本文讨论了任意大于2维的四分体引力的哈密顿公式，当四分体和自旋联系被看作独立变量时，利用它的一阶形式，给出了三维情形的完全解。首次应用约束动力学的方法，明确地导出了系统的哈密顿量和约束，并计算了所有约束之间的泊松括号的代数。第一类次约束的Poisson括号的代数与ISO（2，1）Poincare群的李代数局部重合。所有的第一类约束，这个公式，根据狄拉克猜想和使用卡斯特拉尼程序，使我们能够明确地推导出规范变换的生成元，并找到规范变换的四元组和自旋连接，这原来是相同的发现威滕没有求助于汉密尔顿方法[ \textit{Nucl. Phys. B 311 (1988) 46} ]。本文将约束李代数生成的四元引力的规范对称性与另一种不变性--超同态进行了比较。简要讨论了高维哈密顿公式的一些结论，特别是当四分体和自旋联系作为独立变量时，从任意维四分体引力的哈密顿公式中得到的自同构不变性是\textit{not derivable}作为\textit{gauge symmetry}.
英文摘要：
The Hamiltonian formulation of the tetrad gravity in any dimension higher than two, using its first order form when tetrads and spin connections are treated as independent variables, is discussed and the complete solution of the three dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of the Poisson brackets among all constraints is calculated. The algebra of the Poisson brackets among first class secondary constraints locally coincides with Lie algebra of the ISO(2,1) Poincare group. All the first class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same found by Witten without recourse to the Hamiltonian methods [\textit{Nucl. Phys. B 311 (1988) 46}]. The gauge symmetry of the tetrad gravity generated by Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is \textit{not derivable} as a \textit{gauge symmetry} from the Hamiltonian formulation of tetrad gravity in any dimension when tetrads and spin connections are used as independent variables.
We discuss a simple symplectic formulation for tetrad gravity that leads to the real Ashtekar variables in a direct and transparent way. It also sheds light on the role of the Immirzi parameter and the time gauge.
We use covariant phase space methods to study the metric and tetrad formulations of General Relativity in a manifold with boundary and compare the results obtained in both approaches. Proving their equivalence has been a long-lasting problem that we solve here by using the cohomological approach provided by the relative bicomplex framework. This setting provides a clean and ambiguity-free way to describe the solution spaces and associated symplectic structures. We also compute several relevant charges in both schemes and show that they are equivalent, as expected.
We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero. Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form le…
Hamiltonian linearization of the rest-frame instant form of tetrad gravity in a completely fixed 3-orthogonal gauge: a radiation gauge for background-independent gravitational waves in a post-Minkowskian Einstein spacetime
We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.
Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: II) The Weyl Tensor, Weyl Scalars, the Weyl Eigenvalues and the Problem of the Observables of the Gravitational Field
在研究了在空间无穷远处全局双曲、渐近平坦的时空上正则四元引力的规范变换的哈密顿群之后，我们发现柯西超曲面$Σ_τ$通过规范变换与之相连，而柯西超曲面$Σ_τ$与$R^3, we find the dependence of the cotriads on $ <$_τ $ and of their momenta on the six parameters associated with rotations and space diffeomorphisms. The choice of 3-coordinates on $ <$_τ $ is equivalent to the parametrization of the cotriads with the last three degrees of freedom individuating the 3-geometries. The Shanmugadhasan canonical transformation, corresponding to the choice of 3-orthogonal coordinates on $ <$_τ $ and adapted to 13 of the 14 first class constraints, and the interpretation of the gauge t…
We consider a generalized teleparallel theory of gravitation, where the action contains an arbitrary function of the torsion scalar and a scalar field, $f(T,ϕ)$, thus encompassing the cases of $f(T)$ gravity and nonminimally coupled scalar field as subclasses. The action is manifestly Lorentz invariant when besides the tetrad one allows for flat but nontrivial spin connection. We derive the field equations and demonstrate how the antisymmetric part of the tetrad equations is automatically satisfied when the spin connection equation holds. The spin connection equation is a vital part of the cov…
An explicit, geometric description of the first-class constraints and their Poisson brackets for gravity in the Palatini-Cartan formalism (in space-time dimension greater than three) is given. The corresponding Batalin- Fradkin-Vilkovisky (BFV) formulation is also developed.
We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that in addition to the tetrad or metric, they employ a flat connection as additional field variable, but differ by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanish…
A Hamiltonian approach to the equations of general relativity is proposed using the powerful mathematical language of multivector-valued differential forms. In the approach, the gravitational coordinates are the 12 spatial components of the line interval (the vierbein) including their antisymmetric parts, and their 12 conjugate momenta. A feature of the proposed formalism is that it allows Lorentz gauge freedoms to be imposed on the Lorentz connections rather than on the vierbein, which may facilitate numerical integration in some challenging problems. The 40 Hamilton's equations comprise 12 +…
We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the 'volume' part of simplicity constraints, required for the passage from the topological BF theory to gravity. The crucial role, played by 4d normals in the linear version of constraints, necessitates the extension of the configuration space, and we argue to switch from normal 3-forms directly to tetrads. The requirement of vanishing torsion leads to c…
First order minisuperspace model for the Faddeev formulation of gravity
考虑了由10个向量场或10维四元组构成度规的广义相对论的Faddeev表述。在部分使用的场方程，这个理论的结果在通常的GR。早些时候，我们已经提出了一些minisuperspace模型的Faddeev制定的四元组字段是分段常数的多面体，如4-单纯形，或者说，长方体，其中${\rm I \hspace{-3pt} R}^4$可以分解。现在我们研究这个（离散）理论的某些表示，它是通常连续统GR中希尔伯特-爱因斯坦作用量的Cartan-Weyl连接型形式的类似物。
The constraint algebra is derived in the second order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts, and the Hassan-Rosen transform (characterized as "a complicated redefinition of the shift variable" according to the authors) appears here not as an ansatz but as fixing of a Lagrange multiplier. A comparison of this approach with the other ones is provided.
The teleparallel version of Lewis-Papapetrou space-time is investigated. For this space-time the true discussion of the geometric and physical properties is given. We show that the value of space-like torsion-axial vector does not depend on the choice of tetrad field. Consequently, the spin procession of the Dirac particle and the corresponding Hamiltonian do. The energy and momentum densities for such space-time are calculated. We show that when choosing two sets of tetrad fields the dependence and independence of the aforementioned quantities on the tetrad field depend on the Lorentz factor,…
Varying the gravitational Lagrangian produces a boundary contribution that has various physical applications. It determines the right boundary terms to be added to the action once boundary conditions are specified, and defines the symplectic structure of covariant phase space methods. We study general boundary variations using tetrads instead of the metric. This choice streamlines many calculations, especially in the case of null hypersurfaces with arbitrary coordinates, where we show that the spin-1 momentum coincides with the rotational 1-form of isolated horizons. The additional gauge symme…
The f(T) gravity is nowadays being widely used for cosmological model building, as well as for constructing spherically symmetric solutions. In its classical pure tetrad formulation it violates the local Lorentz symmetry in the space of tetrads. By using an appropriate spin connection it can be brought to a formally Lorentz invariant shape. However, despite some mathematical elegance obtained and new options of looking for non-standard couplings to matter probably made possible, it is fully equivalent in itself to the initial formulation. It seems that this fact, that the covariantisation does…
In this article we explore local Lorentz transformations in theories of gravity based on the teleparallel formalism. For the teleparallel equivalent of general relativity (TEGR), the spin connection plays no role in the equations of motion, and therefore it is possible to simply put it equal to zero with no change in physical quantities, and then the theory is formulated purely in terms of the tetrad field which can be freely chosen in any way. In nonlinear modifications of TEGR, this is a more intricate issue, and vanishing spin connection is then the Weitzenböck gauge choice which imposes re…
The geometric trinity of gravity offers a platform in which gravity can be formulated in three analogous approaches, namely curvature, torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a $3+1$ decomposition of the teleparallel equivalent of general relativity and the symmetric teleparall…
