NUMERICAL RELATIVITY: EVOLVING SPACETIME

1993 ◽  
Vol 04 (04) ◽  
pp. 883-907 ◽  
Author(s):  
C. BONA ◽  
J. MASSÓ
Keyword(s):  
Ad Hoc ◽  
Numerical Solutions ◽  
Numerical Relativity ◽  
Historical Review ◽  
Evolution System ◽  
Field Equations ◽  
Metric Tensor ◽  
New Family ◽  
Mandatory Use ◽  
Evolution Systems

The construction of numerical solutions of Einstein's General Relativity equations is formulated as an initial-value problem. The space-plus-time (3 + 1) decomposition of the spacetime metric tensor is used to discuss the structure of the field equations. The resulting evolution system is shown to depend in a crucial way on the coordinate gauge. The mandatory use of singularity avoiding coordinate conditions (like maximal slicing or similar gauges) is explained. A brief historical review of Numerical Relativity is included, showing the enormous effort in constructing codes based in these gauges, which lead to non-hyperbolic evolution systems, using "ad hoc" numerical techniques. A new family of first order hyperbolic evolution systems for the vacuum Einstein field equations in the harmonic slicing gauge is presented. This family depends on a symmetric 3 × 3 array of parameters which can be used to scale the dynamical variables in future numerical applications.

scholarly journals On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 529-539
Author(s):  
Xianghu Liu
Keyword(s):  
Approximate Controllability ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Evolution System ◽  
Controlled Systems ◽  
Uniqueness Of The Solution ◽  
Fractional Evolution Systems ◽  
Evolution Systems

Abstract The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.

On a non-symmetric theory of the pure gravitational field

1958 ◽  
Vol 54 (1) ◽  
pp. 72-80 ◽  
Author(s):  
D. W. Sciama
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Symmetric Tensor ◽  
Momentum Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Unified Field

ABSTRACTIt is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

scholarly journals Physically Acceptable Embedded Class-I Compact Stars in Modified Gravity with Karmarkar Condition

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 962 ◽  
Author(s):  
Saira Waheed ◽  
Ghulam Mustafa ◽  
Muhammad Zubair ◽  
Asifa Ashraf
Keyword(s):  
Compact Star ◽  
Graphical Analysis ◽  
Specific Model ◽  
Compact Stars ◽  
Fluid Distribution ◽  
Anisotropic Fluid ◽  
Strong Impact ◽  
Metric Tensor ◽  
Physical Conditions ◽  
New Family

The present study is devoted to explore the existence of a new family of compact star solutions by adopting the Karmarkar as well as Pandey–Sharma condition in the background of f ( R , T ) modified gravitational framework. For this purpose, we consider static spherically symmetric spacetime with anisotropic fluid distribution in absence of electric charge. In respect of Karmarkar condition, we assume a specific model of g r r metric potential representing a new family of solutions which is also compatible with the Pandey–Sharma condition. This assumed model permits us to calculate the g t t component of metric tensor by making the use of Karmarkar condition. Further, we investigate the interior solutions for V e l a X − 1 model of compact star by utilizing this new family of solutions for different values of parameter λ . We have tuned the solution for V e l a X − 1 so that the solutions matches the observed mass and radius. For the same star we have extensively discussed the behavior of the solutions. It is found that these solutions fulfill all the necessary conditions under the observational radii and mass attribute data for small values of parameter λ and hence physically well-behaved and promising. Through graphical analysis, it is observed that our obtained analytical solutions are physically acceptable with a best degree of accuracy for n ∈ [ 1.8 , 7 ) − { 2 , 4 , 6 } , where parameter n is involved in the discussed model. It is also noticed the causality condition is violated for all n ≥ 7 and the tangential sound velocity v t is observed as complex valued for all 0 < n < 1.8 . Likewise, we explore these properties by considering large parameter λ values. It is seen that the presented model violates all the physical conditions for n ∈ { 2 , 4 , 6 } , while some of these for large values of λ . Consequently, it can be concluded that the parameters n and λ have a strong impact on the obtained solutions.

scholarly journals ON THE HAMILTONIAN FORMULATION OF THE EINSTEIN–HILBERT ACTION IN TWO DIMENSIONS

2006 ◽  
Vol 21 (11) ◽  
pp. 899-905 ◽  
Author(s):  
N. KIRIUSHCHEVA ◽  
S. V. KUZMIN
Keyword(s):  
Hamiltonian Formulation ◽  
Two Dimensions ◽  
General Covariance ◽  
Sufficient Condition ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Gauge Transformations ◽  
Total Divergence ◽  
Hilbert Action

It is shown that if general covariance is to be preserved (i.e. a coordinate system is not fixed) the well-known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein–Hilbert action to be a total divergence. Consequently, a Hamiltonian formulation is possible without any modification of the two-dimensional Einstein–Hilbert action. We find the resulting constraints and the corresponding gauge transformations of the metric tensor.

scholarly journals Black hole and naked singularity geometries supported by three-form fields

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
Bruno J. Barros ◽  
Bogdan Dǎnilǎ ◽  
Tiberiu Harko ◽  
Francisco S. N. Lobo
Keyword(s):  
Black Hole ◽  
Killing Vector ◽  
Field Potential ◽  
Naked Singularity ◽  
Radial Coordinate ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Metric Tensor ◽  
Killing Horizon ◽  
Form Field

Abstract We investigate static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert–Einstein action with a Lagrangian constructed from a three-form field $$A_{\alpha \beta \gamma }$$Aαβγ, which is related to the field strength and a potential term. The field equations are obtained explicitly for a static and spherically symmetric geometry in vacuum. For a vanishing three-form field potential the gravitational field equations can be solved exactly. For arbitrary potentials numerical approaches are adopted in studying the behavior of the metric functions and of the three-form field. To this effect, the field equations are reformulated in a dimensionless form and are solved numerically by introducing a suitable independent radial coordinate. We detect the formation of a black hole from the presence of a Killing horizon for the timelike Killing vector in the metric tensor components. Several models, corresponding to different functional forms of the three-field potential, namely, the Higgs and exponential type, are considered. In particular, naked singularity solutions are also obtained for the exponential potential case. Finally, the thermodynamic properties of these black hole solutions, such as the horizon temperature, specific heat, entropy and evaporation time due to the Hawking luminosity, are studied in detail.

scholarly journals Hyperbolic evolution system for numerical relativity

1992 ◽  
Vol 68 (8) ◽  
pp. 1097-1099 ◽  
Author(s):  
C. Bona ◽  
J. Massó
Keyword(s):  
Numerical Relativity ◽  
Evolution System

Quadratic Lagrangians and gauge-invariance in covariant field theories

1960 ◽  
Vol 56 (3) ◽  
pp. 247-251 ◽  
Author(s):  
G. Stephenson
Keyword(s):  
General Relativity ◽  
Gauge Transformation ◽  
Gauge Invariance ◽  
Variational Principles ◽  
Scalar Function ◽  
Field Equations ◽  
General Group ◽  
Metric Tensor ◽  
Invariance Properties ◽  
Image Position

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformationwhere is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed assuch that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).

scholarly journals Early dynamical world models: A historical review

2009 ◽  
Vol 5 (S260) ◽  
pp. 182-188
Author(s):  
Helge Kragh
Keyword(s):  
Paradigm Shift ◽  
Historical Review ◽  
Closed Universe ◽  
Field Equations ◽  
History Of ◽  
Evolving Models ◽  
The Universe

AbstractModels of the universe, in the sense of solutions to the cosmological field equations, took their start in 1917 with Einstein's closed universe. During the next two decades they were developed to comprise evolving models, some of them cyclic and some of them with a definite age. The history of this development, as it occurred up to the mid 1930s, is reviewed. It is argued that in 1930-31, cosmology experienced a kind of paradigm shift.

Sign in / Sign up

Export Citation Format

Share Document