We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (“static”) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show (...) the completeness of all timelike and spacelike curves despite the superficial failure of the metric. (shrink)

We show that the big bang is a coordinate singularity for a large class of \ inflationary FLRW spacetimes which we have dubbed ‘Milne-like.’ By introducing a new set of coordinates, the big bang appears as a past boundary of the universe where the metric is no longer degenerate—a result which has already been investigated in the context of vacuum decay. We generalize their results and approach the problem from a more mathematical perspective. Similar to how investigating the (...) geometrical properties of the \ event horizon in Schwarzschild led to a better understanding of black holes, we believe that investigating the geometrical properties of the big bang coordinate singularity for Milne-like spacetimes could lead to a better understanding of cosmology. We show how the mathematics of these spacetimes may help illuminate certain issues associated with dark energy, dark matter, and the universe’s missing antimatter. (shrink)

The article is devoted to the transformations of society in the era of globalization. The global world is seen as a consequence of the successful implementation of the world-historical the Project of Modernity. Its completion results in the loss of its intellectual authority and historical effective- ness. The principal quality of contemporary society became its globality. The paradoxical phenomenon of the world, that had ceased to be a reality, became an integrative shape of the global transformations. Visibility took the privileged (...) place here. The degradation of reality is analyzed in its main forms: dematerialization of reality; decline and destruction of human sensory experience; destruction of the alive contacts with living beings and the capacity to empathize with somebody; the loss of the space of the human existence space features of a stable, predictable, structured se- mantic order. The consequence of that is the destruction of the foundations of the communities and society as a whole, which is fixed by the notion of their «vagueness». The correlation to this social metamorphosis is a new anthropological format — the singularity, which provides for the theoretical abolition of identity as a way for defining a man. It means the loss of an individual’s predictability and its determination as a social character. The network is the way through which the singularities interact, and swarms are the natural form of their association. Swarm is a historically new, unstructured, but subordinated to common impulses union, which consists of points of activity of different intensity. The analysis of society is incomplete without taking into account its discursive-communicative nature. In the global world there is a decline in the discourses. The signs of that process are the lack of legitimacy in classical instances of the meaning (truth, good, beauty, etc.), the degeneration of thinking into mixing, changing criteria of significance (dominance of curiosity and impression), social degradation of the media. The main feature of these changes is the latest phenomenon of information power (the power of depersonalized information space and information technology — those modes of circulation and use of information that became a determining factor in human life in the globalized world). (shrink)

In 1907, Einstein set out to fully relativize all motion, no matter whether uniform or accelerated. After five failed attempts between 1907 and 1918, he finally threw in the towel around 1920, setting himself a new goal. For the rest of his life he searched for a classical field theory unifying gravity and electromagnetism. As he struggled to relativize motion, Einstein had to readjust both his approach and his objectives at almost every step along the way; he got himself hopelessly (...) confused at times; he fooled himself with fallacious arguments and sloppy calculations; and he committed what he later allegedly called the biggest blunder of his career: he introduced the cosmological constant. There is a very uplifting moral to this somber tale. Although Einstein never reached his original destination, the harvest of his thirteen-year odyssey is quite impressive. First of all, what is left of absolute motion in general relativity is far more palatable than absolute motion in special relativity or Newtonian theory. And general relativity does seem to eliminate absolute space. More importantly, from a modern physics point of view, Einstein produced a spectacular new theory of gravity based on what he called the equivalence principle. This principle says that inertial and gravitational effects are due to one and the same structure, the inertio-gravitational field, which in Einstein’s theory is represented by a metric tensor field. In addition to laying the foundations of this theory, Einstein, among other things, launched relativistic cosmology, suggested the possibility of gravitational waves, gave the first sensible definition of a space-time singularity, and caught on to the intimate connection between general covariance and energy-momentum conservation, an example of the general connection between symmetries and conservation laws of Noether’s theorems. These results more than make up for the—at least by the standards of modern philosophy of science—rather opportunistic way in which they were obtained.. (shrink)

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes (...) the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

In this paper, a degenerate parabolic equation u t − div x θ ∇ u = x a u p with p > 1 and θ < 2, a ∈ ℝ, is considered. Based on rescaling arguments combined with a doubling property, the space-time singularity and decay estimates are established. Moreover, a universal and a priori bound of global nonnegative solutions for the corresponding initial boundary value problem is derived.

In accordance with an old suggestion of Asher Peres (1962), we consider the electromagnetic field as fundamental and the metric as a subsidiary field. In following up this thought, we formulate Maxwell’s theory in a diffeomorphism invariant and metric-independent way. The electromagnetic field is then given in terms of the excitation $H = ({\cal H}, {\cal D})$ and the field strength F = (E,B). Additionally, a local and linear “spacetime relation” is assumed between H and F, namely H (...) ~ κ F, with the constitutive tensor κ. The propagation is studied of electromagnetic wave fronts (surfaces of discontinuity) with a method of Hadamard. We find a generalized Fresnel equation that is quartic in the wave covector of the wave front. We discuss under which conditions the waves propagate along the light cone. Thereby we derive the metric of spacetime, up to a conformal factor, by purely electromagnetic methods. (shrink)

For a standard Finsler F on a manifold M, the domain is the whole tangent bundle T M and the fundamental tensor g is positive-definite. However, in many cases, these two conditions hold in a relaxed form only, namely one has either a psuedo-Finsler metric or a conic Finsler metric. Our aim is twofold. First, we want to give an account of quite a few subtleties that appear under such generalizations, say, for conic pseudo-finsler metrics. Second, we (...) aim to provide some criteria that determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler-metric -- or, more precisely, that the conic domain on which g remains positive-definite. Such a combination generalizes the known -metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors. (shrink)

This paper explains why spacetime singularities do not constitute a breakdown of physical laws, and points out that the difference between the metrics at singularities and those outside of singularities is factual, rather than nomological.

f gravity is examined in the context of a five-dimensional Kaluza-Klein theory with degenerate metric. In this theory electromagnetism is described by two vector fields, and there is a reflection symmetry between them which unifies them with gravitation. For matter, it is shown how the Lagrangian may be any function and still generate the same equations of motion, provided that some simple conditions are satisfied. The field equations are derived, and it is found that f gravity is not (...) consistent with the reflection symmetry. (shrink)

In the first person singular of the athematic middle optative in the R̥ gveda, there is strong metrical evidence that the poets knew and used forms in *-iy-a along- side the morphologically regular forms in -īy-a. I argue that the forms in *-iy-a are older and developed from PIE *-ih1-h2e by regular sound change, whereas the younger ones in -īy-a result from morphological regularization. The phonological development of *-ih1-h2e > *-iy-a provides further evidence for the historical phonology of “laryngeals” (...) and geminate consonants in the Indo-Iranian branch of Indo-European. (shrink)

If we live on the weak brane with zero effective cosmological constant in a warped 5D bulk spacetime, gravitational waves and brane fluctuations can be generated by a part of the 5D Weyl tensor and carries information of the gravitational field outside the brane. We consider on a cylindrical symmetric warped FLRW background a U self-gravitating scalar field coupled to a gauge field without bulk matter. It turns out that brane fluctuations can be formed dynamically, due to the modified (...) energy–momentum tensor components of the scalar-gauge field. As a result, we find that the late-time behavior could significantly deviate from the standard evolution of the universe. The effect is triggered by the time-dependent warpfactor with two branches of the form \}\) constants) and the modified brane equations comparable with a dark energy effect. This is a brane-world mechanism, not present in standard 4D FLRW, where the large disturbances are rapidly damped as the expansion proceed. Because gravity can propagate in the bulk, the cosmic string can build up a huge angle deficit by the warpfactor and can induce massive KK-modes felt on the brane. Disturbances in the spatial components of the stress-energy tensor cause cylindrical symmetric waves, amplified due to the presence of the bulk space and warpfactor. They could survive the natural damping due to the expansion of the universe. It turns out that one of the metric components becomes singular at the moment the warp factor develops an extremum. This behavior could have influence on the possibility of a transition from acceleration to deceleration or vice versa. (shrink)

Components of the Ricci and Einstein tensors are expressed in terms of the Gaussian curvatures of elementary two-spaces formed by the orthogonal coordinate planes, and the results are applied to some standard metrics.

The singularity space consists of all germs , with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a complete, separable space for the metric given by the order to which jets agree after base change. In the terminology of descriptive set-theory, the classification of singularities up to analytic extensions of scalars is a smooth problem. Over , the following two classification (...) problems up to isomorphism are then also smooth: analytic germs; and polarized schemes. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

A number of different forms of the Schwarzschild solution are considered. The static forms all have a singularity at the Schwarzschild radius. This Schwarzschild singularity can be eliminated if one goes over to a stationary or time-dependent form of solution. However, the coordinate transformations needed for this have singularities. It is stressed that coordinate systems connected by singular transformations are not equivalent and the corresponding metrics may describe different physical situations.

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the (...) issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a "geometric object", I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy. (shrink)

We construct all cosmic field tensors which are symmetric rank-two tensor concomitants of a metric and a background metric and which have zero divergence when the background metric satisfies the generalized De Donder condition. The resulting background cosmic field represents an Einstein space-time.

A naive introduction of a dependency of the mass of a black hole on the Schwarzschild time coordinate results in singular behavior of curvature invariants at the horizon, violating expectations from complementarity. If instead a temporal dependence is introduced in terms of a coordinate akin to the river time representation, the Ricci scalar is nowhere singular away from the origin. It is found that for a shrinking mass scale due to evaporation, the null radial geodesics that generate the (...) horizon are slightly displaced from the coordinate singularity. In addition, a changing horizon scale significantly alters the form of the coordinate singularity in diagonal (orthogonal) metric coordinates representing the space-time. A Penrose diagram describing the growth and evaporation of an example black hole is constructed to examine the evolution of the coordinate singularity. (shrink)

The cosmic singularity provides negligible evidence for creation in the finite past, and hence theism. A physical theory might have no metric or multiple metrics, so a ‘beginning’ must involve a first moment, not just finite age. Whether one dismisses singularities or takes them seriously, physics licenses no first moment. The analogy between the Big Bang and stellar gravitational collapse indicates that a Creator is required in the first case only if a Destroyer is needed in the second. The (...) need for and progress in quantum gravity and the underdetermination of theories by data make it difficult to take singularities seriously. The singularity exemplifies the sort of gap that is likely to be closed by scientific progress, obviating special divine action. The apparent irrelevance of cardinality to practices of counting infinite sets in classical field theory and Fourier analysis is noted. Introduction The Doctrine of Creation and Its Warrant Cardinality and Sizes of Infinity Modern Cosmology and Creation Tolerance or Intolerance toward Singularities? Leibniz against Incompetent Watchmaker? Induction from Earlier Theories' Breakdown? Stellar Collapse Implies Theistic Destroyer Stacking the Deck for GTR Quantum Gravity Tends to Resolve Singularities Vicious God-of-the-Gaps Character Fluctuating or Inaccessible Warrant Big Bang Cosmology Not Especially Congenial to Faith CiteULike Connotea Del.icio.us What's this? (shrink)

We give a description of the effect of the gravitational field by using the geodesic equation of motion with respect to a first order Finslerian approximation of the Minkowski metric. This motivates linking the physical force of gravity to the non flat nature of space in the Finslerian setting and leads to an anisotropic version of the red shift formula. We solve the linearized Finslerian field equations proposed by S.F. Rutz (Gen. Relativ. Gravit. 25(11):1139–1158, 1993).

A class of metric theories of gravitation on Minkowski spacetime is considered, which is—provided that certain assumptions (staying close to the original ideas of Einstein) are made—the almost most general one that can be considered. In addition to the Minkowskian metric G a dynamical metric H (called the Einstein metric)is defined by means of a second-rank tensor field S (referred to as gravitational potential).The theory is defined by a Lagrangian ℒ, from which the field equations (...) as well as, e.g., the energy-momentum tensor field for the gravitational field follow. The case of weak fields is considered explicitly. The static, spherically and time-inversal symmetric field is calculated, and as a first step to investigate the theory's viability the parameters are fitted to the experimental data of the perihelion advance and the deflection of light at the Sun. Finally the question of gauge freedoms in the gravitational potential is briefly discussed. (shrink)

It has been shown that the orbits of motion for a wide class of non-relativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual by means of a conformal map. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence in this way (...) with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a Kaluza-Klein type theory, and indicate how the TeVeS structure of Bekenstein and Saunders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields ${\mathcal{U}}_{\mu}(x)$ which preserve the Bekenstein-Sanders condition ${\mathcal{U}}_{\mu}{\mathcal{U}}^{\mu}=-1$ . The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e. they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the Yang-Mills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al. (shrink)

The nonadditive properties of mass make it desirable to abandon mass as a basis unit in physics and to replace it by a unit of the dimension of the quantum of action [h]. The ensuing four-unit system of action, charge, length, and time [h, q, l, t] interacts in a much more elucidating fashion with experiment and with the fundamental structure of physics. All space-time differential forms expressing fundamental laws of physics are forms of physical dimensions, h, h/q, or q. (...) Their occurrence as periods (residues) of period integrals of forms makes h and q topological invariants in space-time and stresses in general the global nature of quantization. The coordinate dimensions [l] and [t] only come into play for the form-coefficients, the metric tensor, and their tensorial relatives. One thus obtains a very natural guidance for implementing the principle of general covariance, when coordinate-based descriptions are necessary. The consequences for the general theory of relativity are discussed. (shrink)

A generally covariant theory, written in the spirit of Bohm's theory of quantum potentials, which applies to spinless, non interacting, gravitating systems, is formulated. In this theory the quantum state ψ is coupled to the metric tensor g, and the effect of the “quantum potential” is absorbed in the geometry. At the same time, ψ satisfies a covariant wave equation with respect to the very same g. This provides sufficient constraints to derive 11 coupled equations in the 11 (...) unknowns: ψ and the components of the metric tensor gµv. The states of stable localized particles are identified, and vacuum-state solutions for both the Euclidean and the Lorentzian case are explicitly presented. (shrink)

According to prevailing theory, relativistic degenerate stars with masses beyond the Chandrasekhar and Oppenheimer–Volkoff (OV) limits cannot achieve hydrostatic equilibrium through either electron or neutron degeneracy pressure and must collapse to form stellar black holes. In such end states, all matter and energy within the Schwarzschild horizon descend into a central singularity. Avoidance of this fate is a hoped-for outcome of the quantization of gravity, an as-yet incomplete undertaking. Recent studies, however, suggest the possibility that known quantum processes may (...) intervene to arrest complete collapse, thereby leading to equilibrium states of macroscopic size and finite density. I describe here one such process which entails pairing (or other even-numbered association) of neutrons (or constituent quarks in the event of nucleon disruption) to form a condensate of composite bosons in equilibrium with a core of degenerate fermions. This process is analogous to, but not identical with, the formation of hadron Cooper pairs that give rise to neutron superfluidity and proton superconductivity in neutron stars. Fermion condensation to composite bosons in a star otherwise destined to collapse to a black hole facilitates hydrostatic equilibrium in at least two ways: (1) removal of fermions results in a decrease in the Fermi level which stiffens the dependence of degeneracy pressure on fermion density, and (2) phase separation into a fermionic core surrounded by a self-gravitating condensate diminishes the weight which must be balanced by fermion degeneracy pressure. The outcome is neither a black hole nor a neutron star, but a novel end state, a “fermicon star,” with unusual physical properties. (shrink)

The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} on an additional time variable, named intrinsic time (...) at variance with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative g˙μν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{g}}_{\mu \nu }$$\end{document} of the metric field with respect to intrinsic time, corresponding to the conjugated momentum πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document}. Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional A[gμν,πμν]=K[gμν,πμν]-S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$\end{document}, playing formally the role of a Hamilton function, where S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} is the standard Einstein–Hilbert action while K[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action A[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} and not of energy. Since the Einstein–Hilbert action S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document}. Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document} gives rise to a cosmological constant term in the path-integral approach. (shrink)

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is a (...) geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion. (shrink)

In this paper, we review a general technique for converting the standard Lagrangian description of a classical system into a formulation that puts time on an equal footing with the system's degrees of freedom. We show how the resulting framework anticipates key features of special relativity, including the signature of the Minkowski metric tensor and the special role played by theories that are invariant under a generalized notion of Lorentz transformations. We then use this technique to revisit a (...) classification of classical particle-types that mirrors Wigner's classification of quantum particle-types in terms of irreducible representations of the Poincaré group, including the cases of massive particles, massless particles, and tachyons. Along the way, we see gauge invariance naturally emerge in the context of classical massless particles with nonzero spin, as well as study the massless limit of a massive particle and derive a classical-particle version of the Higgs mechanism. (shrink)

Given a quantum system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs with vertices representing subsystems and edges representing mutual information between subsystems. The second step is to deform the adjacency matrices of the information (...) graphs to that of a low-dimensional lattice using the graph flow equations introduced in the paper. The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze two information graph flows with geometric attractors and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to quantum gravity in which the geometry emerges directly from a quantum state due to the flow of the information graphs. (shrink)

An attempt is made to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe. This is done by introducing a background metric γμν (in addition to gμν) describing a spacetime of constant curvature with positive spatial curvature. The additional terms in the field equations are negligible for the solar system but important for intense fields. Cosmological models are obtained without singular states but (...) simulating the “big bang.” The field of a particle differs from the Schwarzschild field only very close to, and inside, the Schwarzschild sphere. The interior of this sphere is unphysical and impenetrable. A star undergoing gravitational collapse reaches a state in which it fills the Schwarzschild sphere with uniform density (and pressure) and has the geometry of a closed Einstein universe. Any charge present is on the surface of the sphere. Elementary particles may have similar structures. (shrink)

Patients with Wilson’s disease suffer from prospective memory impairment, and some of patients develop cognitive impairment. However, very little is known about how brain structure and function changes effect PM in WD. Here, we employed multimodal neuroimaging data acquired from 22 WD patients and 26 healthy controls who underwent three-dimensional T1-weighted, diffusion tensor imaging, and resting state functional magnetic resonance imaging. We investigated gray matter volumes with voxel-based morphometry, DTI metrics using the fiber tractography method, and RS-fMRI using the (...) seed-based functional connectivity method. Compared with HC, WD patients showed GM volume reductions in the basal ganglia and occipital fusiform gyrus, as well as volume increase in the visual association cortex. Moreover, whiter matter tracks of WD were widely impaired in association and limbic fibers. WM tracks in association fibers are significant related to PM in WD patients. Relative to HC, WD patients showed that the visual association cortex functionally connects to the thalamus and hippocampus, which is associated with global cognitive function in patients with WD. Together, these findings suggested that PM impairment in WD may be modulated by aberrant WM in association fibers, and that GM volume changes in the association cortex has no direct effect on cognitive status, but indirectly affect global cognitive function by its aberrant functional connectivity in patients with WD. Our findings may provide a new window to further study how WD develops into cognitive impairment, and deepen our understanding of the cognitive status and neuropathology of WD. (shrink)

In the United States, approximately 2.53 million people sustain a concussion each year. Relative to adults, youth show greater cognitive deficits following concussion and a longer recovery. An accurate and reliable imaging method is needed to determine injury severity and symptom resolution. The primary objective of this study was to characterize concussions with diffusion tensor imaging. This was performed through a normative Z-scoring analysis of DTI metrics, fractional anisotropy, axial diffusivity, and radial diffusivity, to quantify patient-specific injuries and identify (...) commonly damaged brain regions in paediatric concussion patients relative to healthy controls. It was hypothesized that personalizing the detection analysis through normative Z-scoring would provide an understanding of trauma-induced microstructural damage. Concussion patients were volunteers recruited from the Emergency Department of the McMaster Children’s Hospital with a recent concussion, 9 males and 17 females, mean age 14.22 ± 2.64, while healthy paediatric brain DTI datasets were obtained from an MRI data repository. Significant abnormalities were commonly found in the longitudinal fasciculus, fronto-occipital fasciculus, and corticospinal tract, while unique abnormalities were localized in a number of other areas reflecting the individuality of each child’s injury. Total injury burden, determined by the number of regions containing outliers per DTI metric per patient, was used as the metric to quantify the overall injury severity of each patient. The primary outcome of this analysis found that younger patients experienced a significantly greater injury burden when measured using fractional anisotropy. These results show that DTI was able to detect microstructural changes caused by concussion, on a per-person basis, and has the potential to be a useful tool for improving diagnostic accuracy and prognosis of a concussion. (shrink)

The prediction of general relativity on the gravitational collapse of matter ending in a point is viewed as an absurdity of the kind to be expected in any consistent physical theory due to ultimate conflicts of the axioms of geometry with the properties of physical objects. The necessity to introduce a probability interpretation for the solution of partial differential equations in space time for quantum theory points to similar roots. It is pointed out that quantum theory in the very small (...) is not going to eliminate the problem, but macroscopic quantum effects in the large, modifying the properties of the horizon, may achieve it. Solutions such as wormholes allow as much empirical evidence as any science fiction. The present approach considers successive modifications of the field equations and equations of motion of gravitational theory by admixture of terms with higher derivatives. The rigorous application of a gauge principle combines Einstein's equations with the tensor analog of Maxwell's equations which are of third order for the metric. It is speculated that the natural presence of torsion in such a gauge theory is related to matter sources. (shrink)

Intelligent communication processing in English aims to obtain effective information from unstructured text data using various text processing techniques. Text vector representation and text similarity calculation are important fundamental tasks in the whole field of natural language processing. In response to the shortcomings of existing sentence vector representation models and the singularity of text similarity algorithms, improved models and algorithms are proposed based on a thorough study of related domain technologies. This paper presents an in-depth and comprehensive study of text (...) vectorization representation and text similarity calculation algorithms in the field of natural language processing. The existing text vectorized representation models and text similarity computation algorithms are described, and their shortcomings are summarized to provide a basis for the background and significance of this paper, as well as to provide ideas for improvement directions. It is experimentally verified that the sentence vector model proposed in this paper achieves higher accuracy than the SIF sentence vector model for text classification tasks. In the task of text similarity computation, it achieves better results in three evaluation metrics: accuracy, recall, and F1 value. The algorithm also improves the computational efficiency of the model to a certain extent by removing feature words with low feature contribution. The algorithm first improves the deficiencies of the traditional word-shift distance algorithm by defining multifeature fusion weights and realizes a text similarity calculation algorithm based on multifeature weighted fusion with better similarity calculation results. Then, a linear weighting model is constructed to further combine the similarity calculation results of the hierarchical pooled IIG-SIF sentence vectors to realize the multimodel fusion text similarity calculation algorithm. (shrink)

Examples of hypotheses about the number of planets are frequently used to introduce the topic of (actual) truthlikeness but never analyzed in detail. In this paper we first deal with the truthlikeness of singular quantity hypotheses, with reference to several ‘the number of planets’ examples, such as ‘The number of planets is 10 _versus_ 10 billion (instead of 8).’ For the relevant ratio scale of quantities we will propose two, strongly related, normalized metrics, the proportional metric and the (...) (simplest and hence favorite) fractional metric, to express e.g. the distance from a hypothetical number to the true number of planets, i.e. the distance between quantities. We argue that they are, in view of the examples and plausible conditions of adequacy, much more appropriate, than the standardly suggested, normalized absolute difference, metric. Next we deal with disjunctive hypotheses, such as ‘The number of planets is between 7 and 10 inclusive is much more truthlike than between 1 and 10 billion inclusive.’ We compare three (clusters of) general ways of dealing with such hypotheses, one from Ilkka Niiniluoto, one from Pavel Tichý and Graham Oddie, and a trio of ways from Theo Kuipers. Using primarily the fractional metric, we conclude that all five measures can be used for expressing the distance of disjunctive hypotheses from the actual truth, that all of them have their strong and weak points, but that (the combined) one of the trio is, in view of principle and practical considerations, the most plausible measure. (shrink)

Counterfactuals have become an important area of interdisciplinary interest, especially in logic, philosophy of language, epistemology, metaphysics, psychology, decision theory, and even artificial intelligence. In this study, we propose a new form of analysis for counterfactuals: analysis by algorithmic complexity. Inspired by Lewis-Stalnaker's Possible Worlds Semantics, the proposed method allows for a new interpretation of the debate between David Lewis and Robert Stalnaker regarding the Limit and Singularity assumptions. Besides other results, we offer a new way to answer the problems (...) raised by Goodman and Quine regarding vagueness, context-dependence, and the non- monotonicity of counterfactuals. Engaging in a dialogue with literature, this study will seek to bring new insights and tools to this debate. We hope our method of analysis can make counterfactuals more understandable in an intuitively plausible way, and a philosophically justifiable manner, aligned with the way we usually think about counterfactual propositions and our imaginative reasoning. (shrink)

I consider the direct product algebra formed from two isomorphic Clifford algebras. More specifically, for an element x in each of the two component algebras I consider elements in the direct product space with the form x ⊗ x. I show how this construction can be used to model the algebraic structure of particular vector spaces with metric, to describe the relationship between wavefunction and observable in examples from quantum mechanics, and to express the relationship between the electromagnetic field (...) tensor and the stress-energy tensor in electromagnetism. To enable this analysis I introduce a particular decomposition of the direct product algebra. (shrink)

This essay argues that stable, heritable, habituated semiotics on one scale of life allows for opportunism, origination, and the solving of novel problems on others. This is grounded in how interpretation is neither caused nor determined by its object, such that success at interpretation simply cannot be defined by any comparison between an interpretation and its object. Rather, an interpretation is a reciprocated incorporation of a living thing and its environment, and successful if it furthers the living, interpreting thing. By (...) applying biosemiotic theory to seemingly disparate studies of parasitic infections, autonomic nervous systems, and social change as well as the classical pragmatic notion that biology, psychology and sociology are disparate approaches to the singular, radically continuous, and perennial question of who am I. I argue that the distinction between voluntary and autonomic behavior is but a ghost of older dualisms, the pseudo-contradictions of matter v. mind, body v. soul, but also self v. not self. Moreover, all such pseudo-contradictions are resolved as scale thick, self-similar examples of semiotic transaction wherein degeneration or habituation on one scale of life allows for generative or novel interaction on another. (shrink)

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

With the help of syncretiсs as a new philosophical logic, the philosophy of carriers, the theory of similarity and the theory of Infinite Hierarchical Nesting of Matter, the problems of modern physics are analyzed. We consider the classical and relativistic mechanics, the special and general theories of relativity, the theory of electromagnetic and gravitational fields, of weak and strong interactions. The goal is axiomatization of these theories, building models of elementary particles and of their interactions with each other. The main (...) obtained results are: description of the electrokinetic theory of the origin of magnetic fields in cosmic bodies; calculation of metric in the uniformly accelerated reference frame; the axiomatic construction of electrodynamics, Lorentz invariant (LITG) and covariant theories of gravitation (CTG); comparing CTG with the general theory of relativity and with the results of gravitational experiments. Among other results – the analysis of the properties of ether as the medium responsible for transfer of electromagnetic and gravitational waves, and derivation of the formula for entropy in a tensor form. For students and researchers, as well as for those interested in physical and philosophical problems. (shrink)

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, $W\gamma _{ik} = - G_{ik} + \tfrac{1}{2}g_{ik} g^{mn} G_{mn} - k(T_{ik} - \tfrac{1}{2}g_{ik} g^{mn} T_{mn} )$ together with the covariant Hilbert-gauge condition, $(\gamma _i^h - \tfrac{1}{2}\delta _i^k g^{mn} \gamma _{mn} )_{;k} = 0$ where W denotes the covariant wave operator and G ik the Einstein tensor of the metric g ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions (...) γ ik(z)with g ik Z,i Z,k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,Δp(Δg*)2(Δx)3≥h hG/ c 3 concerning the total nonlinear gravitational field g *ik =g k +γ k. (shrink)

A conservative extension of general relativity is proposed by alleviating the differentiability of the metric and allowing for non-smooth solutions. We show that these metrics break some symmetries of the Riemann tensor, yielding a new scalar curvature invariant besides the usual Ricci scalar. To first order in the curvature, this adds a new piece of information to the action, containing interesting and unexplored physics. The spectrum of the theory reveals the presence of an additional massless spin-1 field apart (...) from the massless spin-2 graviton. We argue that this new contribution violates P and CP symmetries at leading order in the curvature and we discuss the possibility of observing these effects in existing experiments. (shrink)

The geometrical structures implicit in the de Broglie waves associated with a relativistic charged scalar quantum mechanical particle in an external field are analyzed by employing the ray concept of the causal interpretation. It is shown how an osculating Finslerian metric tensor, a torsion tensor, and a tetrad field define respectively the strain, the dislocation density, and the Burgers vector in the “natural state” of the wave, which is a non-Riemannian space of distant parallelism. A quantum torque (...) determined by the quantum potential is introduced and the example of a screw dislocated wave is discussed. (shrink)

In this monograph the author presents the special and general theories of relativity from a geochronometrical viewpoint. The amount of mathematics demanded is not too great, and one can get quite far along on the material on vectors presented early in the book. The first three chapters especially derive from the work of A. A. Robb several decades ago: they treat foundations of geochronometry, one-plus-one geochronometry and its generalization to a one-plus-three system. Later chapters cover such staples as the Lorentz (...) transformation, the metric and dynamical tensors, dynamical relationships of particles, accelerating and curving events, geodesics, and applications to the red-shift of spectral lines and bending of light-rays. The best introduction to the study of the foundation of physics is physics itself; so those philosophers interested in the foundations of relativity theory will find here a lucid and logically respectable exposition of that theory, and in that way an understanding of its fundamentals and problems will be reached.—P. J. M. (shrink)

This paper uncovers the basic reason for the mysterious change of sign from plus to minus in the fourth coordinate of nature's Pythagorean law, usually accepted on empirical grounds, although it destroys the rational basis of a Riemannian geometry. Here we assume a genuine, positive-definite Riemannian space and an action principle which is quadratic in the curvature quantities (and thus scale invariant). The constant σ between the two basic invariants is equated to1/2. Then the matter tensor has the trace (...) zero. In consequence of the constancy of the scalar curvature and the divergence identity of the matter tensor, the perturbation metric has to satisfy a scalar and a vector condition, with a negative sign in the fourth coordinate. These conditions lead to the Lorentz condition and the wave equation for the vector potential. Thus the entire Maxwell-Lorentz type of electrodynamics becomes logically derivable, making no concession to any irrationality. (shrink)

I discuss the idea of relativistic causality, i.e., the requirement that causal processes or signals can propagate only within the light-cone. After briefly locating this requirement in the philosophy of causation, my main aim is to draw philosophers' attention to the fact that it is subtle, indeed problematic, in relativistic quantum physics: there are scenarios in which it seems to fail. I set aside two such scenarios, which are familiar to philosophers of physics: the pilot-wave approach, and the Newton-Wigner representation. (...) I instead stress two unfamiliar scenarios: the Drummond-Hathrell and Scharnhorst effects. These effects also illustrate a general moral in the philosophy of geometry: that the mathematical structures, especially the metric tensor, that represent geometry get their geometric significance by dint of detailed physical arguments. (shrink)

The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, (...) the gravitational and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory. (shrink)

It is shown how to identify potential signatures of noncommutative geometry within the decay spectrum of a muon in orbit near the event horizon of a microscopic Schwarzschild black hole. This possibility follows from a re-interpretation of Moffat’s nonsymmetric theory of gravity, first published in Phys. Rev. D 19:3554, 1979, where the antisymmetric part of the metric tensor manifests the hypothesized noncommutative geometric structure throughout the manifold. It is further shown that for a given sign convention, the predicted (...) signatures counteract the effects of curvature-induced muon stabilization predicted by Singh and Mobed in Phys. Rev. D 79:024026, 2009. While it is unclear whether evidence for noncommutative geometry may become observable anytime soon, this approach at least provides a useful direction for future quantum gravity research based on the ideas presented here. (shrink)

In this paper, we continue the study of the Killing symmetries of an N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space–time rotations in such spaces, by confining ourselves to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space M_4, for which we derive the explicit general form (...) of the finite rotations and boosts in different parametric bases. (shrink)