# Alcubierre ## The warp drive: hyper-fast travel within general relativity \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0009013), [PDF](https://arxiv.org/pdf/gr-qc/0009013.pdf)\] \[Abstract: It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the ''[[0263 Warp drives|warp drive]]'' of science fiction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.\] # Corley, Jacobson ## Collapse of Kaluza-Klein Bubbles \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9403017), [PDF](https://arxiv.org/pdf/gr-qc/9403017.pdf)\] \[Abstract: [[0169 Kaluza-Klein|Kaluza-Klein]] theory admits "[[0168 Bubble of nothing|bubble]]" configurations, in which the circumference of the fifth dimension shrinks to zero on some compact surface. A three parameter family of such bubble initial data at a moment of time-symmetry (some including a magnetic field) has been found by [[BrillHorowitz1991|Brill and Horowitz]], generalizing the (zero-energy) "[[1982#Witten|Witten bubble]]" solution. Some of these data have negative total energy. We show here that all the negative energy bubble solutions start out expanding away from the moment of time symmetry, while the positive energy bubbles can start out either expanding or contracting. Thus it is unlikely that the negative energy bubbles would collapse and produce a naked singularity.\] ## Motivation - we don't want to form naked singularities, but collapsing bubbles might be expected to form them - we don't want to violate [[0005 Black hole second law|BH second law]]. If we have localised bubbles with negative energy, they might be thrown into a black hole and decrease entropy. This is hopefully avoided if the negative-energy bubbles all expand and refuse to be thrown into a black hole. # Dorn, Otto ## Two and three-point functions in Liouville theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9403141), [PDF](https://arxiv.org/pdf/hep-th/9403141.pdf)\] \[Abstract: Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for [[0562 Liouville theory|Liouville]] exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.\] # Hayward ## Gravitational energy in spherical symmetry \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9408002), [PDF](https://arxiv.org/pdf/gr-qc/9408002.pdf)\] \[Abstract: Various properties of the [[0595 Quasi-local energy#Misner-Sharp energy|Misner-Sharp spherically symmetric gravitational energy]] $E$ are established or reviewed. In the Newtonian limit of a perfect fluid, $E$ yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. For test particles, the corresponding Hájíček energy is conserved and has the behavior appropriate to energy in the Newtonian and special-relativistic limits. In the small-sphere limit, the leading term in $E$ is the product of volume and the energy density of the matter. In vacuo, $E$ reduces to the Schwarzschild energy. At null and spatial infinity, $E$ reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies, respectively. The conserved Kodama current has charge $E$. A sphere is trapped if $E>1/2 r$, marginal if $E=1/2r$, and untrapped if $E<1/2r$, where $r$ is the areal radius. A central singularity is spatial and trapped if $E>0$, and temporal and untrapped if $E<0$. On an untrapped sphere, $E$ is nondecreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that $E\ge0$ on an untrapped spatial hypersurface with a regular center, and $E\ge1/2r_0$ on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius $r_0$. All these inequalities extend to the asymptotic energies, recovering the Bondi-Sachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured [[0476 Penrose inequality|Penrose inequality]] for black or white holes. Implications for the [[0221 Weak cosmic censorship|cosmic censorship]] hypothesis and for general definitions of [[0592 Gravitational energy|gravitational energy]] are discussed.\] # Iyer, Wald ## Some properties of Noether charge and a proposal for dynamical black hole entropy \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9403028), [PDF](https://arxiv.org/pdf/gr-qc/9403028.pdf)\] \[Abstract: We consider a general, classical theory of gravity with arbitrary matter fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian, $\textbf{L}$. We first show that $\textbf{L}$ always can be written in a "manifestly covariant" form. We then show that the symplectic potential current ($n-1$)-form, $\mathbf{\Theta}$, and the symplectic current ($n-1$)-form, ${\omega}$, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current ($n-1$)-form, $\textbf{J}$, and corresponding Noether charge ($n-2$)-form, $\textbf{Q}$. We derive a general "decomposition formula" for $\textbf{Q}$. Using this formula for the Noether charge, we prove that the first law of [[0127 Black hole thermodynamics|black hole mechanics]] holds for arbitrary perturbations of a stationary black hole. (For [[0006 Higher-derivative gravity|higher derivative theories]], previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, $S_{dyn}$, of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of $\textbf{L}$, $\mathbf{\Theta}$, and $\textbf{Q}$. However, the issue of whether this dynamical entropy in general obeys a "[[0005 Black hole second law|second law]]" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.\] ## Comments - gives the EOM for arbitrary [[0006 Higher-derivative gravity|HDG]] coupled to matter ## Ambiguities - summarised in this page: [[0018 JKM ambiguity]] ## Dynamical black hole entropy - $S_{d y n}=\frac{1}{4} \operatorname{Area}[\mathcal{C}]+8 \pi \alpha \int_{\mathcal{C}} {}^{(n-2)}R$ - not known whether it satisfied 2nd law - [[2015#Wall (Essay)]] solves this problem # Malec, Murchadha ## Trapped surfaces and the Penrose inequality in spherically symmetric geometries \[Links: [Inspire](https://inspirehep.net/literature/37047)\] \[Abstract: We demonstrate that the [[0476 Penrose inequality|Penrose inequality]] is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trapped surfaces. This formulation can also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of General Relativity.\] # Petkova, Zuber ## On Structure Constants of $sl(2)$ Theories \[Links: [arXiv](https://arxiv.org/abs/hep-th/9410209), [PDF](https://arxiv.org/pdf/hep-th/9410209)\] \[Abstract: Structure constants of [[0599 Minimal models|minimal conformal theories]] are reconsidered. It is shown that *ratios* of structure constants of spin zero fields of a non-diagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjacency matrix of the corresponding Dynkin diagram. This is proved by inspection, which leads us to carefully determine the *signs* of the structure constants that had not all appeared in the former works on the subject. We also present a proof relying on the consideration of lattice correlation functions and speculate on the extension of these identities to more complicated theories.\] # Polchinski ## Combinatorics of Boundaries in String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9407031), [PDF](https://arxiv.org/pdf/hep-th/9407031)\] \[Abstract: We investigate the possibility that stringy nonperturbative effects appear as holes in the world-sheet. We focus on the case of Dirichlet string theory, which we argue should be formulated differently than in previous work, and we find that the effects of boundaries are naturally weighted by $e^{-O(1/g_{\rm st})}$.\] # Sfetsos ## On gravitational shock waves in curved spacetimes \[Links: [arXiv](https://arxiv.org/abs/hep-th/9408169), [PDF](https://arxiv.org/pdf/hep-th/9408169.pdf)\] \[Abstract: Some years ago [[1985#Dray, t'Hooft|Dray and 't Hooft]] found the necessary and sufficient conditions to introduce a [[0117 Shockwave|gravitational shock wave]] in a particular class of vacuum solutions to Einstein's equations. We extend this work to cover cases where non-vanishing matter fields and cosmological constant are present. The sources of gravitational waves are massless particles moving along a null surface such as a horizon in the case of black holes. After we discuss the general case we give many explicit examples. Among them are the $d$-dimensional charged black hole (that includes the 4-dimensional Reissner-Nordström and the $d$-dimensional Schwarzschild solution as subcases), the 4-dimensional De-Sitter and Anti-De-Sitter spaces (and the Schwarzschild-De-Sitter black hole), the 3-dimensional Anti-De-Sitter black hole, as well as backgrounds with a covariantly constant null Killing vector. We also address the analogous problem for string inspired gravitational solutions and give a few examples.\] ## Summary - on [[0117 Shockwave]] - extending [[1985#Dray, t'Hooft]] to include matter fields and cosmological constant # Srednicki ## Chaos and Quantum Thermalization \[Links: [arXiv](https://arxiv.org/abs/cond-mat/9403051), [PDF](https://arxiv.org/pdf/cond-mat/9403051.pdf)\] \[Abstract: We show that a bounded, isolated quantum system of many particles in a specific initial state will [[0541 Thermalisation|approach thermal equilibrium]] if the energy eigenfunctions which are superposed to form that state obey *Berry's conjecture*. Berry's conjecture is expected to hold only if the corresponding classical system is [[0008 Quantum chaos|chaotic]], and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a ==rarefied hard-sphere gas== as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon *[[0040 Eigenstate thermalisation hypothesis|eigenstate thermalization]]*. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an [[0154 Ensemble averaging|ensemble]] of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.\] ## Refs - OG for [[0040 Eigenstate thermalisation hypothesis|ETH]] # Susskind, Uglum ## Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9401070), [PDF](https://arxiv.org/pdf/hep-th/9401070.pdf)\] \[Abstract: In this paper the [[0004 Black hole entropy|entropy]] of an eternal Schwarzschild black hole is studied in the limit of infinite black hole mass. The problem is addressed from the point of view of both canonical quantum gravity and superstring theory. The entropy per unit area of a free scalar field propagating in a fixed black hole background is shown to be quadratically divergent near the horizon. It is shown that such quantum corrections to the entropy per unit area are equivalent to the quantum corrections to the gravitational coupling. Unlike field theory, superstring theory provides a set of identifiable configurations which give rise to the classical contribution to the entropy per unit area. These configurations can be understood as open superstrings with both ends attached to the horizon. The entropy per unit area is shown to be finite to all orders in superstring perturbation theory. The importance of these conclusions to the resolution of the problem of black hole information loss is reiterated.\] ## Refs - [[0004 Black hole entropy]] - explained in detail in [[2022#a]] # Turaev (Book) ## Quantum Invariants of Knots and 3-Manifolds \[Links: [arXiv](https://arxiv.org/abs/hep-th/9409028), [PDF](https://arxiv.org/pdf/hep-th/9409028)\] # Woolgar ## The positivity of energy for AAdS spacetimes \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9404019), [PDF](https://arxiv.org/pdf/gr-qc/9404019.pdf)\] \[Abstract: We use the formulation of asymptotically anti-de Sitter boundary conditions given by Ashtekar and Magnon to obtain a coordinate expression for the general asymptotically AdeS metric in a neighbourhood of infinity. From this, we are able to compute the time delay of null curves propagating near infinity. If the gravitational mass is negative, so will be the time delay (relative to null geodesics at infinity) for certain null geodesics in the spacetime. Following closely an argument given by Penrose, Sorkin, and Woolgar, who treated the asymptotically flat case, we are then able to argue that a negative time delay is inconsistent with non-negative matter-energies in spacetimes having good [[0091 Boundary causality|causal]] properties. We thereby obtain a new [[0116 Positive energy theorem|positive mass theorem]] for these spacetimes. The theorem may be applied even when the matter flux near the boundary-at-infinity falls off so slowly that the mass changes, provided the theorem is applied in a time-averaged sense. The theorem also applies in certain spacetimes having local matter-energy that is sometimes negative, as can be the case in semi-classical gravity.\] ## Related topics - [[0116 Positive energy theorem]] - [[0091 Boundary causality]] - [[0247 Energy conditions]] ## Refs - later paper [[2002#Page, Surya, Woolgar]] ## Summary - use [[0091 Boundary causality|boundary causality]] of AdS to show [[0116 Positive energy theorem|positive energy theorem]]