Showing papers on "Introduction to the mathematics of general relativity published in 2009"

General Relativity and the Einstein Equations

Abstract: FOREWORD ACKNOWLEDGEMENTS 1. Lorentzian Geometry 2. Special Relativity 3. General Relativity and the Einstein Equations 4. Schwarzschild Space-time and Black Holes 5. Cosmology 6. Local Cauchy Problem 7. Constraints 8. Other Hyperbolic-Elliptic systems 9. Relativistic Fluids 10. Kinetic Theory 11. Progressive Waves 12. Global Hyperbolicity and Causality 13. Singularities 14. Stationary Space-times and Black Holes 15. Global Existence Theorems, Asymptotically Euclidean Data 16. Global existence theorems, cosmological case APPENDICES I. Sobolev Spaces II. Elliptic Systems III. Second Order Quasidiagonal Systems IV. General Hyperbolic Systems V. Cauchy Kovalevski and Fuchs theorems VI. Conformal Methods VII. Kaluza Klein Formulas

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity.

Abstract: The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

Structure and evolution of self-gravitating objects and the orthogonal splitting of the Riemann tensor

Abstract: The full set of equations governing the structure and the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, and the active gravitational mass. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through these scalars. Some solutions are exhibited to illustrate this point.

General very special relativity in Finsler cosmology

Abstract: General very special relativity (GVSR) is the curved space-time of very special relativity (VSR) proposed by Cohen and Glashow. The geometry of general very special relativity possesses a line element of Finsler geometry introduced by Bogoslovsky. We calculate the Einstein field equations and derive a modified Friedmann-Robertson-Walker cosmology for an osculating Riemannian space. The Friedmann equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial-spurionic vector field introduced into the metric gives back an estimation of the energy evolution and inflation.

ABC of Relativity

Abstract: Preface Introduction 1. Touch and Sight: The Earth and the Heavens 2. What Happens and What is Observed 3. The Velocity of Light 4. Clocks and Foot-rules 5. Space-Time 6. The Special Theory of Relativity 7. Intervals in Space-Time 8. Einstein's Law of Gravitation 9. Proofs of Einstein's Law of Graviation 10. Mass, Momentum, Energy, and Action 11. The Expanding Universe 12. Conventions and Natural Laws 13. The Abolution of 'Force' 14. What is Matter? 15. Physical Consequences Index

Thermodynamic structure of Lanczos-Lovelock field equations from near-horizon symmetries

Abstract: It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near-horizon field equations again represent a thermodynamic identity in all these models. These results confirm the conjecture that this thermodynamic perspective of gravity extends far beyond Einstein's theory.

Spatially averaged cosmology in an arbitrary coordinate system

Abstract: This paper presents a general averaging procedure for a set of observers which are tilted with respect to the cosmological matter fluid. After giving the full set of equations describing the local dynamics, we define the averaging procedure and apply it to the scalar parts of Einstein's field equations. In addition to the standard backreaction, new terms appear that account for the effect of the peculiar velocity of the matter fluid as well as the possible effect of a shift in the coordinate system.

On quasi-Einstein spacetimes

Abstract: The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study quasi-Einstein spacetimes. Some basic geometric properties of such a spacetime are obtained. The applications of quasi-Einstein spacetimes in general relativity and cosmology are investigated. Finally, the existence of such spacetimes are ensured by several interesting examples.

Bulk viscous string cosmological models with electromagnetic field

Abstract: LRS Bianchi type-I string cosmological models are studied in the frame work of general relativity when the source for the energy momentum tensor is a bulk viscous stiff fluid containing one dimensional strings embedded in electromagnetic field. The bulk viscosity is assumed to be inversely proportional to the scalar expansion. The physical and kinematical properties of the models are discussed. The effects of Viscosity and electromagnetic field on the physical and kinematical properties are also investigated.

The extended relativity theory in clifford spaces

Abstract: An introduction to some of the most important features of the Extended Relativity theory in Clifford-spaces (C-spaces) is presented whose "point" coordinates are non-commuting Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in target Ddimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, non-commuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable dimensions/signatures. It permits to study the dynamics of all (closed) p-branes, for all values of p, on a unified footing. It resolves the ordering ambiguities in QFT, the problem of time in Cosmology and admits superluminal propagation ( tachyons ) without violations of causality. A discussion of the maximalacceleration Relativity principle in phase-spaces follows and the study of the invariance group of symmetry transformations in phase-space allows to show why Planck areas are invariant under acceleration-boosts transformations . This invariance feature suggests that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking different n-dimensional slices through C-space. The conformal group in spacetime emerges as a natural subgroup of the Clifford group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge field of dilations. We finalize by constructing the generalization of Maxwell theory of Electrodynamics of point charges to a theory in C-spaces that involves extended charges coupled to antisymmetric tensor fields of arbitrary rank. In the concluding remarks we outline briefly the current promising research programs and their plausible connections with C-space Relativity.

The Confrontation Between General Relativity and Experiment

Abstract: We review the experimental evidence for Einstein’s general relativity. A variety of high precision null experiments confirm the Einstein Equivalence Principle, which underlies the concept that gravitation is synonymous with spacetime geometry, and must be described by a metric theory. Solar system experiments that test the weak-field, post-Newtonian limit of metric theories strongly favor general relativity. Binary pulsars test gravitational-wave damping and aspects of strong-field general relativity. During the coming decades, tests of general relativity in new regimes may be possible. Laser interferometric gravitational-wave observatories on Earth and in space may provide new tests via precise measurements of the properties of gravitational waves. Future efforts using X-ray, infrared, gamma-ray and gravitational-wave astronomy may one day test general relativity in the strong-field regime near black holes and neutron stars.

Rotation in relativity and the propagation of light

Abstract: We compare and contrast the different points of view of rotation in general relativity, put forward by Mach, Thirring and Lense, and Goedel. Our analysis relies on two tools: (i) the Sagnac effect which allows us to measure rotations of a coordinate system or induced by the curvature of spacetime, and (ii) computer visualizations which bring out the alien features of the Goedel Universe. In order to keep the paper self-contained, we summarize in several appendices crucial ingredients of the mathematical tools used in general relativity. In this way, our lecture notes should be accessible to researchers familiar with the basic elements of tensor calculus and general relativity.

Post-Newtonian Methods: Analytic Results on the Binary Problem

Abstract: A detailed account is given on approximation schemes to the Einstein theory of general relativity where the iteration starts from the Newton theory of gravity. Two different coordinate conditions are used to represent the Einstein field equations, the generalized isotropic ones of the canonical formalism of Arnowitt, Deser, and Misner and the harmonic ones of the Lorentz-covariant Fock-de Donder approach. Conserved quantities of isolated systems are identified and the Poincare algebra is introduced. Post-Newtonian expansions are performed in the near and far (radiation) zones. The natural fitting of multipole expansions to post-Newtonian schemes is emphasized. The treated matter models are ideal fluids, pure point masses, and point masses with spin and mass-quadrupole moments modelling rotating black holes. Various Hamiltonians of spinning binaries are presented in explicit forms to higher post-Newtonian orders. The delicate use of black holes in post-Newtonian expansion calculations and of the Dirac delta function in general relativity find discussions.

The Energy of Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics

Abstract: According to the Einstein, Weinberg, and Moller energy-momentum complexes, we evaluate the energy distribution of the singularity-free solution of the Einstein field equations coupled to a suitable nonlinear electrodynamics suggested by Ayon-Beato and Garcia. The results show that the energy associated with the definitions of Einstein and Weinberg are the same, but Moller not. Using the power series expansion, we find out that the first two terms in the expression are the same as the energy distributions of the Reissner-Nordstrom solution, and the third term could be used to survey the factualness between numerous solutions of the Einstein field equations coupled to a nonlinear electrodynamics.

Spin-spin interaction in general relativity and induced geometries with nontrivial topology

Abstract: We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. It is shown that both these matter distributions can induce a vortex field described by the curl 4-vector of a tetrad: ω i = 1 � iklm e(a)ke (a) l;m ,w heree (a) k are components of the tetrad. The energy-momentum tensor Tik(ω) of this field has been found and shown to violate the strong and weak energy conditions which leads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exact solutions to the equations of general relativity have been found. It is also shown that other vortex fields, e.g., the magnetic field, can also possess such properties.

The static extension problem in General Relativity

Abstract: We prove the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere, provided that the mean curvature is positive and has no critical points on regions of nonpositive Gaussian curvature. This gives a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.

Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint

Abstract: The analytic hyperbolic geometric viewpoint of Einstein's special theory of relativity is presented. Owing to the introductio n of vectors into hyperbolic geometry, where they are called gyrovectors, the use of analytic hyperbolic geometry extends Einstein's unfinished symphony significantly, elev ating it to the status of a mathematical theory that could be emulated to the benefit of t he entire mathematical and physical community. The resulting theory involves a gyrovector space approach to hyperbolic geometry and relativistic mechanics, and could be studied with profit by anyone with a sufficient background in the common vector spac e approach to Euclidean geometry and classical mechanics. Einstein noted in his 1905 paper that founded the special theory of relativity that his velocity addition law satisfies the law of velocity parallelogram only to a first approximation. Within our hype rbolic geometric viewpoint of special relativity it becomes clear that Einstein's velo city addition law leads to a hyperbolic parallelogram addition law of Einsteinian velocities, which is supported experimentally by the cosmological effect known as stellar aberration and its relativistic interpretation. The latter, in turn, is supported experime ntally by the "GP-B" gyroscope experiment developed by NASA and Stanford University. Furthermore, the hyperbolic viewpoint of special relativity meshes extraordinarily well with the Minkowskian four- vector formalism of special relativity, revealing that the seemingly notorious relativistic mass meshes up with the four-vector formalism as well, owing to the natural emergence of dark matter. It is therefore hoped that both special relativity and its u nderlying analytic hyperbolic geometry will become part of the lore learned by all undergraduate and graduate mathematics and physics students.

Dynamics and Relativity

Abstract: Preface. I. INTRODUCTORY DYNAMICS. 1. SPACE, TIME AND MOTION. 1.1 Defining Space and Time. 1.2 Vectors and Co-ordinate Systems. 1.3 Velocity and Acceleration. 1.4 Standards and Units. 2. FORCE, MOMENTUM AND NEWTON'S LAWS. 2.1 Force and Static Equilibrium. 2.2 Force and Motion. 2.3 Applications of Newton's Laws. 3. ENERGY. 3.1 Work, Power and Kinetic Energy. 3.2 Potential Energy. 3.3 Collisions. 3.4 Energy Conservation in Complex Systems. 4. ANGULAR MOMENTUM. 4.1 Angular Momentum of a Particle. 4.2 Conservation of Angular Momentum in Systems of Particles. 4.3 Angular Momentum and Rotation About a Fixed Axis. 4.4 Sliding and Rolling. 4.5 Angular Impulse and the Centre of Percussion. 4.6 Kinetic Energy of Rotation. II. INTRODUCTORY SPECIAL RELATIVITY. 5. THE NEED FOR A NEW THEORY OF SPACE AND TIME. 5.1 Space and Time Revisited. 5.2 Experimental Evidence. 5.3 Einstein's Postulates. 6. RELATIVISTIC KINEMATICS. 6.1 Time Dilation, Length Contraction and Simultaneity. 6.2 Lorentz Transformations. 6.3 Velocity Transformations. 7. RELATIVISTIC ENERGY AND MOMENTUM. 7.1 Momentum and Energy. 7.2 Applications in Particle Physics. III ADVANCED DYNAMICS. 8. NON-INERTIAL FRAMES. 8.1 Linearly Accelerating Frames. 8.2 Rotating Frames. 9. GRAVITATION. 9.1 Newton's Law of Gravity. 9.2 The Gravitational Potential. 9.3 Reduced Mass. 9.4 Motion in a central force. 9.5 Orbits. 10. RIGID BODY MOTION. 10.1 The angular momentum of a rigid body. 10.2 The moment of inertia tensor. 10.3 Principal axes. 10.4 Fixed-axis rotation in the lab frame. 10.5 Euler's equations. 10.6 The free rotation of a symmetric top. 10.7 The stability of free rotation. 10.8 Gyroscopes. IV. ADVANCED SPECIAL RELATIVITY. 11. THE SYMMETRIES OF SPACE AND TIME. 11.1 Symmetry in Physics. 11.2 Lorentz Symmetry. 12. FOUR-VECTORS AND LORENTZ INVARIANTS. 12.1 The Velocity Four-vector. 12.2 The Wave Four-vector. 12.3 The Energy-momentum Four-vector. 12.4 Electric and Magnetic Fields. 13. SPACE-TIME DIAGRAMS AND CAUSALITY. 13.1 Relativity Preserves Causality. 13.2 An Alternative Approach. 14. ACCELERATION AND GENERAL RELATIVITY. 14.1 Acceleration in Special Relativity. 14.2 A glimpse of General Relativity. A DERIVING THE GEODESIC EQUATION. B SOLUTIONS TO PROBLEMS.

Gauge fixing in the tensor model and emergence of local gauge symmetries

Abstract: The tensor model can be regarded as theory of dynamical fuzzy spaces, and gives a way to formulate gravity on fuzzy spaces. It has recently been shown that the low-lying fluctuations around the Gaussian background solutions in the tensor model agree correctly with the metric fluctuations on the flat spaces with general dimensions in the general relativity. This suggests that the local gauge symmetry (the symmetry of local translations) is also emergent around these solutions. To systematically study this possibility, I apply the BRS gauge fixing procedure to the tensor model. The ghost kinetic term is numerically analyzed, and it has been found that there exist some massless trajectories of ghost modes, which are clearly separated from the other higher ghost modes. Comparing with the corresponding BRS gauge fixing in the general relativity, these ghost modes forming the massless trajectories in the tensor model are shown to be identical to the reparametrization ghosts in the general relativity.

On Tensorial Concomitants and the Non-Existence of a Gravitational Stress-Energy Tensor

Abstract: Based on an analysis of what it may mean for one tensor to depend in the proper way on another, 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.

(n + 1)-Dimensional Lorentzian Wormholes in an Expanding Cosmological Background

Abstract: We discuss (n+1)-dimensional dynamical wormholes in an evolving cosmological background with a throat expanding with time. These solutions are examined in the general relativity framework. A linear relation between diagonal elements of an anisotropic energy-momentum tensor is used to obtain the solutions. The energy-momentum tensor elements approach the vacuum case when we are far from the central object for one class of solutions. Finally, we discuss the energy-momentum tensor which supports this geometry, taking into account the energy conditions.

Matter from Space

Abstract: General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its Riemannian geometry. I review this picture of `Geometrodynamics' and comment on various developments after Einstein.

Tube dislocations in gravity

Abstract: We consider static massive thin cylindrical shells (tubes) as the sources in Einstein’s equations. They correspond to δ- and δ′-function-type energy-momentum tensors. The corresponding metric components are found explicitly. They are not continuous functions, in general, and lead to ambiguous curvature tensor components. Nevertheless all ambiguous terms in Einstein’s equations safely cancel. The interplay between elasticity theory, geometric theory of defects, and general relativity is analyzed. The elasticity theory provides a simple picture for defect creation and a new look on general relativity.

The gödel engine - an interactive approach to visualization in general relativity

Abstract: We present a methodical new approach to visualize the aspects of general relativity from a self-centered perspective. We focus on the visualization of the Godel universe, which is an exact solution to Einstein's field equations of general relativity. This model provides astounding features such as the existence of an optical horizon and the possibility of time travel. Although we know that our universe is not of Godel type, we can -- using this solution to Einstein's equations -- visualize and understand the effects resulting from the theory of relativity, which itself has been verified on the large scale in numerous experiments over the last century. We derive the analytical solution to the geodesic equations of Godel's universe for special initial conditions. Along with programmable graphics hardware we achieve a tremendous speedup for the visualization of general relativity. This enables us to interactively explore the physical aspects and optical effects of Godel's universe. We also demonstrate how the analytical solution enables dynamic lighting with local illumination models. Our implementation is tailored for Godel's universe and five orders of magnitude faster than previous approaches. It can be adapted to manifolds for which an analytical expression of the propagation of light is available.

Outer Boundary Conditions in Numerical Relativity

Abstract: This thesis is concerned with outer boundary conditions in numerical relativity. In numerical simulations, the spatially infinite universe is typically modelled using a finite spatial domain, on the edge of which boundary conditions are imposed. These boundary conditions should mirror the unbounded physical domain as closely as possible. They should be transparent to outgoing gravitational radiation and should not introduce spurious incoming radiation via reflections of outgoing radiation off the boundary. The concepts of incoming and outgoing gravitational radiation are only well understood in certain specific charts and tetrads. The first half of this thesis investigates the relationship between these charts and tetrads and those used in numerical relativity. We begin by studying a previous calculation [134], in which quantities such as the Bondi mass and the news function were expressed in terms of the Newman-Penrose scalars in an axisymmetric spacetime. The calculation is generalized to spacetimes with no symmetries. The results above still require a specific choice of tetrad. By supposing that the region of spacetime far from an isolated gravitating source is in some sense Minkowskian, we demonstrate how to transform between the charts and tetrads used in theoretical studies of gravitational radiation and the charts and tetrads used in numerical relativity. This enables us to provide “numerical relativity recipes” in which the Weyl scalars, the Bondi mass and news function are expressed in terms of the metric variables in a numerical chart. The second half of this thesis addresses the problem of absorbing boundary conditions in numerical relativity. Using Hertz potentials, the far-field region of a spacetime can be expressed as a linear perturbation about Minkowski, Schwarzschild or Kerr backgrounds. The resulting field equations enable us to investigate the propagation of linearized gravitational radiation. On a Minkowski background, incoming and outgoing waves propagate independently. The presence of a curved background creates a “gravitational tail” whose behaviour near future null infinity we are able to estimate. This enables us to formulate absorbing boundary conditions for numerical relativity. Finally, we link the two threads mentioned above. The boundary conditions are expressed in terms of the metric variables in a numerical relativity chart.

New analytical stellar models in general relativity.

Abstract: We present new exact solutions to the Einstein and Einstein-Maxwell field equations that model the interior of neutral, charged and radiating stars. Several new classes of solutions in static spherically symmetric interior spacetimes are found in the presence of charge. These correspond to isotropic matter with a specified electric field intensity. Our solutions are found by choosing different rational forms for one of the gravitational potentials and a particular form for the electric field. The models generated contain results found previously including Finch and Skea (1989) neutron stars, Durgapal and Bannerji (1983) dense stars, Tikekar (1990) superdense stars in the limit of vanishing charge. Then we study the general situation of a compact relativistic object with anisotropic pressures in the presence of the electromagnetic field. We assume the equation of state is linear so that the model may be applied to strange stars with quark matter and dark energy stars. Several new classes of exact solutions are found, and we show that the densities and masses are consistent with real stars. We regain as special cases the Lobo (2006) dark energy stars, the Sharma and Maharaj (2007) strange stars and the realistic isothermal universes of Saslaw et al (1996). In addition, we consider relativistic radiating stars undergoing gravitational collapse when the fluid particles are in geodesic motion. We transform the governing equation into Bernoulli, Riccati and confluent hypergeometric equations. These admit an infinite family of solutions in terms of simple elementary functions and special functions. Particular models contain the Minkowski spacetime and the Friedmann dust spacetime as limiting cases. Finally, we model the radiating star with shear, acceleration and expansion in the presence of anisotropic pressures. We obtain several classes of new solutions in terms of arbitrary functions in temporal and radial coordinates by rewriting the junction condition in the form of a Riccati equation. A brief physical analysis indicates that these models are physically reasonable.

The Fully Covariant Energy Momentum Stress Tensor For Gravity and the Einstein Equation in General Relativity

Abstract: We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter and fields must be divergence free, when spacetime is 4 dimensional. Moreoverr, if the total source matter fields are assumed to be divergence free, then either spacetime is dimension 4 or the spacetime has constant scalar curvature.

General relativity extended to non-Riemannian space-time geometry

Abstract: The gravitation equations of the general relativity, written for Riemannian space-time geometry, are extended to the case of arbitrary (non-Riemannian) space-time geometry. The obtained equations are written in terms of the world function in the coordinateless form. These equations determine directly the world function, (but not only the metric tensor). As a result the space-time geometry appears to be non-Rieamannian. Invariant form of the obtained equations admits one to exclude influence of the coordinate system on solutions of dynamic equations. Anybody, who trusts in the general relativity, is to accept the extended general relativity, because the extended theory does not use any new hypotheses. It corrects only inconsequences and restrictions of the conventional conception of general relativity. The extended general relativity predicts an induced antigravitation, which eliminates existence of black holes.

Covariant energy-momentum and an uncertainty principle for general relativity

Abstract: We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy-momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers comoving with the matter is seen to compute the intrinsic global energy of a system. The new localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving. It is suggested that in the extreme of strong gravity, the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy-momentum.

Extended Palatini action for general relativity

Abstract: In the Palatini action of general relativity, the connection and the metric are treated as independent dynamical variables. Instead of assuming a relation between these quantities, the desired relation between them is derived through the Euler-Lagrange equations of the Palatini action. In this manuscript we construct an extended Palatini action, where we do not assume any a priori relationship between the connection, the covariant metric tensor, and the contravariant metric tensor. Instead we treat these three quantities as independent dynamical variables. We show that this action reproduces the standard Einstein field equations depending on a single metric tensor. We further show that in vacuum and in the absence of cosmological constant this theory has an enhanced symmetry.