We present a new variational framework for dissipative general relativistic fluid dynamics. The model extends the convective variational principle for multi-fluid systems to account for a range of dissipation channels. The key ingredients in the construction are i) the use of a lower dimensional matter space for each fluid component, and ii) an extended functional dependence for the associated volume forms. In an effort to make the concepts clear, the formalism is developed in steps with the model example of matter coupled to heat considered at each level. Thus we discuss a model for heat flow, derive the relativistic Navier-Stokes equations and discuss why the individual dissipative stress tensors need not be spacetime symmetric. We argue that the new formalism, which notably does not involve an expansion away from an assumed equilibrium state, provides a conceptual breakthrough in this area of research and provide an ambitious list of directions in which one may want to extend it in the future. This involves an exciting set of problems, relating to both applications and foundational issues.
We prove that, for any theory defined over a space-time with boundary, the symplectic form derived in the covariant phase space is equivalent to the one derived from the canonical formalism.
In this short note we study unimodular gravity in Weyl-De Donder formalism. We find corresponding Hamiltonian and study consequence of the unimodular constraint on the conjugate covariant momenta. We also find covariant Hamiltonian for Henneaux-Teitelboim unimodular action and study corresponding equations of motion.
The Hamiltonian formalism of Einstein--Cartan (EC) gravity is a starting point for canonical quantum gravity. The existing formalisms are at most Lorentz covariant, or diffeomorphism covariant. Here we analyze the Hamiltonian EC gravity in a 5d covariant way, with the gauge group being the de Sitter (dS) group, which unifies the Lorentz transformations and translation in an elegant manner, and also coincides with the acceleration of the universe. We reformulate the EC equations into a dS-covariant form, then find out the dS-covariant constraints of the phase space, and make all the constraint …
In this paper we introduce a new procedure, termed by us \emph{energy formalism}, to deal with dissipative systems in metric theories of gravity. This approach aims at determining the analytic expression of Rayleigh dissipation function in the context of the inverse problem in the calculus of variations. We describe our method in detail, presenting a simple example. After, we consider as first extensive application the general relativistic Poynting-Robertson effect. The obtained results and future implications are discussed.
I present a covariant approach to developing 1+3 formalism without an introduction of any basis or coordinates. In the formalism, a spacetime which has a timelike congruence is assumed. Then, tensors are split into temporal and spatial parts according to the tangent direction to the congruence. I make use of the natural derivatives to define the kinematical quantities and to investigate their properties. They are utilized in the splitting of covariant derivatives. In this way, the Riemann curvature is split into the temporal and spatial part, i.e. Gauss, Codazzi, and Ricci relation. Finally, t…
A general proof of the equivalence between the δN and covariant formalisms
最近，δN和协变形式主义之间的等价性已经被证明（Suyama et al. 2012），但他们在证明中基本上假设了爱因斯坦引力。他们指出，在均匀能量密度切片的协变形式中曲率协向量的演化方程与δN形式中曲率微扰的演化方程一致，假设均匀能量和均匀膨胀（哈勃）切片的一致性，这是爱因斯坦引力超视界尺度的情况。在这个简短的说明中，我们明确地证明了δN和协变形式主义之间的等价性，而没有指定切片条件和相关的切片重合，换句话说，不管引力理论如何。
We investigate generators of local gauge transformations in the covariant canonical formalism (CCF) for matter fields, gauge fields and the second order formalism of gravity. The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms $π_A$ are defined as derivatives of the Lagrangian $d$-form $L(ψ^A, dψ^A)$ with respect to $dψ^A$, namely $π_A := \partial L/\partial dψ^A$, where $ψ^A $ are $p$-form dynamical fields. The form-canonical equations are derived from the form-Legendre transformation of the Lagrangian form $H:=dψ^A \w…
It is well-known that Dirac particles gain geometric phase, namely Berry phase, while moving in an electromagnetic field. Researchers have already shown covariant formalism for the Berry connection due to an electromagnetic field. A similar effect is expected to happen due to the presence of Gravity. We use WKB approximation to develop a covariant formalism of Berry-like connection in the presence of Einstein gravity, which can be further used to describe the Berry-like phase or simply Berry phase. We also extend this formalism for massless Dirac particles (Weyl particles).Then we further show…
Recently, a new strategy to the reconstruction of $f(R)$-gravity models based on the Raychaudhuri equation has been suggested by Choudhury et al. In this paper, utilizing this method, the reconstruction of anisotropic Einstein-Maxwell equation in the $1+3$ covariant formalism of $f(R)$-gravity is investigated in four modes: $i.$ Reconstruction from a negative constant deceleration parameter refereeing to an ever-accelerating universe; $ii.$ Reconstruction from a constant jerk parameter $j=1$ which recovers celebrated $Λ\text{CDM}$ mode of evolution; $iii.$ Reconstruction from a variable jerk p…
We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group $\mathrm{SO}(4,1)$ under the covariant Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the covariant formulation. The decomposition of the internal algebra $\mathfrak{so}(4,1)\simeq\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}$ allows the symmetry breaking $\mathrm{SO}(4,1)\to\mathrm{SO}(3,1)$, which reduces the original action to the Palatini action without the topologi…
Within the general framework of spatially covariant theories of gravity, we study the conditions for having only the two tensorial degrees of freedom. Generally, there are three degrees of freedom propagating in the theory, of which two are tensorial and one is of the scalar type. Through a detailed Hamiltonian analysis, we find two necessary and sufficient conditions to evade the scalar type degree of freedom. The first condition implies that the lapse-extrinsic curvature sector must be degenerate. The second condition ensures that the dimension of the phase space at each spacetime point is e…
Covariant eigenmode overlap formalism for gravitational wave signals in electromagnetic cavities
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology（高能物理研究所（IFAE），巴塞罗那科学与技术研究院）；Universität Hamburg（汉堡大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究所（DESY））
We present Hamilton's equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism…
Models of turbulent flows require the resolution of a vast range of scales, from large eddies to small-scale features directly associated with dissipation. As the required resolution is not within reach of large scale numerical simulations, standard strategies involve a smoothing of the fluid dynamics, either through time averaging or spatial filtering. These strategies raise formal issues in general relativity, where the split between space and time is observer dependent. To make progress, we develop a new covariant framework for filtering/averaging based on the fibration of spacetime associa…
This paper is devoted to the analysis of the covariant canonical formalism of $F(R)$ gravity in Einstein frame. We also find canonical transformation between covariant canonical formulation of F(R) gravity in Jordan frame and Einstein frames and we also determine corresponding generating function.
