed using fundamental string theory, as advocated by Susskind.
The coupling of a string to gravity allows for Schwarzschild black holes whose entropy to area relation is $S=(A/4)(1-4μ)$, where $μ$ is the string tension. This departure from the A/4 universality results from a string instanton generating a black hole with smaller entropy at a temperature exceeding the Hawking value. The temperature is sensitive to the presence of classical matter outside the black hole horizon but the entropy is not. The horizon materializes at the quantum level. It is conjectured that such a macroscopic non local effect may be operative in retrieving information from incip…
We study the entropy of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) charged black hole, originated from the effective action that emerges in the low-energy of string theory, beyond semiclassical approximations. Applying the properties of exact differentials for three variables to the first law thermodynamics we derive the quantum corrections to the entropy of the black hole. The leading (logarithmic) and non leading corrections to the area law are obtained.
The one-loop contribution to the entropy of a black hole from field modes near the horizon is computed in string theory. It is modular invariant and ultraviolet finite. There is an infrared divergence that signifies an instability near the event horizon of the black hole. It is due to the exponential growth of the density of states and the associated Hagedorn transition characteristic of string theory. It is argued that this divergence is indicative of a tree level contribution, and the Bekenstein-Hawking-Gibbons formula for the entropy should be understood in terms of string states stuck near…
Using adiabatic invariance and the Bohr-Sommerfeld quantization rule we investigate the entropy spectroscopy of a charged black hole of heterotic string theory. It is shown that the entropy spectrum is equally spaced identically to the Schwarzschild, Reissner-Nordström and Kerr black holes. Since the adiabatic invariance method does not use quasinormal mode analysis, there is no need to impose the small charge limit and no confusion on whether the real part or imaginary part is responsible for the entropy spectrum.
We discuss further our proposed modification of the Susskind-Horowitz-Polchinski scenario in which black hole entropy goes over to string entropy as one scales the string length scale up and the string coupling constant down, keeping Newton's constant unchanged. In our approach, based on our 'matter-gravity entanglement hypothesis', 'string entropy' here should be interpreted as the likely entanglement entropy between (approximately) the single long string and the stringy atmosphere which, as we argue, arise in a pure multistring state of given energy in a (rescaled) box. In a previous simple …
Exploiting a recently constructed target space action for the exact string black hole, logarithmic corrections to the leading order entropy are studied. There are contributions from thermal fluctuations and from corrections due to alpha'>0 which for the microcanonical entropy appear with different signs and therefore may cancel each other, depending on the overall factor in front of the action. For the canonical entropy no such cancellation occurs. Remarks are made regarding the applicability of the approach and concerning the microstates. As a byproduct a formula for logarithmic entropy corre…
I review some recent work in which the quantum states of string theory which are associated with certain black holes have been identified and counted. For large black holes, the number of states turns out to be precisely the exponential of the Bekenstein-Hawking entropy. This provides a statistical origin for black hole thermodynamics in the context of a potential quantum theory of gravity.
We show that the entropy resulting from the counting of microstates of non extremal black holes using field theory duals of string theories can be interpreted as arising from entanglement. The conditions for making such an interpretation consistent are discussed. First, we interpret the entropy (and thermodynamics) of spacetimes with non degenerate, bifurcating Killing horizons as arising from entanglement. We use a path integral method to define the Hartle-Hawking vacuum state in such spacetimes and discuss explicitly its entangled nature and its relation to the geometry. If string theory on …
We argue that Bekenstein entropy can be interpreted as the entropy of an effective string with a rescaled tension. Using the AdS/CFT correspondence we show that the Bekenstein entropy on the boundary CFT is given by the entropy of a string at the stretched horizon of the AdS black hole in the bulk. The gravitationally redshifted tension and energy of the string match those required to reproduce Bekenstein entropy.
We discuss whether black hole entropy counts short or long range microstates in quantum gravity. In brick wall and induced gravity models the entropy arises due to short distance correlations across the event horizon cut off at the Planck length. However, the energy of these short range degrees of freedom is too high compared to the black hole energy. We argue that the long string phenomenon, which naturally appears in matrix quantum mechanics, resolves this issue by lowering the excitation energy per degree of freedom. This mechanism also reduces the total number of microscopic degrees of fre…
The quantum corrections to the counting of statistical entropy for the 5+1-dimensional extremal black string in type-IIB supergravity with two observers are studied using anomalous Wess-Zumino actions for the corresponding intersecting D-brane description. The electric-magnetic duality symmetry of the anomalous theory implies a new symmetry between D-string and D-fivebrane sources and renders opposite sign for the RR charge of one of the intersecting D-branes relative to that of the black string. The electric-magnetic symmetric Hilbert space decomposes into subspaces associated with interior a…
We formulate the Rindler space description of rotating black holes in string theories. We argue that the comoving frame is the natural frame for studying thermodynamics of rotating black holes and statistical analysis of rotating black holes gets simplified in this frame. We also calculate statistical entropies of general class of rotating black holes in heterotic strings on tori by applying D-brane description and the correspondence principle. We find at least qualitative agreement between the Bekenstein-Hawking entropies and the statistical entropies of these black hole solutions.
We study the conjectured connection between AdS bubbles (AdS solitons) and closed string tachyon condensations. We confirm that the entanglement entropy, which measures the degree of freedom, decreases under the tachyon condensation. The entropies in supergravity and free Yang-Mills agree with each other remarkably. Next we consider the tachyon condensation on the AdS twisted circle and argue that its endpoint is given by the twisted AdS bubble, defined by the double Wick rotation of rotating black 3-brane solutions. We calculated the Casimir energy and entropy and checked the agreements betwe…
We make some observations regarding string/black hole correspondence with a view to understanding the nature of the quantum degrees of freedom of a black hole in string theory. In particular, we compare entropy change in analogous string and black hole processes in order to support the interpretation of the area law entropy as arising from stringy constituents.
The classical Bekenstein entropy of a black hole is argued to arise from configurations of strings with ends which are frozen on the horizon. Quantum corrections to this entropy are probably finite unlike the case in quantum field theory. Finally it is speculated that all black holes are single string states. The level density of strings is of the right order of magnitude to reproduce the Bekenstein entropy.
During the last few years, exact solutions that describe black holes that are bound to a two-brane in a four dimensional anti-de Sitter bulk have been constructed. In situations wherein there is a negative cosmological constant on the brane, for large masses, these solutions are exactly the rotating BTZ black holes on the brane and, in fact, describe rotating BTZ black strings in the bulk. We evaluate the canonical entropy of a free and massless scalar field (at the Hawking temperature) around the rotating BTZ black string using the brick wall model. We explicitly show that the Bekenstein-Hawk…
We show that the entropy of de Sitter space in any dimension can be understood as the entropy of a highly excited string located near the horizon. The string tension is renormalized to $T \sim Λ$ due to the large gravitational redshift near the horizon. The de Sitter temperature is given by the Hagedorn temperature of the string.
In this paper, we study the entanglement entropy in string theory in the simplest setup of dividing the nine dimensional space into two halves. This corresponds to the leading quantum correction to the horizon entropy in string theory on the Rindler space. This entropy is also called the conical entropy and includes surface term contributions. We first derive a new simple formula of the conical entropy for any free higher spin fields. Then we apply this formula to computations of conical entropy in open and closed superstring. In our analysis of closed string, we study the twisted conical entr…
By using BPS closed string, the entropy is calculated of the extremal five dimensional black hole consisting of Dirichlet onebranes, Dirichlet fivebranes and Kaluza-Klein momentum in the flat background approximation. In our formulations we consider two kinds of BPS closed strings with or without a winding number. In the former case heavy excitation modes of closed strings are used to derive the entropy. In the latter case we have no oscillator modes and consider collective motion of such massless closed strings. The entropy is given by the number of the ways how we divide the Kaluza-Klein mom…
We compute the entropy and the corresponding central charge of the self-dual string soliton in the supergravity regime using the blackfold description of the fully localized M2-M5 intersection.
In this paper, we will compute the holographic complexity (dual to a volume in AdS), holographic fidelity susceptibility and the holographic entanglement entropy (dual to an area in AdS) in a two-dimensional version of $AdS$ which is dual to open strings. We will explicitly demonstrate that these quantities are well defined, and then argue that a relation for fidelity susceptibility and time should hold in general due to the $AdS_2$ version of the classical Kepler's principle. We will demonstrate that it holds for $AdS_2$ solution as well as conformal copies metrics in bulk theory of a prescri…
This is a short review summarizing the current status of the comparison between microscopic and macroscopic entropy of extremal BPS black holes in string theory.
A generalized second law in string cosmology accounts for geometric and quantum entropy in addition to ordinary sources of entropy. The proposed generalized second law forbids singular string cosmologies, under certain conditions, and forces a graceful exit transition from dilaton-driven inflation by bounding curvature and dilaton kinetic energy.
In a previous paper, we proposed a probability interpretation for higher genus amplitudes of BMN (Berenstein-Maldacena-Nastase) strings in a pp-wave background with infinite negative curvature. This provides a natural definition of the entropy of a BMN string as the Shannon entropy of its corresponding probability distribution. We prove a universal upper bound that the entropy grows at most logarithmically in the strong string coupling limit. We also study the entropy by numerical methods and discuss some interesting salient features.
The method of topological renormalization in anti-de Sitter (AdS) gravity consists in adding to the action a topological term which renders it finite, defining at the same time a well-posed variational problem. Here, we use this prescription to work out the thermodynamics of asymptotically locally anti-de Sitter (AlAdS) spacetimes, focusing on the physical properties of the Misner strings of both the Taub-NUT-AdS and Taub-Bolt-AdS solutions. We compute the contribution of the Misner string to the entropy by treating on the same footing the AdS and AlAdS sectors. As topological renormalization …
This is an expanded version of a talk given at ``{\em IInd Recontre du Vietnam}'' held at Ho Chi Minh City in October, 1995. We discuss several aspects of black hole entropy in string theory. We first explain why the geometric entropy in two dimensional noncritical string theory is nonperturbatively finite. We then explain the philosophy of regarding massive string states as black branes and how the Beckenstein-Hawking entropy for extremal BPS black holes may be understood as coming from degeneracy of string states. This is then discussed in the context of D-strings in Type IIB superstrings. W…
The coupling of a Nambu-Goto string to gravity allows for Schwarzschild black holes whose entropy to area relation is $S=(A/4)(1-4μ)$, where $μ$ is the string tension.
We investigate the inclusion of 10-dimensional string loop corrections to the entropy function of two-charge extremal small black holes of the heterotic string theory compactified on S^1 x T^5 and show that the entropy is given by π\sqrt{a q_1 q_2+b q_1} where q_1 and q_2 are the charges with q_1 >> q_2 >> 1 and a and b are constants. Incorporating certain multi-string states into the microstate counting, we show that the new statistical entropy is consistent with the macroscopic scaling for one and two units of momentum (winding) and large winding (momentum). We discuss our scaling from the p…
Using the spectral representation approach to the Zamolodchikov's c-function and the Maldacena conjecture for the D1-branes, we compute the entropy of type IIB strings. An agreement, up to a numerical constant which cannot be determined using this approach, with the Bekenstein-Hawking entropy is found.
We examine the BPS and low energy non-BPS excitations of the D-string, in terms of open strings that travel on the D-string. We use this to study the energy thresholds for exciting a long D-string, for arbitrary winding number. We also compute the leading correction to the entropy from non-BPS states for a long D-string, and observe the relation of all these quantities with the corresponding quantities for the elementary string.
We investigate logarithmic corrections to the entropy of supersymmetric, rotating, asymptotically AdS$_5$ black holes and black strings. Within the framework of the AdS/CFT correspondence, the entropy of these black objects is determined, on the field theory side, by the superconformal index and the refined topologically twisted index of $\mathcal{N}=4$ supersymmetric Yang-Mills theory, respectively. We read off the logarithmic correction from those field-theoretic partition functions. On the gravity side, we take the near-horizon limit and apply the Kerr/CFT correspondence whose associated ch…
We show that the entropy of Schwarzschild black holes in any dimension can be described by a gas of free string bits at the stretched horizon. The number of string bits is equal to the black hole entropy and energy dependent. For an asymptotic observer the bit gas is at the Hawking temperature. We show that the same description is also valid for de Sitter space--times in any dimension.
In local quantum field theory, the entanglement entropy of a region is divergent due to the arbitrary short-wavelength correlations near the boundary of the region. Quantum gravitational fluctuations are expected to cut off the entropy of the ultraviolet modes. Guided by this, we study the entanglement entropy in closed string theory. We factorize the configuration space of closed strings based on the position of the center of mass. We then compute the one-loop Rényi partition functions corresponding to this factorization using the replica method and string field theory. The short-wavelength m…
In this article we derive the Bekenstein-Hawking formula of black hole entropy from a single string. We consider a open string in the Rindler metric which can be obtained in the large mass limit from the Schwarzschild black hole metric. By solving the field equations we find a nontrivial solution with the exact value of the Hawking temperature. We see that this solution gives us the Bekenstein-Hawking formula of black hole entropy to leading order of approximation. This string has effectively a rescaled string tension having a relation with a redshift factor. It is also pointed out that this f…
After discussing some old (and not-so-old) entropy bounds both for isolated systems and in cosmology, I will argue in favour of a "Hubble entropy bound" holding in the latter context. I will then apply this bound to recent developments in string cosmology, show that it is naturally saturated throughout pre-big bang inflation, and claim that its fulfilment at later times has interesting implications for the exit problem of string cosmology.
Using a boundary counterterm prescription motivated by the AdS/CFT conjecture, I evaluate the energy, entropy and angular momentum of the class of Kerr-NUT/bolt-AdS spacetimes. As in the non-rotating case, when the NUT charge is nonzero the entropy is no longer equal to one-quarter of the area due to the presence of the Misner string. When the cosmological constant is also non-zero, the entropy is bounded from above.
The entropy of the states associated to the solutions of the equations of motion of the bosonic open string with combinations of Neumann and Dirichlet boundary conditions is given. Also, the entropy of the string in the states $| A^i > = α^{i}_{-1} |0>$ and $| ϕ^a > = α^{a}_{-1} |0>$ that describe the massless fields on the world-volume of the Dp-brane is computed.
We calculate the quantum corrections to the entropy of a very large black hole, coming from the theory of a $D$-dimensional, non-critical bosonic string. We show that, for $D >2$, as a result of modular invariance the entropy is ultraviolet finite, although it diverges in the infrared (while for $D=2$ the entropy contains both ultraviolet and infrared divergences). The issue of modular invariance in field theory, in connection with black-hole entropy, is also investigated.
Entanglement entropy for spatial subregions is difficult to define in string theory because of the extended nature of strings. Here we propose a definition for Bosonic open strings using the framework of string field theory. The key difference (compared to ordinary quantum field theory) is that the subregion is chosen inside a Cauchy surface in the "space of open string configurations". We first present a simple calculation of this entanglement entropy in free light-cone string field theory, ignoring subtleties related to the factorization of the Hilbert space. We reproduce the answer expected…
The Euclidean action and entropy are computed in string-generated gravity models with quadratic curvatures, and used to argue that a negative mass extremal metric is the background for hyperbolic (k=-1) black hole spacetimes, k being the curvature constant of the event horizon. The entropy associated with a black hole is always positive for k=(0,1) family. The positivity of energy condition also ensures that the k=-1 (extremal) entropy is non-negative.
We construct a four-dimensional BPS saturated heterotic string solution from the Taub-NUT solution. It is a non-extremal black hole solution since its Euler number is non-zero. We evaluate its black hole entropy semiclassically. We discuss the relation between the black hole entropy and the degeneracy of string states. The entropy of our string solution can be understood as the microscopic entropy which counts the elementary string states without any complications.
We extend our recent analysis of the entropy of extremal black strings with traveling waves. We previously considered waves carrying linear momentum on black strings in six dimensions. Here we study waves carrying angular momentum on these strings, and also waves carrying linear momentum on black strings in five dimensions. In both cases, we show that the horizon remains homogeneous and compute its area. We also count the number of BPS states at weak string coupling with the same distribution of linear and angular momentum, and find complete agreement with the black string entropy.
We generalize the entropy function formalism to five-dimensional and four-dimensional non-extremal black holes in string theory. In the near horizon limit, these black holes have BTZ metric as part of the spacetime geometry. It is shown that the entropy function formalism also works very well for these non-extremal black holes and it can reproduce the Bekenstein-Hawking entropy of these black holes in ten dimensions and lower dimensions.
