Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a "Lagrangian one-form" to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein Cartan gravity, to which we apply the method explicitly. We find that one can demonstrate the closure of the constraints without gauge fixing the Lorentz group or introducing primary constraints on the phase space variables. Finally, using geometric quantization techniques, we briefly discuss implications of the formalism for the quantum theory.
By using the Gitman-Lyakhovich-Tyutin canonical analysis for higher-order theories a four-dimensional Chern-Simons modification of general relativity is analyzed. The counting of physical degrees of freedom, the symmetries, and the fundamental Dirac brackets are reported. Additionally, we report the complete structure of the constraints and its Dirac algebra is developed.
We present a manifestly Lorentz-covariant description of the phase space of general relativity with the Immirzi parameter. This formulation emerges after solving the second-class constraints arising in the canonical analysis of the Holst action. We show that the new canonical variables give rise to other Lorentz-covariant parametrizations of the phase space via canonical transformations. The resulting form of the first-class constraints in terms of new variables is given. In the time gauge, these variables and the constraints become those found by Barbero.
Gauge fields associated to the Dirac matrix algebra used with the standard quadratic gauge field Lagrangian lead to an extended gravitational Lagrangian which includes the Einstein-Hilbert one, plus quadratic, cosmological constant and torsion terms. This note looks at three cases : the static central symmetric field, the isotropic expanding universe, and the asymptotic field of a rotating body, and show that, in weak gravitational fields, there is no contradiction with General Relativity results.
We extend the Non-Relativistic formulation of General Relativity (NRGR) given in Goldberger et al. (Phys Rev D 73:104029, 2006) to incorporate the effects of electromagnetic charge of the constituents of the binary. We incorporate the photon field in NRGR by giving the field decomposition and power counting rules. Using these, we develop the Feynman rules to describe photon and graviton interactions with point particle worldline. We then find the corrections to the Einstein-Infeld-Hoffmann Lagrangian (Einstein et al. in Ann Math 39:65-100, 1938) due to the presence of the photon field and elec…
Ising models of emergent geometry are well known to possess ground states with many of the desired features of a low dimensional, Ricci flat vacuum. Further, excitations of these ground states can be shown to replicate the quantum dynamics of a free particle in the continuum limit. It would be a significant next step in the development of emergent Ising models to link them to an underlying physical theory that has General Relativity as its continuum limit. In this work we investigate how the canonical formulation of General Relativity can be used to construct such a discrete Hamiltonian using …
In this paper we review the canonical analysis of constrained systems and apply it in the case of teleparallel equivalent of general relativity and teleparallel gravity. For each of them we find all the Poisson brackets, generators of gauge symmetries, as well as the number of degrees of freedom.
We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by $n$-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local $SO(n)$ or $SO(n-1,1)$ transformations, `improved diffeomorphisms', and the `generalization of local translations' of the orthonormal frame and the connection. A remarkable aspect of our …
We discuss the relation between the canonical Hamilton-Jacobi theory and the De Donder-Weyl Hamilton-Jacobi theory known in the calculus of variations using the examples of a scalar field in curved space-time and general relativity. By restricting ourselves to the Gaussian coordinates we show how the canonical Hamilton-Jacobi equation of general relativity can be derived from the De Donder-Weyl Hamilton-Jacobi formulation of the Einstein equations.
In the main article [CQG 38 (2021) 055003], a new "canonical" form for the Lewis metrics of the Weyl class has been obtained, depending only on three parameters -- Komar mass and angular momentum per unit length, plus the angle deficit -- corresponding to a coordinate system fixed to the "distant stars" and an everywhere timelike Killing vector field. Such form evinces the local but non-global static character of the spacetime, and striking parallelisms with the electromagnetic analogue. We discuss here its generality, main physical features and important limits (the Levi-Civita static cylinde…
In the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid, we take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. We then evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters. Afte…
General Relativity and Quantum Jumps: The Existence of Nondiffeomorphic Solutions to the Cauchy Problem in Nonempty Spacetime and Quantum Jumps as a Provider of a Canonical Spacetime Structure
Using a suitable rescaling of the independent variable, a Lagrangian is found for the nonlinear Vialov equation ruling the longitudinal profiles of glaciers and ice caps in the shallow ice approximation. This leads to a formal analogy between the (rescaled) Vialov equation and the Friedmann equation of relativistic cosmology, which is explored. This context provides a new symmetry of the (rescaled) Vialov equation and gives, at least formally, all its solutions using a generating function, which is the Nye profile for the degenerate case of perfectly plastic ice.
The Vacuum Structure, Special Relativity and Quantum Mechanics Revisited: a Field Theory No-Geometry Approach within the Lagrangian and Hamiltonian Formalisms. Part 2
We describe a novel procedure to map the field equations of nonlinear Ricci-based metric-affine theories of gravity, coupled to scalar matter described by a given Lagrangian, into the field equations of General Relativity coupled to a different scalar field Lagrangian. Our analysis considers examples with a single and $N$ real scalar fields, described either by canonical Lagrangians or by generalized functions of the kinetic and potential terms. In particular, we consider several explicit examples involving $f(R)$ theories and the Eddington-inspired Born-Infeld gravity model, coupled to differ…
On canonical transformations between equivalent Hamiltonian formulations of General Relativity
两个广义相对论的哈密顿公式，由于皮拉尼，Schild和斯金纳（物理Rev. 87，452，1952）和狄拉克（Proc。Soc. A 246，333，1958）。这两个公式，尽管有不同的表达式的约束，允许一个推导四维的同构不变性。这两个配方之间的关系在所有阶段的狄拉克方法的约束哈密顿系统进行了分析。它示出，他们的相空间变量的完整的集合是由一个变换，满足普通的条件，规范已知的无约束哈密顿，此外，转换成另一个总的哈密顿量，从而保持形式不变性的广义哈密顿方程的约束系统。
The problem of formulating synchronous variational principles in the context of General Relativity is discussed. Based on the analogy with classical relativistic particle dynamics, the existence of variational principles is pointed out in relativistic classical field theory which are either asynchronous or synchronous. The historical Einstein-Hilbert and Palatini variational formulations are found to belong to the first category. Nevertheless, it is shown that an alternative route exists which permits one to cast these principles in terms of equivalent synchronous Lagrangian variational formul…
