Weak solutions of the massless field equations

1982 ◽  
Vol 384 (1787) ◽  
pp. 403-425 ◽  
Keyword(s):  
Minkowski Space ◽  
Weak Solutions ◽  
Rest Mass ◽  
Analytic Solutions ◽  
Alternative Proof ◽  
Field Equations ◽  
Distribution Solution ◽  
Massless Field ◽  
Space Of Distributions ◽  
Twistor Methods

We consider weak solutions of the zero-rest-mass (z. r. m.) equations described in Eastwood et al . (1981). The space of hyperfunctions, which contains the space of distributions, is defined and we consider hyperfunction solutions of the equations on real Minkowski space M I and its conformal compactification M . We define a hyperfunction z. r. m. field to be future or past analytic if it is the boundary value of a holomorphic z. r. m. field on the future or past tube of complex Minkowski space respectively; and we demonstrate that any field on M I that is the sum of future and past analytic fields extends as a hyperfunction z. r. m. field to all of M . It is shown that any distribution solution on M I splits as required and hence extends as a hyperfunction solution to M . Twistor methods are then used to show that the same applies in the more general case of hyperfunction solutions on M I . This leads to an alternative proof of the main result of Wells (1981): a hyperfunction z. r. m. field on compactified real Minkowski space is a unique sum of future and past analytic solutions.

Massless fields on Minkowski space-time with quantized, Kerr sources

1993 ◽  
Vol 441 (1911) ◽  
pp. 191-200 ◽  
Keyword(s):  
Minkowski Space ◽  
Rest Mass ◽  
Space Time ◽  
Maxwell Field ◽  
Field Equations ◽  
Self Energy ◽  
Minkowski Space Time ◽  
Massless Fields ◽  
Mass Field

Solutions to the zero rest mass field equations generated by spinning Kerr sources are reviewed. It is shown that the act of quantization removes the singularity when the source has spin ½ . For a Maxwell field, the integrated self-energy is also finite.

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.

On Conformally Covariant Spinor Field Equations

1972 ◽  
Vol 50 (18) ◽  
pp. 2100-2104 ◽  
Author(s):  
Mark S. Drew
Keyword(s):  
Minkowski Space ◽  
Invariant Mass ◽  
Spinor Field ◽  
Dimensional Space ◽  
Flat Space ◽  
Field Equations ◽  
First Order ◽  
Conformally Invariant ◽  
Spinor Fields

Conformally covariant equations for free spinor fields are determined uniquely by carrying out a descent to Minkowski space from the most general first-order rotationally covariant spinor equations in a six-dimensional flat space. It is found that the introduction of the concept of the "conformally invariant mass" is not possible for spinor fields even if the fields are defined not only on the null hyperquadric but over the entire manifold of coordinates in six-dimensional space.

scholarly journals Relativistic equations for elementary particles

1948 ◽  
Vol 192 (1029) ◽  
pp. 195-218 ◽  
Keyword(s):  
Elementary Particle ◽  
Wave Equation ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Wave Form ◽  
Total Charge ◽  
Field Equations ◽  
Relativistic Approximation ◽  
Special Cases ◽  
Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

scholarly journals A MAXWELL-LIKE FORMULATION OF GRAVITATIONAL THEORY IN MINKOWSKI SPACE–TIME

2007 ◽  
Vol 16 (06) ◽  
pp. 1027-1041 ◽  
Author(s):  
EDUARDO A. NOTTE-CUELLO ◽  
WALDYR A. RODRIGUES
Keyword(s):  
Gravitational Field ◽  
Minkowski Space ◽  
Gravitational Theory ◽  
Space Time ◽  
Lorentzian Geometry ◽  
Field Equations ◽  
Yang Mills ◽  
Minkowski Space Time ◽  
Clifford Bundle ◽  
Gauge Fixing

Using the Clifford bundle formalism, a Lagrangian theory of the Yang–Mills type (with a gauge fixing term and an auto interacting term) for the gravitational field in Minkowski space–time is presented. It is shown how two simple hypotheses permit the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries. In the case of a Lorentzian geometry interpretation of the theory, the field equations are shown to be equivalent to Einstein's equations.

scholarly journals New cosmological solutions in hybrid metric-Palatini gravity from dynamical symmetries

2021 ◽  
pp. 2150100
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Conservation Laws ◽  
Integrable Systems ◽  
Functional Form ◽  
Analytic Solutions ◽  
Momentum Conservation ◽  
Field Equations ◽  
Dynamical Symmetries ◽  
Cosmological Solutions ◽  
Liouville Integrable

We investigate the existence of Liouville integrable cosmological models in hybrid metric-Palatini theory. Specifically, we use the symmetry conditions for the existence of quadratic in the momentum conservation laws for the field equations as constraint conditions for the determination of the unknown functional form of the theory. The exact and analytic solutions of the integrable systems found in this study are presented in terms of quadratics and Laurent expansions.

Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

2018 ◽  
Vol 61 (3) ◽  
pp. 705-733 ◽  
Author(s):  
Athanasios N. Lyberopoulos
Keyword(s):  
Scalar Field ◽  
Weak Solutions ◽  
Bound States ◽  
Critical Sobolev Exponent ◽  
Field Equations ◽  
Critical Sobolev Exponents ◽  
Sobolev Exponent ◽  
Image Position ◽  
Scalar Field Equations

AbstractWe are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.

Sign in / Sign up

Export Citation Format

Share Document