scholarly journals An optimal transport formulation of the Einstein equations of general relativity

2022 ◽  
Author(s):  
Andrea Mondino ◽  
Stefan Suhr
Keyword(s):  
General Relativity ◽  
Optimal Transport ◽  
Einstein Equations

scholarly journals CAN f(R) GRAVITY MIMIC GENERAL RELATIVITY?

2012 ◽  
Vol 10 ◽  
pp. 63-68 ◽  
Author(s):  
JE-AN GU
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Early Time ◽  
Einstein Equations ◽  
Modified Theories Of Gravity ◽  
Cosmological Perturbations ◽  
Long Run ◽  
Theories Of Gravity ◽  
The Stability ◽  
Higher Order Differential Equations

We discuss the stability of the general-relativity (GR) limit in modified theories of gravity, particularly the f(R) theory. The problem of approximating the higher-order differential equations in modified gravity with the Einstein equations (2nd-order differential equations) in GR is elaborated. We demonstrate this problem with a heuristic example involving a simple ordinary differential equation. With this example we further present the iteration method that may serve as a better approximation for solving the equation, meanwhile providing a criterion for assessing the validity of the approximation. We then discuss our previous numerical analyses of the early-time evolution of the cosmological perturbations in f(R) gravity, following the similar ideas demonstrated by the heuristic example. The results of the analyses indicated the possible instability of the GR limit that might make the GR approximation inaccurate in describing the evolution of the cosmological perturbations in the long run.

scholarly journals Lagrangian formulation of Newtonian cosmology

2014 ◽  
Vol 36 (3) ◽  
Author(s):  
H.S. Vieira ◽  
V.B. Bezerra
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Classical Mechanics ◽  
Lagrangian Formulation ◽  
Einstein Equations ◽  
Lagrangian Formalism ◽  
Newtonian Cosmology

In this paper, we use the Lagrangian formalism of classical mechanics and some assumptions to obtain cosmological differential equations analogous to Friedmann and Einstein equations, obtained from the theory of general relativity. This method can be used to a universe constituted of incoherent matter, that is, the cosmologic substratum is comprised of dust.

scholarly journals Kodama Time, Entropy Bounds, the Raychaudhuri  Equation, and the Quantum Interest Conjecture

2021 ◽  
Author(s):  
◽  
Gabriel Abreu
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Negative Energy ◽  
Specific Form ◽  
Einstein Equations ◽  
Surface Gravity ◽  
Quantum Field ◽  
Raychaudhuri Equation ◽  
Dimensional Minkowski Space ◽  
Entropy Bounds

<p>General Relativity, while ultimately based on the Einstein equations, also allows one to quantitatively study some aspects of the theory without explicitly solving the Einstein equations. These geometrical notions of the theory provide an insight to the nature of more general spacetimes. In this thesis, the Raychaudhuri equation, the choice of the coordinate system, the notions of surface gravity and of entropy, and restrictions on negative energy densities on the form of the Quantum Interest Conjecture, will be discussed. First, using the Kodama vector, a geometrically preferred coordinate system is built. With this coordinate system the usual quantities, such as the Riemann and Einstein tensors, are calculated. Then, the notion of surface gravity is generalized in two different ways. The first generalization is developed considering radial ingoing and outgoing null geodesics, in situations of spherical symmetry. The other generalized surface gravity is a three-vector obtained from the spatial components of the redshifted four acceleration of a suitable set of fiducial observers. This vectorial surface gravity is then used to place a bound on the entropy of both static and rotating horizonless objects. This bound is obtain mostly by classical calculations, with a minimum use of quantum field theory in curved spacetime. Additionally, several improved versions of the Raychaudhuri equation are developed and used in different scenarios, including a two congruence generalization of the equation. Ultimately semiclassical quantum general relativity is studied in the specific form of the Quantum Inequalities, and the Quantum Interest Conjecture. A variational proof of a version of the Quantum Interest Conjecture in (3 + 1)–dimensional Minkowski space is provided.</p>

scholarly journals Kodama Time, Entropy Bounds, the Raychaudhuri  Equation, and the Quantum Interest Conjecture

2021 ◽  
Author(s):  
◽  
Gabriel Abreu
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Negative Energy ◽  
Specific Form ◽  
Einstein Equations ◽  
Surface Gravity ◽  
Quantum Field ◽  
Raychaudhuri Equation ◽  
Dimensional Minkowski Space ◽  
Entropy Bounds

<p>General Relativity, while ultimately based on the Einstein equations, also allows one to quantitatively study some aspects of the theory without explicitly solving the Einstein equations. These geometrical notions of the theory provide an insight to the nature of more general spacetimes. In this thesis, the Raychaudhuri equation, the choice of the coordinate system, the notions of surface gravity and of entropy, and restrictions on negative energy densities on the form of the Quantum Interest Conjecture, will be discussed. First, using the Kodama vector, a geometrically preferred coordinate system is built. With this coordinate system the usual quantities, such as the Riemann and Einstein tensors, are calculated. Then, the notion of surface gravity is generalized in two different ways. The first generalization is developed considering radial ingoing and outgoing null geodesics, in situations of spherical symmetry. The other generalized surface gravity is a three-vector obtained from the spatial components of the redshifted four acceleration of a suitable set of fiducial observers. This vectorial surface gravity is then used to place a bound on the entropy of both static and rotating horizonless objects. This bound is obtain mostly by classical calculations, with a minimum use of quantum field theory in curved spacetime. Additionally, several improved versions of the Raychaudhuri equation are developed and used in different scenarios, including a two congruence generalization of the equation. Ultimately semiclassical quantum general relativity is studied in the specific form of the Quantum Inequalities, and the Quantum Interest Conjecture. A variational proof of a version of the Quantum Interest Conjecture in (3 + 1)–dimensional Minkowski space is provided.</p>

scholarly journals Using Algebraic Computing To Teach General Relativity And Cosmology

2012 ◽  
Vol 56 (1) ◽  
pp. 139-144
Author(s):  
Dumitru N. Vulcanov ◽  
Remus-Ştefan Ş. Boată
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Undergraduate Students ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Einstein Equations ◽  
Algebraic Programming ◽  
Algebraic Computing

AbstractThe article presents some new aspects and experience on the use of computer in teaching general relativity and cosmology for undergraduate students (and not only) with some experience in computer manipulation. Some years ago certain results were reported [1] using old fashioned computer algebra platforms but the growing popularity of graphical platforms as Maple and Mathematica forced us to adapt and reconsider our methods and programs. We will describe some simple algebraic programming procedures (in Maple with GrTensorII package) for obtaining and the study of some exact solutions of the Einstein equations in order to convince a dedicated student in general relativity about the utility of a computer algebra system.

scholarly journals Deformed General Relativity and Quantum Black Holes Interior

Universe ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 39 ◽  
Author(s):  
Denis Arruga ◽  
Jibril Ben Achour ◽  
Karim Noui
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Einstein Equations ◽  
Gravity Models ◽  
Effective Theory ◽  
Hamiltonian Constraint ◽  
Spherically Symmetric ◽  
Full Generality ◽  
Full Theory ◽  
Effective Theories

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance cannot be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

scholarly journals Traversable Wormholes, Regular Black Holes, and Black-Bounces

2021 ◽  
Author(s):  
◽  
Alexander Simpson
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Black Hole ◽  
Einstein Equations ◽  
Time Dependent ◽  
Alternative Theories Of Gravity ◽  
Regular Black Hole ◽  
Traversable Wormhole ◽  
Regular Black Holes ◽  
Traversable Wormholes

<p>Various spacetime candidates for traversable wormholes, regular black holes, and ‘black-bounces’ are presented and thoroughly explored in the context of the gravitational theory of general relativity. All candidate spacetimes belong to the mathematically simple class of spherically symmetric geometries; the majority are static (time-independent as well as nonrotational), with a single dynamical (time-dependent) geometry explored. To the extent possible, the candidates are presented through the use of a global coordinate patch – some of the prior literature (especially concerning traversable wormholes) has often proposed coordinate systems for desirable solutions to the Einstein equations requiring a multi-patch atlas. The most interesting cases include the so-called ‘exponential metric’ – well-favoured by proponents of alternative theories of gravity but which actually has a standard classical interpretation, and the ‘black-bounce’ to traversable wormhole case – where a metric is explored which represents either a traversable wormhole or a regular black hole, depending on the value of the newly introduced scalar parameter a. This notion of ‘blackbounce’ is defined as the case where the spherical boundary of a regular black hole forces one to travel towards a one-way traversable ‘bounce’ into a future reincarnation of our own universe. The metric of interest is then explored further in the context of a time-dependent spacetime, where the line element is rephrased with a Vaidya-like time-dependence imposed on the mass of the object, and in terms of outgoing/ingoing EddingtonFinkelstein coordinates. Analysing these candidate spacetimes extends the pre-existing discussion concerning the viability of non-singular black hole solutions in the context of general relativity, as well as contributing to the dialogue on whether an arbitrarily advanced civilization would be able to construct a traversable wormhole.</p>

scholarly journals Inflation without a trace of lambda

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
John D. Barrow ◽  
Spiros Cotsakis
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Cosmological Constant ◽  
Conformal Transformation ◽  
Vacuum Energy ◽  
Einstein Equations ◽  
Scale Invariant ◽  
Field Source ◽  
Scale Theory ◽  
Wide Range

AbstractWe generalise Einstein’s formulation of the traceless Einstein equations to f(R) gravity theories. In the case of the vacuum traceless Einstein equations, we show that a non-constant Weyl tensor leads via a conformal transformation to a dimensionally homogeneous (‘no-scale’) theory in the conformal frame with a scalar field source that has an exponential potential. We then formulate the traceless version of f(R) gravity, and we find that a conformal transformation leads to a no-scale theory conformally equivalent to general relativity and a scalar field $$\phi $$ ϕ with a potential given by the scale-invariant form: $$V(\phi )=\frac{D-2}{4D}Re^{-\phi }$$ V ( ϕ ) = D - 2 4 D R e - ϕ , where $$\phi =[2/(D-2)]\ln f^{\prime }(R)$$ ϕ = [ 2 / ( D - 2 ) ] ln f ′ ( R ) . In this theory, the cosmological constant is a mere integration constant, statistically distributed in a multiverse of independent causal domains, the vacuum energy is another unrelated arbitrary constant, and the same is true of the height of the inflationary plateau present in a huge variety of potentials. Unlike in the conformal equivalent of full general relativity, flat potentials are found to be possible in all spacetime dimensions for polynomial lagrangians of all orders. Hence, we are led to a novel interpretation of the cosmological constant vacuum energy problem and have accelerated inflationary expansion in the very early universe with a very small cosmological constant at late times for a wide range of no-scale theories. Fine-tunings required in traceless general relativity or standard non-traceless f(R) theories of gravity are avoided. We show that the predictions of the scale-invariant conformal potential are completely consistent with microwave background observational data concerning the primordial tilt and the tensor-to-scalar ratio.

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