Diffeomorphism Symmetry in the Lagrangian Formulation of Gravity

Samanta, Saurav

doi:10.1007/s10773-008-9914-8

Cited by 19 publications

(51 citation statements)

References 26 publications

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“…There is another statement in [41] that can also be found in many places "it is well known that this decomposition plays a central role in all Hamiltonian formulations of general relativity". This sentence combined with Hawking's "spiritual" statement forces one to conclude that the Hamiltonian formulation by itself contradicts the spirit of GR.…”

Section: Introductionmentioning

confidence: 86%

“…Soon after appearance of [32], Samanta [41] posed the question "whether it is possible to describe the diffeomorphism symmetries without recourse to the ADM decomposition". To answer this question, he derived the transformation (1) starting from the Einstein-Hilbert (EH) Lagrangian (not the ADM Lagrangian) and applying the Lagrangian method for recovering gauge symmetries based on the use of certain gauge identities that appear in [13].…”

Section: Introductionmentioning

confidence: 99%

“…It is important that (1) follows exactly from this procedure without the need of a field-dependent and non-covariant redefinition of the gauge parameters, which would be necessary in [25,32,[34][35][36][37][38][39] where the ADM Hamiltonian is used. The question of the equivalence of (1) and (2) does not even arise in the approach of [41]. In [41] the dif- 4 This point is explicitly stated in the historical papers of Salisbury and comments, for example, in [40]: "Had he [Rosenfeld] expressed this orthonormal tetrad in terms of the lapse and shift functions introduced by Arnowitt, Deser and Misner he would have obtained the triad form of their ADM Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

“…The question of the equivalence of (1) and (2) does not even arise in the approach of [41]. In [41] the dif- 4 This point is explicitly stated in the historical papers of Salisbury and comments, for example, in [40]: "Had he [Rosenfeld] expressed this orthonormal tetrad in terms of the lapse and shift functions introduced by Arnowitt, Deser and Misner he would have obtained the triad form of their ADM Hamiltonian. [11]" (The Salisbury's reference [11] is our [22] where neither the tetrad nor the triad form of the ADM Hamiltonian is discussed.)…”

Section: Introductionmentioning

confidence: 99%

“…Comparison of [32] and [41] allows us to conclude that ADM decomposition is not only inessential but incorrect because the EH Lagrangian leads directly to diffeomorphism invariance without the need of any field-dependent and non-covariant redefinition of gauge parameters; and a correct Hamiltonian formulation should give the same result. This discrepancy between these two recent results vindicates Hawking's old statement [42] "the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity", the more recent statements of Pons [37]: "Being non-intrinsic, the 3 + 1 decomposition is somewhat at odds with a generally covariant formalism, and difficulties arise for this reason", and Rovelli [43]: "The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning".…”

Section: Introductionmentioning

confidence: 99%

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The Hamiltonian formulation of general relativity: myths and reality

Kiriushcheva¹,

Kuzmin²

2011

Open Physics

197

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Abstract: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 0 = 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 0 = 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 fourdimensional 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 µν to lapse and shift functions and the three-metric , 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. PACS

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Hamiltonian formulation of general relativity: myths and reality

Kiriushcheva¹,

Kuzmin²

2011

Open Physics

197

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Symmetries of topological gravity with torsion in the hamiltonian and lagrangian formalisms

Banerjee

Gangopadhyay

Mukherjee

et al. 2010

J. High Energ. Phys.

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A systematic analysis of the symmetries of topological 3D gravity with torsion and a cosmological term, in the first order formalism, has been performed in details -both in the hamiltonian and lagrangian formalisms. This illuminates the connection between the symmetries of curved spacetime (diffeomorphisms plus local Lorentz transformations) with the Poincare gauge transformations. The Poincare gauge symmetries of the action are shown to be inequivalent to its gauge symmetries. Finally, the complete analysis is compared with the metric formulation where the diffeomorphism symmetry is shown to be equivalent to the gauge symmetry. *

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Gauge symmetry and W-algebra in higher derivative systems

Banerjee

Mukherjee

Paul

2011

J. High Energ. Phys.

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The problem of gauge symmetry in higher derivative Lagrangian systems is discussed from a Hamiltonian point of view. The number of independent gauge parameters is shown to be in general {\it{less}} than the number of independent primary first class constraints, thereby distinguishing it from conventional first order systems. Different models have been considered as illustrative examples. In particular we show a direct connection between the gauge symmetry and the W-algebra for the rigid relativistic particle.Comment: 1+22 pages, 1 figure, LaTeX, v2; title changed, considerably expanded version with new results, to appear in JHE

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Diffeomorphism Symmetry in the Lagrangian Formulation of Gravity

Cited by 19 publications

References 26 publications

The Hamiltonian formulation of general relativity: myths and reality

The Hamiltonian formulation of general relativity: myths and reality

Symmetries of topological gravity with torsion in the hamiltonian and lagrangian formalisms

Gauge symmetry and W-algebra in higher derivative systems

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