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Showing papers on "Introduction to the mathematics of general relativity published in 2012"


BookDOI
01 Jan 2012

314 citations


Journal ArticleDOI
TL;DR: This Letter addresses the implications of consistent nonlinear gravity-matter coupling and argues that such a completion of general relativity is viable from both an experimental and theoretical point of view through energy conditions, consistency, and singularity-avoidance perspectives.
Abstract: The coupling between matter and gravity in general relativity is given by a proportionality relation between the stress tensor and the geometry. This is an oriented assumption driven by the fact that both the stress tensor and the Einstein tensor are divergenceless. However, general relativity is in essence a nonlinear theory, so there is no obvious reason why the coupling to matter should be linear. On another hand, modified theories of gravity usually affect the vacuum dynamics, yet keep the coupling to matter linear. In this Letter, we address the implications of consistent nonlinear gravity-matter coupling. The Eddington-inspired Born-Infeld theory recently introduced by Banados and Ferreira provides an enlightening realization of such coupling modifications. We find that this theory coupled to a perfect fluid reduces to general relativity coupled to a nonlinearly modified perfect fluid, leading to an ambiguity between modified coupling and modified equation of state. We discuss observational consequences of this degeneracy and argue that such a completion of general relativity is viable from both an experimental and theoretical point of view through energy conditions, consistency, and singularity-avoidance perspectives. We use these results to discuss the impact of changing the coupling paradigm.

134 citations


Book
17 Sep 2012
TL;DR: A review of special Relativity four-vectors index notations can be found in this paper, where the Schwarzschild metric particle orbits in Kerr spacetime Ergoregion and Horizon Negative-Energy Orbits index are discussed.
Abstract: Introduction Review of Special Relativity Four-Vectors Index Notation Arbitrary Coordinates Tensor Equations Maxwell's Equations Geodesics The Schwarzschild Metric Particle Orbits Precession of the Perihelion Photon Orbits Deflection of Light Event Horizon Alternative Coordinates Black Hole Thermodynamics The Absolute Gradient Geodesic Deviation The Riemann Tensor The Stress-Energy Tensor The Einstein Equation Interpreting the Equation The Schwarzschild Solution The Universe Observed A Metric for the Cosmos Evolution of the Universe Cosmic Implications The Early Universe CMB Fluctuations and Inflation Gauge Freedom Detecting Gravitational Waves Gravitational Wave Energy Generating Gravitational Waves Gravitational Wave Astronomy Gravitomagnetism The Kerr Metric Particle Orbits in Kerr Spacetime Ergoregion and Horizon Negative-Energy Orbits Index

60 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the self-torque of a self-gravitating body can be approximated by an appropriate power series, and that the instantaneous force and torque exerted on it can be identical to the force exerted on an appropriate test body moving in the effective metric.
Abstract: The ‘external’ or ‘bulk’ motion of extended bodies is studied in general relativity. Compact material objects of essentially arbitrary shape, spin, internal composition and velocity are allowed as long as there is no direct (non-gravitational) contact with other sources of stress–energy. Physically reasonable linear and angular momenta are proposed for such bodies and exact equations describing their evolution are derived. Changes in the momenta depend on a certain ‘effective metric’ that is closely related to a non-perturbative generalization of the Detweiler–Whiting R-field originally introduced in the self-force literature. If the effective metric inside a self-gravitating body can be adequately approximated by an appropriate power series, the instantaneous gravitational force and torque exerted on it is shown to be identical to the force and torque exerted on an appropriate test body moving in the effective metric. This result holds to all multipole orders. The only instantaneous effect of a body’s self-field is to finitely renormalize the ‘bare’ multipole moments of its stress–energy tensor. The MiSaTaQuWa expression for the gravitational self-force is recovered as a simple application. A gravitational self-torque is obtained as well. Lastly, it is shown that the effective metric in which objects appear to move is approximately a solution to the vacuum Einstein equation if the physical metric is an approximate solution to Einstein’s equation linearized about a vacuum background.

48 citations


Journal ArticleDOI
TL;DR: In this article, a detailed reconstruction of Einstein's rather sketchy accounts of the twins and the bucket and examine the role of these two relativity principles play a role in these accounts: (a) the relativity of non-uniform motion, in the weak sense that the laws of physics are the same in the two space-time coordinate systems involved; (b) what Einstein in 1920 called the gravitational field, the notion that there is a unified inertio-gravitational field that splits differently into inertial and gravitational components in different coordinate systems.
Abstract: In publications in 1914 and 1918, Einstein claimed that his new theory of gravity in some sense relativizes the rotation of a body with respect to the distant stars (a stripped-down version of Newton's rotating bucket experiment) and the acceleration of the traveler with respect to the stay-at-home in the twin paradox What he showed was that phenomena seen as inertial effects in a space-time coordinate system in which the non-accelerating body is at rest can be seen as a combination of inertial and gravitational effects in a (suitably chosen) space-time coordinate system in which the accelerating body is at rest Two different relativity principles play a role in these accounts: (a) the relativity of non-uniform motion, in the weak sense that the laws of physics are the same in the two space-time coordinate systems involved; (b) what Einstein in 1920 called the relativity of the gravitational field, the notion that there is a unified inertio-gravitational field that splits differently into inertial and gravitational components in different coordinate systems I provide a detailed reconstruction of Einstein's rather sketchy accounts of the twins and the bucket and examine the role of these two relativity principles I argue that we can hold on to (b) but that (a) is either false or trivial

38 citations


Journal ArticleDOI
TL;DR: In this paper, a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvatures in the Friedmann equation was investigated by fitting to a combination of HST, CMB, type Ia supernovae (SNIa), and baryon acoustic oscillation (BAO) data sets.
Abstract: Averaging in general relativity is a complicated operation, due to the general covariance of the theory and the nonlinearity of Einstein's equations. The latter of these ensures that smoothing spacetime over cosmological scales does not yield the same result as solving Einstein's equations with a smooth matter distribution, and that the smooth models we fit to observations need not be simply related to the actual geometry of spacetime. One specific consequence of this is a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvature in the Friedmann equation. Here we investigate the consequences of this decoupling by fitting to a combination of Hubble Space Telescope (HST), CMB, type Ia supernovae (SNIa), and baryon acoustic oscillation (BAO) data sets. We find that only the geometrical spatial curvature is tightly constrained and that our ability to constrain dark energy dynamics will be severely impaired until we gain a thorough understanding of the averaging problem in cosmology.

30 citations


Journal ArticleDOI
TL;DR: In this article, the curvature invariants between the Ricci tensor and the Riemann tensor have been studied for a general class of analytic functions and the authors analyze the dynamics of a photon embedded in a gravitational field of a generic f!R;R!" R!" )-gravity.
Abstract: For a general class of analytic functionsf!R;R !" R !" ;R !"#$ R !"#$ "we discuss the gravitational lensing in the Newtonian limit of theory. From the properties of the Gauss-Bonnet invariant it is enough to consider only one curvature invariant between the Ricci tensor and the Riemann tensor. Then, we analyze the dynamics of a photon embedded in a gravitational field of a genericf!R;R !" R !" "gravity. The metric is time independent and spherically symmetric. The metric potentials are Schwarzschild-like, but there are two additional Yukawa terms linked to derivatives offwith respect to two curvature invariants. Considering first the case of a pointlike lens, and after the one of a generic matter distribution of the lens, we study the deflection angle and the angular position of images. Though the additional Yukawa terms in the gravitational potential modifies dynamics with respect to general relativity, the geodesic trajectory of the photon is unaffected by the modification if we consider onlyf!R"gravity. We find different results (deflection angle smaller than the angle of general relativity) only due to the introduction of a generic function of the Ricci tensor square. Finally, we can affirm that the lensing phenomena for all f!R"gravities are equal to the ones known for general relativity. We conclude the paper by showing and comparing the deflection angle and position of images forf!R;R !" R !" "gravity with respect to the gravitational lensing of general relativity.

25 citations


Journal ArticleDOI
TL;DR: In this paper, a model of the gravitational field based on two symmetric tensors is presented, and the equations of motion of test particles are derived: massive particles do not follow a geodesic but massless particles trajectories are null geodesics of an effective metric.

20 citations


Journal ArticleDOI
TL;DR: In this article, the authors recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime, and showed that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state.
Abstract: We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress?energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ?Ai. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.

17 citations


Journal ArticleDOI
TL;DR: In this article, the authors discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment.
Abstract: We briefly discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein's theory of gravitation can be locally embedded in a Ricci-flat 5D manifold such that the 4D energy-momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein's general relativity is presented.

17 citations


Posted Content
TL;DR: In this article, a local version of Shape Dynamics that is equivalent to General Relativity is constructed, in the sense that the algebras of Dirac observables weakly coincide.
Abstract: In this conceptual paper we construct a local version of Shape Dynamics that is equivalent to General Relativity in the sense that the algebras of Dirac observables weakly coincide. This allows us to identify Shape Dynamics observables with General Relativity observables, whose observables can now be interpreted as particular representative functions of observables of a conformal theory at fixed York time. An application of the observable equivalence of General Relativity and Shape Dynamics is to define the quantization of General Relativity through quantizing Shape Dynamics and using observable equivalence. We investigate this proposal explicitly for gravity in 2+1 dimensions.

Journal ArticleDOI
TL;DR: In this article, an approximate coordinate transformation to an accelerated frame was proposed, which turns out to be closely related to Rindler coordinates, widely used in modern general relativity, leading him directly to interpret gravitation in terms of spacetime curvature.
Abstract: On his way to general relativity, Einstein used the equivalence principle to formulate a theory of the static gravitational field. In this context he introduced an approximate coordinate transformation to an accelerated frame which turns out to be closely related to Rindler coordinates, widely used in modern general relativity. This work, published in the Annalen, led him directly to interpret gravitation in terms of spacetime curvature.

Posted Content
TL;DR: In this article, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these curvature singularities are studied in arbitrary dimension.
Abstract: In the first part of this thesis, Kerr-Schild metrics and extended Kerr-Schild metrics are analyzed in the context of higher dimensional general relativity Employing the higher dimensional generalizations of the Newman-Penrose formalism and the algebraic classification of spacetimes based on the existence and multiplicity of Weyl aligned null directions, we establish various geometrical properties of the Kerr-Schild congruences, determine compatible Weyl types and in the expanding case discuss the presence of curvature singularities We also present known exact solutions admitting these Kerr-Schild forms and construct some new ones using the Brinkmann warp product In the second part, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these quadratic gravities are studied in arbitrary dimension We investigate classes of Einstein spacetimes and spacetimes with a null radiation term in the Ricci tensor satisfying the vacuum field equations of quadratic gravity and provide examples of these metrics

Journal ArticleDOI
TL;DR: A review of the treatment of boundaries in general relativity is presented in this article with the emphasis on application to the formulations of Einstein's equations used in numerical relativity, and the underlying problems that make the initial-boundary-value problem much more complicated than the Cauchy problem.
Abstract: A review of the treatment of boundaries in general relativity is presented with the emphasis on application to the formulations of Einstein's equations used in numerical relativity. At present, it is known how to treat boundaries in the harmonic formulation of Einstein's equations and a tetrad formulation of the Einstein–Bianchi system. However, a universal approach valid for other formulations is not in hand. In particular, there is no satisfactory boundary theory for the 3+1 formulations which have been highly successful in binary black hole simulation. I discuss the underlying problems that make the initial-boundary-value problem much more complicated than the Cauchy problem. I review the progress that has been made and the important open questions that remain.Science is a differential equation. Religion is a boundary condition. (Alan Turing, quoted in J D Barrow, ‘Theories of Everything’)

Book
28 Jan 2012
TL;DR: In this article, the authors presented the hypothesis that there is no speed barrier in the universe -thus refuting the speed of light postulate, and they obtained a Parameterized Special Theory of Relativity (PSTR) for the same thought experiment, i.e. considering non-constant accelerations and arbitrary 3D-curves.
Abstract: In this book we present the hypothesis that there is no speed barrier in the universe - thus refuting the speed of light postulate. While Einstein considered a relative space and relative time but the ultimate speed of light, we do the opposite: we consider an absolute time and absolute space but no ultimate speed, and we call it the Absolute Theory of Relativity (ATR). We then parameterize Einstein’s thought experiment with atomic clocks, supposing that we know neither if the space and time are relative or absolute, nor if the speed of light is ultimate speed or not. We obtain a Parameterized Special Theory of Relativity (PSTR). Our PSTR generalizes not only Einstein’s Special Theory of Relativity, but also our ATR, and introduces three more possible Relativities to be studied in the future. Afterwards, we extend our research considering not only constant velocities but constant accelerations too. Eventually we propose a Noninertial Multirelativity for the same thought experiment, i.e. considering non-constant accelerations and arbitrary 3D-curves.

Book
02 Mar 2012
TL;DR: In this article, the Atom and Brownian Motion were used to set the stage for the development of quantum mechanics, and the evolution of Quantum Mechanics was discussed. But the main focus of the paper was on the evolution and history of quantum physics.
Abstract: 1. Setting the Stage for 1905 2. Radiation and the Quanta 3. The Atom and Brownian Motion 4. The Special Theory of Relativity 5. The General Theory of Relativity 6. Einstein and the Evolution of Quantum Mechanics 7. Epilogue

Journal ArticleDOI
TL;DR: In this paper, the authors delineate the roles played by non-dynamical background structures and Killing symmetries in the construction of stress-energy-momentum tensors generated from a diffeomorphism invariant action density.
Abstract: This article attempts to delineate the roles played by non-dynamical background structures and Killing symmetries in the construction of stress-energy-momentum tensors generated from a diffeomorphism invariant action density. An intrinsic coordinate independent approach puts into perspective a number of spurious arguments that have historically lead to the main contenders, viz the Belinfante-Rosenfeld stress-energy-momentum tensor derived from a Noether current and the Einstein-Hilbert stress-energy-momentum tensor derived in the context of Einstein's theory of general relativity. Emphasis is placed on the role played by non-dynamical background (phenomenological) structures that discriminate between properties of these tensors particularly in the context of electrodynamics in media. These tensors are used to construct conservation laws in the presence of Killing Lie-symmetric background fields.

Journal ArticleDOI
09 Feb 2012
TL;DR: In this article, a return to the mothers theorem of the Schwarzschild metric of general relativity is proposed to better understand the gothic-R theorem of GRS, which is already implicit in Einstein's equivalence principle of 1907 and hence in special relativity (with acceleration included).
Abstract: General relativity is notoriously difficult to interpret. A "return to the mothers" is proposed to better understand the gothic-R theorem of the Schwarzschild metric of general relativity. It is shown that the new finding is already implicit in Einstein's equivalence principle of 1907 and hence in special relativity (with acceleration included). The TeLeMaCh theorem, named onomatopoetically after Telemachus, is bound to transform metrology if correct. Key words: Equivalence principle, Telemach theorem, Schwarzschild metric, metrology, Large Hadron Collider (LHC).

Proceedings ArticleDOI
TL;DR: In this article, the relativistic field equations for a gas in special and general relativity are determined from the Boltzmann equation, and constitutive equations are obtained from the Chapman-Enskog methodology applied to a relativism model equation proposed by Anderson and Witting.
Abstract: Relativistic field equations for a gas in special and general relativity are determined from the Boltzmann equation The constitutive equations are obtained from the Chapman-Enskog methodology applied to a relativistic model equation proposed by Anderson and Witting Two applications in general relativity are considered: one refers to a gas in a homogeneous and isotropic Universe where irreversible processes are present during its evolution; in the other it is analyzed a gas under the influence of a spherically symmetrical non-rotating and uncharged source of the gravitational field

Proceedings ArticleDOI
05 Oct 2012
TL;DR: In this article, it is argued that by slight modifications of standard gravitation theory, our notion of the sources of gravity-the right hand side of Einstein's equations-, could be something radically different from what is usually expected.
Abstract: Some of the assumptions of cosmology, as based on the simplest version of General Relativity, are discussed. It is argued that by slight modifications of standard gravitation theory, our notion of the sources of gravity-the right hand side of Einstein's equations-, could be something radically different from what is usually expected. One example is exhibited to prove the point, and some consequences are discussed.

Posted Content
TL;DR: In this article, the authors prove existence of solutions of the vacuum Einstein equations with initial data induced by a smooth metric on a light-cone, based on which they construct a probabilistic model.
Abstract: We prove existence of solutions of the vacuum Einstein equations with initial data induced by a smooth metric on a light-cone.

Posted Content
TL;DR: In this article, the authors define Newtonian cosmology and general relativity, both in its standard and covariant formulations, and show how the two theories deal with inhomogeneous cosmological models and introduce the backreaction conjecture.
Abstract: Numerical N-body simulations of large scale structure formation in the universe are based on Newtonian gravity. However, according to our current understanding, the most correct theory of gravity is general relativity. It is therefore important to understand which degrees of freedom and which features are lost when the relativistic universe is approximated, or rather replaced, by a Newtonian one. This is the main purpose of our investigation. We first define Newtonian cosmology and we give an overview on general relativity, both in its standard and covariant formulations. We show how the two theories deal with inhomogeneous cosmological models and we introduce the backreaction conjecture. Then we review on how Newtonian gravity and general relativity relate to each other in the fully non-linear regime. For this purpose we discuss frame theory. We carry out the same investigation also in the weak-field, small-velocity limit of general relativity, and we derive the Newtonian limit resorting to the framework of post-Newtonian cosmology. Finally we remark that there are solutions of Newtonian gravity which do not have any relativistic counterpart.

Dissertation
01 Jan 2012
TL;DR: A study of shear-free perfect fluids in general relativity (rotating and/or expanding) and a characterisation of the orthogonal Bianchi class A perfect fluids as geodesic, non-rotating fluids with vanishing divergence of the electric and magnetic part of the Weyl tensor are presented in this paper.
Abstract: A study of shear-free perfect fluids in general relativity (rotating and/or expanding) and a characterisation of the orthogonal Bianchi class A perfect fluids as geodesic, non-rotating fluids with vanishing divergence of the electric and magnetic part of the Weyl tensor.

Journal ArticleDOI
TL;DR: In this article, the Einstein tensor is shown to appear naturally from the Bianchi identities, thus emphasizing the pure geometrical nature of the left hand side of the equations of general relativity.
Abstract: In the context of the theory of fiber bundles and connections, mainly restricted to the frame and associated bundles of a Riemannian or pseudo-Riemannian differentiable manifold, we present the global and local versions of the concepts of covariant derivative, parallel transport, geodesics, metric compatible connections with (Riemann-Cartan) or without (Levi-Civita) torsion, and curvature and torsion with their geometric interpretations. The Einstein tensor is shown to appear naturally from the Bianchi identities, thus emphasizing the pure geometrical nature of “the left hand side” of the equations of general relativity.

Dissertation
01 Feb 2012
TL;DR: Conboye et al. as mentioned in this paper proposed Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity, a transverse trace-free tensors in numerical relativity.
Abstract: Axial symmetry and transverse trace-free tensors in numerical relativity Author(s) Conboye, Rory P. A. Publication date 2012-02 Original citation Conboye, Rory P.A. 2012. Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity. PhD Thesis, University College Cork. Type of publication Doctoral thesis Rights © 2012, Rory Patrick Albert Conboye http://creativecommons.org/licenses/by-nc-nd/3.0/

Posted Content
TL;DR: Einstein's 1912 theory of static fields finally led him to reject the generally covariant field equations and to develop limited general covariant fields as mentioned in this paper, which was based on his 1911 June conclusion about a relationship between the velocity of light and the gravitational potential.
Abstract: In December 1911, Max Abraham published a paper on gravitation at the basis of which was Albert Einstein's 1911 June conclusion about a relationship between the velocity of light and the gravitational potential In February 1912, Einstein published his work on static gravitational fields, which was based on his 1911 June theory In March 1912, Einstein corrected his paper, but Abraham claimed that Einstein borrowed his equations; however, it was actually Abraham who needed Einstein's ideas and not the other way round Einstein thought that Abraham converted to his theory of static fields while Abraham presumed exactly the opposite Einstein then moved to Zurich and switched to new mathematical tools He examined various candidates for generally covariant field equations, and already considered the field equations of his general theory of relativity about three years before he published them in November 1915 However, he discarded these equations only to return to them more than three years later Einstein's 1912 theory of static fields finally led him to reject the generally covariant field equations and to develop limited generally covariant field equations

Book ChapterDOI
01 Jan 2012
TL;DR: In this paper, the curving of spacetime in the general theory of gravity is discussed and a new aspect of physics brought about by Einstein's Relativity is discussed. But not a word was mentioned about gravity and it is with the inclusion of gravity that the theory takes on a new and exciting complexion.
Abstract: In the preceding chapters, we witnessed the new aspects of physics brought about by Einstein’s Relativity. All of this arose because: a) there is a speed limit in nature, \(c\), the maximum speed at which influences can be propagated and b) because all inertial observers are physically equivalent, they must all agree on this speed limit. However, not a word was mentioned about gravity and it is with the inclusion of gravity that Einstein’s Relativity takes on a whole new and exciting complexion. Einstein’s Relativity without gravity is called “Special Relativity” to distinguish it from Relativity with gravity which is called “General Relativity”. In brief, General Relativity is Einstein’s theory of gravity. In what follows, we will delve into General Relativity, showing how it is the curving of spacetime in the general theory that replaces the old Newtonian idea of gravity being just another force.

Journal ArticleDOI
TL;DR: In this article, the system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered and the general solution to a problem representing two classes of line elements with arbitrary functions g00 and g22 is obtained.
Abstract: The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g00 and g22 is obtained. The properties of the found solutions are analyzed.

Journal ArticleDOI
TL;DR: This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functions 𝑔(𝑟) and 𝓂(𝓚) and also discussing their physical and geometric properties.
Abstract: This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functions 𝑔(𝑟) and 𝑤(𝑟) and also discussing their physical and geometric properties.

Journal Article
TL;DR: In this paper, the authors classified space-time curvature tensors and energy and momentum tensors into various groups based on which objects they are attributed to and obtained the values of the Einstein tensor and momentum energy tensor.
Abstract: The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity, which describe the fundamental interaction (e&eb) of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with (=) the energy and momentum tensor within that spacetime (expressed by the stress–energy tensor).Both space time curvature tensor and energy and momentum tensor is classified in to various groups based on which objects they are attributed to. It is to be noted that the total amount of energy and mass in the Universe is zero. But as is said in different context, it is like the Bank Credits and Debits, with the individual debits and Credits being conserved, holistically, the conservation and preservation of Debits and Credits occur, and manifest in the form of General Ledger. Transformations of energy also take place individually in the same form and if all such transformations are classified and written as a Transfer Scroll, it should tally with the total, universalistic transformation. This is a very important factor to be borne in mind. Like accounts are classifiable based on rate of interest, balance standing or the age, we can classify the factors and parameters in the Universe, be it age, interaction ability, mass, energy content. Even virtual particles could be classified based on the effects it produces. These aspects are of paramount importance in the study. When we write A+b+5, it means that we are adding A to B or B to A until we reach 5. Similarly, if we write A-B=0, it means we are taking away B from A and there may be time lag until we reach zero. There may also be cases in which instantaneous results are reached, which however do not affect the classification. By means of such a classification we obtain the values of Einstein Tensor and Momentum Energy Tensor, which are in fact the solutions to the Einstein’s Field Equation. Terms “e” and “eb” are used for better comprehension of the lay reader. It has no other attribution or ascription whatsoever in the context of the paper. For the sake of simplicity, we shall take the equality case of Heisenberg’s Principle Of Uncertainty for easy consolidation and consubstantiation process. The “greater than” case can be attended to in a similar manner, with the symbolof”greater than” incorporated in the paper series .