Chapter 5 Relationship between the Metric Tensor and the Strain Tensor in Low Gravitational Field

2021 ◽  
pp. 71-78
Keyword(s):  
Gravitational Field ◽  
Strain Tensor ◽  
Metric Tensor

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.

Stresses Analysis on Coarse Grain Zn Film during Tensile Loading

2006 ◽  
Vol 524-525 ◽  
pp. 723-728
Author(s):  
Wen Jun Huang ◽  
Vincent Ji ◽  
Wilfrid Seiler
Keyword(s):  
Steel Substrate ◽  
Strain Tensor ◽  
Tensile Loading ◽  
New Method ◽  
Coarse Grain ◽  
Initial State ◽  
Metric Tensor ◽  
Coarse Grains ◽  
Deformed State ◽  
Galvanized Coating

The advance of the XRD technique allows us to reach the properties of each coarse grain. This paper has demonstrated a new method to determine stress in a single crystal for multicrystal material and this new method could be specially applied for any symmetric crystalline systems. The strain tensor ε is determined by the change of the metric tensor G before the initial state and after the deformed state in the crystal reference system. Then stress tensor at grain scale is calculated by the Hooks law. The stress evaluations are carried out in coarse grains of a thin galvanized coating on a steel substrate during tensile loading. This study allows us to link the microstructure evolution to the elastic heterogeneity at grain scale or between the grains.

scholarly journals Einstein’s Equation of motion for exterior test particles with spherical mass distribution having varied field, time and radial distance; Golden metric tensors approach.

2021 ◽  
Keyword(s):  
Gravitational Field ◽  
Polar Motion ◽  
Equations Of Motion ◽  
Radial Distance ◽  
Equation Of Motion ◽  
Metric Tensor ◽  
Test Particles ◽  
Einstein's Equation ◽  
Spherical Mass ◽  
Metric Tensors

Golden metric tensors exterior to hypothetical distribution of mass whose field varies with time and radial distance have been used to construct the coefficient of affine connections that invariably was used to obtained the Einstein’s equations of motion for test particles of non-zero rest masses. The expression for the variation of time on a clock moving in this gravitational field was derived using the time equation of motion. The test particles in this field under the condition of pure polar motion have an inverse square dependence velocity which depends on radial distance. Our result indicates that despite using the golden metric tensor, the inverse square dependence of the velocity on radial distance has not been changed.

scholarly journals Gravitational Fields of Conical Mass Distributions

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Chifu Ebenezer Ndikilar
Keyword(s):  
Gravitational Field ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Prolate Spheroidal ◽  
Radial Equation ◽  
Metric Tensor ◽  
Gravitational Fields ◽  
Mass Distributions ◽  
Oblate Spheroidal ◽  
Test Particles

The gravitational field of conical mass distributions is formulated using the general theory of relativity. The gravitational metric tensor is constructed and applied to the motion of test particles and photons in this gravitational field. The expression for gravitational time dilation is found to have the same form as that in spherical, oblate spheroidal, and prolate spheroidal gravitational fields and hence confirms an earlier assertion that this gravitational phenomena is invariant in form with various mass distributions. It is shown using the pure radial equation of motion that as a test particle moves closer to the conical mass distribution along the radial direction, its radial speed decreases.

ON TOLMAN'S MASS-ENERGY RELATION AND A NEW TOLMAN-TYPE RELATION

1989 ◽  
Vol 04 (02) ◽  
pp. 327-334
Author(s):  
B. M. BARKER ◽  
R. F. O'CONNELL
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Energy Relation ◽  
Mass Energy ◽  
Schwarzschild Solution ◽  
Metric Tensor ◽  
Harmonic Coordinates ◽  
Isotropic Coordinates ◽  
Special Cases ◽  
Type Relation

Tolman derived the mass-energy relation [Formula: see text] using a particular choice of coordinates, viz. the Schwarzschild solution for the metric tensor in isotropic coordinates for a body of mass m at rest at the origin. Here we show that this relation retains the same form for the case of a very general coordinate system. The latter includes the Schwarzschild and harmonic coordinates as special cases. In addition, we give a new Tolman-type relation [Formula: see text]. The quantities [Formula: see text] and [Formula: see text] are the energy-momentum densities for matter and the gravitational field, respectively.

scholarly journals GRAVITATIONAL FIELD OF A GLOBAL MONOPOLE IN A MODIFIED GRAVITY

2011 ◽  
Vol 03 ◽  
pp. 446-454 ◽  
Author(s):  
T. R. P. CARAMÊS ◽  
E. R. BEZERRA DE MELLO ◽  
M. E. X. GUIMARÃES
Keyword(s):  
Gravitational Field ◽  
Modified Gravity ◽  
Field Approximation ◽  
Weak Field ◽  
Symmetric System ◽  
Radial Coordinate ◽  
Spherically Symmetric ◽  
Global Monopole ◽  
Field Equations ◽  
Metric Tensor

In this paper we analyze the gravitational field of a global monopole in the context of f(R) gravity. More precisely, we show that the field equations obtained are expressed in terms of [Formula: see text]. Since we are dealing with a spherically symmetric system, we assume that F(R) is a function of the radial coordinate only. Moreover, adopting the weak field approximation, we can provide all components of the metric tensor. A comparison with the corresponding results obtained in General Relativity and in the Brans-Dicke theory is also made.

Conceptual Pitfalls in Teaching the Linearized Approximation in Introductory General Relativity

2020 ◽  
Vol 02 (04) ◽  
pp. 2020005
Author(s):  
Valerio Faraoni
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Test Field ◽  
Metric Tensor ◽  
Small Perturbations ◽  
Flat Background ◽  
Course Material ◽  
Generally Covariant

The decomposition of the metric tensor into a flat background plus small perturbations used in linearized general relativity is often a source of confusion for the student because these two parts are only Lorentz-invariant but not generally covariant. The underlying, crucial, conceptual switch from a dynamical gravitational field to a test field on a fixed background is often omitted in presenting this course material. This issue is clarified and an improved presentation is proposed.

scholarly journals Dark matter density profile and galactic metric in Eddington-inspired Born–Infeld gravity

2014 ◽  
Vol 29 (09) ◽  
pp. 1450049 ◽  
Author(s):  
Tiberiu Harko ◽  
Francisco S. N. Lobo ◽  
M. K. Mak ◽  
Sergey V. Sushkov
Keyword(s):  
Dark Matter ◽  
Gravitational Field ◽  
Density Profile ◽  
Galactic Center ◽  
Matter Density ◽  
Dark Matter Halos ◽  
Field Equations ◽  
Metric Tensor ◽  
Dark Matter Density ◽  
Gravitational Field Equations

We consider the density profile of pressureless dark matter in Eddington-inspired Born–Infeld (EiBI) gravity. The gravitational field equations are investigated for a spherically symmetric dark matter galactic halo, by adopting a phenomenological tangential velocity profile for test particles moving in stable circular orbits around the galactic center. The density profile and the mass distribution, as well as the general form of the metric tensor is obtained by numerically integrating the gravitational field equations, and in an approximate analytical form by using the Newtonian limit of the theory. In the weak field limit, the dark matter density distribution is described by the Lane–Emden equation with polytropic index n = 1, and is nonsingular at the galactic center. The parameter κ of the theory is determined so that the theory could provide a realistic description of the dark matter halos. The gravitational properties of the dark matter halos are also briefly discussed in the Newtonian approximation.

Sign in / Sign up

Export Citation Format

Share Document