Area Theorem Research Articles

Rényi second laws for black holes

Hawking’s black hole area theorem provides a geometric realization of the second law of thermodynamics and constrains gravitational processes. In this work we explore a one-parameter extension of this constraint formulated in terms of the monotonicity properties of Rényi entropies. We focus on black hole mergers in asymptotically AdS space and determine new restrictions which these Rényi second laws impose on the final state. We evaluate the entropic inequalities starting from the thermodynamic ensembles description of black hole geometries, and find that for many situations they set more stringent bounds than those imposed by the area increase theorem.

Area theorem for surface plasmons interacting with resonant atoms

Area theorem for surface plasmons interacting with resonant atoms

Sky marginalization in black hole spectroscopy and tests of the area theorem

Sky marginalization in black hole spectroscopy and tests of the area theorem

A generalization of the Hawking black hole area theorem

Hawking’s black hole area theorem was proven using the null energy condition (NEC), a pointwise condition violated by quantum fields. The violation of the NEC is usually cited as the reason that black hole evaporation is allowed in the context of semiclassical gravity. Here we provide two generalizations of the classical black hole area theorem: first, a proof of the original theorem with an averaged condition, the weakest possible energy condition to prove the theorem using focusing of null geodesics. Second, a proof of an area-type result that allows for the shrinking of the black hole horizon but provides a bound on it. This bound can be translated to a bound on the black hole evaporation rate using a condition inspired from quantum energy inequalities. Finally, we show how our bound can be applied to two cases that violate classical energy conditions.

A possible quantum gravity hint in binary black hole merger

We present a semi-rigorous justification of Bekenstein's Generalized Second Law of Thermodynamics applicable to a universe with black holes present, based on a generic quantum gravity formulation of a black hole spacetime, where the bulk Hamiltonian constraint plays a central role. Specializing to Loop Quantum Gravity, and considering the inspiral and post-ringdown stages of binary black hole merger into a remnant black hole, we show that the Generalized Second Law implies a lower bound on the non-perturbative LQG correction to the Bekenstein-Hawking area law for black hole entropy. This lower bound itself is expressed as a function of the Bekenstein-Hawking area formula for entropy. Results of the analyses of LIGO-VIRGO-KAGRA data recently performed to verify the Hawking Area Theorem for binary black hole merger are shown to be entirely consistent with this Loop Quantum Gravity-induced inequality. However, the consistency is independent of the magnitude of the LQG corrections to black hole entropy, depending only on the negative algebraic sign of the quantum correction. We argue that results of alternative quantum gravity computations of quantum black hole entropy, where the quantum entropy exceeds the Bekenstein-Hawking value, may not share this consistency.

A Set-Theoretic Analysis of the Black Hole Entropy Puzzle

Motivated by the known mathematical and physical problems arising from the current mathematical formalization of the physical spatio-temporal continuum, as a substantial technical clarification of our earlier attempt (Etesi in Found Sci 25:327–340, 2020), the aim in this paper is twofold. Firstly, by interpreting Chaitin’s variant of Gödel’s first incompleteness theorem as an inherent uncertainty or fuzziness present in the set of real numbers, a set-theoretic entropy is assigned to it using the Kullback–Leibler relative entropy of a pair of Riemannian manifolds. Then exploiting the non-negativity of this relative entropy an abstract Hawking-like area theorem is derived. Secondly, by analyzing Noether’s theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the set of real numbers in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its “instantaneous” event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too.

Area theorem in a ring laser cavity

Area theorem in a ring laser cavity

Self-induced transparency in a semiconductor quantum dot medium at ultracold temperatures

We investigate the feasibility of minimum absorption and minimum broadening of pulse propagation in an inhomogeneously broadened semiconductor quantum dot medium. The phonon interaction is inevitable in studying any semiconductor quantum dot system. We have used the polaron transformation technique to deal with quantum dot phonon interaction in solving system dynamics. We demonstrate that a short pulse can propagate inside the medium with minimal absorption and broadening in pulse shape. The stable pulse area becomes slightly higher than the prediction of the pulse area theorem and is also dependent on the environment temperature. The change in the final pulse shape is explained very well by numerically solving the propagation equation supported by the susceptibility of the medium. Our system also exhibits the pulse breakup phenomena for higher input pulse areas. Therefore, the considered scheme can have important applications in quantum communication, quantum information, and mode-locking with the advantage of scalability and controllability.

Hierarchical triple mergers: testing Hawking’s area theorem with the inspiral signals

ABSTRACT Hawking’s area theorem is one of the fundamental laws of black holes (BHs), which has been tested at a confidence level of $\sim 95~{{\ \rm per\ cent}}$ with gravitational wave (GW) observations by analysing the inspiral and ringdown portions of GW signals, independently. In this work, we propose to carry out the test in a new way with the hierarchical triple merger (i.e. two successive BH mergers occurred sequentially within the observation window of GW detectors), for which the properties of the progenitor BHs and the remnant BH of the first coalescence can be inferred from the inspiral portions of the two mergers. As revealed in our simulations, the BH area law can be well confirmed for some plausible hierarchical triple merger events detectable in LIGO/Virgo/KAGRA/LIGO-India’s O4/O5 runs. Our proposed method provides significant facilitation for testing the area law and complements previous methods.

Pulse-area theorem in a single-mode waveguide and its application to photon echo and optical memory in Tm3+:Y3Al5O12

We derive the area theorem for light pulses interacting with an inhomogeneously broadened ensemble of two-level atoms in a single-mode optical waveguide and present its analytical solution for Gaussian-type modes, which demonstrates the significant difference from the formation of $2\ensuremath{\pi}$ pulses by plane waves. We generalize this theorem to the description of photon echo and apply it to the two-pulse (primary) echo and the revival of silenced echo (ROSE) protocol of photon echo quantum memory. We implemented ROSE protocol in a single-mode laser-written waveguide made of an optically thin crystal ${\mathrm{Tm}}^{3+}$:${\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}$ . The experimental data obtained are satisfactorily explained by the developed theory. Finally, we discuss the obtained experimental results and possible applications of the derived pulse-area approach.

PROOFS OF THE FUNDAMENTAL THEOREMS OF THE POINT CALCULUS

The manuscript includes proofs of the fundamental theorems of the point calculus, on the basis of which the basic tools for modeling and parameterization of geometrical objects in the point calculus are developed. The basic theorem of simple relations or O-theorem is a generalization of Menelaus’ theorem for a polygon. Its proof in the point calculus is based on a particular case of the triangle area theorem, the essence and proof of which are also given. The triangle area theorem allows us to establish the relationship between the ratio of triangle areas and the simple ratios of three points obtained on the sides of the triangle. Another consequence of the theorem on the areas of triangles can be obtained as a result of using the theorem on the areas of triangles with equal altitudes, which confirms the reliability of the obtained results and their unambiguous relationship with the known theorems. The proof of the theorem on the ratio of the areas of triangles or S-theorem is based on the use of the properties of the determinants of the matrix and obtained by transforming them. This theorem has a generalization in the form of the theorem on the relation of tetrahedron volumes or the V-theorem of the point calculus with a further generalization to the multidimensional space, the proof of which is similar to that given in the manuscript. This makes it possible to determine areas, volumes and hyper-volumes of geometric objects using the denominator of the ratio as a unit of measure by means of simple relations of three points.

Testing the black hole area law with Event Horizon Telescope

Hawking's black hole area theorem can be tested by monitoring the evolution of a single black hole over time. Using current imaging observations of two supermassive black holes M87* and Sgr A* from the Event Horizon Telescope (EHT), we find their horizon area variation fractions are consistent with the prediction of the black hole area law at the confidence level. We point out that whether the black hole area law is valid or not could be determined by future high-precision EHT observations of Sgr A*.

Self-starting coherent mode locking in a two-section laser with identical gain and absorber media

Coherent mode-locking (CML) represents a method for ultrashort pulse generation in lasers based on the self-induced transparency (SIT) phenomena, which allows us to overcome the principal limitations on the pulse duration imposed by the active medium's gain bandwidth in standard passively mode-locked lasers. So far, this regime was only studied in lasers with transition dipole moments in the absorber medium being twice larger than in the gain medium. This fact ensures that the $2\ensuremath{\pi}$-SIT-soliton in the absorber also forms the stable $\ensuremath{\pi}$-soliton in the gain but can be barely realized in experiments. In this paper we demonstrate theoretically that the self-starting stable coherent mode-locking regime can also arise if the same medium is used both in the gain and absorber laser sections, i.e., the transition dipole moments are equal. These results are both obtained analytically using the area theorem and confirmed by the numerical solution of the Maxwell-Bloch equations. Our findings can enable new opportunities in the area of ultrafast lasers.

Constraining the topological Gauss-Bonnet coupling from GW150914

Recent gravitational wave observation based on the data from GW150914 has confirmed Hawking's area theorem and estimated the increase in total horizon area during a merger process of two Kerr black holes. We use this result and the validity of the second law to obtain the first observational bound on the 4D topological Gauss-Bonnet coupling as $\gamma \lesssim 2.804^{+7.946}_{-1.169} \times 10^{9}\, m^2$ with $95 \%$ credibility.

Settling the Sharp Reconstruction Thresholds of Random Graph Matching

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rényi graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {G}}(n,p)$ </tex-math></inline-formula> . For dense Erdős-Rényi graphs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p=n^{-o(1)}$ </tex-math></inline-formula> , we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the “all-or-nothing” phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erdős-Rényi graphs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p=n^{-\Theta (1)}$ </tex-math></inline-formula> , we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erdős-Rényi graphs. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation and an “area theorem” that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.

Photon/spin echo in a Fabry-Perot cavity.

The pulse area theorem is a well-known versatile analytical tool for capturing the general nonlinear nature of light propagation in a two-level medium. Here we derive the pulse area theorem for the photon/spin echo signal generated in a one-side cavity. The obtained analytical solutions for primary and secondary echo pulse areas allow us to describe the nonlinear patterns of the photon/spin echo signals in an atomic ensemble in a cavity. The developed approach and the obtained results constitute an important step in the study of the general properties of the photon/spin echo in optical and microwave cavities expanding applications of the photon echo in coherent spectroscopy, quantum memory, and processing.

Analyses of COVID-19 in Japan by the Mathematical Model of SIR

The COVID-19 in Japan is analyzed by means of the SIR equation. Contents are listed up below as 1~12. In Sec. 4, the basic reproduction numbers for the 5 and 6 waves in Japan are determined as 2.5 and 5, respectively. Here we make use of data of the removed number, rather than that of the infective number, because the latter seems to be ambiguous. Other important contents are discussed about the stronger the infection power, the smaller the number of deaths in Sec5, an area theorem of infection curves for large basic reproduction number in Sec.9, introducions of vaccine effect and a rate of people with self-control into the SIR model in Sec.11, and finally a logistic formula for organic extinctions in Sec.12.

Holographic entanglement density for spontaneous symmetry breaking

We investigate the properties of the holographic entanglement entropy of the systems in which the U(1) or the translational symmetry is broken spontaneously. For this purpose, we define the entanglement density of the strip-subsystems and examine both the first law of entanglement entropy (FLEE) and the area theorem. We classify the conditions that FLEE and/or the area theorem obey and show that such a classification may be useful for characterizing the systems. We also find universalities from both FLEE and the area theorem. In the spontaneous symmetry breaking case, FLEE is always obeyed regardless of the type of symmetry: U(1) or translation. For the translational symmetry, the area theorem is always violated when the symmetry is weakly broken, independent of the symmetry breaking patterns (explicit or spontaneous). We also argue that the log contribution of the entanglement entropy from the Goldstone mode may not appear in the strongly coupled systems.

Optical resonance in the inversion [formula omitted] channel of a silicon MOS structure

The features of the manifestation of the self-induced transparency (SIT) effect at intersubband resonance in the inversion n− layer of silicon semiconductor systems (in field-effect transistors (100)Si−MOSFET) are analyzed. The course of the interaction of ultrashort laser pulses with the quantum states of the inversion layer of the MOS structure varies, depending on the concentration of the induced charge, by changing the voltage across the metal gate. The evolution of a pulse in the transient stage of deformation as it propagates in a resonant medium of an inversion channel is discussed. Possible consequences for a non-degenerate one-photon resonance are considered, with controlled changes in the detuning between a fixed pulse frequency and the resonant energy of the intersubband transition, caused by changes in the charge density. The parametric dependence on the concentration of the induced charges in the “area theorem” for the propagation process in the inversion channel is revealed.

Model systematics in time domain tests of binary black hole evolution

We perform several consistency tests between different phases of binary black hole dynamics; the inspiral, the merger, and the ringdown on the gravitational wave events GW150914 and GW170814. These tests are performed explicitly in the time domain, without any spectral leakage between the different phases. We compute posterior distributions on the mass and spin of the initial black holes and the final black hole. We also compute the initial areas of the two individual black holes and the final area from the parameters describing the remnant black hole. This facilitates a test of Hawking's black hole area theorem. We use different waveform models to quantify systematic waveform uncertainties for the area increase law with the two events. We find that these errors may lead to overstating the confidence with which the area theorem is confirmed. For example, we find $>99\%$ agreement with the area theorem for GW150914 if a damped sinusoid consisting of a single-mode is used at merger to estimate the final area. This is because this model overestimates the final mass. Including an overtone of the dominant mode decreases the confidence to $\sim94\%$; using a full merger-ringdown model further decreases the confidence to $\sim 85-90\%$. We find that comparing the measured change in the area to the expected change in area yields a more robust test, as it also captures over estimates in the change of area. We find good agreement with GR when applying this test to GW150914 and GW170814.