An official website of the United States government
We propose a new formula for the entropy of a dynamical black hole—valid to leading order for perturbations off of a stationary black hole background—in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In particular, in general relativity, our formula for the entropy of a dynamical black hole differs from the standard Bekenstein-Hawking formula by a term involving the integral of the expansion of the null generators of the horizon. We show that, to leading perturbative order, our dynamical entropy in general relativity is equal to of the area of the apparent horizon. Our formula for entropy in a general theory of gravity is obtained from the requirement that a “local physical process version” of the first law of black hole thermodynamics hold for perturbations of a stationary black hole. It follows immediately that for first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading-order change in entropy occurs at second order in perturbation theory, and the second law is obeyed at leading order if and only if the modified canonical energy flux is positive (as is the case in general relativity but presumably would not hold in more general theories of gravity). Our formula for the entropy of a dynamical black hole differs from a formula proposed independently by Dong and by Wall. We obtain the general relationship between their formula and ours. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter. Published by the American Physical Society2024
more »
« less
Formalizing Human Ingenuity: A Quantitative Framework for Copyright Law’s Substantial Similarity
A central notion in U.S. copyright law is judging the substantial similarity between an original and an (allegedly) derived work. Capturing this notion has proven elusive, and the many approaches offered by case law and legal scholarship are often ill-defined, contradictory, or internally-inconsistent. This work suggests that key parts of the substantial-similarity puzzle are amendable to modeling inspired by theoretical computer science. Our proposed framework quantitatively evaluates how much "novelty" is needed to produce the derived work with access to the original work, versus reproducing it without access to the copyrighted elements of the original work. "Novelty" is captured by a computational notion of description length, in the spirit of Kolmogorov-Levin complexity, which is robust to mechanical transformations and availability of contextual information. This results in an actionable framework that could be used by courts as an aid for deciding substantial similarity. We evaluate it on several pivotal cases in copyright law and observe that the results are consistent with the rulings, and are philosophically aligned with the abstraction-filtration-comparison test of Altai.
more »
« less
- PAR ID:
- 10358595
- Date Published:
- Journal Name:
- 2nd ACM Symposium on Computer Science and Law
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.more » « less
-
Lattice symmetries are central to the characterization of electronic topology. Recently, it was shown that Green's function eigenvectors form a representation of the space group. This formulation has allowed the identification of gapless topological states even when quasiparticles are absent. Here we demonstrate the profundity of the framework in the extreme case, when interactions lead to a Mott insulator, through a solvable model with long-range interactions. We find that both Mott poles and zeros are subject to the symmetry constraints, and relate the symmetry-enforced spectral crossings to degeneracies of the original noninteracting eigenstates. Our results lead to new understandings of topological quantum materials and highlight the utility of interacting Green's functions toward their symmetry-based design. Published by the American Physical Society2024more » « less
-
We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws, $$h_i(x) = c_i x^n$$, $i = 1,2$, where $$c_2 > c_1$$ for $$n \geq 1$$. Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as $$t^{1/3}$$, a result which is universal, i.e., applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for $n>1$, there exists a scaling and a similarity transformation that are independent of $$c_i$$ and $$n$$, which gives rise to the universal $$t^{1/3}$$ power-law for capillary rise. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of $$n$$, and it is shown to be bounded and monotonically decreasing as $$n\to \infty$$. Accordingly, the profile of the interface radius as a function of altitude is also independent of $$c_i$$ and exhibits slight variations with $$n$$. Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude.more » « less
-
Assessing similarity between design ideas is an inherent part of many design evaluations to measure novelty. In such evaluation tasks, humans excel at making mental connections among diverse knowledge sets and scoring ideas on their uniqueness. However, their decisions on novelty are often subjective and difficult to explain. In this paper, we demonstrate a way to uncover human judgment of design idea similarity using two dimensional idea maps. We derive these maps by asking humans for simple similarity comparisons of the form “Is idea A more similar to idea B or to idea C?” We show that these maps give insight into the relationships between ideas and help understand the domain. We also propose that the novelty of ideas can be estimated by measuring how far items are on these maps. We demonstrate our methodology through the experimental evaluations on two datasets of colored polygons (known answer) and milk frothers (unknown answer) sketches. We show that these maps shed light on factors considered by raters in judging idea similarity. We also show how maps change when less data is available or false/noisy ratings are provided. This method provides a new direction of research into deriving ground truth novelty metrics by combining human judgments and computational methods.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
