A static spherically symmetric solution of the Einstein—Maxwell—Yukawa field equations

1962 ◽  
Vol 58 (3) ◽  
pp. 521-526 ◽  
Author(s):  
G. Stephenson
Keyword(s):  
Gravitational Constant ◽  
Spherically Symmetric ◽  
Field Tensor ◽  
Field Equations ◽  
Relativistic Field ◽  
Metric Tensor ◽  
Spherically Symmetric Solution ◽  
Electromagnetic Field Tensor ◽  
General Relativistic ◽  
Image Position

The general relativistic field equations for regions of space containing electromagnetic fields but no matter are,where,and κ = 8ΠG/c4 is the gravitational constant. Here giκ is the metric tensor and Fiκ is the electromagnetic field tensor which satisfies Maxwell's equations ifand,where Ai is the electromagnetic four-potential.

Approximate solutions of the general relativity field equations with a scalar meson field

1963 ◽  
Vol 59 (4) ◽  
pp. 739-741 ◽  
Author(s):  
J. Hyde
Keyword(s):  
General Relativity ◽  
Scalar Meson ◽  
Empty Space ◽  
Approximate Solutions ◽  
Spherically Symmetric ◽  
Coordinate Transformations ◽  
Field Equations ◽  
Spherically Symmetric Solution ◽  
Image Position ◽  
Scalar Meson Field

It was shown by Birkhoff ((1), p. 253) that every spherically symmetric solution of the field equations of general relativity for empty space,may be reduced, by suitable coordinate transformations, to the static Schwarzschild form:where m is a constant. This result is known as Birkhoff's theorem and excludes the possibility of spherically symmetric gravitational radiation. Different proofs of the theorem have been given by Eiesland(2), Tolman(3), and Bonnor ((4), p. 167).

The General Static Spherically Symmetric Solution of the "Weak" Unified Field Equations

1962 ◽  
Vol 14 ◽  
pp. 568-576 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Ricci Tensor ◽  
Spherically Symmetric ◽  
Dimensional Manifold ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Spherically Symmetric Solution ◽  
Geometric Objects ◽  
Symmetric Connection ◽  
Image Position

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where(1)Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:(2a)(2b)In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notationfor the symmetric and skew-symmetric parts of geometric objects Q is employed.

On the isometric motion of a charged fluid in the Einstein-Maxwell theory

1974 ◽  
Vol 336 (1606) ◽  
pp. 273-284 ◽  
Keyword(s):  
Killing Vector ◽  
Fluid Motion ◽  
Fluid Pressure ◽  
Field Equations ◽  
Maxwell Theory ◽  
Charged Fluid ◽  
Spherically Symmetric Solution ◽  
Electromagnetic Field Tensor ◽  
Discrete Values ◽  
Full Solution

It is shown that in the Einstein-Maxwell theory a class of four-dimensional charged fluid space-times exists, with non-zero fluid pressure, satisfying the conditions that (i) the fluid motion is isometric, (ii) the dual of the electromagnetic field tensor has no projection in the direction of a Killing vector - equivalent to the condition that in a static space time the local field of an observer moving with the fluid is purely electric - and (iii) the ratio of charge to mass is constant. For the case of a diagonal static metric it is seen that a group of quasi-conformal transformations may be determined which leaves the field equations unchanged. This may be used to obtain a full solution of the field equations, in three independent variables, from a given solution in one independent variable. A spherically symmetric solution of this kind is obtained which is seen to be expressible in terms of hypergeometric functions. An interesting aspect of this is that the charge/mass ratio can only have discrete values depending on the eigenvalues of a linear boundary-value problem.

Study of gravitational decoupled anisotropic solution

2019 ◽  
Vol 28 (16) ◽  
pp. 2040004
Author(s):  
M. Sharif ◽  
Sobia Sadiq
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Energy Conditions ◽  
Spherically Symmetric ◽  
Anisotropic Model ◽  
Field Equations ◽  
Anisotropic Solution ◽  
Spherically Symmetric Solution ◽  
Gravitational Source ◽  
The Stability

This paper formulates the exact static anisotropic spherically symmetric solution of the field equations through gravitational decoupling. To accomplish this work, we add a new gravitational source in the energy–momentum tensor of a perfect fluid. The corresponding field equations, hydrostatic equilibrium equation as well as matching conditions are evaluated. We obtain the anisotropic model by extending the known Durgapal and Gehlot isotropic solution and examined the physical viability as well as the stability of the developed model. It is found that the system exhibits viable behavior for all fluid variables as well as energy conditions and the stability criterion is fulfilled.

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.

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).

Exact solution of a static charged sphere in general relativity

1969 ◽  
Vol 47 (21) ◽  
pp. 2401-2404 ◽  
Author(s):  
S. J. Wilson
Keyword(s):  
Exact Solution ◽  
General Relativity ◽  
Gravitational Constant ◽  
The Self ◽  
Spherically Symmetric ◽  
Charged Sphere ◽  
Field Equations ◽  
First Order ◽  
Self Energy ◽  
Spherically Symmetric Distribution

An exact solution of the field equations of general relativity is obtained for a static, spherically symmetric distribution of charge and mass which can be matched with the Reissner–Nordström metric at the boundary. The self-energy contributions to the total gravitational mass are computed retaining only the first order terms in the gravitational constant.

scholarly journals THE MAXWELL LAGRANGIAN IN PURELY AFFINE GRAVITY

2008 ◽  
Vol 23 (03n04) ◽  
pp. 567-579 ◽  
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
Electromagnetic Field ◽  
Ricci Tensor ◽  
Maxwell Equations ◽  
Legendre Transformation ◽  
Conformal Transformations ◽  
Field Tensor ◽  
Metric Tensor ◽  
Hamiltonian Density ◽  
Electromagnetic Field Tensor ◽  
Affine Gravity

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein–Maxwell equations in the metric–affine and metric formulation. We show that this equivalence is related to the invariance of the Maxwell Lagrangian under conformal transformations of the metric tensor. We also apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the Maxwell–Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.

scholarly journals Discussion of a Possible Corrected Black Hole Entropy

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Miao He ◽  
Ziliang Wang ◽  
Chao Fang ◽  
Daoquan Sun ◽  
Jianbo Deng
Keyword(s):  
Black Hole ◽  
Electromagnetic Field ◽  
Black Hole Entropy ◽  
Einstein Gravity ◽  
Matter Field ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Corrected Entropy ◽  
Entropy Of Black Hole ◽  
Spherically Symmetric Solution

Einstein’s equation could be interpreted as the first law of thermodynamics near the spherically symmetric horizon. Through recalling the Einstein gravity with a more general static spherical symmetric metric, we find that the entropy would have a correction in Einstein gravity. By using this method, we investigate the Eddington-inspired Born-Infeld (EiBI) gravity. Without matter field, we can also derive the first law in EiBI gravity. With an electromagnetic field, as the field equations have a more general spherically symmetric solution in EiBI gravity, we find that correction of the entropy could be generalized to EiBI gravity. Furthermore, we point out that the Einstein gravity and EiBI gravity might be equivalent on the event horizon. At last, under EiBI gravity with the electromagnetic field, a specific corrected entropy of black hole is given.

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