Constraints Of General Relativity Research Articles

Hyperbolic compactification of M-theory and de Sitter quantum gravity

We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.

Asymptotically Flat Boundary Conditions for the U(1)3 Model for Euclidean Quantum Gravity

A generally covariant U(1)3 gauge theory describing the GN→0 limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we the study boundary conditions and asymptotic symmetries of the U(1)3 model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that the non-Abelian part of the SU(2) Gauss constraint, which is absent in the U(1)3 model, plays a crucial role in obtaining boost and rotation generators.

On the Evolutionary Form of the Constraints in Electrodynamics

The constraint equations in Maxwell theory are investigated. In analogy with some recent results on the constraints of general relativity, it is shown, regardless of the signature and dimension of the ambient space, that the “divergence of a vector field”-type constraint can always be put into linear first order hyperbolic form for which the global existence and uniqueness of solutions to an initial-boundary value problem are guaranteed.

De Sitter spacetime with a Becchi–Rouet–Stora quartet

We generalize the topological model recently proposed and investigate the cosmological perturbations of the model. The model has an exact de Sitter background solution associated with the Becchi–Rouet–Stora (BRS) quartet terms which are regarded as the Lagrangian density of the topological field theory. The de Sitter solution can be selected without spontaneously breaking the BRS symmetry, and be interpreted as a gauge fixing of de Sitter spacetime. The BRS symmetry is preserved for the perturbations around the de Sitter background before we solve the constraints of general relativity. We derive action to the second-order of the perturbations and confirm that even after solving the constraints, we have the BRS symmetry at least for the second-order action. We construct the cosmological perturbation theory involving the BRS sector, and obtain the two-point correlation functions for the curvature perturbation and the isocurvature perturbations which compose the BRS sector. Our result gives a new description for de Sitter spacetime and the quantum field theory in de Sitter spacetime.

General covariance from the quantum renormalization group

The Quantum renormalization group (QRG) is a realisation of holography through a coarse graining prescription that maps the beta functions of a quantum field theory thought to live on the `boundary' of some space to holographic actions in the `bulk' of this space. A consistency condition will be proposed that translates into general covariance of the gravitational theory in the $D + 1$ dimensional bulk. This emerges from the application of the QRG on a planar matrix field theory living on the $D$ dimensional boundary. This will be a particular form of the Wess--Zumino consistency condition that the generating functional of the boundary theory needs to satisfy. In the bulk, this condition forces the Poisson bracket algebra of the scalar and vector constraints of the dual gravitational theory to close in a very specific manner, namely, the manner in which the corresponding constraints of general relativity do. A number of features of the gravitational theory will be fixed as a consequence of this form of the Poisson bracket algebra. In particular, it will require the metric beta function to be of gradient form.

Decoupling the momentum constraints in general relativity

We present a 2+1 decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy-Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.

On the Chern-Simons State in General Relativity and Modified Gravity Theories

The Chern-Simons state is one solution to quantum constraints of gravity in the context of general relativity (GR) theory if we use Ashtekar's variables and if one orders the constraints with the triads to the left. Six years ago Krasnov introduced a certain class of modified gravity theories by replacing the cosmological constant by a "cosmological function of the curvature". If this function is a constant we come back to GR. In this note we review how the Chern-Simons state is one solution to the constraints of GR and we state the problem to face if we wish a generalized Chern-Simons state for the modified Krasnov's theories.

On the initial state and consistency relations

We study the effect of the initial state on the consistency conditions for adiabatic perturbations. In order to be consistent with the constraints of General Relativity, the initial state must be diffeomorphism invariant. As a result, we show that initial wavefunctional/density matrix has to satisfy a Slavnov-Taylor identity similar to that of the action. We then investigate the precise ways in which modified initial states can lead to violations of the consistency relations. We find two independent sources of violations: i) the state can include initial non-Gaussianities; ii) even if the initial state is Gaussian, such as a Bogoliubov state, the modified 2-point function can modify the q⃗→ 0 analyticity properties of the vertex functional and result in violations of the consistency relations.

GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold [Formula: see text] of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

Note on flat foliations of spherically symmetric spacetimes

It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.

Mach's principle, time, quantum gravity, and the origin of the arrow of time

Since my three talks at Peyresq were all based on already published papers, I give here only an abstract of my talks, which were essentially a review of the work done by Bruno Bertotti and myself (Barbour and Bertotti, 1977, 1982) on the creation of Machian dynamical models in which both time and motion are treated relationally. This work eventually led us to conclude that Einstein's general theory of relativity is one of a large class of possible Machian dynamical theories (Barbour and Bertotti (1982), but with a very special and distinguished structure. The Machian nature of general relativity is expressed through constraints on the initial data for the Cauchy problem; these constraints play the central role in all attempts to quantize general relativity in the canonical approach. Indeed, because they reflect the deep dynamical structure of general relativity, they must be central to all attempts to quantize gravity. The constraint that has proved the most difficult to handle and understand conceptually is the so-called Hamiltonian constraint. In my talks I pointed out that this constraint arises because of the fact (which I believe has not been properly appreciated) that general relativity treats time in a very Machian manner. Indeed, there is a well-defined sense in which time does not exist in general relativity at all. These issues are discussed in some length in (Barbour, 1994). See also the discussions in Barbour and Pfister (1995), especially the final discussion on the problem of time in quantum gravity, in which I propose the notion of "time capsules" in order to explain how the static wave function of the universe that follows from quantization of the constraints of general relativity can still describe a universe experienced as full of motion and memories. A time capsule is a static configuration of part or all of the universe containing structures which suggest they are mutually consistent records of processes that took place in a past

Generalized lines of force as the gauge invariant degrees of freedom for general relativity and Yang-Mills theory.

We present a set of equations that determine (with certain qualifications) what is (i) a complete canonical set of gauge-invariant variables for Yang-Mills (YM) theory that generalize Faraday's electric lines of force and (ii) a complete canonical set of the diffeomorphism and gauge invariant observables for general relativity (GR). Both results come from solving the Hamilton-Jacobi version of the constraint equations: the Gauss law constraints of YM theory and the Gauss law and diffeomorphism constraint of GR in the Ashtekar formalism. These results are local in physical space as well as in phase space.

A lattice approach to spinorial quantum gravity

The authors present a new lattice regularisation of quantum general relativity based on Ashtekar's reformulation (1986) of Hamiltonian general relativity. In this form, quantum states of the gravitational field are represented within the physical Hilbert space of a Kogut-Susskind lattice gauge theory. The gauge field of the theory is a complexified SU(2) connection which is the gravitational connection for left-handed spinor fields. The physical states of the gravitational field are those which are annihilated by additional constraints which correspond to the four constraints of general relativity. The authors construct lattice versions of these constraints. Those corresponding to the three-dimensional diffeomorphism generators move states associated with Wilson loops around on the lattice. The lattice Hamiltonian constraint has a simple form, and a correspondingly simple interpretation: it is an operator which cuts and joins Wilson loops at points of intersection.

Integral constraints in general relativity

Using a covariant formalism we derive the conditions for there to exist integral constraints on energy-momentum perturbations in an arbitrary background spacetime.

Global solutions of the Lichnerowicz equation in General Relativity on an asymptotically Euclidean complete manifold

We prove some existence and uniqueness theorems for the solution of the Lichnerowicz equation:\(8\Delta _{g\phi } - R_\phi + M_\phi ^{ - 7} + Q_\phi ^{ - 3} + \tau _\phi ^5 = 0\), on an asymptotically Euclidian manifold. This equation governs the conformai factor of a metric solution of the constraints in General Relativity. In the first part we prove existence and uniqueness under the simple assumptionR⩾0,M⩾0,Q⩾0, τ⩾0, which insures the monotony of the non-differentiated part. In the second part we obtain an existence theorem under more general hypothesis on the coefficients, by use of the Leray-Schauder degree theory. The results of this paper have been announced in [4].

Symmetry constraints in general relativity and the Schwarzchild interior metric

Einstein's equations with a perfect fluid source are subjected to compatibility conditions in the context of a space-time that contains symmetric subspaces. These conditions constitute, in some cases, a powerful tool for exhibiting the solutions to a given problem. The Schwarzchild interior metric in conformally flat coordinates is derived using these methods.

On nonmaximal solutions to the initial-value constraints of general relativity

This paper investigates a new series of solutions to the initial-value constraints of general relativity which is obtained by perturbing the trace of the extrinsic curvature. This analysis is important for two reasons. Firstly, it completes the proof that flat space is a local energy minimum. Secondly, it casts light on the extent to which this variable can be identified with the gauge degree of freedom in the initial-value problem.

Momentum Constraints as Integrability Conditions for the Hamiltonian Constraint in General Relativity

It is shown that if the Hamiltonian constraint of general relativity is imposed as a restriction on the Hamilton principal functional in the classical theory, or on the state functional in the quantum theory, then the momentum constraints are automatically satisfied. This result holds both for closed and open spaces and it means that the full content of the theory is summarized by a single functional equation of the Tomonaga-Schwinger type.