The Einstein equation solution inside a ball with uniform density

We discuss the relation between the gravitational and electromagnetic fields as governed by the Einstein-Maxwell field equations. It is emphasized that the tendency of the gravitational field to induce electromagnetic effects increases as the size of the system decreases. This is because the charge-to-mass ratio $Q/M$ is typically larger in smaller systems. For most astrophysical systems, $Q/M$ is $\ll 1$ while for a Millikan oil drop, $Q/M \sim 10^6$. Going all the way down to elementary particles, the value for the electron is $Q/M \sim 10^{21}$. For subatomic systems there is an additional phenomenon which comes into play. In fact, according to general relativity, the gravitational field tends to become dominated by the spin at distances of the order of the Compton wavelength.The relevant quantity which governs this behavior is the ratio $S/M^2$ where $S$ is the (spin) angular momentum. For an electron, $S/M^2 \sim 10^{44}$.As a consequence, the gravitational field becomes dominated by gravitomagnetic effects in the subatomic domain.This fact has important consequences for the electromagnetic fields of spinning charged particles.To analyze this situation we use the asymptotic structure in the form of the multipole fields.%Such an approach avoids the pitfalls should one try to use a near-field approach using some kind of semi-classical formulation of the Einstein-Maxwell equations for example. To obtain more exact results however, one must take quantum effects into account including radiative contributions. Although such effects are not considered in this work, the order of magnitude of the considered effects are not expected to change drastically when going to a quantum mechanical treatment. The most relevant solution of the Einstein-Maxwell equations in this context is the Kerr-Newman metric. It is the preferred solution which is in accord with all the four known multipole moments of the electron to an accuracy of one part in a thousand. Our main result is that general relativity predicts corrections to the Coulomb field for charged spinning sources. Experimentally verifiable consequences include a predicted electric quadrupole moment for the electron, possible quasi-bound states in positron-heavy ion scattering with sizes corresponding to observed anomalous peaks, as well as small corrections to energy levels in microscopic bound systems such as the hydrogen atom

In the Einstein field equations, the geometry or the curvature of space-time defined as depended on the distribution of mass and energy principally resides on the left-hand side is set identical to a non-geometrical tensorial representation of matter on the right-hand side. In one or another form, general relativity accords a direct geometrical significance only to the gravitational field while the other physical fields are not of space time. They reside only in space time. Less well known, though of comparable importance is Einstein’s dissatisfaction with the fundamental asymmetry between gravitational and non-gravitational fields and his contributions to develop a completely relativistic geometrical field theory of all fundamental interactions, a unified field theory. Of special note in this context and equally significant is Einstein’s demand to replace the symmetrical tensor field by a non-symmetrical one and to drop the condition gik = gki for the field components. Historically, many other attempts were made too, to extend the general theory of relativity’s geometrization of gravitation to non-gravitational interactions, in particular, to electromagnetism. Still, progress has been very slow. It is the purpose of this publication to provide a unified field theory in which the gravitational field, the electromagnetic field and other fields are only different components or manifestations of the same unified field by mathematizing the relationship between cause and effect under conditions of general theory of relativity.

Effect of interconvertibility of electromagnetic and gravitational waves in strong external electromagnetic fields and the propagation of waves in the field of a charged “black hole”: PMM vol. 38, n≗ 6, 1974, pp. 1122–1129

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term $${{\bf A} \wedge {\it d}{\bf A}}$$ that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.

We investigate the gravitational and electromagnetic fields on the generalized Lagrange space endowed with the metricgij(x, y) = γij(x) + {1 + 1/n2(x, y)}yiyj. The generalized Lagrange spacesMm do not reduce to Lagrange spaces. Consequently, they cannot be studied by methods of symplectic geometry. The restriction of the spacesMm to a sectionSν(M) leads to the Maxwell equations and Einstein equations for the electromagnetic and gravitational fields in dispersive media with the refractive indexn(x, V) endowed with the Synge metric. Whenn(x, V) = 1 we have the classical Einstein equations. If 1/n2=1−1/c2 (c being the light velocity), we get results given previously by the authors. The present paper is a detailed version of a work in preparation.

Some unified theories on the gravitational and electromagnetic fields are researched. We investigate mainly two new geometric unified theories. A method is that the gravitational field and the source-free electromagnetic field can be unified by the equations Rklmi = κTklmi* in the Riemannian geometry, both are contractions of im and ik, respectively. If Rklmi = κTklmi* =constant, it will be equivalent to the Yang’s gravitational equations Rkm;l –Rkl;m = 0, which include Rlm= 0. From Rlm= 0 we can obtain the Lorentz equations of motion, the first system and second source-free system of Maxwell’s field equations. This unification can be included in the gauge theory, and the unified gauge group is SL(2,C) × U(1)=GL(2,C), which is just the same as the gauge group of the Riemannian manifold. Another unified method on the general nonsymmetric metric field with high-dimensional space-time and its matrix representations are analyzed mathematically. Further, the general unified theory of five-dimensional space-time combined quantum theory and four interactions is researched. Some possible unification ways on the gravitational and electromagnetic fields are discussed. The general matrix and various corresponding theories may decompose to a sum of symmetry and antisymmetry. Moreover, we proposed an imaginative representation on the ten dimensional space-time.

FFT gravity field calculation method and super-ellipsoid generated field

Einstein's gravitational field equations in empty space outside a massive plane with infinite extension give a class of solutions describing a field with flat spacetime giving neutral, freely moving particles an acceleration. This points to the necessity of defining the concept “gravitational field” not simply by the nonvanishing of the Riemann curvature tensor, but by the nonvanishing of certain elements of the Christoffel symbols, called the physical elements, or the nonvanishing of the Riemann curvature tensor. The tidal component of a gravitational field is associated with a nonvanishing Riemann tensor, while the nontidal components are associated with nonvanishing physical elements of the Christoffel symbols. Spacetime in a nontidal gravitational field is flat. Such a field may be separated into a homogeneous and a rotational component. In order to exhibit the physical significance of these components in relation to their transformation properties, coordinate transformations inside a given reference frame are discussed. The mentioned solutions of Einstein's field equations lead to a metric identical to that obtained as a result of a transformation from an inertial frame to a uniformly accelerated frame. The validity of the strong principle of equivalence in extended regions for nontidal gravitational fields is made clear. An exact calculation of the weight of an extended body in a uniform gravitational field, from a global point of view, gives the result that its weight is independent of the position of the scale on the body.

The rapid growth in the use of teams and multiple team membership has led more organizations to structure work around dynamic teams, characterized by short lifespans and fluid membership boundaries. This approach to organizing work makes it difficult for teams to work efficiently or to support the learning and growth of members. Given the importance of productivity and learning, we ask how these processes can be supported in dynamic teams. We do so in a randomized controlled field experiment conducted with 91 teams in a teaching hospital, a context in which physician trainees must learn while providing patient care in dynamic teams of care providers. We randomly assigned physician teams, or “core” teams, to launch their work with an intervention focused either on internal coordination among the core team members or on external coordination with a changing cast of external contributors such as nurses and specialists. We measured resulting levels of internal and external coordination over teams’ week-long lifespans and observed that external coordination was associated with improved team efficiency, whereas internal coordination was associated with improved individual learning. Post hoc exploratory analysis suggested that individual learning was highest when teams achieved high levels of both internal and external coordination, and it was positively correlated with improvement in the team’s patients’ average length of hospital stay. We discuss the implications of the demonstrated causal effects of team launches, along with our exploratory findings, for the management of dynamic teams and the opportunities to mitigate potential trade-offs between learning and productivity. Supplemental Material: The online appendix is available at https://doi.org/10.1287/orsc.2022.16729 .

The Indonesian manufacturing industry contributes significantly to economic growth in the globalization era. The capability of company leaders to manage excellently determines manufacturing's competitiveness and sustainable performance to survive in current conditions. This study examines the impact of management style and supply chain coordination on operational performance through lean project adoption. The research surveyed manufacturing companies in Indonesia by distributing questionnaires with Google form links to industry practitioners. The survey obtained 172 manufacturing companies with purposive sampling, which is at least the staff level in the company and permanent employees. Data processing is carried out by structural equation modeling and meets the goodness of fitness. The processing results showed that management style influenced internal supply chain coordination by 0.729, external supply chain coordination by 0.221, and lean project manufacture by 0.152. The company's internal supply chain coordination impacts increased external supply chain coordination by 0.512, lean project manufacture by 0.375, and operational performance by 0.405. External supply chain coordination impacted the increase in lean project manufacturing by 0.396. and operational performance of 0.234. Lean project manufacturing in the company impacts operational performance by 0.172. The practical contribution of the research provides insight into supervisors and managers always to be consistent and committed to lean manufacturing practices and build strong internal and external coordination. The theoretical contribution of research can enrich the theory of a resource-based view with sustainable supply chain and leadership transformation.

Weyl’s Gauge Principle of 1929 has been used to show that a five‐dimensional gauge field wherein the fifth dimension is conserved in the same mathematical sense as the conservation of mass embeds a four‐dimensional hyper surface into the five dimensional manifold. The equations specifying the geometry of the embedded hyper surface are similar in form to Einstein’s field equations of his General Theory of Relativity. These field equations become Einstein’s field equations when one takes the fifth dimension to be mass density. This means the predictions of Einstein’s General Theory may be achieved in two different ways. One by using Einstein’s method and the other by using a five dimensional manifold of space, time and mass density while restricting mass to be conserved. The five dimensional gauge fields associated with these phenomena are converted into a four dimensional curved hyper surface by the conservation of mass. The mathematical equivalence of the two methods provides the justification to seek what predictions might be made considering the five dimensional gauge field wherein the electromagnetic fields are inductively coupled to the gauge gravitational field. This presentation will present the logic showing how Einstein’s General Theory of Relativity may be derived from the five dimensional manifold using the conservation of mass. Comparison of predictions of perihelion advance by using the view from the embedded four‐dimensional hyper surface and from the five dimensional manifold shows the equivalence of the two methods of viewing physical phenomena. The inductive coupling between the electromagnetic and gravitational fields may be given experimental support by showing that predictions of the Earth’s magnetic moment due to its spinning gravitational mass is within 7% of the experimentally measured value. Such an inductively coupling between the gravitational gauge field and the electromagnetic gauge field provides an opportunity to seek predictions of the resulting five dimensional wave equations. Whereas in Maxwellian electromagnetism there are two, coupled, vector wave equations in the five dimensional gauge field there are three, coupled, vector wave equations that are also coupled to a scalar wave equation. The presentation will discuss the resulting transverse and longitudinal solutions to this system of wave equations. It will show that the inductive coupling with the gravitational field provides a very weak gravitational component to the transverse waves. It will also show that the longitudinal wave solutions may be independent of the transverse solutions and that the longitudinal solutions consist of only electric and gravitational vector components with an accompanying scalar wave. Longitudinal electrogravitic waves do not interact with the propagation medium as do transverse waves and make good candidates for space communication. Not only will longitudinal waves pass through intervening material, but they may be made very directive and, thereby, avoid a 1/r2 loss. The presentation will also present an antenna design for converting transverse electromagnetic waves into longitudinal electrogravitic waves for using current transmitting laser communication equipment and the reciprocal antenna design for converting longitudinal waves back into transverse waves to use current receivers.

Information transmittal, relativity and gravitation

The combined action functional for the electromagnetic and gravitational fields in the presence of charged matter is given by $$ S\equiv S_{\rm G}[g_{\mu\nu}]+ S_{\rm E}[g_{\mu\nu},A_\mu] + S_{\rm M}[g_{\mu\nu},A_\mu,\Phi^A], $$where S M is the action functional for the material system and the ? A are the matter dynamical variables. (Here we are treating the electromagnetic field separately from the material system.) Note that S M is a functional of the g ?? and the A ? as well as of the ? A. In quantum field theory the ? A are the components of a complex tensor (or spinor) field ? whose coupling to the electromagnetic field is determined by the principle of minimal coupling (akin to the strong equivalence principle): Every ordinary derivative ?,? appearing in the matter field Lagrangian when no electromagnetic field is present, is replaced by the combination ?,? ? i eA ??, where e is the unit charge of the matter field quanta (typically, the charge on the electron). Because ordinary derivatives become covariant derivatives when a gravitational field is present, this means that in general relativity matter field derivatives occur only in the combination KeywordsGauge TransformationMatter FieldWorld LineGauge ParameterFlat SpacetimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor with respect to the coordinate system xu. After done this, xu is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, the concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein’s equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The essential significance of the de Donder conditions (the harmonic conditions or gauge) is to desorb the tensor field of gravitation from the Riemannian space-time to the Minkowski space-time with the Cartesian coordinates. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock’s works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.

Partons and gravitation: Some puzzling questions