Hamiltonian Formulation Of General Relativity Research Articles

Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Introduction to Hamiltonian formulation of general relativity and homogeneous cosmologies

We give a pedagogical introduction to the Hamiltonian formalism of general relativity at an advanced undergraduate and graduate levels. After covering the mathematical pre-requisites as well as the 3+1-decomposition of spacetime, we proceed to discuss the Arnowitt-Deser-Misner (ADM) formalism (a Hamiltonian approach) of general relativity. Then we proceed to give a brief but self-contained introduction to homogeneous (but not necessarily isotropic) universes and discuss the associated Bianchi classification. We first study their dynamics in the Lagrangian formulation, followed by the Hamiltonian formulation to show the equivalence of both approaches. We present a variety of examples to illustrate the ADM formalism: (i) free & massless scalar field coupled to homogeneous (in particular, Bianchi IX) universe, (ii) scalar field with a potential term coupled to Bianchi IX universe, (iii) electromagnetic field coupled to gravity in general, and (iv) electromagnetic field coupled to Bianchi IX universe.

On the breakdown of space–time in general relativity

Based on the canonical quantization of d > 2-dimensional general relativity (GR) via the Dirac constraint formalism (also termed as ‘constraint quantization’), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein–Hilbert action, whereby the loss of diffeomorphism invariance, besides first-class constraints, second-class constraints also exist leading to non-standard ghost fields that render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR, for which only first-class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we first establish, for the first time, that the breakdown in space–time is a property of GR itself as its limitation, indicating it to be an effective field theory (EFT). We further propose that the breakdown of space–time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang–Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes. Our findings are in contrast to the asymptotic safety program. They support emergent theories of space–time and gravity, though are unique, as they do not require thermodynamics such as the entropic gravity program. From an EFT view, these indicate that new degrees of freedom and(or) principles in the non-perturbative sector of the full theory are a requirement, whereby covariance as a symmetry is broken in the high-energy (strong field) sector of GR.

Linking the ADM formulation to other Hamiltonian formulations of general relativity

We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$ dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$ Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$ [$SO(n)$] requires no gauge fixing whereas $SO(n-1)$ involves the time gauge]. Similarly, the Hamiltonian formulation of general relativity in terms of Ashtekar-Barbero variables can also be directly obtained from the Arnowitt-Deser-Misner Hamiltonian formulation and vice versa, which is an alternative approach to the way followed by Barbero. We give the relevant maps among the phase-space variables and relate the corresponding symplectic structures and the first-class constraints.

Covariant formulation of non-equilibrium thermodynamics in General Relativity

We construct a generally-covariant formulation of non-equilibrium thermodynamics in General Relativity. We find covariant entropic forces arising from gradients of the entropy density, and a corresponding non-conservation of the energy momentum tensor in terms of these forces. We also provide a Hamiltonian formulation of General Relativity in the context of non-equilibrium phenomena and write the Raychaudhuri equations for a congruence of geodesics. We find that a fluid satisfying the strong energy condition could avoid collapse for a positive and sufficiently large entropic-force contribution. We then study the forces arising from gradients of the bulk entropy of hydrodynamical matter, as well as the entropy of boundary terms in the action, like those of black hole horizons. Finally, we apply the covariant formulation of non-equilibrium thermodynamics to the expanding universe and obtain the modified Friedmann equations, with an extra term corresponding to an entropic force satisfying the second law of thermodynamics.

The Principle of Covariance and the Hamiltonian Formulation of General Relativity.

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.

Local metric with parameterized evolution

AbstractWe present a canonical Hamiltonian formulation of general relativity in which τ, the parameter of system evolution, is external to spacetime, playing a role similar to what we call time in nonrelativistic mechanics. This approach, known as Stueckelberg–Horwitz–Piron (SHP) theory, inherits the full computational power of classical analytical mechanics while maintaining manifest covariance throughout and eliminating possible conflict with general diffeomorphism invariance. In particular, SHP theory simplifies the initial value problem with potential applications in highly dynamical interactions, such as black hole collisions. By allowing the energy–momentum tensor and metric to depend explicitly on τ, we may describe particle motion in geodesic form with respect to a dynamically evolving background metric. As a toy model, we consider a τ‐dependent mass M(τ), first as a perturbation in the Newtonian approximation and then for a Schwarzschild‐like metric. As expected, the extended Einstein equations imply a nonzero energy–momentum tensor, proportional to dM/dτ, representing a flow of mass and energy into spacetime that corresponds to the changing source mass. In τ‐equilibrium, this system becomes a generalized Schwarzschild solution for which the extended Ricci tensor and energy–momentum tensor vanish.

A Covariant Canonical Quantization of General Relativity

A Hamiltonian formulation of General Relativity within the context of the Nexus Paradigm of quantum gravity is presented. We show that the Ricci flow in a compact matter free manifold serves as the Hamiltonian density of the vacuum as well as a time evolution operator for the vacuum energy density. The metric tensor of GR is expressed in terms of the Bloch energy eigenstate functions of the quantum vacuum allowing an interpretation of GR in terms of the fundamental concepts of quantum mechanics.

Extending general covariance: Moyal-type noncommutative manifolds

In the Hamiltonian formulation of general relativity, Einstein's equation is replaced by a set of four constraints. Classically, the constraints can be identified with the generators of the hypersurface-deformation Lie algebroid (HDA) that belongs to the groupoid of finite evolutions in space-time. Taken over to deformed general relativity, this connection allows one to study possible Drinfeld twists of space-time diffeomorphisms with Hopf-algebra techniques. After a review of noncommutative differential structures, two cases --- twisted diffeomorphisms with standard action and deformed (or $\star$-) diffeomorphisms with deformed action --- are considered in this paper. The HDA of twisted diffeomorphisms agrees with the classical one, while the HDA obtained from deformed diffeomorphisms is modified due to the explicit presence of $\star$-products in the brackets. The results allow one to distinguish between twisted and deformed symmetries, and they indicate that the latter should be regarded as the relevant symmetry transformations for noncommutative manifolds. The algebroid brackets maintain the same general structure regardless of space-time noncommutativity, but they still show important consequences of non-locality.

A practical approach to the Hamilton-Jacobi formulation of holographic renormalization

We revisit the subject of holographic renormalization for asymptotically AdS spacetimes. For many applications of holography, one has to handle the divergences associated with the on-shell gravitational action. The brute force approach uses the Fefferman-Graham (FG) expansion near the AdS boundary to identify the divergences, but subsequent reversal of the expansion is needed to construct the infinite counterterms. While in principle straightforward, the method is cumbersome and application/reversal of FG is formally unsatisfactory. Various authors have proposed an alternative method based on the Hamilton-Jacobi equation. However, this approach may appear to be abstract, difficult to implement, and in some cases limited in applicability. In this paper, we clarify the Hamilton-Jacobi formulation of holographic renormalization and present a simple algorithm for its implementation to extract cleanly the infinite counterterms. While the derivation of the method relies on the Hamiltonian formulation of general relativity, the actual application of our algorithm does not. The work applies to any $D$-dimensional holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and arbitrary slicing. We illustrate the method in several examples, including the FGPW model, a holographic model of 3d ABJM theory, and cases with marginal scalars such as a dilaton-axion system.

Fokker action of nonspinning compact binaries at the fourth post-Newtonian approximation

The Fokker action governing the motion of compact binary systems without spins is derived in harmonic coordinates at the fourth post-Newtonian approximation (4PN) of general relativity. Dimensional regularization is used for treating the local ultraviolet (UV) divergences associated with point particles, followed by a renormalization of the poles into a redefinition of the trajectories of the point masses. Effects at the 4PN order associated with wave tails propagating at infinity are included consistently at the level of the action. A finite part procedure based on analytic continuation deals with the infrared (IR) divergencies at spatial infinity, which are shown to be fully consistent with the presence of near-zone tails. Our end result at 4PN order is Lorentz invariant and has the correct self-force limit for the energy of circular orbits. However, we find that it differs from the recently published result derived within the ADM Hamiltonian formulation of general relativity [T. Damour, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 89, 064058 (2014)]. More work is needed to understand this discrepancy.

Spontaneously broken Lorentz symmetry for Hamiltonian gravity

In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y. This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3,1) to SO(3) is not in conflict with Lorentz covariance.

Solution of the noncanonicity puzzle in General Relativity: A new Hamiltonian formulation

We study the transformation leading from Arnowitt, Deser, Misner (ADM) Hamiltonian formulation of General Relativity (GR) to the ΓΓ metric Hamiltonian formulation derived from the Lagrangian density which was firstly proposed by Einstein. We classify this transformation as gauged canonical – i.e. canonical modulo a gauge transformation. In such a study we introduce a new Hamiltonian formulation written in ADM variables which differs from the usual ADM formulation mainly in a boundary term firstly proposed by Dirac. Performing the canonical quantization procedure we introduce a new functional phase which contains an explicit dependence on the fields characterizing the 3+1 splitting. Given a specific regularization procedure our new formulation privileges the symmetric operator ordering in order to: have a consistent quantization procedure, avoid anomalies in constraints algebra, be equivalent to the Wheeler–DeWitt (WDW) quantization. Furthermore we show that this result is consistent with a path-integral approach.

On canonical transformations between equivalent hamiltonian formulations of general relativity

Two Hamiltonian formulations of general relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for the constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for the constrained systems.

Hamiltonian analysis of Lagrange multiplier modified gravity

We develop a Hamiltonian formalism for Lagrange multiplier modified gravity. We further calculate the Poisson brackets between constraints and we show that they coincide with the algebra of constraints in the Hamiltonian formulation of general relativity.

Zero-point quantum fluctuations and dark energy

In the Hamiltonian formulation of General Relativity the energy associated to an asymptotically flat space-time with metric $g_{\mu\nu}$ is related to the Hamiltonian $H_{GR}$ by $E=H_{GR}[g_{\mu\nu}]-H_{\rm GR}[\eta_{\mu\nu}]$, where the subtraction of the flat-space contribution is necessary to get rid of an otherwise divergent boundary term. This classic result indicates that the energy associated to flat space does not gravitate. We apply the same principle to study the effect of zero-point fluctuations of quantum fields in cosmology, proposing that their contribution to the cosmic expansion is obtained computing the vacuum energy of quantum fields in a FRW space-time with Hubble parameter $H(t)$ and subtracting from it the flat-space contribution. [...] After renormalization, this produces a renormalized vacuum energy density $\sim M^2H^2(t)$, where $M$ is the scale where quantum gravity sets is, so for $M$ of order of the Planck mass a vacuum energy density of the order of the critical density can be obtained without any fine tuning. The counterterms can be chosen so that the renormalized energy density and pressure satisfy $p=w\rho$, with $w$ a parameter that can be fixed by comparison to the observed value, so in particular one can chose $w=-1$. An energy density evolving in time as $H^2(t)$ is however observationally excluded as an explanation for the dominant dark energy component which is responsible for the observed acceleration of the universe. We rather propose that zero-point vacuum fluctuations provide a new subdominant "dark" contribution to the cosmic expansion that, for a UV scale $M$ slightly smaller than the Planck mass, is consistent with existing limits and potentially detectable.

The Hamiltonian formulation of general relativity: myths and reality

Abstract A conventional wisdom often perpetuated in the literature states that: (i) a 3 + 1 decomposition of spacetime into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [21] and of ADM [22] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i–iii) have been shown to be incorrect in [45] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac’s references to space-like surfaces are inessential and that such surfaces do not enter his calculations. In addition, we show that his assumption g 0k = 0, used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption g 0k = 0. The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary (also known as the “Hamiltonian” and “diffeomorphism” constraints). The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints. Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points (i–iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric g μν to lapse and shift functions and the three-metric g km, which is not canonical. This proves that point (iv) is incorrect. Points (i–iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein’s theory itself.

On canonical transformations of gravitational variables in extended phase space

In the recent years, a certain attention was attracted to the statement that Hamiltonian formulations of general relativity (GR), in which different parametrizations of gravitational variables were used, may not be related by a canonical transformation. An example was given by Dirac’s Hamiltonian formulation and that of Arnowitt-Deser-Misner. It might be indicative of a non-equivalence of these formulations and the original (Lagrangian) formulation of GR. The problem is believed to be of importance since many authors make use of various representations of the gravitational field as a starting point in searching for a way to reconcile the theory of gravity with quantum principles. It can be shown that the above-mentioned conclusion on non-equivalence of different Hamiltonian formulations is based on a consideration of canonical transformations in the phase space of physical degrees of freedom only, while the transformations also involve the gauge degrees of freedom. We shall give a clear proof that the Hamiltonian formulations corresponding to different parametrizations of the gravitational variables are related by canonical transformations in extended phase space embracing the gauge degrees of freedom on equal footing with the physical ones. It will be demonstrated for the full gravitational theory in a wide enough class of parametrizations and gauge conditions.

The Hamiltonian of Einstein affine-metric formulation of General Relativity

It is shown that the Hamiltonian of the Einstein affine-metric (first-order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as in the case of the second-order formulation. In the second-order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables (arXiv:0809.0097). For the first-order formulation, the necessity of such a redefinition “to correspond to diffeomorphism invariance” (reported by Ghalati, arXiv:0901.3344) is just an artifact of using the Henneaux–Teitelboim–Zanelli ansatz (Nucl. Phys. B 332:169, 1990), which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani (Ann. Phys. 143:357, 1982) is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second-order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second- and first-order formulations of metric GR. The first-order formulation of Einstein–Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.

Strongly hyperbolic Hamiltonian systems in numerical relativity: formulation and symplectic integration

We consider two strongly hyperbolic Hamiltonian formulations of general relativity and their numerical integration with a free and a partially constrained symplectic integrator. In those formulations, we use hyperbolic drivers for the shift and in one case also for the densitized lapse. A system where the densitized lapse is an external field allows us to enforce the momentum constraints in a holonomically constrained Hamiltonian system and to turn the Hamilton constraint function from a weak to a strong invariant. These schemes are tested in a perturbed Minkowski and a Schwarzschild space-time. In those examples, we find advantages of the strongly hyperbolic formulations over the Arnowitt–Deser–Misner (ADM) system presented in [28]. Furthermore, we observe stabilizing effects of the partially constrained evolution in Schwarzschild spacetime as long as the momentum constraints are enforced.