scholarly journals Unit-Lapse Forms of Various Spacetimes

2021 ◽  
Author(s):  
Joshua Baines
Keyword(s):  
Initial Velocity ◽  
Mathematical Object ◽  
Kerr Metric ◽  
Coordinate Systems ◽  
Relativity Principle ◽  
Spatial Infinity ◽  
Kerr Spacetime ◽  
Spacetime Curvature ◽  
Stationary Spacetimes ◽  
Better Than

<p><b>Every spacetime is defined by its metric, the mathematical object which further defines the spacetime curvature. From the relativity principle, we have the freedom to choose which coordinate system to write our metric in. Some coordinate systems, however, are better than others. In this text, we begin with a brief introduction into general relativity, Einstein's masterpiece theory of gravity. We then discuss some physically interesting spacetimes and the coordinate systems that the metrics of these spacetimes can be expressed in. More specifically, we discuss the existence of the rather useful unit-lapse forms of these spacetimes. Using the metric written in this form then allows us to conduct further analysis of these spacetimes, which we discuss. </b></p><p>Overall, the work given in this text has many interesting mathematical and physical applications. Firstly, unit-lapse spacetimes are quite common and occur rather naturally for many specific analogue spacetimes. In an astrophysical context, unit-lapse forms of stationary spacetimes are rather useful since they allow for very simple and immediate calculation of a large class of timelike geodesics, the rain geodesics. Physically these geodesics represent zero angular momentum observers (ZAMOs), with zero initial velocity that are dropped from spatial infinity and are a rather tractable probe of the physics occurring in the spacetime. Mathematically, improved coordinate systems of the Kerr spacetime are rather important since they give a better understanding of the rather complicated and challenging Kerr spacetime. These improved coordinate systems, for example, can be applied to the attempts at finding a "Gordon form" of the Kerr spacetime and can also be applied to attempts at upgrading the "Newman-Janis trick" from an ansatz to a full algorithm. Also, these new forms of the Kerr metric allows for a greater observational ability to differentiate exact Kerr black holes from "black hole mimickers".</p>

scholarly journals Unit-Lapse Forms of Various Spacetimes

2021 ◽  
Author(s):  
Joshua Baines
Keyword(s):  
Initial Velocity ◽  
Mathematical Object ◽  
Kerr Metric ◽  
Coordinate Systems ◽  
Relativity Principle ◽  
Spatial Infinity ◽  
Kerr Spacetime ◽  
Spacetime Curvature ◽  
Stationary Spacetimes ◽  
Better Than

<p><b>Every spacetime is defined by its metric, the mathematical object which further defines the spacetime curvature. From the relativity principle, we have the freedom to choose which coordinate system to write our metric in. Some coordinate systems, however, are better than others. In this text, we begin with a brief introduction into general relativity, Einstein's masterpiece theory of gravity. We then discuss some physically interesting spacetimes and the coordinate systems that the metrics of these spacetimes can be expressed in. More specifically, we discuss the existence of the rather useful unit-lapse forms of these spacetimes. Using the metric written in this form then allows us to conduct further analysis of these spacetimes, which we discuss. </b></p><p>Overall, the work given in this text has many interesting mathematical and physical applications. Firstly, unit-lapse spacetimes are quite common and occur rather naturally for many specific analogue spacetimes. In an astrophysical context, unit-lapse forms of stationary spacetimes are rather useful since they allow for very simple and immediate calculation of a large class of timelike geodesics, the rain geodesics. Physically these geodesics represent zero angular momentum observers (ZAMOs), with zero initial velocity that are dropped from spatial infinity and are a rather tractable probe of the physics occurring in the spacetime. Mathematically, improved coordinate systems of the Kerr spacetime are rather important since they give a better understanding of the rather complicated and challenging Kerr spacetime. These improved coordinate systems, for example, can be applied to the attempts at finding a "Gordon form" of the Kerr spacetime and can also be applied to attempts at upgrading the "Newman-Janis trick" from an ansatz to a full algorithm. Also, these new forms of the Kerr metric allows for a greater observational ability to differentiate exact Kerr black holes from "black hole mimickers".</p>

scholarly journals Exact Axially Symmetric Solution inf(T)Gravity Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Gamal G. L. Nashed
Keyword(s):  
Vacuum Solution ◽  
Gravity Theory ◽  
Kerr Metric ◽  
Radial Coordinate ◽  
Lorentz Transformations ◽  
Axially Symmetric ◽  
Field Equations ◽  
Tetrad Field ◽  
Kerr Spacetime ◽  
Euler’S Angles

A general tetrad field with sixteen unknown functions is applied to the field equations off(T)gravity theory. An analytic vacuum solution is derived with two constants of integration and an angleΦthat depends on the angle coordinateϕand radial coordinater. The tetrad field of this solution is axially symmetric and the scalar torsion vanishes. We calculate the associated metric of the derived solution and show that it represents Kerr spacetime. Finally, we show that the derived solution can be described by two local Lorentz transformations in addition to a tetrad field that is the square root of the Kerr metric. One of these local Lorentz transformations is a special case of Euler’s angles and the other represents a boost when the rotation parameter vanishes.

2.3. Working Group 3: Practical Realization of the Reference Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 27-38
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
Fundamental Constants ◽  
Practical Realization ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Measuring Techniques ◽  
Group 3 ◽  
Computational Procedures ◽  
Better Than

For a reference coordinate system to be useful to Blarth dynamics it must clearly display the phenomena of interest in a systematic and unambiguous way, free of detailed assumptions. For a clear display, it is absolutely essential that the system be realized to an accuracy substantially better than has been obtained heretofore. This demands not only improved measuring techniques and instruments, but also precise specification of computational procedures, assumptions, fundamental constants, etc., and meticulous implementation.

scholarly journals A tale of two velocities: Threading versus slicing

2018 ◽  
Vol 15 (03) ◽  
pp. 1850047 ◽  
Author(s):  
Razieh Gharechahi ◽  
Mohammad Nouri-Zonoz ◽  
Alireza Tavanfar
Keyword(s):  
Ad Hoc ◽  
Rotation Parameter ◽  
Spatial Distance ◽  
Local Velocity ◽  
Axially Symmetric ◽  
Kerr Spacetime ◽  
Gravitational Fields ◽  
Circular Particle ◽  
Stationary Spacetimes ◽  
Definition Of

One of the important quantities in cosmology and astrophysics is the 3-velocity of an object. Specifically, when the gravitational fields are strong, one should require the employment of general relativity both in its definition and measurement. Looking into the literature for GR-based definitions of 3-velocity, one usually finds different ad hoc definitions applied according to the case under consideration. Here, we introduce and analyze systematically the two principal definitions of 3-velocity assigned to a test particle following the timelike trajectories in stationary spacetimes. These definitions are based on the [Formula: see text] (threading) and [Formula: see text] (slicing) spacetime decomposition formalisms and defined relative to two different sets of observers. After showing that Synge’s definition of spatial distance and 3-velocity is equivalent to those defined in the [Formula: see text] (threading) formalism, we exemplify the differences between these two definitions by calculating them for particles in circular orbits in axially symmetric stationary spacetimes. Illustrating its geometric nature, the relative linear velocity between the corresponding observers is obtained in terms of the spacetime metric components. Circular particle orbits in the Kerr spacetime, as the prototype and the most well known of stationary spacetimes, are examined with respect to these definitions to highlight their observer-dependent nature. We also examine the Kerr-NUT spacetime in which the NUT parameter, contributing to the off-diagonal terms in the metric, is mainly interpreted not as a rotation parameter but as a gravitomagnetic monopole charge. Finally, in a specific astrophysical setup which includes rotating black holes, it is shown how the local velocity of an orbiting star could be related to its spectral line shifts measured by distant observers.

scholarly journals Gaussian coordinate systems for the Kerr metric

2011 ◽  
Vol 17 (3) ◽  
pp. 230-241 ◽  
Author(s):  
M. Novello ◽  
E. Bittencourt
Keyword(s):  
Kerr Metric ◽  
Coordinate Systems

On separable solutions of Dirac’s equation for the electron

1982 ◽  
Vol 383 (1785) ◽  
pp. 247-278 ◽  
Keyword(s):  
Dirac Operator ◽  
Jacobi Equation ◽  
Flat Space ◽  
Kerr Metric ◽  
Coordinate Systems ◽  
Systematic Account ◽  
Jacobi Operators ◽  
Flat Spacetime ◽  
Matrix Coefficients ◽  
Hamilton Jacobi Equation

The properties of coordinate systems that admit separation of the Laplacian and Hamilton–Jacobi operators have been thoroughly explored so that the nature of solutions in separable form of Laplace’s equation, the wave equation, Schrödinger’s equation and the Hamilton–Jacobi equation are well understood. The corresponding problems for the Dirac operator in flat spacetime have been less completely examined and this paper contains studies intended to produce a more systematic account of possible solutions of Dirac’s equation. Because the Dirac operator differs from the Laplacian in being a first-degree differential operator and in having matrix coefficients, it is not possible to discuss possible solutions in as general a way and the separable solutions are far less rich than for equations with the Laplacian. In particular, the forms of the potentials for which separable solutions are possible are not for the most part of physical interest. Although the discussion is confined to coordinates in flat space–time, some of the procedures are derived from those developed to solve Dirac’s equation in coordinates with a Kerr metric.

scholarly journals NEWTON–HOOKE LIMIT OF BELTRAMI–DE SITTER SPACETIME, PRINCIPLE OF GALILEI–HOOKE'S RELATIVITY AND POSTULATE ON NEWTON–HOOKE UNIVERSAL TIME

2007 ◽  
Vol 22 (14n15) ◽  
pp. 2535-2562 ◽  
Author(s):  
CHAO-GUANG HUANG ◽  
HAN-YING GUO ◽  
YU TIAN ◽  
ZHAN XU ◽  
BIN ZHOU
Keyword(s):  
Cosmological Constant ◽  
Group Algebra ◽  
Transformation Group ◽  
Universal Time ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Coordinate Systems ◽  
Relativity Principle ◽  
Sitter Spacetime ◽  
Straight Lines

Based on the Beltrami–de Sitter spacetime, we present the Newton–Hooke model under the Newton–Hooke contraction of the BdS spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton–Hooke space–time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton–Hooke group. In order to determine uniquely the Newton–Hooke limit, we propose the Galilei–Hooke's relativity principle as well as the postulate on Newton–Hooke universal time. All results are readily extended to the Newton–Hooke model as a contraction of Beltrami–anti-de Sitter spacetime with negative cosmological constant.

scholarly journals Painlevé–Gullstrand form of the Lense–Thirring Spacetime

Universe ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 105
Author(s):  
Joshua Baines ◽  
Thomas Berry ◽  
Alex Simpson ◽  
Matt Visser
Keyword(s):  
The Body ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Slow Rotation ◽  
Type I ◽  
Observational Astronomy ◽  
Kerr Spacetime ◽  
Schwarzschild Geometry ◽  
Distance Approximation ◽  
Curvature Tensors

The standard Lense–Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense–Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense–Thirring metric, carefully chosen for simplicity, clarity, and various forms of improved mathematical and physical behaviour, (to be more carefully defined in the body of the article). We shall see that this Lense–Thirring variant can be viewed as arising from the linearization of a suitably chosen tetrad representing the Kerr spacetime. In particular, we shall construct an explicit unit-lapse Painlevé–Gullstrand variant of the Lense–Thirring spacetime, one that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense–Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the “rain” geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs (innermost stable circular orbits) and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r; verifying that for sufficiently slow rotation we explicitly recover slowly rotating Schwarzschild geometry. This Lense–Thirring variant can be viewed, in its own right, as a “black hole mimic”, of direct interest to the observational astronomy community.

scholarly journals Disforming the Kerr metric

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Timothy Anson ◽  
Eugeny Babichev ◽  
Christos Charmousis ◽  
Mokhtar Hassaine
Keyword(s):  
Event Horizon ◽  
Compact Object ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Candidate Event ◽  
Kerr Spacetime ◽  
Constant Degree ◽  
Tensor Theory ◽  
Timelike Hypersurface ◽  
Null Congruence

Abstract Starting from a recently constructed stealth Kerr solution of higher order scalar tensor theory involving scalar hair, we analytically construct disformal versions of the Kerr spacetime with a constant degree of disformality and a regular scalar field. While the disformed metric has only a ring singularity and asymptotically is quite similar to Kerr, it is found to be neither Ricci flat nor circular. Non-circularity has far reaching consequences on the structure of the solution. As we approach the rotating compact object from asymptotic infinity we find a static limit ergosurface similar to the Kerr spacetime with an enclosed ergoregion. However, the stationary limit of infalling observers is found to be a timelike hypersurface. A candidate event horizon is found in the interior of this stationary limit surface. It is a null hypersurface generated by a null congruence of light rays which are no longer Killing vectors. Under a mild regularity assumption, we find that the candidate surface is indeed an event horizon and the disformed Kerr metric is therefore a black hole quite distinct from the Kerr solution.

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