scholarly journals New Hamiltonian formalism and quasilocal conservation equations of general relativity

2004 ◽  
Vol 70 (8) ◽  
Author(s):  
Jong Hyuk Yoon
Keyword(s):  
General Relativity ◽  
Hamiltonian Formalism ◽  
Conservation Equations

scholarly journals Classification of primary constraints for new general relativity in the premetric approach

2021 ◽  
pp. 2140003
Author(s):  
María-José Guzmán ◽  
Shymaa Khaled Ibraheem
Keyword(s):  
General Relativity ◽  
Hessian Matrix ◽  
Hamiltonian Formalism ◽  
Temporal Part ◽  
Tetrad Field ◽  
Mathematical Properties ◽  
Constitutive Tensor ◽  
Hamiltonian Analysis

We introduce a novel procedure for studying the Hamiltonian formalism of new general relativity (NGR) based on the mathematical properties encoded in the constitutive tensor defined by the premetric approach. We derive the canonical momenta conjugate to the tetrad field and study the eigenvalues of the Hessian tensor, which is mapped to a Hessian matrix with the help of indexation formulas. The properties of the Hessian matrix heavily rely on the possible values of the free coefficients [Formula: see text] appearing in the NGR Lagrangian. We find four null eigenvalues associated with trivial primary constraints in the temporal part of the momenta. The remaining eigenvalues are grouped in four sets, which have multiplicity 3, 1, 5 and 3, and can be set to zero depending on different choices of the coefficients [Formula: see text]. There are nine possible different cases when one, two, or three sets of eigenvalues are imposed to vanish simultaneously. All cases lead to a different number of primary constraints, which are consistent with previous work on the Hamiltonian analysis of NGR by Blixt et al. (2018).

Hamiltonian formalism for perfect fluids in general relativity

1980 ◽  
Vol 21 (10) ◽  
pp. 2785-2793 ◽  
Author(s):  
Jacques Demaret ◽  
Vincent Moncrief
Keyword(s):  
General Relativity ◽  
Hamiltonian Formalism ◽  
Perfect Fluids

scholarly journals The Boltzmann Equations in General Relativity

1936 ◽  
Vol 4 (4) ◽  
pp. 238-253 ◽  
Author(s):  
A. G. Walker
Keyword(s):  
General Relativity ◽  
Continuous Medium ◽  
Equations Of Motion ◽  
Free Particle ◽  
Energy Tensor ◽  
Principle Of Equivalence ◽  
Boltzmann Equations ◽  
Conservation Equations ◽  
Proper Mass ◽  
Conservation Of Momentum

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

scholarly journals Spinning bodies in general relativity

2016 ◽  
Vol 13 (08) ◽  
pp. 1640002 ◽  
Author(s):  
J. W. van Holten
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Momentum Tensor ◽  
Curved Space ◽  
Constants Of Motion ◽  
Spinning Bodies ◽  
Compact Spinning ◽  
Independent Derivation

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

scholarly journals On the nonperturbative graviton propagator

2018 ◽  
Vol 33 (36) ◽  
pp. 1850220 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Functional Integration ◽  
Hamiltonian Formalism ◽  
Planck Scale ◽  
Functional Integral ◽  
Perturbative Expansion ◽  
Length Scale ◽  
Leading Order ◽  
Starting Point ◽  
Graviton Propagator

To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the expansion over a scale of the discrete lapse-shift functions and has maxima at finite (Planck scale) areas/lengths and rapidly decreases at large areas/lengths, as we have mainly considered previously; the phase arises in the leading order (Regge action) of the stationary phase expansion. Now we consider the possibility of confining ourselves to these leading terms in a certain region of the parameters of the theory; consider background edge lengths as an optimal starting point for the perturbative expansion of the theory; estimate the background length scale and consider the form of the graviton propagator. In parallel with the general simplicial structure, we consider the simplest periodic simplicial structure with a part of the variables frozen (“hypercubic”), for which also the propagator in the leading approximation over metric variations can be written in a closed form.

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