Letter: Generation of Solutions of the Einstein Equations by Means of the Kaluza–Klein Formulation

It is shown that starting from a solution of the Einstein–Maxwell equations coupled to a scalar field given by the Kaluza–Klein theory, invariant under a one-parameter group, one can obtain a one-parameter family of solutions of the same equations.     

The symmetries of the Manton superconductivity model

The symmetries and conserved quantities of Manton’s modified superconductivity model with non-relativistic Maxwell–Chern–Simons dynamics (also related to the Quantized Hall Effect) are obtained in the “Kaluza–Klein type” framework of Duval et al.     

5D Dirac Equation in Induced-Matter Theory

According to an induced-matter approach, Liu and Wesson obtained the rest mass of a typical particle from the reduction of a 5D Klein–Gordon equation to a 4D one. Introducing an extra-dimension momentum operator identified with the rest mass eigenvalue operator, we consider a way to generalize the 4D Dirac equation to 5D. An analogous normal Dirac equation is gained when the generalization reduces to 4D. We find the rest mass of a particle in curved space varies with spacetime coordinates and check this for the case of exact solitonic and cosmological solution of the 5D vacuum gravitational field equations.     

A Relativistic Gauge Theory of Nonlinear Quantum Mechanics and Newtonian Gravity

A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schrödinger picture of a given field theory. While, for simplicity, we study the example of a \(\mathcal{U}(1)\) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schrödinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. A probabilistic interpretation (Born’s rule) holds, provided the underlying model is scale free.     

Xanthopoulos Theorem in the Kaluza–Klein Theory

It is shown that if _AB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^ _AB = 0, and l _A is a null vector field, then _AB + l _A l _B is also an exact solution of the Einstein equations R^ _AB = 0 if and only if the perturbation l _A l _B satisfies the Einstein equations linearized about _AB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations.     

Universal features of dimensional reduction schemes from general covariance breaking

Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors, that successfully generalize to one side Kaluza–Klein gauge fields and to the other side extrinsic curvature and torsion of embedded spaces, thus fully characterizing the geometry of dimensional reduction. We obtain general formulas for the reduction of the main tensors and operators of Riemannian geometry. In particular, we provide what is probably the maximal possible generalization of Gauss, Codazzi and Ricci equations and various other standard formulas in Kaluza–Klein and embedded spacetimes theories. After general covariance breaking, part of the residual covariance is perceived by effective lower dimensional observers as an infinite dimensional gauge group. This reduces to finite dimensions in Kaluza–Klein and other few remarkable backgrounds, all characterized by the vanishing of appropriate lower dimensional tensors.     

A New Solution with Torsion in Model of dS Gauge Theory of Gravity

A new static de Sitter solution with torsion in the model of de Sitter gauge theory of gravity is obtained. The torsion only contains O(3)-symmetric tensor part according to irreducible decomposition. Some properties of the new solution are discussed.     

Beyond the relativistic point particle: A reciprocally invariant system and its generalisation

We investigate a reciprocally invariant system proposed by Low and Govaerts et al., whose action contains both the orthogonal and the symplectic forms and is invariant under global O(2,4)∩Sp(2,4) transformations. We find that the general solution to the classical equations of motion has no linear term in the evolution parameter, τ, but only the oscillatory terms, and therefore cannot represent a particle propagating in spacetime. As a remedy, we consider a generalisation of the action by adopting a procedure similar to that of Bars et al., who introduced the concept of a τ derivative that is covariant under local Sp(2) transformations between the phase space variables x^μ(τ) and p^μ(τ). This system, in particular, is similar to a rigid particle whose action contains the extrinsic curvature of the world line, which turns out to be helical in spacetime. Another possible generalisation is the introduction of a symplectic potential proposed by Montesinos. We show how the latter approach is related to Kaluza–Klein theories and to the concept of Clifford space, a manifold whose tangent space at any point is Clifford algebra Cl(8), a promising framework for the unification of particles and forces.     

Equation of Motion of a Mass Point in Gravitational Field and Classical Tests of Gauge Theory of Gravity

A systematic method is developed to study the classical motion of a mass point in gravitational gauge field.First,by using Mathematica,a spherical symmetric solution of the field equation of gravitational gauge field is obtained,which is just the traditional Schwarzschild solution.Combining the principle of gauge covariance and Newton's second law of motion,the equation of motion of a mass point in gravitational field is deduced.Based on the spherical symmetric solution of the field equation and the equation of motion of a mass point in gravitational field,we can discuss classical tests of gauge theory of gravity,including the deflection of light by the sun,the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun.It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity.     

Topological Five-Dimensional Chern–Simons Gravity Theory in the Canonical Covariant Formalism

The canonical covariant formalism (CCF) of the topological five-dimensional Chern–Simons gravity is constructed. Because this gravity model naturally contains a Gauss–Bonnet term, the extended CCF valid for higher curvature gravity must be used. In this framework, the primary constraint and the total Hamiltonian are found. By using the equations of the CCF, it is shown that the bosonic five-form which defines the total Hamiltonian is a first-class dynamical quantity strongly conserved. In this context the equations of motion are also analyzed. To determine the effective interactions of the model, the toroidal dimensional reduction of the five-dimensional Chern–Simons gravity is carried out. Finally the first-order CCF and the usual canonical vierbein formalism (CVF) are related and the Hamiltonian as generator of time evolution is constructed in terms of the first-class constraints of the coupled system.     

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

Finitary, Causal, and Quantal Vacuum Einstein Gravity

We continue recent work (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001; in press) and formulate the gravitational vacuum Einstein equations over a locally finite space-time by using the basic axiomatics, techniques, ideas, and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and explored in (Mallios and Raptis, International Journal of Theoretical Physics 40, 1885, 2001), is a curved principal finitary space-time sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a nontrivial locally finite spin-Loretzian connection on it which lays the structural foundation for the formulation of a covariant dynamics of quantum causality in terms of sheaf morphisms. Our scheme is innately algebraic and it supports a categorical version of the principle of general covariance that is manifestly independent of a background -smooth space-time manifold M. Thus, we entertain the possibility of developing a fully covariant path integral-type of quantum dynamical scenario for these connections that avoids ab initio various problems that such a dynamics encounters in other current quantization schemes for gravity—either canonical (Hamiltonian) or covariant (Lagrangian)—involving an external, base differential space-time manifold, namely, the choice of a diffeomorphism-invariant measure on the moduli space of gauge-equivalent (self-dual) gravitational spin-Lorentzian connections and the (Hilbert space) inner product that could in principle be constructed relative to that measure in the quantum theory—the so-called inner product problem, as well as the problem of time that also involves the Diff(M) structure group of the classical -smooth space-time continuum of general relativity. Hence, by using the inherently algebraico—sheaf—theoretic and calculus-free ideas of Abstract Differential Geometry, we are able to draw preliminary, albeit suggestive, connections between certain nonperturbative (canonical or covariant) approaches to quantum general relativity (e.g., Ashtekar's new variables and the loop formalism that has been developed along with them) and Sorkin et al.'s causal set program. As it were, we noncommutatively algebraize, differential geometrize and, as a result, dynamically vary causal sets. At the end, we anticipate various consequences that such a scenario for a locally finite, causal and quantal vacuum Einstein gravity might have for the obstinate (from the viewpoint of the smooth continuum) problem of -smooth space-time singularities.     

Effects of an additional dimension in the Young experiment

The results of the Young experiment can be analyzed either by classical or Quantum Physics. The later one though leads to a more complete interpretation, based on two different patterns that appear when one works either with single or double slits. Here we show that the two patterns can be derived from a single principle, in the context of General Relativity, if one assumes an additional spatial dimension to the four known today. The found equations yield the same results as those in Quantum Mechanics.     

Black Holes in Bose–Einstein Condensates

It is shown that there exist both dynamically stable and unstable dilute-gas Bose–Einstein condensates that, in the hydrodynamic limit, exhibit a behavior completely analogous to that of gravitational black holes. The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy states. We illustrate these features in two qualitatively different one-dimensional models. We have also simulated the creation of a stable sonic black hole by solving the Gross–Pitaevskii equation numerically for a condensate subject to a trapping potential that is adiabatically deformed. A sonic black hole could in this way be created experimentally with state-of-the-art or planned technology.     

RG flows of Quantum Einstein Gravity in the linear-geometric approximation

We construct a novel Wetterich-type functional renormalization group equation for gravity which encodes the gravitational degrees of freedom in terms of gauge-invariant fluctuation fields. Applying a linear-geometric approximation the structure of the new flow equation is considerably simpler than the standard Quantum Einstein Gravity construction since only transverse-traceless and trace part of the metric fluctuations propagate in loops. The geometric flow reproduces the phase-diagram of the Einstein–Hilbert truncation including the non-Gaussian fixed point essential for Asymptotic Safety. Extending the analysis to the polynomial f(R)$f\left(R\right)$-approximation establishes that this fixed point comes with similar properties as the one found in metric Quantum Einstein Gravity; in particular it possesses three UV-relevant directions and is stable with respect to deformations of the regulator functions by endomorphisms.     

Accretion onto Charged Black Holes in Einstein and Massive Theories of Gravity *

The accretion process is being investigated onto some important black holes such as Born-Infeld-AdS black hole, non-linear charged black hole solution in AdS space-time and Einstein-Yang-Mills massive gravity in the presence of Born-Infeld nonlinear electrodynamics. We find out the relations of radial velocity, energy density and change of mass for mention black holes and analyze their behavior graphically for different values of equation of state parameters $\omega$. We also examine the relations for critical speed for these black holes. It is observed that for different state parameters different fluids exhibit different evolutions in black holes backgrounds. The energy density of some fluids is negative or positive near the black hole while other fluids become cause to increase or decrease in black hole mass.     

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.     

Mechanism of Gravity Impulse

It is well known that energy-momentum is the source of gravitational field. For a long time, it is generally believed that only stars with huge masses can generate strong gravitational field. Based on the unified theory of gravitational interactions and electromagnetic interactions, a new mechanism of the generation of gravitational field is studied. According to this mechanism, in some special conditions, electromagnetic energy can be directly converted into gravitational energy, and strong gravitational field can be generated without massive stars. Gravity impulse found in experiments is generated by this mechanism.     

Solution to the Cosmological Constant Problem by Gauge Theory of Gravity

Based on geometry picture of gravitational gauge theory, the cosmological constant is determined theoreti-cally. The cosmological constant is related to the average energy density of gravitational gauge field. Because the energydensity of gravitational gauge field is negative, the cosmological constant is positive, which generates repulsive force onstars to make the expansion rate of the Universe accelerated. A rough estimation of it gives out its magnitude of theorder of about 10-52m-2, which is well consistent with experimental results.     

Spin-Spin Interactions in Gauge Theory of Gravity, Violation of Weak Equivalence Principle and New Classical Test of General Relativity

For a long time, it has been generally believed that spin-spin interactions can only exist in a theory where Lorentz symmetry is gauged, and a theory with spin-spin interactions is not perturbatively renormalizable. But this is not true. By studying the motion of a spinning particle in gravitational field, it is found that there exist spin-spin interactions in gauge theory of gravity. Its mechanism is that a spinning particle will generate gravitomagnetic field in space-time, and this gravitomagnetic field will interact with the spin of another particle, which will cause spin-spin interactions. So, spin-spin interactions are transmitted by gravitational field. The form of spin-spin interactions in post Newtonian approximations is deduced. This result can also be deduced from the Papapetrou equation. This kind of interaction will not affect the renormalizability of the theory. The spin-spin interactions will violate the weak equivalence principle, and the violation effects are detectable. An experiment is proposed to detect the effects of the violation of the weak equivalence principle.