Formulation Of General Relativity Research Articles

An equivalent system of Einstein Equations

We provide the full set of equations governing the evolution of self-gravitating systems by using a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor. We apply this formalism to spherical case and found that: the only static solution with homogeneous energy density, under the physical reasonable considerations, is the Schwarzchild solution; the conditions under which a non-dissipative fluid is static and finally that shear-free and isotropic pressure conditions are equivalent, for non-dissipative fluids with homogeneous energy density.

E(2) gauge theory model of effective gravitational theory at large scale

At the cosmological scale, there exist many anisotropic anomalies in the low-l multipoles of the CMB angular power spectrum. Especially, the normals to the octopole and quadrupole planes are aligned with the direction of the cosmological dipole at a level inconsistent with Gaussian random. The inconsistency indicates that the anomalies may not be boost effect from the CMB rest frame to the peculiar frame. It hints us that the boost invariance might be violated on a cosmological scale. There are some discrepancies between the astronomical and cosmological observations, and the predictions are solely based on general relativity and the standard model for elementary particle physics. The solutions are the introduction of dark matter and dark energy. However, all the experiments aiming at finding dark matter particles give negative result and it is still a mystery:what the dark energy is comprised of. We suppose that the Lorentz symmetry begins to be violated partly from the scale of galaxy and utilize the very special relativity symmetry group E(2) as an example to illustrate the Lorentz violation effect on the large-scale effective gravity. A local E(2) but Lorentz invariant gauge theory can be constructed based on the equivalence principle and the gauge principle. To realize the E(2) symmetry, the closure requirement of Maurer-Cartan eqnarray on E(2) algebra needs to be satisfied by postulating constraint conditions among the components of the Lorentz connection. The local Lorentz invariant gauge theory with a Hilbert-Einstein action is a theory with torsion in general case. However in the case of scalar matter source, the theory is exactly the theory of general relativity with Levi-Civita connection and zero torsion. In the E(2) gauge theory case, the closure requirement of Maurer-Cartan eqnarray for E(2) algebra postulates 12 constraint eqnarrays among the components of the Lorentz connection and the eqnarrays of motion for connection reduce the number of independent components of connection to 12. The eqnarrays of motion for the tetrad field do not contain only the involved tetrad field components nor these relevant independent components. So the whole number of variables needed to be solved is 12 more than that in general relativity while there are 12 more eqnarrays in the meantime. The torsion or the contortion field of the E(2) gauge theory is non-trivial even in the case of scalar matter source distribution. Decompose the connection into Levi-Civita one and the contortion part and rewrite the eqnarrays for tetrad field in the formalism of general relativity, then there will appear an effective energy-momentum tensor contributed by the contortion distribution, in addition to the ordinary matter source distribution even for the case of scalar matter source. We expect it to contribute at least part of the dark matter effect. We also examine the holding of the first and second Bianchi identities induced by Jacobi identity of the E(2) gauge theory. The approach of our modified gravity is different from other approach of modified gravity in the sense that we construct the modified gravity by modifying the spacetime symmetry on a large scale and the emergence of effective energy-momentum tensor caused by Lorentz violation effect is due to a purely large scale effect.

General Relativistic Study of Thin Disk with Radial Flow in Kerr – Newman Geometry

A detailed general relativistic formulation of the thin disk with the radial flow in Kerr – Newman geometry. The effects of rotation through an angular momentum a and charge <i>Q</i> were obtained on pressure, velocity and density for co rotating and counter rotating thin disk.

Global versus local — Mach’s principle versus the equivalence principle

The equivalence principle is the conceptual basis for general relativity. In contrast, Mach’s principle, although said to have been influential on Einstein in his formulation of general relativity, has not been shown to be central to the structure of general relativity. In this paper, we suggest that the quantum effects of Hawking and Unruh radiation are a manifestation of a thermal Mach’s principle, where the local thermodynamic properties of the system are determined by the nonlocal structure of the quantum fields which determine the vacuum of a given spacetime. By comparing Hawking and Unruh temperatures for the same local acceleration we find a violation of the Einstein elevator version of the equivalence principle, which vanishes in the limit that the horizon is approached.

Radial modes of slowly rotating compact stars in the presence of magnetic field

Compact stars are composed of very high-density hadron matter. When the matter is above nuclear matter density, then there is a chance of different phases of matter such as hadron matter to quark matter. There is a possible phase which, having the quark core surrounded by a mixed phase followed by hadronic matter, may be considered as a hybrid phase inside the stars called hybrid star (HS). The star which consists of only u, d and s quarks is called quark star (QS) and the star which has only hadronic matter is called neutron star (NS). For the equation of state (EOS) of hadronic matter, we have considered the Relativistic Mean Field (RMF) theory and we incorporated the effect of strong magnetic fields. For the EOS of the quark phase we use the simple MIT bag model. We have assumed Gaussian parametrization to make the density dependent for both bag pressure in quark matter and magnetic field. We have constructed the intermediate mixed phase by using the Glendenning conjecture. Eigenfrequencies of radial pulsations of slowly rotating magnetized compact stars (NS, QS, HS) are calculated in a general relativistic formalism given by Chandrasekhar and Friedman. We have studied the effect of central density on the square of the frequencies of the compact stars in the presence of zero and strong magnetic field.

Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity

Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach. The intention throughout is to provide a survey accessible to anyone, equipped only with an introductory knowledge of GA, whether in maths, physics or engineering.

A rotating Michelson interferometer from the co-rotating point of view

The phase shift induced by a uniform rotation in a Michelson interferometer is re-derived in the geometrical framework of the coordinate-free formalism of general relativity from the co-rotating point of view. The effect is second order in the ratio of the interferometer’s speed to the speed of light and further suppressed by the ratio of the length of the interferometer’s arms to the radius of rotation. The relation of the effect to gravitational time dilation is discussed.

Mass gap in the critical gravitational collapse of a kink

We study the gravitational collapse of a kink within spherical symmetry and the characteristic formulation of General Relativity. We explore some expected but elusive gravitational collapse issues which have not been studied before in detail, finding new features. The numerical one-parametric solution and the structure of the spacetime are calculated using finite differences, Galerkin collocation techniques, and some scripting for automated grid coverage. We study the threshold of black hole formation and confirm a mass gap in the phase transition. In the supercritical case we find a mass scaling power law $M_{BH}={M^*_{BH}}+K[\lambda-\lambda^*]^{2\gamma}+f(K[\lambda-\lambda^*]^{2\gamma})$, with $\gamma\approx 0.37$ independent of the initial data for the cases considered, and $M^*_{BH}$, $K$ and $\lambda^*$ each depending on the initial datum. The spacetime has a self-similar structure with a period of $\Delta\approx 3.4$. In the subcritical case the Bondi mass at null infinity decays in cascade with $\Delta/2$ interval as expected.

General relativity and cosmic structure formation

Numerical simulations are a versatile tool providing insight into the complicated process of structure formation in cosmology. This process is mainly governed by gravity, which is the dominant force on large scales. To date, a century after the formulation of general relativity, numerical codes for structure formation still employ Newton's law of gravitation. This approximation relies on the two assumptions that gravitational fields are weak and that they are only sourced by non-relativistic matter. While the former appears well justified on cosmological scales, the latter imposes restrictions on the nature of the "dark" components of the Universe (dark matter and dark energy) which are, however, poorly understood. Here we present the first simulations of cosmic structure formation using equations consistently derived from general relativity. We study in detail the small relativistic effects for a standard {\Lambda}CDM cosmology which cannot be obtained within a purely Newtonian framework. Our particle-mesh N-body code computes all six degrees of freedom of the metric and consistently solves the geodesic equation for particles, taking into account the relativistic potentials and the frame-dragging force. This conceptually clean approach is very general and can be applied to various settings where the Newtonian approximation fails or becomes inaccurate, ranging from simulations of models with dynamical dark energy or warm/hot dark matter to core collapse supernova explosions.

A quantum theory of spacetime in spinor formalism and the physical reality of cross-ratio representation: the equation of density parameters of dark energy, matter, and ordinary matter is derived: ΩM2 = 4 Ωb ΩΛ

ABSTRACT By theorizing the physical reality through the deformation of an arbitrary cross-ratio, we leverage Galois differential theory to describe the dynamics of isomonodromic integratable system. We found a new description of curvature of spacetime by the equivalency of isomonodromic integratable system and Penrose’s spinor formalism of general relativity. Using such description, we hypothetically quantize the curvature of spacetime (gravity) and apply to the problem of the evolution of the universe. The Friedmann equation is recovered and compared so that the mathematical relationship among dark energy, matter (dark matter + ordinary matter), and ordinary matter, Ω M 2 ≃ 4 Ω b Ω Λ , is derived; the actual observed results are compared to this equation (calculated Ω M = 0.33 vs. observed Ω M = 0.31); the model might explain the origin of dark energy and dark matter of the evolution of the universe.

General relativity as a two-dimensional CFT

The tree-level scattering amplitudes of general relativity (GR) encode the full nonlinearity of the Einstein field equations. Yet remarkably compact expressions for these amplitudes have been found which seem unrelated to a perturbative expansion of the Einstein–Hilbert action. This suggests an entirely different description of GR which makes this on-shell simplicity manifest. Taking our cue from the tree-level amplitudes, we discuss how such a description can be found. The result is a formulation of GR in terms of a solvable two-dimensional conformal field theory (CFT), with the Einstein equations emerging as quantum consistency conditions.

Constraint algebra of general relativity from a formal continuum limit of canonical tensor model

Canonical tensor model (CTM for short below) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. In the classical case, the constraints form a first-class constraint Poisson algebra with structures similar to that of the ADM formalism of general relativity, qualifying CTM as a possible discrete formalism for quantum gravity. In this paper, we show that, in a formal continuum limit, the constraint Poisson algebra of CTM with no cosmological constant exactly reproduces that of the ADM formalism. To this end, we obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism. On the other hand, the cosmological constant term of CTM seems to induce non-local dynamics, and is inconsistent with an assumption about locality of the continuum limit.

Closure constraints for hyperbolic tetrahedra

We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant gauge theory. Its classical states are graphs provided with algebraic data. In particular, closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space . One then glues them, allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group would allow us to account for a non-vanishing cosmological constant , and in particular that deforming the loop gravity phase space with real parameter would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in ). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

Path integral for (2+1)-dimensional Shape Dynamics

We studied the path integral quantization for the Shape Dynamics formulation of General Relativity in the 2+1 torus universe. We show that the Shape Dynamics path integral on the reduced phase space is equivalent with the previous results obtained for the ADM 2+1 gravity and we found that the Shape Dynamics Hamiltonian allows us to establish a straightforward relation between reduced systems in the (τ, V)-form and the τ-form through the York time gauge fixing.

Instanton Representation of Plebanski Gravity. The Classical Theory

This paper is a self-contained introduction to the instanton representation of Plebanski gravity (IRPG), a formulation of General Relativity (GR) where the basic variables are a spacetime gauge connection and a three by three matrix valued in the Lie algebra of so(3,C). We present a classical analysis of the IRPG from various perspectives, noting some of its interesting features and motivations.

An Eidetic Reflex and Moment of Breakthrough in Time and Scientific Creation: 10 Years of Progress in Physics, 100 Years of General Relativity, and the Zelmanov Cosmological Group

We celebrate the first 10-year momentous span of the solid body of critical scientific results and efforts delivered by the visionary editorial and founding team of the pioneering open new-millennium journal for advanced studies in theoretical and experimental physics, mathematics, astronomy, and cosmology, Progress in Physics (see the Editor-in-Chief’s message: “Progress in Physics: 10 years in Print”), behind which is the core scientists and guardians of universal scientific creation, scientific revolution, and scientific-intellectual freedom and ethics: the few core scientists of the quintessential Zelmanov Cosmological Group, such as the founding editors and scientific creators Dmitri Rabounski and Larissa Borissova. The Zelmanov Cosmological Group, which is also behind The Abraham Zelmanov Journal for General Relativity, gravitation, and cosmology, dedicates itself to the profound and extensive scope and depth of the works of the master theoretician “par excellence” of the Soviet-era general relativistic and cosmological school, Abraham Leonidovich Zelmanov, and to the most unique problems and possible extensions of General Relativity in general. Abraham Zelmanov’s profundity “sine qua non” is reflected in the singular creation of the theories of chronometric, kinemetric, and orthometric (monad) formalism in General Relativity, the Infinite Relativity Principle, the Anthropic Principle, the extensive classification of all possible cosmologicalmodels in the space-time of General Relativity (the Zelmanov Classification, including the possibility of absolute reference frames in a deforming, rotating, gravitating closed finite Universe), and many others (see the website of The Abraham Zelmanov Journal for details, and in particular the 2012 foreword to the book Particles Here and Beyond the Mirror). So, Zelmanov’s theoretical mastery singularly encompasses the general fully non-linear, anisotropic, inhomogeneous, anholonomic, non-simply-connected spacetime structure (and sub-structure) of General Relativity and the fabric of the cosmos, achieving the unification of the underlying structure of space-time, reference frame systems, and the fundamental observer. Zelmanov’s few students and theoretical inheritors — such as Dmitri Rabounski and Larissa Borissova — have thereby preserved and extended his scientific and philosophical ideals as a whole, comprehensive, unitive scientific legacy: a singular univocity — “ZelmanovianUniversum”— in the form of an ideologicallymost unique and versatile platform for the most singular kind of meta-science and scientific creation, which is the embryo of the present Zelmanov Cosmological Group. In the background of such unique origination, the general fundamental physics journal Progress in Physics, with a substantial portion of publications in General Relativity and differential geometry — in common with The Abraham Zelmanov Journal, is dedicated mostly to original, profound, critical, and challenging scientific works that potentially engage with the overall, far-reaching horizons and verizons of theoretical and experimental physics, mathematics, astronomy/cosmology, and of science as a whole, thereby expanding and synthesizing new scientific landscapes for both the present and the future. This is done mostly by identifying the pertinent objective quality and originality of the idea(s) in a submitted scientific work and the first and foremost crucial identification of the author as an essentially independent creative mind (whether specifically affiliated or not) and as a true person of integrity and clarity, therefore isolating the process of scientific judgement infinitely and decisively from the pervasively corruption-mongering, business-minded, pseudoscientific (so, pseudo-objective) politics of typical modern academic practice and science administration (i.e., “big-wig scientism”). In specific cases where the editors and expert peer reviewers (who dare be non-anonymous) do not agree with the ideology and content of a submitted paper, a fidelity to pure scientific-intellectual freedom is still maintained as much as possible in the publication of the said work, as long as the basic technicality and competence (such as the mathematics and logical reasoning) is fulfilled. This is also true for some tremendous-looking extremely short papers that can subtly serve as an impetus for reflection and future scientific inspiration: they can be so short and still publishable in view of inspiring some pertinent new ideas in the future. A word on a better peer-review system is at hand: above all, the journal categorically and distinctly promotes original thinkers and original scientific creators, along with fundamentally improving and transcending the largely deficient anonymous peer-review system, thus often allowing a work to be published with the potential for an on-going open peerreview (in the full critical vastness of time and space as regards judgement and validation): such as witnessed in the

The Newman–Penrose formalism for Riemannian 3-manifolds

We adapt the Newman–Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field k with geodesic flow, if an integral curve of k is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if k is complete, hypersurface-orthogonal, and satisfies Ric(k,k)≥0, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length k with geodesic flow satisfies Ric(k,k)>0, then k cannot be hypersurface-orthogonal, thus recovering a result in Harris and Paternain (2013). Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields k that satisfy the condition R(k,⋅,⋅,⋅)=0. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field k satisfies R(k,⋅,⋅,⋅)=0 if and only if it is a Killing vector field.

Radial modes of oscillations of slowly rotating magnetized compact hybrid stars

Stars having a quark core surrounded by a mixed phase followed by hadronic matter are termed as hybrid stars(HS). For the equation of state(EOS) of hadronic matter, we have considered RMF(Relativistic Mean Field) theory, in the presence of strong magnetic fields leading to Landau quantization of the charged particles. For the EOS of quark phase we have used the simple MIT bag model also in presence of strong magnetic field. The bag pressure and magnetic field are parametrized to make density dependant. We have constructed the intermediate mixed phase by using Glendenning conjecture. The EOSs are constructed comprising the hadron, mixed and quark phases. Eigenfrequencies of radial pulsations of slowly rotating magnetised hybrid star are calculated using constructed EOSs in a general relativistic formalism given by Chandrasekhar and Friedman.

Canonical ADM tetrad gravity: From metrological inertial gauge variables to dynamical tidal Dirac observables

In this updated review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations I show which is the status-of-the-art in the search of a canonical basis adapted to the first-class Dirac constraints and of the Dirac observables of general relativity (GR) describing the tidal degrees of freedom of the gravitational field. In these space-times the asymptotic ADM Poincaré group replaces the Poincaré group of particle physics, there is a York canonical basis diagonalizing the York–Lichnerowicz approach and a post-Minkowskian linearization is possible with the associated description of gravitational waves in the family of non-harmonic 3-orthogonal Schwinger time gauges. Moreover I show that every fixation of the inertial gauge variables (i.e. the choice of a non-inertial frame) of every generally covariant formulation of GR is equivalent to a set of conventions for the metrology of the space-time (like the GPS ones near the Earth): for instance the freedom in clock synchronization is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle are connected with the dark side of the universe and could explain the presence of dark matter or at least part of it by means of the adoption of suitable metrical conventions for the ICRS celestial reference system. Also some comments on a canonical quantization of GR coherent with this viewpoint are done.

Investigation of plasma waves propagation around traversible wormhole’s throat

The formalism of general relativity is used in a preliminary investigation of waves propagating in a plasma around the throat of traversible wormhole. The relativistic two-fluid equations are used to take account of gravitational effects due to the throat of traversible wormhole. Here, a local approximation is used to investigate the one-dimensional electromagnetic waves radial propagation near the throat of wormhole. We use approximation near by the throat of wormhole for one-dimensional radial propagation. The dispersion relations for the transverse and longitudinal electromagnetic waves are obtained. Finally, the components of stress-energy tensor and the zero tidal force, and so flaring-out condition for traversible wormhole metric are calculated.