The gravitational field in the wave zone I. The radiating terms of the metric tensor

We consider an asymptotically flat space-time generated by a perfect fluid source of compact spatial support. Using the de Donder gauge conditions, the Einstein equations are reduced to a new form of Poisson-type equations. A formal iterative scheme is set up to solve these equations by expanding the components of the metric tensor in powers ofc ^–1. The coefficient of each power ofc ^–1 depends on the asymptotically retarded timeu andx, y, z and satisfies a Poisson-type equation. Assuming asymptotic flatness the solution is carried out in the first orders. The results are explicit expressions of the metric up to orderc ^–4 in terms of the source functions. These expressions hold over all space-time. A further expansion in powers ofr ^–1 gives the first terms of the metric that contribute to gravitational radiation.     

Scale Transformation,Modified Gravity,and Brans-Dicke Theory

A model of Einstein-Hilbert action subject to the scale transformation is studied. By introducing a dilaton field as a means of scale transformation a new action is obtained whose Einstein field equations are consistent with traceless matter with non-vanishing modified terms together with dynamical cosmological and gravitational coupling terms. The obtained modified Einstein equations are neither those in f(R) metric formalism nor the ones in f(ℛ) Palatini formalism, whereas the modified source terms are formally equivalent to those of f(R)=\frac12R²f({\mathcal{R}})=\frac{1}{2}{\mathcal{R}}^{2} gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with w = -\frac32\omega=-\frac{3}{2} is explicitly shown, provided the dilaton field is condensated to its vacuum state.     

Rest Energy of Classical Self Fields of Point Gravitating Sources

A variant of Brans-Dicke theory is discussed in which the singularities of electric, scalar and metric sector of classical self fields of a point gravitating source are in Jordan frame suppressed and their energy - momentum tensor is integrable. The total energy of the classical electron Coulomb field is finite and in accordance with special relativistic expression m ₀ c ². The same may be said with respect to total rest energy of the quasi-Coulomb field, i.e. the scalar self-field of the source in the case of electron and in the case of a source with zero electric charge. Although (pseudo-)Einstein equations in (pseudo-)Pauli frame are modified, all experimental predictions concerning gravitational effects of macroscopic (celestial) bodies are in accordance with that of GRT.     

Gravitational Field Equations of Fourth Order and Supersymmetry

We discuss the field equations which stem from a variational principle containing the quadratic terms αR_μνR_μν and βR² besides the Einstein-Hilbert Lagrangian R. Comparison of this theory with a pure theory of fourth order shows that R must necessarily be included if we wish to interpret the field equations as gravitational equations. The Einstein-Bach-Weyl theory (α = ?3β) has the property of being a theory of “supergravitation”. Apart from gravitons without rest-mass, we have here only one additional kind of particles with rest-mass. Their mass may be determined by Planck' slength (hG/c³)^1/2. The occurrence of those particles results from the breakdown of a “supersymmetry”, that is of the conform invariance. The Einstein tensor E_μν ? R_μν ?1/2g_μνR can be regarded as a source of the gravitons without rest-mass.     

Kolmogorov’s Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $${\mathbb {R}^3}$$

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in \mathbb R³{\mathbb {R}^3} . We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the α ^th -order fractional derivatives of the velocity for some α > 0 in the space variables in L ², which is independent of the viscosity μ > 0. Then it is shown that this key observation yields the L ²-equicontinuity in the time variable and the uniform bound in L ^q , for some q > 2, of the velocity independent of μ > 0. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in \mathbb R³{\mathbb {R}^3} . We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of μ > 0, that is in the high Reynolds number limit.     

Spin-1/2 Maxwell Fields

Requiring covariance of Maxwell's equations without a priori imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, ²–A ², E ²–B ²+2i EB–(₀ +A)². The associated Maxwell Lagrangian density is also the same for both spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein–Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein–Gordon mass, and thus one complex Klein–Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass.     

Application of energy and angular momentum balance to gravitational radiation reaction for binary systems with spin-orbit coupling

We study gravitational radiation reaction in the equations of motion for binary systems with spin-orbit coupling, at order (v/c)⁷ beyond Newtonian gravity, or O(v/c)² beyond the leading radiation reaction effects for non-spinning bodies. We use expressions for the energy and angular momentum flux at infinity that include spin-orbit corrections, together with an assumption of energy and angular momentum balance, to derive equations of motion that are valid for general orbits and for a class of coordinate gauges. We show that the equations of motion are compatible with those derived earlier by a direct calculation.     

Liénard-Wiechertsche Potentiale für Multipolquellen

Liénard-Wiechert Potentials for Multipole Sources The Liénard-Wiechert solution is generalized to the case of a point-like multipole source possessing time-dependent multipole moments of arbitrary orders and moving along any time-like world line. The resulting potentials are exact solutions to the field equations of tensor fields of arbitrary rank s. If s ? 2, the motion of the source is constrained by the field equations.     

Solutions of the Einstein-Maxwell equations with many black holes

In Newtonian gravitational theory a system of point charged particles can be arranged in static equilibrium under their mutual gravitational and electrostatic forces provided that for each particle the charge,e, is related to the mass,m, bye=G ^1/2 m. Corresponding static solutions of the coupled source free Einstein-Maxwell equations have been given by Majumdar and Papapetrou. We show that these solutions can be analytically extended and interpreted as a system of charged black holes in equilibrium under their gravitational and electrical forces.We also analyse some of stationary solutions of the Einstein-Maxwell equations discovered by Israel and Wilson. If space is asymptotically Euclidean we find that all of these solutions have naked singularities.Alfred P. Sloan Research Fellow, supported in part by the National Science Foundation.     

Quaternion analyticity and conformally Kählerian structure in Euclidean gravity

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kähler manifold H ²×S ², we show the Euclidean Schwarzschild metric to be conformally related to another Kähler manifold M ²×S ² with M ² being conformal to H ² in two dimensions. Both metrics which are conformally Kählerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.     

Homogeneous anisotropic cosmological models with variable gravitational and cosmological “Constants”

The Einstein field equations with perfect fluid source and variable andG for Bianchi-type universes are studied under the assumption of a power-law time variation of the expansion factor, achieved via a suitable power-law assumption for the Hubble parameter suggested by M. S. Berman. All the models have a power-law variation of pressure and density and are singular at the epocht=0. The variation ofG(t) as 1/t and (t) as 1/t ² is consistent with these models.     

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities ² and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions E. The functions E are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. ³ These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.     

Hole mobility in a homogeneous nucleotide chain

The conductivity of molecular DNA-based conductors has been calculated. Charge motion is described by quantum-mechanical equations, and macromolecular vibrations are described by classical equations of motion with dissipation and a source of temperature fluctuations. In a homogeneous sequence of G-C nucleotide pairs, the calculated hole mobility at T=300 K equals ≈2cm² V^?1 s^?1.     

Proper-Time Relativistic Dynamics on Hyperboloid

The dynamics of a point-like relativistic particle with respect of the proper time is formulated on the hyperboloid p ² ₀–p ²=M ² c ². The Hamilton-Jacobi equations on the hyperboloid are derived for the particles with mass (M ²=m ², m>0), for the particles with zero-mass (M=0, m>0), and for the neutrino. It is shown, in a certain factorization of the momentum, the model can be identified with Nambu's three-dimensional phase space formalism. A first quantized version of the model is formulated according to a canonical scheme of quantization (Schrödinger quantization scheme).     

A Way To Discover Maxwell's Equations Theoretically

The Coulomb force, established in the rest frame of a source-charge Q, when transformed to a new frame moving with a velocity V has a form F = q E + q v × B, where E = E′_∥ + γE′_⊥ and B = (1/c ²)v × E and E′ is the electric field in the rest frame of the source. The quantities E and B are then manifestly interdependent. We prove that they are determined by Maxwell's equations, so they represent the electric and magnetic fields in the new frame and the force F is the well known from experiments Lorentz force. In this way Maxwell's equations may be discovered theoretically for this particular situation of uniformly moving sources. The general solutions of the discovered Maxwell's equations lead us to fields produced by accelerating sources.     

Relativistic three-quark equations and low-lying baryon spectroscopy

The relativistic three-quark equations are found in the framework of the dispersion relations technique. The approximate solutions of these equations using the method based on the extraction of leading singularities of the amplitude are obtained. The calculated mass values of two baryonic multipletsJ ^P =1/2⁺, 3/2⁺ are in good agreement with the experimental ones.     

Bubble dynamics in the expanding universe

Using the Gauss-Codazzi equations, the behavior of a singular hypersurface, which divides the universe into two Friedmann-Robertson-Walker space-time regionsV ⁺ andV ^–, is investigated. The equation of motion for a spherical bubble in the expanding universe is presented and the physical meaning of the equation is clarified. The equations of state for fluids inV ^± and on the boundary shell, which should be determined by microscopic physics, are arbitrary in the present geometrical approach. The derived equations are quite similar to those for a shell in a vacuum and can be applied to the case that one ofV ^± or both are Schwarzschild-de Sitter space-time too.     

The gravitational field in the wave zone III. Elimination of the gauge condition

The gravitational field of a bounded source is studied as a formal series expansion of powers ofc ^–1 without the use of a gauge condition. The conditions imposed on the metric by the asymptotic flatness and some mathematical properties of the field equations at each step of the expansion are proved to be sufficient for the unique determination of those combinations of the metric components that describe the emission of gravitational radiation.     

Self-gravitating general-relativistic cosmic strings

We examine the coupled Einstem-Euler-Lagrange equations for nonstationary cosmic strings. Self-consistent solutions to all the equations are found under the assumption that the energy-momentum tensor is of the formT _t ^t =T _z ^z while all other components vanish. It is shown that the strings are necessarily static in this case and that the scalar field potential must be of the usual quartic form with the coupling constants satisfying e²=8.