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We first define energy conditions, which are gravitational analogs of the positivity of the energy in nongravitational theories. After defining the notion of singularity more precisely, we state (without proof) the singularity theorems of Hawking and Penrose, and a “counterexample,” which evades all of their assumptions. Then we define wormholes, traversable wormholes, and give as example the Morris–Thorne wormhole, with its embedding diagram.
We first describe the Newtonian limit for gravity, in particular in the case of spherical symmetry. We use it to help with the ansatz in the case of the vacuum solution of the Einstein’s equations with spherical symmetry. After calculating the equations of motion on the ansatz, we find the most general such solution, the Schwarzschild solution. We also define what a Schwarzschild black hole is.
We first define the notion of Wald gravitational entropy, defined in a more general setting than the Bekenstein–Hawking one. Then we define Sen’s entropy function formalism, in a general gravity theory, that defines the entropy function, whose minimization at the event horizon gives the entropy. Finally, we define the effective potential of the event horizon, defined in a theory with scalars, and show that the horizon is an attractor for the equations of motion involving scalars.
We first describe the classification of three-dimensional Lie algebras. Then we show how that implies a classification of three-dimensional Lie invariant Riemann spaces, as Bianchi spaces, associated to cosmologies. Then, we describe examples of homogenous Bianchi cosmologies, the Kasner spacetime and the Mixmaster Universe.
We first review general concepts of quantum field theory, like Feynman diagrams and path integrals. Then we define the worldline formalism for quantum field theory, and apply it to gravity, in the case of classical gravity with sources. We describe first an electromagnetic analogy of how to write down the worldline action and obtain physics from it, and then use it in the gravity case, for a black hole or neutron star. We then consider tidal forces in the nonrelativistic regime and find how to describe Love numbers and radiated power, for instance. As a simple example, we consider scalar gravity interacting with a source current and how to obtain information from Feynman diagrams. Then, define Non-Relativistic General Relativity (NRGR), a formalism for the gravitational inspiral of two bodies (e.g., two black holes) and show how to calculate the two-body (or many-body) action from Feynman diagrams and how to obtain the radiated power via gravitational waves. We find the Einstein–Infeld–Hofman Lagrangian for many bodies in the NRGR case.
We define the ADM parametrization of gravitational actions. Then, we define extrinsic curvature for both a spacelike hypersurface in the ADM parametrization and for a general surface embedded in a larger space. The Gauss–Codazzi equations for the embedding of a surface are defined and explained, as well as the Israel junction conditions. After defining the notion of Killing vectors, we consider asymptotically flat spacetimes, with the possible definitions of mass (Kumar, ADM, linearized) and the BMS group of asymptotic symmetries. Finally, we calculate and define the boundary term of the gravitational action, the Gibbons–Hawking–York term.
We describe the geometric formulation for gravity, without using an inverse metric, only the vielbein and spin connection as gauge fields. We define the general Lanczos–Lovelock Lagrangian, in particular Chern–Simons (in odd dimensions) and Born–Infeld (in even dimensions) gravity, and the topological gravity terms and their dimensional extensions. We end by an application to anomalies, specifically gravitational anomalies.
We define Penrose diagrams, which keep the causal and topological properties of gravitational spacetimes, while moving infinity to a finite distance on the diagram. We use the examples of Minkowski space, in two dimensions and dimensions greater than two, then describe Anti-de Sitter spacetime in Poincaré coordinates (the Poincaré patch), and finally consider the Schwarzschild black hole.
We define the vielbein–spin connection formulation of general relativity and describe what happens in the presence of fermions (which can only be described in this formulation). Then, we see how close is general relativity to being described by a gauge theory, in three dimensions, which is special, and in four or higher dimensions.
We describe the parametrized post-Newtonian (PPN) formalism for expansion around the Newtonian limit. First we describe the formalism for generic gravity theories, for the equations of motion and the energy-momentum tensor. Then we consider it for many-body systems (like binary inspirals) and the coefficients for PPN in general relativity in this case. We describe the effective field theory approach in this case, show how to calculate the post-Newtonian (PN) order, and how it fits with the EIH Lagrangian. Finally, we consider metric frames, the usual Einstein frame versus the generic Jordan frame, and the original Brans–Dicke theory.
This research investigated the potential improvement of IVR on procedural practical knowledge in maritime safety education in a lifeboat case study. Participants were divided into three groups: a VR, control and VR+ group. A practical test exam with a real lifeboat was conducted to evaluate the differences in number of students passed/failed and overall performances in the execution of lifeboat launching procedures between groups. There was no evidence that the VR students had a higher success rate than students in the control group in correctly and safely performing a lifeboat drill. However, VR students’ overall performances on procedural correctness were significantly better than those of the control group, despite that the VR students never practiced with a physical lifeboat. Given the importance of safety in the maritime industry, it is worthwhile to further investigate to what extent IVR can provide a solution to the current limitations in maritime safety education.
The unmanned surface vehicle (USV) is deemed with significant potential to deal with the maritime search and rescue (SAR) missions. This paper investigates the path planning of the USV with SAR tasks, and proposes a novel algorithm based on combined convolutional neural network rapid-exploration random tree and improved artificial potential field (CRRT-IAPF). The proposed scheme can be divided into the global and the local path plannings. The rapid-exploration random tree (RRT) method is employed to generate the global path in the sea chart, which is further discriminated to be optimal or non-optimal through a well-trained convolutional neural network (CNN). The artificial potential field (APF) method is adopted to plan the local path in the environment with small obstacles and SAR task points. To facilitate the path convergence and avoid the oscillation, the potential field function is improved in a more efficient way. In addition, the evaluation functions of search success rate and rescue success rate are established to evaluate the completeness of SAR tasks. Through the simulation, it is verified that the proposed CRRT-IAPF scheme has the superiority over the others.
The article is devoted to the mathematical theories and algorithms necessary for the implementation of a software package that fully automates the calculations necessary in Nautical Astronomy. The article describes a method for calculating the equatorial and horizontal coordinates of the celestial bodies at any moment of time. The authors describe the calculation of the time of the apparent rising (setting) of the Sun, solar illumination and events of other celestial bodies. A formula for calculating astronomical refraction is proposed. A matrix method for implementing the method of least squares for determining the coordinates of a place along the lines of position is described. An algorithm for identifying navigational planets is also described and a method for estimating the error for it is proposed. Based on this, the results of the development of the software package ‘Astronomy Package’ for Nautical Astronomy are presented.
The need for Global Navigation Satellite System (GNSS) receiver testing increases with the advent of widespread Internet of Things (IoT) technologies and other electronic devices dependent on position determination. In this paper, a low-cost GNSS multiband L1+L5 signal recorder and replayer for equipment testing purposes is proposed. It is implemented using Software-Defined Radio (SDR) modules HackRF One with proper time and phase synchronisation. The recorder–replayer has been tested with GPS, GALILEO, BEIDOU and GLONASS satellites and several commercial GNSS receivers. Reduced GNSS signal bandwidth of approximately 10 MHz is sufficient for efficient reception of recorded signals. Performed tests with a driving car show applicability of this GNSS recorder–replayer in dynamic settings.
The first globalisation of the world occurred under the motivation of the Avis dynasty of Portugal, aimed at discovering new lands and wealth, exploring ocean routes, especially with the successful epic journey to India. The political decisions of the Avis dynasty kings, along with Christianity’s interest in expanding into Africa and Asia, were key factors in the success of these maritime explorations. However, the Coriolis force was a significant force of nature for the outcome of this journey. Here we investigate Caminha’s letter, the scribe of Pedro Álvares Cabral’s fleet when he found the lands of Brazil. This letter contains detailed scientific data, distances travelled, dates, geographical features, fauna and flora, initial anthropological information on indigenous peoples, and records of coastal depths. Analysis of these elements and facts lead to a new proposal for the location of Mount Pascoal and the so-called safe harbour, where Cabral’s fleet landed.
Terrain-aided navigation with a three-dimensional (3D) map has both high accuracy and high reliability, which is crucial for applications in the global navigation satellite system (GNSS)-denied scenarios. In this paper, a new terrain matching algorithm with 3D Zernike moments (3D ZMs) is proposed. The redundant items in the even-order 3D ZMs are analysed in theory. The 3D ZMs are also correlated with the standard deviations of terrain further to identify the redundant items. The new 3D ZM descriptors are proposed for the feature vector of the matching algorithm by excluding the redundant items from the descriptors. The simulation results demonstrate that the algorithm with the revised descriptors achieves a higher matching success rate than both that with the existing descriptors and that with the odd-order descriptors under the same conditions.