A modification of Einstein–Schrödinger theory that contains both general relativity and electrodynamics

The field generated by a time‐harmonic pure irrotational current distributed in a finite region of free space is derived analytically. Only an electric field is produced and is nonzero even merely within the finite interior of the current distribution, though the problem space itself is infinite. To consider the obtained field solution nonphysical according to our current physical concepts, however, lacks evidence or reasoning. It satisfies the original first‐order Maxwell's equations (MEs) and the related continuity conditions exactly, hence cannot be rejected mathematically. It is also shown that a conceptual denial of such a different kind of solution based on physical interpretations of other known solutions of the same MEs is logically not acceptable. A direct experiment study is necessary to conclude convincingly whether or not the above solution is physical. The completeness problem of MEs and their consistency with the Lorentz force law and the equation of motion are discussed. Free space MEs are incomplete or underdetermined due to the absence of a curl equation for the current density. If the solution is nonphysical, ME‐based electromagnetic (EM) theory is not self‐consistent. On the other hand, if it is physical, this type of inconsistency within the frame of the ME‐based EM theory disappears, yet it is inconsistent with the Lorentz force law and the equation of motion in this case.

Hyperdimensional unified field theories date back to 1921, when Kaluza reported that the Einstein equations of general relativity written in five dimensions (5D) would yield both the standard four‐dimensional Einstein equations as well as the Maxwell equations, thus unifying general relativity and electrodynamics. Subsequent research focused on the field equations, with little attention paid to the equations of motion. While unified field theories have since evolved to address quantum forces with many extra compactified dimensions, the original classical 5D theory of Kaluza makes some compelling predictions about a macroscopic fifth dimension. These include identification of electric charge with the space/time manifestation of motion in the fifth dimension, emergence of the gravitational constant as a cosmic charge‐to‐mass ratio, and the prediction of hyperluminal 5D proper velocities. This paper provides the mathematical derivation of these and other results with a focus on the equations of motion, a critical consideration of the underpinnings and assumptions of the 5D theory, and a discussion of the historical context of the development of the 5D Kaluza theory and how its serious consideration could have been neglected for so long.

We give an exact solution of the gravitational and electromagnetic field equations with a charged rotating source in new general relativity. The solution has three parameters Q, h and a, and it gives a charged Kerr metric space-time. The parallel vector fields and the electromagnetic vector potential are axially symmetric. In this space-time, we cannot discriminate new general relativity from general relativity, so far as scalar, the Dirac and the Yang-Mills fields and macroscopic bodies are used as probes. The space-time does not have singularities at all, although it has an “effective singularity”. Two kinds of Reissner-Nordström metric solutions, one is our solution with h = 0 and the other is a solution given by Hayashi and Shirafuji, are physically equivalent with each other. Nevertheless, these are markedly different from each other with regard to the asymptotic behavior of the torsion tensor for r →∞ and the space-time singularities.

We review the electromagnetic scalar and vector potentials from the point of view of the Generalized Fredholm Integral Equations. Then we revise two sensitive concepts as the retarded and advanced Green functions. Provided we have developed the appropriate desired equations; we go to the problem of left-hand materials conditions for optimized broadcasting. We give also an academic but simple example. Key Words — Electromagnetic potentials, Generalized Fredholm Integral Equations, left-hand materials conditions, Green functions, resonances.

In recent years there had been a growing interest in analog models of general relativity, with certain superfluid solutions simulating black hole solutions of Einstein's gravitational field equation. The quantization of a superfluid, composed of discrete particles (helium atoms), treated as a nonrelativistic many body problem does not lead to divergencies as the quantization of Einstein's field equations. Quantization of gravity is possible in string theory, but only if one introduces the daring hypothesis of higher dimensions. But if the gravitational field is made up of discrete elements as superfluid helium is made up of helium atoms, then gravity can be quantized without difficulty in three space and one time dimension. Such a hypothesis, of course, implies that Lorentz invariance is a dynamic symmetry caused by real rod and clock deformations, as it was assumed in the pre-Einstein theory of relativity by Lorentz and Poincare, which required the existence of an aether. Making the hypothesis that this aether is a kind of superfluid plasma made up of positive and negative Planck mass particles interacting with the Planck force over a Planck length, one obtains an analog of the standard model, including gravity, which can be quantized as a nonrelativistic many body problem. In this model nonrelativistic vortex rings in three space dimensions and one time dimension simulate the relativistic theory of closed strings in ten space-time dimensions. But because in the vortex lattice, one obtains a large dimensionless number conceivably advancing our understanding of the finestructure constant.

In recent years there had been a growing interest in analog models of general relativity, with certain superfluid solutions simulating black hole solutions of Einstein's gravitational field equation. The quantization of a superfluid, composed of discrete particles (helium atoms), treated as a nonrelativistic many body problem does not lead to divergencies as the quantization of Einstein's field equations. Quantization of gravity is possible in string theory, but only if one introduces the daring hypothesis of higher dimensions. But if the gravitational field is made up of discrete elements as superfluid helium is made up of helium atoms, then gravity can be quantized without difficulty in three space and one time dimension. Such a hypothesis, of course, implies that Lorentz invariance is a dynamic symmetry caused by real rod and clock deformations, as it was assumed in the pre-Einstein theory of relativity by Lorentz and Poincaré, which required the existence of an aether. Making the hypothesis that this aether is a kind of superfluid plasma made up of positive and negative Planck mass particles interacting with the Planck force over a Planck length, one obtains an analog of the standard model, including gravity, which can be quantized as a nonrelativistic many body problem. In this model nonrelativistic vortex rings in three space dimensions and one time dimension simulate the relativistic theory of closed strings in ten space-time dimensions. But because in the vortex lattice, one obtains a large dimensionless number conceivably advancing our understanding of the finestructure constant.

Maxwell's equations, Lorentz's equations of motion, and Dirac's equations are investigated on a cosmological background. Maxwell's equations are conform invariant and the equations of motion can easily be made conform invariant. Then, without introducing any new assumptions, Dirac's equations are seen to be conform invariant. In open universes the solutions of Maxwell's and Dirac's equations are the same as in flat Minkowski space. The behavior of these equations is different in closed universes by reasons of topology. In closed universes both Maxwell's and Dirac's equations provide new eigenvalue problems. Dirac's equations exhibit a marked difference between an elliptic and a spherical closed universe.

For a best broadcasting efficiency we give in this paper an alternative treatment with electromagnetic potentials (EP) and propose a powerful tool based on the resonance properties. Recently, we have developed some methods for studying the change on broadcasting conditions and explained theoretically as the brake of confinement conditions for the so-called evanescent waves. We also named the former as left-hand material conditions because the event that occurs inside a left-hand material can be analyzed with exactly the same tool as the broadcasting case. So we have proposed that the equivalent mathematical behavior in case of electromagnetically broadcasting is really a transformation of the confined near field into travelling waves. In addition, we know that if we make a time reversal, the quality of the signal becomes much better and can be focused in a more restricted area. In other words, the theory of the generalized Fredholm integral equations (GFE) has been used to first introduce a source for the time reversal procedure but then we transform the inhomogeneous equations (GIFE) into generalized homogeneous ones (GHFE) for describing the resonances, that is the frequencies for which we can see the brake of confinement conditions and because of their singular properties, prevent information lose. But in the past we never talked about the electromagnetic potentials neither the retarded Green functions and applying them to the mixing of signals in the same thin band but with different recording times. In this paper we built the equivalent formalism that involves both the electromagnetic potentials and the retarded (and advanced) Green functions. We must underline that, for convenience, the system considered is a discrete one, but we can recover a continuum version by including the discrete indexes as continuum variables. To convert evanescent waves into travelling waves we must also locate the sink in the near field region, restriction that we have explained as a consequence of orthogonality rules followed by the resonances. Once we have obtained the inhomogeneous vector-matrix equation we inquire ourselves about a singular condition that is a situation similar to that of scattering quantum mechanics theory when we have an outgoing wave but we don't have an incident one. That condition on scattering theory describes an unstable system known as resonance. If we demand the same condition on our vector-matrix equation, we obtain also resonances but which describes travelling waves that come not for an incident wave but for evanescent waves. In addition, in accordance with Communication Theory we have shown that it is possible to introduce the recording time as an additive contribution to the so-called power. The fact that we are using the same band but different cutting limits, also suggests that we can design an appropriate filter that can distinguish between signals according to the recording time that is we can superpose signals with the same frequency range but with different recording times. In a previous work we have sketched a filter, but now we give a better-defined device. In addition we introduce the so-called Plasma Sandwich Model (PSM) with the purpose to optimize the broadcasting, which propose that regions or zones with a fluctuating refraction index constitute the transmission media and that strongly depends on the magnetization and local values of the EP. Finally we give a simple example for obtaining the resonant frequencies for a particular form for the Fourier transform of the kernel in the GHFE.

The conference brought together people working in mathematical general relativity, a field which lies at the interface of analysis, differential geometry and physics. There was a mix of established workers in the subject with participants at the start of their careers and researchers from neighbouring fields. The Einstein equations are at the heart of general relativity theory. They form a system of evolution equations and their solutions are conveniently parametrized by initial data. These initial data are required to satisfy constraint equations and these were the topic of several lectures at the conference. The talks of Corvino and Pollack were concerned with recent progress in methods for constructing solutions of the constraints. An important concept for solutions of the constraints is that of mass, including quasilocal mass. This was a central theme of the talks of Degeratu, Huisken and Shi. Huisken introduced a striking new approach to the definition of mass based on isoperimetric inequalities. Dain presented a variational characterization of the extreme Reissner–Nordström solution, thus relating the study of the constraint equations to the theory of black holes. The talk of Wohlfarth was rather outside the main area of the conference. He described a new concept of geometry motivated by string theory which may come to enrich the circle of ideas within mathematical relativity. Many of the talks at the conference were on evolution equations related to general relativity. Dafermos and Finster presented results related to the dynamical stability of black holes. Dafermos described new results on the rate of decay of solutions of the wave equation on the Schwarzschild spacetime while Finster explained applications of methods of functional analysis to the wave equation on the Kerr spacetime. Struwe's talk concerned uniqueness for supercritical nonlinear wave equations which from the point of view of the Einstein equations are an important example to compare with. The Maxwell equations are another important comparison system and Bauer showed how they can be used to understand more about radiation formulae. Velázquez gave an introduction to singularity formation in the Keller–Segel model, a parabolic system coming from mathematical biology. This is a possible source of insight for obtaining a rigorous understanding of critical collapse in general relativity. The talks of Andréasson, Choptuik and Lindblom dealt with various aspects of the application of numerical techniques to the study of the Einstein evolution equations. Choptuik showed impressive new simulations of coalescing black holes due to Pretorius which could hardly have been imagined just a year ago. Cosmology is at present a very active area of research in general relativity. This is in part due to the challenge of understanding the observed accelerated expansion of our universe. Mathematics is beginning to make its mark in this subject and this was reflected by talks of Heinzle, Rendall and Tod. Mathematical relativity is a meeting point for many ideas and the abstracts which follow give some idea of the variety of the subject. In fact the spectrum of topics discussed by the participants at the conference was much wider than those for which talks could be scheduled. By limiting the number of presentations it was possible to leave plenty of time for people to exchange insights. The lively interactions observed make us hopeful that this conference has given a boost to the development of the subject in the next few years.

A wave equation complying with Galilean transformations is proposed for electromagnetic and matter waves. A fundamental feature entirely different from the principle of relativity is that the wave equation is referred specifically to the proposed local-ether frame. For electromagnetic wave, the local-ether wave equation accounts for a wide variety of experiments of propagation or interference. For matter wave, the wave equation leads to modifications of Schrodinger's equation which in turn lead to a unified quantum theory of electromagnetic and gravitational forces in conjunction with the identity of inertial and gravitational mass. Further, it leads to modifications of the Lorentz force law and of Maxwell's equations. Moreover, the wave equation leads to dispersion of matter wave, from which the speed-dependences in the mass of a particle and in the wavelength, angular frequency, and quantum energy of a matter wave are derived. The consequences of the wave equation are in accord with a wide variety of experiments which are commonly ascribed to the special relativity, the general relativity, the Lorentz mass-variation law, or to the de Broglie matter wavelength. Thereby, the local-ether wave equation unifies quantum mechanics, electromagnetics, and gravitation.

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Recent attempts to obtain the force of radiative reaction in the classical equations of motion of point charges have proceeded from two different viewpoints. Each of these has to introduce one basic assumption in addition to Maxwell's equations, namely the conservation law for the electromagnetic energy-momentum tensor in field theory and the relation between the Lorentz force and the momentum of the particle in action-at-a-distance theory. In previous field-theoretical derivations the Lorentz-Dirac equations including radiation damping are obtained only if one takes the field produced by the particle to be the retarded field; but equations of motion without the damping term are obtained if one uses half the sum of retarded and advanced fields. On the other hand, the theory of action at a distance as developed by Wheeler and Feynman was able to obtain the radiation damping using fields symmetric in time. It is noted that the need for the exclusive use of retarded fields arose only in the field-theoretical derivations for the one-particle problem. The considerations of Wheeler and Feynman on the total field due to all particles of the universe are, however, applicable to field theory in the symmetric form as well as to action-at-a-distance theory. The acceptance of their condition of "complete absorption" again leads to the radiation damping term in the equations of motion of the many-body problem. Some implications of this result are discussed.

The gauge-covariant theory of gravitation (GCTG) is a successful approach to cosmology that has been developed in the last three years by the authors of ref. (1). We disregard the cosmological applications of such a theory in this letter, and we address ourselves to examining the basic aspects of the theory as well as the consistency with their own postulates. The GCTG begins by examining the idea of different systems of units. Theories which exhibit ful] invariance under Weyl's scale transformations do not define a particular scale of length, so they are valid in any system of units. On the contrary, theories which break scale invarianee do set a particular system o /un i t s . In the case of gravitation (or general relativity) Einstein 's equations of motion are known not to be Weyl's scale invariant and therefore they immediately set the system of gravitational units. Conversely, such a system is defined as the one in which Einstein's equations hold.

Cartan's first and second structure equations together with first and second Bianchi identities can be interpreted as equations of motion for the tetrad, the connection and a set of two-form fields ${T}^{I}$ and ${R}_{J}^{I}$. From this viewpoint, these equations define by themselves a field theory. Restricting the analysis to four-dimensional spacetimes (keeping gravity in mind), it is possible to give an action principle of the $BF$ type from which these equations of motion are obtained. The action turns out to be equivalent to a linear combination of the Nieh-Yan, Pontrjagin, and Euler classes, and so the field theory defined by the action is topological. Once Einstein's equations are added, the resulting theory is general relativity. Therefore, the current results show that the relationship between general relativity and topological field theories of the $BF$ type is also present in the first-order formalism for general relativity.