Conformal Hamiltonian dynamics of general relativity

The General Relativity formulated with the aid of the spin connection coefficients is considered in the finite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution of the General Relativity describes the vacuum creation of the matter in the empty Universe at the electroweak epoch and the dilaton vacuum energy plays a role of the dark energy.     

A canonical dynamics view of the Newtonian limit of general relativity

The Newtonian limit of general relativity was Jürgen Ehlers favourite model for limit relations between theories of physics. In this contribution, for the case of isolated systems, the Newtonian limit of general relativity will be illuminated from a canonical dynamics point of view. The canonical dynamics approach naturally supplies a post-Newtonian expansion of general relativity.     

Geodesically equivalent metrics in general relativity

We discuss whether it is possible to reconstruct a metric from its nonparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general relativity. We show how to understand whether all curves from a sufficiently big family are nonparameterized geodesics of a certain affine connection, and how to reconstruct algorithmically a generic 4-dimensional metric from its nonparameterized geodesics. The algorithm works most effectively if the metric is Ricci-flat. We also prove that almost every metric does not allow nontrivial geodesic equivalence, and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorentz signature.     

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.     

A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy

In this Letter, a generalized Tu formula is firstly presented to construct Hamiltonian structures of fractional soliton equations. The obtained results can be reduced to the classical Hamiltonian hierarchy of AKNS in ordinary calculus.     

On the inertial mass concept in special and general relativity

Inertial mass in relativity theory is discussed from a conceptual view. It is shown that though relativistic dynamics implies a particular dependence of the momentum of a free particle on its velocityin special relativity, which diverges as v approaches c, the inertial mass itself of a moving body remains constant, from any frame of observation. However, extension to general relativity does conceptually introduce variability of the inertial mass of a body, through a necessarily generally covariant field theory of inertia, when the Mach principle is incorporated into the theory of general relativity, as a theory of matter.     

The Mössbauer rotor experiment and the general theory of relativity

In the recent paper Yarman et al. (2015), the authors claim that our general relativistic analysis in Corda (2015), with the additional effect due to clock synchronization, cannot explain the extra energy shift in the Mössbauer rotor experiment. In their opinion, the extra energy shift due to the clock synchronization is of order 10⁻¹³ and cannot be detected by the detectors of γ$\gamma$-quanta which are completely insensitive to such a very low order of energy shifts. In addition, they claim to have shown that the extra energy shift can be explained in the framework of the so-called YARK gravitational theory. They indeed claim that such a theory should replace the general theory of relativity (GTR) as the correct theory of gravity.     

Scalar-tensor cosmologies with a potential in the general relativity limit: Time evolution

We consider Friedmann–Lemaître–Robertson–Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity with arbitrary coupling function and potential. For the era when the cosmological energy density of the scalar potential dominates over the energy density of ordinary matter, we use a nonlinear approximation of the decoupled scalar field equation for the regime close to the so-called limit of general relativity where the local weak field constraints are satisfied. We give the solutions in cosmological time with a particular attention to the classes of models asymptotically approaching general relativity. The latter can be subsumed under two types: (i) exponential convergence, and (ii) damped oscillations around general relativity. As an illustration we present an example of oscillating dark energy.     

Exact solutions for the massless plane symmetric scalar field in general relativity,with cosmological constant

The Klein–Gordon equations are solved for the case of a plane-symmetric static massless scalar field in general relativity with cosmological constant, generalizing the solutions found by Taub, Novotny and Horsky, and Singh. A separate class of solutions is obtained in which the metrics reduce to flat space in the limit that .The static solutions can be used to generate time-dependent cosmological solutions, one of which exhibits rapid inflation followed by continued exponential expansion at all later times.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

General relativity and quantum mechanics in five dimensions

In 5D, I take the metric in canonical form and define causality by null-paths. Then spacetime is modulated by a factor equivalent to the wave function, and the 5D geodesic equation gives the 4D Klein-Gordon equation. These results effectively show how general relativity and quantum mechanics may be unified in 5D.     

Distance and the conventionality of simultaneity in special relativity

The conventionality of simultaneity within inertial frames is presented in a general formalism that clarifies the relationship of spatial measures to the choice of simultaneity. A number of claims that such measures undermine the conventional nature of simultaneity are presented and shown to be unfounded. In particular, a recent claim by Coleman and Korte [9] that such measures empirically establish a unique simultaneity relationship is shown to be in error. In addition, the general formalism enables the empirical status of simultaneity within an inertial frame to be clarified by presenting the choice of simultaneity as a gauge choice.1. Recent introductions to the literature have been given by Redhead [35], Ungar [47], Havas [21], and Vetharaniam and Stedman [48].2. The conventionalist position is by no means a uniform one, and in particular, it is worth noting an important distinction exemplified in the respective positions of Reichenbach and Grünbaum. For Reichenbach [37, p. 144f.] we have no empirical access to the one-way speed of light due to the nature of light as a first signal, and the conventionality comes from our absence ofknowledge about the one-way speed of light. For Grünbaum the one-way speed of light is actually objectively undetermined, and the physical attributes that sustain a speed in a given direction are non-existent. See, for example, [16, p. 87] and [17, p. 352]. Discussions of the differences between the positions of Reichenbach and Grünbaum may be found in [14] and [35]. Naturally, one may adhere to a position espoused by Reichenbach without the added ontological commitment of Grünbaum.3. Our is equivalent to (1 - 2), where is the symbol introduced by Reichenbach and customarily used in the discussions of the conventionality of simultaneity.4. An exposition of this argument may be found in the recent text by Lucas and Hodgson [28].5. Schrödinger [42, p. 78] has aptly labeled this quantity the distance of simultaneity.6. Examples of previous uses space-dependent synchrony parameters may be found in studies by Clifton [8], Havas [21], Anderson and Stedman [1], and Stedman [43; 44, § 2].7. This approach has been reviewed by Basri in [4] and [3].8. A number of faulty assessments of the empirical status of the conventionality of simultaneity may be similarly traced at least in part to overly simplistic assumptions on the nature of as Havas [21] and Clifton [8], for example, have had occasion to point out.9. See, for example, [1]. Kinematic formula relating other quantities in a treatment of STR without the standard convention on the one-way speed of light were first derived by Winnie [53].10. In comparison to other space dependent treatments of the synchrony parameter, ourh is analogous to defined by Clifton in Eq. (15) of [8], and equivalent to -f defined by Havas in Eq. (A1) of [21] and to defined in Eq. (6) of our earlier treatment in [1]. We take this opportunity to mention that the irrotational property ofh was inadvertently referred to as solenoidal in this work.11. Equation (26) is equivalent to Møller's expression in § 8.8 of [32] for the speed of light in terms of the metric components where our-h _i is equivalent to Møller's _i (g _i0)/ .12. Note as well, the expression of this operation in standard texts on STR by Rindler [38, pp. 27–28] and Mermin [30, p. 79] respectively: To measure the rod's length in any inertial frame in which it moves longitudinally, its end-point must be observed simultaneously... and, ...a measurement of the length of a moving meter stick involves determining how far apart the two ends areat the same time. In the same context of determining the length of moving rods, Mermin [30, p. 185] proposes that the sense of length entailing the concept of being determined at simultaneous times is inherent in the notion of rods: ...it is precisely the lines of constant time that determine whatA orB means by the stick. For the notion of the stick includes implicitly the assumption that all the points of matter making up the stick exist at the same moment.13. In many ways the claim that the special properties of proper lengths with Einstein synchronization undermines the conventionality of simultaneity is analogous to the claim that the correspondence of the slow-clock transport method of synchronization with that of Einstein synchronization provides an empirical determination of synchronization. The use of clock transport as a means for synchronization was discussed by Reichenbach [37, p. 133f], while the proposal that slow transport of clocks provides a unique form of synchronization was first argued for by Eddington [10]. Arguments that it undermines any significant sense of the conventionality in the one-way speed of light have been given by Ellis and Bowman [13] with responses by Grünbaum [19] and Salmon [41, 40].14. Coleman and Korte [9, pp. 423–425] claim their method is free from any assumptions on the one-way speed of light; however, they assume that is a constant 3-vector.15. Reichenbach explicitly mentioned in [36, § 43] that a condition equivalent to Eq. (13) is a sufficient condition for a constant roundtrip speed of light.16. The remarks of one of the referees have served to alert us to the need to emphasize both of these points.17. The manner in which gravity may be viewed as a gauge theory has been the subject of considerable discussion (see, for example, the discussion in [23] and [24]). We note that the manner in which we are takingh as a potential differs from the sense in which the Christoffel symbols as affine connections may be seen to play a role of gauge potentials in GTR.18. A discussion of the significance of Weyl's work and the importance of the round-trip measurements may be found in works by Yang [56] and Mills [31].19. In the context only of time orthogonal coordinates, an example of the fiber structure we are imposing on space and time may be found in [26, p. 71f]. Again we note that in a more general treatment, where the Christoffel symbols are considered as connections, the fiber structure instead consists of a bundle of linear frames of Riemannian spacetime (see, for example, the presentations in [46] and [23]).20. Our position is not unlike Göckeler and Schücker's [15, p. 75] claim that Einstein's particular choice of coordinates in GTR masks the general gauge structure of the theory.     

Initial-value problems and singularities in general relativity

Linearized solution of Datta in a non-symmetric and isentropic motion of a perfect fluid is studied by dealing with a Cauchy problem in co-moving coordinates in the framework of general relativity. The problem of singularities is discussed from the standpoint of a local observer both for rotating and non-rotating fluids. It is shown that, whatever the distribution of matter, a singularity which occurred in the past in both the rotating and non-rotating parts of the universe must occur again later after some finite proper time, if the universe is closed. A modification is incorporated in Penrose’s theorem by explicitly exhibiting that the universe defined by Penrose can possess a Cauchy hypersurface.     

Differentiability of invariants and Liapunov equations in Hamiltonian systems

In this communication, we investigate the behavior of the derivatives of invariants for Hamiltonian systems, using information derived from an analysis of the Liapunov exponents of the system. We show that under certain conditions on the analyticity properties of the solutions of the equations of motion, it is possible to construct 2n invariants of motion which are guaranteed to beC ^∞ as functions of phase space and time in a suitably defined domainD.     

Radiation bursts from particles in the field of compact, impenetrable, astrophysical objects

The radiation emitted by charged, scalar particles in a Schwarzschild field with maximal acceleration corrections is calculated classically and in the tree approximation of quantum field theory. In both instances the particles emit radiation that has characteristics similar to those of gamma-ray bursters.     

Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.