Theory Of Reciprocal Relations Research Articles

On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the deformed quaplectic group that is given by the semi-direct product of U(1,3) with the deformed (noncommutative) Weyl–Heisenberg group corresponding to noncommutative fiber coordinates and momenta [Xa,Xb]≠0; [Pa,Pb]≠0. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [Xa1a2⋯an,Xb1b2⋯bn]≠0; [Pa1a2⋯an,Pb1b2⋯bn]≠0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting xμ,pμ operator variables (associated to an 8D curved phase space) to the canonical YA,ΠA operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, the embedding functions YA(x,p),ΠA(x,p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric gμν(x,p), the fiber metric of the vertical space hab(x,p), and the nonlinear connection Naμ(x,p) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

On maximal acceleration, strings with dynamical tension, and Rindler worldsheets

Starting with a different action and following a different procedure than the construction of strings with dynamical tensions described by Guendelman [1], a variational procedure of our action leads to a coupled nonlinear system of D+4 partial differential equations for the D string coordinates Xμ and the quartet of scalar fields φ1,φ2,ϕ,T, including the dilaton ϕ(σ) and the tension T(σ) field. Trivial solutions to this system of complicated equations lead to a constant tension and to the standard string equations of motion. One of the most relevant features of our findings is that the Weyl invariance of the traditional Polyakov string is traded for the invariance under area-preserving diffeomorphisms. The final section is devoted to the physics of maximal proper forces (acceleration), minimal length within the context of Born's Reciprocal Relativity theory [6] and to the Rindler world sheet description of accelerated open and closed strings from a very different approach and perspective than the one undertaken by [7].

Born’s Reciprocal Relativity theory, curved phase space, Finsler geometry and the cosmological constant

A very brief introduction of the history of Born’s Reciprocal Relativity Theory, Hopf algebraic deformations of the Poincare algebra, de Sitter algebra, and noncommutative spacetimes paves the road for the exploration of gravity in curved phase spaces within the context of the Finsler geometry of the cotangent bundle T∗M of spacetime. A scalar-gravity model is duly studied, and exact nontrivial analytical solutions for the metric and nonlinear connection are found that obey the generalized gravitational field equations, in addition to satisfying the zero torsion conditions for all of the torsion components. The curved base spacetime manifold and internal momentum space both turn out to be (Anti) de Sitter type. The most salient feature is that the solutions capture the very early inflationary and very-late-time de Sitter phases of the Universe. A regularization of the 8-dim phase space action leads naturally to an extremely small effective cosmological constant Λeff, and which in turn, furnishes an extremely small value for the underlying four-dim spacetime cosmological constant Λ, as a direct result of a correlation between Λeff and Λ resulting from the field equations. The rich structure of Finsler geometry deserves to be explored further since it can shine some light into Quantum Gravity, and lead to interesting cosmological phenomenology.

Thermal relativity, corrections to black hole entropy, Born’s reciprocal relativity theory, and quantum gravity

Starting with a brief description of Born’s reciprocal relativity theory (BRRT), based on a maximal proper force, maximal speed of light, and inertial and non-inertial observers, we derive the exact thermal relativistic corrections to the Schwarzschild, Reissner–Nordstrom, and Kerr–Newman black hole entropies and provide a detailed analysis of the many novel applications and consequences to the physics of black holes, quantum gravity, minimal area, minimal mass, Yang–Mills mass gap, information paradox, arrow of time, dark matter, and dark energy. We finish by outlining our proposal towards a space–time–matter unification program where matter can be converted into spacetime quanta and vice versa.

Is dark matter and black hole cosmology an effect of Born’s reciprocal relativity theory?

Born’s reciprocal relativity theory (BRRT), based on a maximal proper-force, maximal speed of light, and inertial and non-inertial observers, is re-examined in full detail. Relativity of locality and chronology are natural consequences of this theory, even in flat phase space. The advantage of BRRT is that Lorentz invariance is preserved and there is no need to introduce Hopf algebraic deformations of the Poincaré algebra, de Sitter algebra, nor non-commutative space–times. After a detailed study of the notion of generalized force, momentum, and mass in phase space, we explain that what one may interpret as “dark matter” in galaxies, for example, is just an effect of observing ordinary galactic matter in different accelerating frames of reference than ours. Explicit calculations are provided that explain these novel relativistic effects due to the accelerated expansion of the Universe, and which may generate the present-day density parameter value ΩDM ∼ 0.25 of dark matter. The physical origins behind the numerical coincidences in black hole cosmology are also explored. We finalize with a rigorous study of the curved geometry of (co)tangent bundles (phase space) within the formalism of Finsler geometry, and provide a short discussion on Hamilton spaces.

A gauge theory of gravity in curved phase-spaces

After a cursory introduction of the basic ideas behind Born’s Reciprocal Relativity theory, the geometry of the cotangent bundle of spacetime is studied via the introduction of nonlinear connections associated with certain nonholonomic modifications of Riemann–Cartan gravity within the context of Finsler geometry. A novel gauge theory of gravity in the [Formula: see text] cotangent bundle [Formula: see text] of spacetime is explicitly constructed and based on the gauge group [Formula: see text] which acts on the tangent space to the cotangent bundle [Formula: see text] at each point [Formula: see text]. Several gravitational actions involving curvature and torsion tensors and associated with the geometry of curved phase-spaces are presented. We conclude with a brief discussion of the field equations, the geometrization of matter, quantum field theory (QFT) in accelerated frames, T-duality, double field theory, and generalized geometry.

On Superluminal Particles and the Extended Relativity Theories

Superluminal particles are studied within the framework of the Extended Relativity theory in Clifford spaces (C-spaces). In the simplest scenario, it is found that it is the contribution of the Clifford scalar component π of the poly-vector-valued momentum which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass \(\mathcal{M} = \sqrt{ M^{2} - \pi^{2} }\) is imaginary (tachyonic). However, from the point of view of C-space, there is no superluminal (tachyonic) behavior because the true physical mass still obeys M2>0. Therefore, there are no violations of the Clifford-extended Lorentz invariance and the extended Relativity principle in C-spaces. It is also explained why the charged muons (leptons) are subluminal while its chargeless neutrinos may admit superluminal propagation. A Born’s Reciprocal Relativity theory in Phase Spaces leads to modified dispersion relations involving both coordinates and momenta, and whose truncations furnish Lorentz-violating dispersion relations which appear in Finsler Geometry, rainbow-metrics models and Double (deformed) Special Relativity. These models also admit superluminal particles. A numerical analysis based on the recent OPERA experimental findings on alleged superluminal muon neutrinos is made. For the average muon neutrino energy of 17 GeV, we find a value for the magnitude \(|\mathcal{M } | = 119.7\mbox{~MeV}\) that, coincidentally, is close to the mass of the muon mμ=105.7 MeV.

SOME CONSEQUENCES OF BORN'S RECIPROCAL RELATIVITY IN PHASE-SPACES

We explore some novel consequences of Born's reciprocal relativity theory in flat phase-space and generalize the theory to the curved space–time scenario. We provide, in particular, six specific results resulting from Born's reciprocal relativity and which are not present in special relativity. These are: momentum-dependent time delay in the emission and detection of photons; energy-dependent notion of locality; superluminal behavior; relative rotation of photon trajectories due to the aberration of light; invariance of areas-cells in phase-space and modified dispersion relations. We finalize by constructing a Born reciprocal general relativity theory in curved space–time which requires the introduction of a complex Hermitian metric, torsion and nonmetricity. The latter procedure can be extended to the curved phase-space scenario.

Current-voltage reciprocity in the magnetoresistance of simple metals

High-field magnetoresistance measurements have been made on unoriented single crystals of potassium and polycrystalline samples of indium and aluminium in fields up to 7.5 T. The measurements show that a reversal of the applied magnetic field may be replaced by an interchange of current and potential leads on the sample, no matter what the position of the contacts and the shape of the sample. Onsager's theory of reciprocal relations in irreversible processes provides a proof of this phenomenon.

The Onsager conference

In 1931, while he was a research instructor in chemistry at Brown University, Lars Onsager published two papers on the theory of reciprocal relations for irreversible processes, and thereby laid the foundations for the now flourishing subject of irreversible thermodynamics. As a most fitting recognition of the thirtieth anniversary of this achievement, Brown University played host, this past June, to an International Conference on Irreversible Thermodynamics and the Statistical Mechanics of Phase Transitions (sponsored by the International Union of Pure and Applied Physics, the National Science Foundation, and the Air Force Office of Scientific Research). The conference began with a suitably impressive ceremonial session at which President Barnaby Keney of Brown conferred the University's honorary degree of Doctor of Science upon Onsager, who is now J. Willard Gibbs Professor at Yale.

Extension of Onsager's Theory of Reciprocal Relations. II

The extension of Onsager's theory of reciprocal relations between irreversible processes, developed previously by the authors, is applied to electric conduction in anisotropic crystals, and the symmetry properties of the conduction tensor are derived.

Extension of Onsager's Theory of Reciprocal Relations. I

Extension of Onsager's Theory of Reciprocal Relations. I

Thermodynamical theory of galvanomagnetic and thermomagnetic phenomena II: Reciprocal relations for moving anisotropic mixtures

Synopsis The extension of Onsager's theory of reciprocal relations between irreversible processes, developed by De Groot and Mazur, is applied to heat conduction, diffusion and cross-effects and to viscosity in anisotropic mixtures of charged components in an electromagnetic field. Reciprocal relations are derived for these phenomena.

On the theory of reciprocal relations between irreversible processes: Statistical foundations of thermodynamical fluctuation theory in moving media

Recently S. R. de Groot and P. Mazur have extended Onsager's proof of reciprocal relations between irreversible processes to the case of vectorial and tensorial phenomena. In the derivation expressions are used for certain averages of fluctuations of the variables a μ i and β μ k , which describe the state of a great number of cells μ , into which the system is divided. These averages are calculated with the help of the probability distribution P ~ exp { Δ S ( a i μ , β k μ ) / k }, ( a ) where ΔS is the deviation of the entropy from its equilibrium value. The use of formula ( a ) in the case, where the state variables include barycentric velocities of the material elements, should be justified. This is possible, but in the present paper we preferred a slightly simpler method to find the averages of the fluctuations, necessary for the proof of reciprocal relations. It turns out, that for their derivation distribution functions for the state variables of one or two cells suffice. From statistical calculations these are found to be P 1 ~ exp ( Δ S 1 0 / k ) , ( b ) P 12 ~ exp { ( Δ S 1 0 + Δ S 2 0 ) / k }, ( c ) where Δ 0 i is the quadratic form in the state variables of the cell i (= 1, 2), including barycentric velocities. In the theory independent state variables are used in a natural way, in contrast with De Groot and Mazur's (and Casimir's) treatment, where dependencies exist amongst the state variables as a consequence of the conservation of mass and energy. One could of course eliminate a number of variables, so as to obtain a set of independent state variables, but this complication can be avoided by using the distribution functions ( b ) and ( c ) instead of ( a ), as is done in this paper.

Some aspects of Onsager's theory of reciprocal relations in irreversible processes

Some aspects of Onsager's theory of reciprocal relations in irreversible processes