Lattice formulation of general relativity

It is shown that almost the entire excitation energy acquired by the fission fragments during the descent from the saddle to the scission point comes from Landau-Zener transitions. The states tractable in the first order adiabatic approximation carry an excitation energy of a few hundred keV.     

Canonical formulation of spin in general relativity

The present article aims at an extension of the canonical formalism of Arnowitt, Deser, and Misner from self‐gravitating point‐masses to objects with spin. This would allow interesting applications, e.g., within the post‐Newtonian (PN) approximation. The extension succeeded via an action approach to linear order in the single spins of the objects without restriction to any further approximation. An order‐by‐order construction within the PN approximation is possible and performed to the formal 3.5PN order as a verification. In principle both approaches are applicable to higher orders in spin. The PN next‐to‐leading order spin(1)‐spin(1) level was tackled, modeling the spin‐induced quadrupole deformation by a single parameter. All spin‐dependent Hamiltonians for rapidly rotating bodies up to and including 3PN are calculated.     

爱因斯坦与广义相对论

文章介绍了爱因斯坦建立相对论,特别是广义相对论的伟大贡献。爱因斯坦提出了光速不变原理、广义相对性原理、马赫原理和等效原理。他不仅首先指出万有引力本质上是时空弯曲的几何效应,而且首先给出了广义相对论的基本方程。文章还讨论了为什么爱因斯坦是狭义相对论和广义相对论的唯一创建者。     

On general and restricted covariance in general relativity

We reconsider the principle of general covariance and give a rigorous formulation of a principle ofrestricted covariance. We give a number of examples of preferred coordinate systems, considered in the literature, and in each case demonstrate the applicability of the notion of restricted covariance proposed.     

Linear derivative Cartan formulation of general relativity

Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a “bi-complex” algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincaré algebra is superior to earlier similar attempts.Received: 24 January 2005, Published online: 8 June 2005     

广义相对论的几个问题

介绍了爱因斯坦如何建立广义相对论,以及人们应该如何理解这一关于时间、空间和引力的理论.     

The covariant derivative in the affine approach to general relativity

The affine presentation of general relativity is considered and a possible generalisation of the definition of covariant derivative is proposed. Under certain weak symmetry conditions it is shown that the only theories resulting from this generalisation are general relativity and Weyl's theory, of which general relativity arises in the most natural way.     

Spontaneously induced general relativity with holographic interior and general exterior

We study the spontaneously induced general relativity (GR) from the scalar-tensor gravity. We demonstrate by numerical methods that a novel inner core can be connected to the Schwarzschild exterior with cosmological constants and any sectional curvature. Deriving an analytic core metric for a general exterior, we show that all the nontrivial features of the core, including the locally holographic entropy packing, are universal for the general exterior in static spacetimes. We also investigate whether the f(R)$f(R)$ gravity can accommodate the nontrivial core.     

Hamiltonian formulation of general relativity in terms of Dirac spinors

The Dirac spinors and matrices are used in combination with the Arnowitt-Deser-Misner formalism in order to obtain yet another formulation of Hamiltonian general relativity, together with a new form of the Gauss-Codazzi equations. The relation with Ashtekar's variables is analyzed; it is shown, for instance, that the matrices are equivalent to the electric field variable. The electric and magnetic decomposition of the gravitational field is also studie using Dirac matrices.     

Fixed mesh refinement in the characteristic formulation of general relativity

We implement a spatially fixed mesh refinement under spherical symmetry for the characteristic formulation of General Relativity. The Courant–Friedrich–Levy condition lets us deploy an adaptive resolution in (retarded-like) time, even for the nonlinear regime. As test cases, we replicate the main features of the gravitational critical behavior and the spacetime structure at null infinity using the Bondi mass and the News function. Additionally, we obtain the global energy conservation for an extreme situation, i.e. in the threshold of the black hole formation. In principle, the calibrated code can be used in conjunction with an ADM 3+1 code to confirm the critical behavior recently reported in the gravitational collapse of a massless scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios studied, the fixed mesh refinement offers improved runtime and results comparable to code without mesh refinement.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

On a new formulation of the many-body problem in general relativity

A generalized Riemannian geometry is studied where the metric tensor is replaced by a matrix g of metrics. In this context new geometric quantities arise, which are trivial in ordinary Riemannian geometry. An application of this formalism to many-body alignments in general relativity is proposed, where the sub-constituents of the overall gravitational field are described by the components of g. The mutual gravitational interactions between the individual particles are encoded in specific tensors. In particular, very specific approximation schemes for Einstein’s field equations may be considered, which exclusively approximate those terms in the field equations which are due to interactions. The Newtonian limit as well as the first post-Newtonian approximation of the presented formalism is studied in order to display the interpretability of the presented formalism in terms of many-body alignments and in order to deduce a physical interpretation of the new geometric quantities.     

The Hamiltonian formulation of general relativity: myths and reality

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.     

Conservation laws in the tetrad formulation of the general theory of relativity

The covariant consequences of a weak conservation law in the tetrad formulation of general relativity that do not contain noncovariant complexes of energy-momentum or external and internal spin momenta are considered. The relationship between a group of arbitrary tetrad Lorentz transformations and a generally covariant definition of the spin angular momentum of nongravitational matter is outlined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 101–105, February, 1979.     

On the Rayski formulation of the laws of conservation in general relativity

The Rayski method is used to calculate the energy of a static and spherically symmetrical gravitational system.In conclusion, the author thanks Dr. P. Burcev and Dr. K. Kucha for kindly pointing out some shortcomings which could thus be eliminated in time.     

Reconsiderations on the formulation of general relativity based on Riemannian structures

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BF-like field theory structure of the Einstein–Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern–Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten’s formulation of three-dimensional gravity as a Chern–Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.     

Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory

Thermodynamics plays an important role in gravitational theories. It is a principle that is independent of gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system with the general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations, which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of a perfect fluid inside a compact region for a fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. This means that the entropy principle of the fluid stores the same information as the gravitational equation in a static configuration. Our proof is universal and holds for any diffeomorphism-covariant purely gravitational theories, such as Einstein gravity, \begin{document}$ f(R)$\end{document} gravity, Lovelock gravity, f(Gauss-Bonnet) gravity and Einstein-Weyl gravity. Our result indicates the consistency between ordinary thermodynamics and gravitational dynamics.