scholarly journals The Minkowski metric in non-inertial observer radar coordinates

2005 ◽  
Vol 73 (12) ◽  
pp. 1117-1121 ◽  
Author(s):  
E. Minguzzi
Keyword(s):  
Minkowski Metric ◽  
Inertial Observer

scholarly journals Spectral properties of the equation (∇ + ieA) × u = ±mu

2001 ◽  
Vol 131 (5) ◽  
pp. 1065-1089
Author(s):  
Daniel M. Elton
Keyword(s):  
Spectral Properties ◽  
Time Variable ◽  
Two Dimensional ◽  
Vector Product ◽  
Relativistic Invariance ◽  
Speed Of Light ◽  
Large Parameter ◽  
Real Vector ◽  
Pauli Equation ◽  
Minkowski Metric

We develop a spectral theory for the equation (∇ + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

scholarly journals An Analysis of Gravitational waves

2021 ◽  
Author(s):  
Vaibhav Kalvakota
Keyword(s):  
Gravitational Wave ◽  
Gravitational Waves ◽  
Mathematical Structure ◽  
Gauge Condition ◽  
Field Tensor ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Minkowski Metric ◽  
Wave Detector ◽  
The Masses

The September 14, 2015 gravitational wave observations showed the inspiral of two black holes observed from Hanford and Livingston LIGO observatories. This detection was significant for two reasons: firstly, it coupled the result and avoided the possibility of a false alarm by 5σ , meaning that the detected “noise” was indeed from an astronomical source of gravitational waves. We will discuss the primary landscape of gravitational waves, their mathematical structure and how they can be used to predict the masses of the merger system. We will also discuss gravitational wave detector optimisations, and then we will consider the results from the detected merger GW150914. We will consider a straight-forward mathematical approach, and we will primarily be interested in the mathematical modelling of gravitational waves from General Relativity (Section 1). We will first consider a “perturbed” Minkowski metric, and then we will discuss the properties of the perturbation addition tensor. We will then discuss on the gravitational field tensor, and how it arises from the perturbation tensor. We will then talk about the gauge condition, essentially the gauge “freedom” , and then we will talk about the curvature tensor, leading eventually to the effect of gravitational waves on a ring of particles. We will consider the polarisation tensor, which maps the amplitude and polarisation details. The polarisation splits into plus polarised and cross polarised waves, which is technically the effect of a propagating gravitational wave through a ring of particles. We will then talk about the linearized Einstein Field Equations, and how the physical system of merger is encoded into the mathematical structural unity of the metric. We will then talk about the detection of these gravitational waves and how the detector can be optimised, or how the detector can be set so that any “noise” detected can fall in the error margins, and how the detector can prevent the interferometric “photon-noise” from being detected (Section 2.2). Then, we will discuss data results from the source GW150914 detection by LIGO (Section 3).

scholarly journals Lorentz violation with a universal minimum speed as foundation of de Sitter relativity

2018 ◽  
Vol 27 (02) ◽  
pp. 1850011 ◽  
Author(s):  
Cláudio Nassif Cruz ◽  
Rodrigo Francisco dos Santos ◽  
A. C. Amaro de Faria
Keyword(s):  
Cosmological Constant ◽  
Extra Dimension ◽  
Negative Curvature ◽  
Lorentz Violation ◽  
Conformal Factor ◽  
De Sitter ◽  
Minkowski Metric ◽  
Minimum Speed ◽  
New Space ◽  
Cosmological Constants

We aim to investigate the theory of Lorentz violation with an invariant minimum speed called Symmetrical Special Relativity (SSR) from the viewpoint of its metric. Thus, we should explore the nature of SSR-metric in order to understand the origin of the conformal factor that appears in the metric by deforming Minkowski metric by means of an invariant minimum speed that breaks down Lorentz symmetry. So, we are able to realize that there is a similarity between SSR and a new space with variable negative curvature ([Formula: see text]) connected to a set of infinite cosmological constants ([Formula: see text]), working like an extended de Sitter (dS) relativity, so that such extended dS-relativity has curvature and cosmological “constant” varying in time. We obtain a scenario that is more similar to dS-relativity given in the approximation of a slightly negative curvature for representing the current universe having a tiny cosmological constant. Finally, we show that the invariant minimum speed provides the foundation for understanding the kinematics origin of the extra dimension considered in dS-relativity in order to represent the dS-length.

Application of the Cluster Analysis in Computational Paleography

2016 ◽  
pp. 525-543
Author(s):  
Loránd Lehel Tóth ◽  
Raymond Eliza Ivan Pardede ◽  
György András Jeney ◽  
Ferenc Kovács ◽  
Gábor Hosszú
Keyword(s):  
Cluster Analysis ◽  
Unknown Origin ◽  
Canonical Decomposition ◽  
Minkowski Metric ◽  
Clustering Technique ◽  
Future Work

This chapter presents a method to determine the actual version of a script used in constructing of a script relic from unknown origin. The glyphs belong to graphemes as models are realized in the relics as symbols. Some group of glyphs may transform their shape (shapeshifting) through time which produces various versions of scripts that use different glyphs to express the same grapheme. These glyph variants can be identified from extant relics, mainly from historical abecedaries that are used as references. Our algorithm can determine whether or not an abecedary is related to the symbols of a relic from unknown origin by means of the canonical decomposition of the glyphs and symbols. From there an aggregated value called fingerprint is created and it is unique for each relic. The fingerprints then are evaluated by clustering technique using various metrics. As the result of performing comparative evaluations the Minkowski metric provides the most interpretable clustering structure. The results of the evaluations, conclusions, and future work are also presented.

The motion of a charged black hole in an electromagnetic field

1980 ◽  
Vol 371 (1746) ◽  
pp. 429-438 ◽  
Keyword(s):  
Exact Solution ◽  
Black Hole ◽  
Electromagnetic Field ◽  
Approximate Solution ◽  
Electric Field ◽  
Hamiltonian Formalism ◽  
Charged Black Hole ◽  
Uniform Electric Field ◽  
Inertial Observer ◽  
Gravitational Perturbations

The motion of a charged black hole in a weak, asymptotically uniform electric field is analysed by using the Hamiltonian formalism for coupled electromagnetic and gravitational perturbations of the Reissner-Nordstrom space-time. The hole is shown to accelerate with respect to a distant inertial observer according to Newton’s law. The relation of the approximate solution obtained to the exact solution of Ernst, representing the charged C-metric without nodal singularity, is then clarified.

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