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 Generalizing the coupling between geometry and matter: $$f\left( R,L_m,T\right) $$ gravity

2021 ◽  
Vol 81 (7) ◽  
Author(s):  
Zahra Haghani ◽  
Tiberiu Harko
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Newtonian Limit ◽  
Gravity Models ◽  
Momentum Balance ◽  
Field Equations ◽  
Gravitational Fields ◽  
Generalized Poisson

AbstractWe generalize and unify the $$f\left( R,T\right) $$ f R , T and $$f\left( R,L_m\right) $$ f R , L m type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R, of the trace of the energy–momentum tensor T, and of the matter Lagrangian $$L_m$$ L m , so that $$ L_{grav}=f\left( R,L_m,T\right) $$ L grav = f R , L m , T . We obtain the gravitational field equations in the metric formalism, the equations of motion for test particles, and the energy and momentum balance equations, which follow from the covariant divergence of the energy–momentum tensor. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equations of motion is also investigated, and the expression of the extra acceleration is obtained for small velocities and weak gravitational fields. The generalized Poisson equation is also obtained in the Newtonian limit, and the Dolgov–Kawasaki instability is also investigated. The cosmological implications of the theory are investigated for a homogeneous, isotropic and flat Universe for two particular choices of the Lagrangian density $$f\left( R,L_m,T\right) $$ f R , L m , T of the gravitational field, with a multiplicative and additive algebraic structure in the matter couplings, respectively, and for two choices of the matter Lagrangian, by using both analytical and numerical methods.

scholarly journals WHY DOES GRAVITY IGNORE THE VACUUM ENERGY?

2006 ◽  
Vol 15 (12) ◽  
pp. 2029-2058 ◽  
Author(s):  
T. PADMANABHAN
Keyword(s):  
Gravitational Field ◽  
Cosmological Constant ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Integration Constant ◽  
Field Equations ◽  
Physical Context ◽  
Gravitational Field Equations ◽  
Higher Order Corrections

The equations of motion for matter fields are invariant under the shift of the matter Lagrangian by a constant. Such a shift changes the energy–momentum tensor of matter by [Formula: see text]. In the conventional approach, gravity breaks this symmetry and the gravitational field equations are not invariant under such a shift of the energy–momentum tensor. We argue that until this symmetry is restored, one cannot obtain a satisfactory solution to the cosmological constant problem. We describe an alternative perspective to gravity in which the gravitational field equations are [Gab - κTab]nanb = 0 for all null vectors na. This is obviously invariant under the change [Formula: see text] and restores the symmetry under shifting the matter Lagrangian by a constant. These equations are equivalent to Gab = κTab + Cgab, where C is now an integration constant so that the role of the cosmological constant is very different in this approach. The cosmological constant now arises as an integration constant, somewhat like the mass M in the Schwarzschild metric, the value of which can be chosen depending on the physical context. These equations can be obtained from a variational principle which uses the null surfaces of space–time as local Rindler horizons and can be given a thermodynamic interpretation. This approach turns out to be quite general and can encompass even the higher order corrections to Einstein's gravity and suggests a principle to determine the form of these corrections in a systematic manner.

The equations of motion in the non-symmetric unified field theory

1954 ◽  
Vol 226 (1166) ◽  
pp. 366-377 ◽  
Keyword(s):  
Equations Of Motion ◽  
Coulomb Force ◽  
Unified Field Theory ◽  
Field Equations ◽  
Original Theory ◽  
Unified Field ◽  
Theory Of Gravitation ◽  
The One ◽  
A Charge ◽  
Weak Fields

The field equations of the non-symmetric unified theory of gravitation and electromagnetism are changed so that they imply the existence of the Coulomb force between electric charges. It is shown that the equations of motion of charged masses then follow correctly to the order of approximation considered. The equations for weak fields in the modified theory are derived and shown to lead to Maxwell’s equations together with a restriction on the current density. This restriction is different from that in the original theory, and in the static, spherically symmetric case permits a charge distribution more likely to correspond to a particle. The failure of the original theory to lead to the equations of motion is related to the structure of the quantities appearing in it, and reasons are given for supposing that no nonsymmetric theory simpler than the one put forward is likely to give these equations in their conventional form.

scholarly journals Unification of Gravity and Electromagnetism

2021 ◽  
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Mathematical Structure ◽  
Momentum Tensor ◽  
Dirac Field ◽  
Field Equations ◽  
Mathematical Structures ◽  
Gravitational Field Equations ◽  
Definition Of ◽  
Two Sides

Gravity and electromagnetism are two sides of the same coin, which is the clue of this unification. Gravity and electromagnetism are representing by two mathematical structures, symmetric and antisymmetric respectively. Einstein gravitational field equation is the symmetric mathematical structure. Electrodynamics Lagrangian is three parts, for electromagnetic field, Dirac field and interaction term. The definition of canonical energy momentum tensor was used for each term in Electrodynamics Lagrangian to construct the antisymmetric mathematical structure. Symmetric and antisymmetric gravitational field equations are two sides of the same Lagrangian

Field theories with high derivatives

1948 ◽  
Vol 44 (1) ◽  
pp. 76-86 ◽  
Author(s):  
T. S. Chang
Keyword(s):  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Functional Derivative ◽  
Momentum Tensor ◽  
Field Theories ◽  
Field Equations ◽  
Gauge Invariant ◽  
Image Position ◽  
Derivatives Of

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative ofwith respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

scholarly journals PALATINI FORMULATION OF MODIFIED GRAVITY WITH A NON-MINIMAL CURVATURE-MATTER COUPLING

2011 ◽  
Vol 26 (20) ◽  
pp. 1467-1480 ◽  
Author(s):  
TIBERIU HARKO ◽  
TOMI S. KOIVISTO ◽  
FRANCISCO S. N. LOBO
Keyword(s):  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Scalar Fields ◽  
Momentum Tensor ◽  
Nonlinear Dependence ◽  
Modified Theories Of Gravity ◽  
Field Equations ◽  
Test Particles ◽  
Theories Of Gravity ◽  
Minimal Curvature

We derive the field equations and the equations of motion for scalar fields and massive test particles in modified theories of gravity with an arbitrary coupling between geometry and matter by using the Palatini formalism. We show that the independent connection can be expressed as the Levi–Cività connection of an auxiliary, matter Lagrangian dependent metric, which is related with the physical metric by means of a conformal transformation. Similarly to the metric case, the field equations impose the nonconservation of the energy–momentum tensor. We derive the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force is obtained in terms of the matter-geometry coupling functions and of their derivatives. Generally, the motion is non-geodesic, and the extra force is orthogonal to the four-velocity. It is pointed out here that the force is of a different nature than in the metric formalism. We also consider the implications of a nonlinear dependence of the action upon the matter Lagrangian.

The foundations of a generalization of gravitation theory

1957 ◽  
Vol 53 (2) ◽  
pp. 473-488 ◽  
Author(s):  
John Moffat
Keyword(s):  
Electromagnetic Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Charged Particles ◽  
Gravitational Theory ◽  
Momentum Tensor ◽  
Original Form ◽  
Geometric Description ◽  
Field Equations ◽  
Electrically Charged

1. Introduction. Among the more notable attempts to derive a generalization of Einstein's gravitational theory is the recent one of Einstein and Schrodinger ((1)–(8)). This was formulated by dropping the symmetry of the fundamental tensor gμν and the components of the affine connexion. The most serious defect of these non-symmetric theories is that the field equations, in their original form, do not determine the motion of electrically charged particles in an electromagnetic field, as has been proved by Infeld(9), Callaway (10) and Bonnor (n). Together with the lack of an energy-momentum tensor and a geometric description of the paths of charged particles, this seems to indicate that the concept of motion is missing in this type of theory. It is clear that one of the most important results which should follow from a generalization of Einstein's gravitational theory is the correct equations of motion of charged particles in an electromagnetic field.

scholarly journals Particle paths of general relativity as geodesics of an affine connection

1970 ◽  
Vol 3 (3) ◽  
pp. 325-335 ◽  
Author(s):  
R. Burman
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Momentum Tensor ◽  
Field Equations ◽  
Test Particles ◽  
Special Cases ◽  
Particle Paths ◽  
Arbitrary Energy

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

scholarly journals On the trace of the energy-momentum tensor of fields associated with particles of zero rest mass

1959 ◽  
Vol 1 (1) ◽  
pp. 99-105 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Energy Momentum Tensor ◽  
Rest Mass ◽  
Momentum Tensor ◽  
Conformal Transformations ◽  
Field Equations ◽  
Photon Field ◽  
Neutrino Field ◽  
Two Component ◽  
Spin Fields

The traceTof the metrical energy-momentum tensorTklof fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property ofTklis correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular thatTvanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.

scholarly journals FIELD EQUATIONS AND EQUATIONS OF MOTION IN POST-NEWTONIAN APPROXIMATION OF THE PROJECTIVE UNIFIED FIELD THEORY

2012 ◽  
Vol 21 (02) ◽  
pp. 1250012
Author(s):  
ALEXANDER GORBATSIEVICH ◽  
ERNST SCHMUTZER
Keyword(s):  
Field Theory ◽  
Experimental Data ◽  
Equations Of Motion ◽  
Inertial Mass ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Newtonian Approximation

The equations of motion of N gravitationally bound bodies are derived from the field equations of Projective Unified Field Theory. The Newtonian and the post-Newtonian approximations of the field equations and of the equations of motion of this system of bodies are studied in detail. In analyzing some experimental data we performed some numeric estimates of the ratio of the inertial mass to the scalaric mass of matter.

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