New ideas regarding the geometry of space-time and gravitation

In the present survey on the basis of an analysis of the problem of the energy-momentum of the gravitational field in the general theory of relativity it is shown that this theory is unsatisfactory as a concrete realization of Einstein's idea of the connection of the geometry of space-time with matter. A new theory of gravitation is proposed which alters established ideas of space-time, makes it possible to describe all present gravitational experiments, and predicts a number of fundamental consequences.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 21, pp. 3–215, 1982.     

Non-differentiable variational principles

We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schrödinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time.     

Scale relativity in Cantorian E(∞) space-time

The present note highlights some mathematical and formal connections between the theory of scale relativity and the Cantorian space-time approach to particle physics.     

Modification of special relativity and local structures of gravity-free space and time

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the one-way speed of light in all inertial frames of reference, the special theory of relativity uses the assumption about the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the so-called primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the flat structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine this assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite speed c^′ in the primed inertial coordinate system and finite speed c in the usual inertial coordinate system. It also involves the c^′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between any two usual inertial coordinate systems. The physical principle is: the c^′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed to the usual inertial coordinate systems. Evidently, the modified special relativity theory and the quantum mechanics theory together found a convergent and invariant quantum field theory.     

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999.     

Is quantum space-time infinite dimensional

The stringy uncertainty relations, and corrections thereof, were explicitly derived recently from the new relativity principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that quantum space-time may very well be infinite dimensional. A discussion of the repercussions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the new relativity principle: The cosmological constant is not a constant, in the same vein that energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D=4 world might just be an average dimension over the infinite possible values of the quantum space-time and why the compactification mechanisms from higher to four dimensions in string theory may not be actually the right way to look at the world at Planck scales.     

Modification of special relativity and formulation of convergent and invariant quantum field theory

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference, the special theory of relativity uses another assumption. This other assumption concerns the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the Euclidean structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine the alternative assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite c′ in the primed inertial coordinate system and finite c in the usual inertial coordinate system. It also involves the c′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. Since all our experimental data are collected and expressed in the usual inertial coordinate system, the physical principle is: the c′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed inertial coordinate system to the usual inertial coordinate system. This principle is applied to a reformulation of mechanics, field theory and quantum field theory. Relativistic mechanics in the usual inertial coordinate system is unchanged, while field theory is developed and divergence-free. Any c′-type Galilean-invariant field system can be quantized by using the canonical quantization method in the primed inertial coordinate system. We establish a transformation law for quantized field systems as they are transformed from the primed to the usual inertial coordinate system. It is shown that the modified special relativity theory, together with quantum mechanics, leads to a convergent and invariant quantum field theory, in full agreement with experimental facts. The formulation of this quantum field theory does not demand departures from the concepts such as local Lorentz invariance in the usual inertial coordinate system, locality of interactions, and local or global gauge symmetries.     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D_T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s ε^(∞) theory through the cluster–cluster coherence (El Naschie global coherence). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s ε^(∞) theory (El Naschie local coherence). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s ε^(∞) theory interferes through El Naschie global and local coherences.     

Some Paradoxes in Special Relativity and the Resolutions

The special theory of relativity is the foundation of modern physics, but its unusual postulate of invariant vacuum speed of light results in a number of plausible paradoxes. This situation leads to radical criticisms and suspicions against the theory of relativity. In this paper, from the perspective that the relativity is nothing but a geometry, we give a uniform resolution to some famous and typical paradoxes such as the ladder paradox, the Ehrenfest’s rotational disc paradox. The discussion shows that all the paradoxes are caused by misinterpretation of concepts. We misused the global simultaneity and the principle of relativity. As a geometry of Minkowski space-time, special relativity can never result in a logical contradiction.     

Quantum Physics and Fractal Space Time

We argue for a new fractal space-time which is different from that of Nottale, Ord and El Naschie. The fractal here is a deterministic fractal, where a fractal seed on an M+1 th scale, let us say, is about 10⁴⁰ times the diameter of the fractal seed on the M th fractal scale. At each scale, the fractal seed is the most fundamental particle on that scale and is the single Z₀₀ source in the Einstein equations, all other particles with mass are either excited states or multibody states of this particle. We take the M th fractal scale to be the electron and the M+1 th scale to be the Hubble universe. Next, we use the selfsimilarity between these two objects to solve for the metric, comoving with the Hubble expansion of the M+1 scale (The Dirac equation can be derived here also) . Plug these metric elements back into the metric equation (which then also derives a 1/ (1−r) potential) and then back into the Dirac equation. You then get tauon, muon, electron masses and QED results, but with only one vertex. The S matrix for this Dirac equation gives the W and Z masses as resonance energies. Applying the M+2 perturbation in the same way as the M+1 perturbation, gives the three neutrino masses. Combining the M+1 th and M+2 solution into the zero energy vacuum, gives the beta decay left, handedness and SU (2) XU (1) gauge. Applying this theory within the r<1 region, gives the quarks and QCD assymptotic freedom is implied by the 1/ (1−r) potential. We do a radial coordinate transformation of the Z_αβ source to the expanding cosmological metric and get a new term added to Z₀₀. From this new term, we calculate the value of the Newtonian gravitational constant.     

Effective fractal dimensions

Classical fractal dimensions (Hausdorff dimension and packing dimension) have recently been effectivized by (i) characterizing them in terms of real‐valued functions called gales, and (ii) imposing computability and complexity constraints on these gales. This paper surveys these developments and their applications in algorithmic information theory and computational complexity theory. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spin and non-commutative geometry

The divide between quantum theory and general relativity, more generally classical physics, has persisted. It is argued here that this is because the latter has, in the words of Witten, a Bosonic space-time underpinning, while the former has a Fermionic or non-commutative space-time underpinning: the purely quantum mechanical spin-half implies non-commutative space-time and vice versa.     

Space-time as a random heap

In this paper, we demonstrate how space-time is, rather than a differentiable manifold, a random heap, and how this ties up with fractal dimension 2 of a Quantum Mechanical path. In this light, we can see that there is a harmonious convergence between the stochastic approach of Nelson and the de Broglie–Bohm approach. These considerations are shown to lead to the emergence of special relativity and Quantum Mechanics.     

Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form that requires imposing additional constraints, which are a part of Einstein’s equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein’s general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 271–288, November, 2007.     

一个非解析复映射的广义Julia集

推广了Michelitsch和Rossler所提出的由一个简单非解析映射所构造Julia集的方法,并由推广的复映射,构造出一系列实数阶的广义Julia集(简称广义J集)． 利用复变函数理论和计算机制图相结合的实验数学的方法,对广义J集的结构和演化进行了研究,结果表明： ①广义J集的几何结构依赖于参数α、R和c; ②广义J集具有对称性和分形特征; ③小数阶广义J集出现了错动和断裂,且其演化过程依赖于相角主值范围的选取.     

Some recent advances in theory and simulation of fractional diffusion processes

To offer an insight into the rapidly developing theory of fractional diffusion processes, we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag–Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.     

Predicting dispersion in porous media

We present a fundamental theory of solute dispersion in porous using (i) critical path analysis and cluster statistics of percolation theory far from the percolation threshold and (ii) the tortuosity and structure of large clusters near the percolation threshold. We use the simplest possible model of porous media, with a single length scale of heterogeneity in which the statistics of local conductances are uncorrelated. This combination of percolation‐based techniques allows comprehensive investigation and predictions concerning the process of dispersion. Our predictions, which ignore molecular diffusion and make minimal use of unknown parameters, account for results obtained in a comprehensive set of nearly 1100 experiments performed on systems ranging in size from centimeters to 100 km. The success of our simple treatment overturns many existing notions about transport in porous media, such as (1) multiscale heterogeneity must be accounted for in predictions (single scale is sufficient), (2) geologic correlations are of great importance (the randomness of percolation theory is more appropriate for prediction than the most complicated models in other frameworks), (3) geologic complexity is more important than statistical physics (exactly the reverse), (4) knowledge of the subsurface is more important than knowledge of the initial conditions of the plume (the latter is critical, the former may be virtually irrelevant), (5) diffusion is dominant over advection (diffusion appears seldom to be relevant at all), (6) fracture networks are fundamentally different, and more complex, than porous media (the two are mostly equivalent), (7) the fractal structure of the medium is relevant to power‐law behavior of the dispersion (in fact, at short times it is the heterogeneity of the medium, while at long times it is the fractal structure of the critical paths), and (8) there is a relation between an increase in dispersion with scale and a similar increase in the hydraulic conductivity (in fact the present model is consistent with both a diminishing hydraulic conductivity and a diminishing solute velocity with increasing spatial scale). © 2009 Wiley Periodicals, Inc. Complexity, 16,43–55, 2010