Quantum Regge Calculus of Einstein–Cartan theory

We study the Quantum Regge Calculus of Einstein–Cartan theory to describe quantum dynamics of Euclidean space–time discretized as a 4-simplices complex. Tetrad field eμ(x)$e_{\mu }(x)$ and spin-connection field ωμ(x)${\omega }_{\mu }(x)$ are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and local   gauge-invariant Einstein–Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)$\mathit{SO}(4)$-group valued ωμ(x)${\omega }_{\mu }(x)$ fields and Dirac-matrix valued eμ(x)$e_{\mu }(x)$ fields over 4-simplices complex.     

The phase structure of Einstein–Cartan theory

In the Einstein–Cartan theory of torsion-free gravity coupling to massless fermions, the four-fermion interaction is induced and its strength is a function of the gravitational and gauge couplings, as well as the Immirzi parameter. We study the dynamics of the four-fermion interaction to determine whether effective bilinear terms of massive fermion fields are generated. Calculating one-particle-irreducible two-point functions of fermion fields, we identify three different phases and two critical points for phase transitions characterized by the strength of four-fermion interaction: (1) chiral symmetric phase for massive fermions in strong coupling regime; (2) chiral symmetric broken phase for massive fermions in intermediate coupling regime; (3) chiral symmetric phase for massless fermions in weak coupling regime. We discuss the scaling-invariant region for an effective theory of massive fermions coupled to torsion-free gravity in the low-energy limit.     

A correction to the law of universal gravitation derived from the Einstein–Hilbert equations

A new form of the law of universal gravitation, more exact than that derived from the three Kepler laws, is derived from the Einstein–Hilbert equations. Based on this result, a genesis of a secular displacement of the elliptic orbit perihelia is explained.     

Republication of: 5. Comment on the axially symmetric solutions to Einstein’s equations of gravitation

This is the English translation of the second of a series of 3 papers by Hermann Weyl (the third one jointly with Rudolf Bach), first published in 1917–1922, in which the authors derived and discussed the now-famous Weyl two-body static axially symmetric vacuum solution of Einstein’s equations. The English translations of the other two papers are published alongside this one. The papers have been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Gernot Neugebauer, David Petroff and Bahram Mashhoon, and by a brief biography of R. Bach, written by H. Goenner.     

On Wilson’s theory of confinement

According to recent results, the Gell-Mann-Low function β(g) of four-dimensional φ⁴ theory is nonalternating and has a linear asymptotics at infinity. According to the Bogoliubov and Shirkov classification, it means the possibility of constructing a continuous theory with finite interaction at large distances. This conclusion is in visible contradiction to the lattice results indicating triviality of φ⁴ theory. This contradiction is resolved by a special character of renormalizability in φ⁴ theory: to obtain the continuous renormalized theory, there is no need to eliminate a lattice from the bare theory. In fact, such kind of renormalizability is not accidental and can be understood in the framework of Wilson’s many-parameter renormalization group. Application of these ideas to QCD shows that Wilson’s theory of confinement is not purely illustrative, but has a direct relation to a real situation. As a result, the problem of analytical proof of confinement and a mass gap can be considered solved, at least on the physical level of rigor.     

The electrostatics of Einstein’s unified field theory

When sources are added at their right-hand sides, and g_(ik) is a priori assumed to be the metric, the equations of Einstein’s Hermitian theory of relativity were shown to allow for an exact solution that describes the general electrostatic field of n point charges. Moreover, the injunction of spherical symmetry of g_(ik) in the infinitesimal neighbourhood of each of the charges was proved to yield the equilibrium conditions of the n charges in keeping with ordinary electrostatics. The tensor g_(ik), however, cannot be the metric of the theory, since it enters neither the eikonal equation nor the equation of motion of uncharged test particles. A physically correct metric that rules both the behaviour of wave fronts and of uncharged matter is the one indicated by Hély.In the present paper it is shown how the electrostatic solution predicts the structure of the n charged particles and their mutual positions of electrostatic equilibrium when Hély’s physically correct metric is adopted.     

On the theory of temperature distribution over the Earth’s atmosphere

The steady-state temperature distribution over the Earth’s atmosphere is described in terms of the law of degradation of energy (or the principle of entropy increase). The distribution is in satisfactory agreement with meteorological measurements. Using constructed dissipative function \(\dot Q\), a nonlinear differential equation is rigorously derived that describes the synergetics of the steady-state temperature distribution over the atmosphere, from troposphere to exosphere. A detailed analysis of solutions of this equation makes it possible to qualitatively explain nontrivial dependence T(z) over an atmospheric area inhomogeneous in chemical composition. This fact substantiates the correctness of formal introduction and necessity to consider interaction between blackbody radiation and convective stream. It is argued that hard gamma quanta far from the violet part of the spectrum are responsible for heat radiation coming from the Sun.     

On stable variants of Einstein’s static universe

The investigation of the stability properties of certain variants of Einstein’s static universe performed by Carneiro and Tavakol (in Stability of the Einstein static universe in the presence of vacuum energy) is generalized. It is shown that all versions of Einstein’s static universe without interaction between the two fluids it contains are unstable. Interaction between the fluids may stabilize the universe. The condition for stability by perturbation of the scale factor from its static value is deduced for a class of universe models containing those investigated by Carneiro and Tavakol. Stability of the static state requires that energy is transformed to matter during such a perturbation.     

Mysteries of R^ik=0： A novel paradigm in Einstein＇s theory of gravitation

Despite a century-long effort, a proper energy-stress tensor of the gravitational field, could not have been discovered. Furthermore, it has been discovered recently that the standard formulation of the energy-stress tensor of matter, suffers from various inconsistencies and paradoxes, concluding that the tensor is not consistent with the geometric formulation of gravitation [Astrophys. Space Sci., 2009, 321： 151; Astrophys. Space Sei., 2012, 340： 373]. This perhaps hints that a consistent theory of gravitation should not have any bearing on the energy-stress tensor. It is shown here that the so-called ＂vacuum＂ field equations Rik = 0 do not represent an empty spacetime, and the energy, momenta and angular momenta of the gravitational and the matter fields are revealed through the geometry, without including any formulation thereof in the field equations. Though, this novel discovery appears baffling and orthogonal to the usual understanding, is consistent with the observations at all scales, without requiring the Moreover, the resulting theory circumvents the besides explaining some unexplained puzzles. hypothetical dark matter, dark energy or inflation long-standing problems of the standard cosmology     

Translational invariance of the Einstein–Cartan action in any dimension

We demonstrate that from the first order formulation of the Einstein– Cartan action it is possible to derive the basic differential identity that leads to translational invariance of the action in the tangent space. The transformations of fields is written explicitly for both the first and second order formulations and the group properties of transformations are studied. This, combined with the preliminary results from the Hamiltonian formulation (Kiriushcheva and Kuzmin in arXiv:0907.1553 [gr-qc]), allows us to conclude that without any modification, the Einstein–Cartan action in any dimension higher than two possesses not only rotational invariance but also a form of translational invariance in the tangent space. We argue that not only a complete Hamiltonian analysis can unambiguously give an answer to the question of what a gauge symmetry is, but also the pure Lagrangian methods allow us to find the same gauge symmetry from the basic differential identities.     

Comparison of String Fluid Dynamical Models in General Relativity and Einstein–Cartan theory

Dynamical models of string fluids areconstructed from the general energy-momentum tensor forstring fluids in general relativity and theEinstein-Cartan theories obtained from the Ray-Hilbertvariational principle. Examples of solutions to the fieldequations for general relativistic spacetimes are givenand compared with solutions obtained from the postulatedenergy-momentum tensor of Letelier. Solutions to the field equations in Riemann-Cartanspacetimes are compared with an extended Leteliersolution. All calculations are given for both thestandard and the extended thermodynamics versions inwhich the latter includes the string as thermodynamicvariables. In general relativity, it is shown for blackhole solutions that the general feature of strings(through the string vector) is to produce a shrinkage of the black hole horizon. In RiemannCartanspacetimes, the torsion field equation shows that stringvector can be identified with the torsion vector. Themost striking feature of strings in Riemann-Cartan spacetimes is that in the Reissner-Nordstromsolution, the addition of torsional strings produces thecorrect asympototic behavior of the metric necessary tomatch the experimental galactic rotationcurves.     

Robinson–Trautman solutions to Einstein’s equations

Solutions to Einstein’s equations in the form of a Robinson–Trautman metric are presented. In particular, we derive a pure radiation solution which is non-stationary and involves a mass m, The resulting spacetime is of Petrov Type II A special selection of parametric values throws up the feature of the particle ‘rocket’, a Type D metric. A suitable transformation of the complex coordinates allows the metrics to be expressed in real form. A modification, by setting m to zero, of the Type II metric thereby converting it to Type III, is then shown to admit a null Einstein–Maxwell electromagnetic field.     

Limits of the validity of the mass ratio independence of the Stokes—Einstein relation: molecular dynamics calculations and comparison with the Enskog theory

Detailed molecular dynamics simulations have been carried out to investigate the mass ratio dependence of the tracer self-diffusion coefficient D ₂ in binary, atomic Lennard-Jones mixtures as a function of density n and length diameter ratio σ₂₂/σ₁₁. For a compact representation of our results an exponential approach is employed. The results are compared with the Stokes-Einstein relation, which predicts no mass ratio dependence, and the Enskog theory. The validity ranges regarding the mass ratio dependence for Enskog and Stokes-Einstein like behaviour are given as well as the mass ratio dependence in the crossover regions between these two cases. For length parameter ratios σ₂₂/₁₁ <1, the Stokes-Einstein prediction is not valid for mass ratios 1/16≤m₂/m₁ ≤50. For length parameter ratios σ₂₂/σ₁₁>2 and for mass ratios m₂/m₁ < 1, the Stokes-Einstein regime (regarding the mass ratio dependence) is reached for smaller densities than for the same system but m₂/m₁ >1.     

On the origin of Kaluza’s idea of unification and its relation to earlier work by Thirring

We argue that the starting point of Kaluzas idea of unifying electrodynamics and gravity was the analogy between gravitation and electromagnetism which was pointed out by Einstein and Thirring. It seems that Kaluzas attention was turned to this point by the three papers on the Lense–Thirring effect and the analogy between gravitation and electromagnetism which were published a short time before Kaluzas paper was submitted. We provide here also an English translation of the third of these papers (Phys. Zeits. 19: 204, 1918).     

Cosmological Consequences of Two Sources of Torsion in the Einstein–Cartan Theory

A modification of the Einstein–Cartan theory is considered in which sources of torsion are an ideal fluid and a nonminimally coupled scalar field. Exact solutions are derived for spatially homogeneous cosmological models of the fluid that obeys an extremely rigorous equation of state. It is demonstrated that in some cases, nonsingular models can be used. The role of each source of torsion in the evolution of models is elucidated.     

On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.