The Calabi complex and Killing sheaf cohomology

Khavkine, Igor

doi:10.1016/j.geomphys.2016.06.009

Cited by 22 publications

(51 citation statements)

References 74 publications

(175 reference statements)

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“…It is important to notice that the Einstein operator Ω → E ij = R ij − 1 2 ω ij ω rs R rs is self-adjoint with 6 terms though the Ricci operator is not with only 4 terms. Recently, many physicists (See [1], [2], [8], [9], [24]) have tried to construct the compatibility conditions (CC) of the Killing operator for various types of background metrics, in particular the three ones already quoted, namely an operator D 1 : S 2 T * → F 1 such that D 1 Ω = 0 generates the CC of Dξ = Ω. We have proved in the above references the following crucial results:…”

Section: ) Introductionmentioning

confidence: 91%

Generating Compatibility Conditions and General Relativity

Pommaret¹

2019

JMP

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The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with a kind of differential transcendence degree, also called degree of generality by A. Einstein in his 1930 letters to E. Cartan.The motivating examples that we provide are among the rare ones known in the literature and could be used as testing examples for future applications of computer algebra.

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Section: ) Introductionmentioning

confidence: 91%

Generating Compatibility Conditions and General Relativity

Pommaret¹

2019

JMP

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“…Completeness refers to the ability to express any local gauge-invariant observable in terms of linear combinations of derivatives of a given set. For technical reasons [19,20], it also becomes important to identify complete sets of differential relations between them, complete sets of differential relations between these differential relations, and so on. Phrased in mathematical terms, given a background metric g, we are interested in constructing a (full) compatibility complex for the corresponding Killing operator K [v], where full refers to the continuation of the sequence of differential relations until it terminates (becomes identically zero), a property that is usually required implicitly.…”

Section: Introductionmentioning

confidence: 99%

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

Khavkine

2019

Class. Quantum Grav.

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In linearized gravity, two linearized metrics are considered gauge-equivalent, hµν ∼ hµν + Kµν [v], when they differ by the image of the Killing operator, Kµν [v] = ∇µvν + ∇ν vµ. A universal (or complete) compatibility operator for K is a differential operator K1 such that K1 • K = 0 and any other operator annihilating K must factor through K1. The components of K1 can be interpreted as a complete (or generating) set of local gauge-invariant observables in linearized gravity. By appealing to known results in the formal theory of overdetermined PDEs and basic notions from homological algebra, we solve the problem of constructing the Killing compatibility operator K1 on an arbitrary background geometry, as well as of extending it to a full compatibility complex Ki (i ≥ 1), meaning that for each Ki the operator Ki+1 is its universal compatibility operator. Our solution is practical enough that we apply it explicitly in two examples, giving the first construction of full compatibility complexes for the Killing operator on these geometries. The first example consists of the cosmological FLRW spacetimes, in any dimension. The second consists of a generalization of the Schwarzschild-Tangherlini black hole spacetimes, also in any dimension. The generalization allows an arbitrary cosmological constant and the replacement of spherical symmetry by planar or pseudo-spherical symmetry.

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“…In many recent technical papers, a few physicists working on General relativity (GR) are trying to construct high order differential sequences while starting with the Kiling operator for a given metric (Minkowski, Schwarzschild, Kerr, ...) ( [1,2], [15,16], [38]). The (technical) methods involved are ranging from Killing/Killing-Yano tensors, Penrose spinors, Teukolski scalars or compexified frames.…”

Section: ) Introductionmentioning

confidence: 99%

Minkowski, Schwarzschild and Kerr Metrics Revisited

Pommaret¹

2018

JMP

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In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (10, order 1), Riemann (20, order 2), Bianchi (20, order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. However, even if they exhibited a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars, Killing-Yano tensors or the 2+2-formalism separating the two variables (t,r) from the two other angular variables of space-time. Using the formal theory of systems of partial differential equations and Lie pseudogroups, the purpose of this difficult computational paper is to provide new intrinsic differential homological methods involving the Spencer operator in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. Most of these new tools are now available as computer algebra packages.There are many different ways to look at such a system. The first natural one is to say that the only solution is y = 0 when u = v = 0. The second one is to look for the CC that must be satisfyed by u, v and we may adopt two possible presentations:• Substitute y and obtain the 2 fourth order CC:which are not differentially independent because one can easily check:• However, we also have:that is a second order CC. Finally, we obtain:and we discover that the CC of D = (P, Q) are generated by (A, B) but also by C alone, though any student will hesitate between the two possibilities !. Refering to differential homological algebra as in ([25]) while indicating the order of an operator below its arrow, the same trivial differential module M = 0 (care) defined by D has therefore two split resolutions:

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The Calabi complex and Killing sheaf cohomology

Cited by 22 publications

References 74 publications

Generating Compatibility Conditions and General Relativity

Generating Compatibility Conditions and General Relativity

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

Minkowski, Schwarzschild and Kerr Metrics Revisited

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