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

Abstract

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.

IDEAL characterization of higher dimensional spherically symmetric black holes

“…Next, we find a way to express the projector r 2 Ω µν onto the warped factor in terms of the curvature. Here we find a slight dimension dependence (as already noted in [15,Sec.3.3]). In dimension n ≥ 5, one can find a formula that involves only products and contractionsḡ ofT :…”

“…In the recent work [15], we have explicitly exhibited (by a different method) complete sets of linear invariants for each geometry in the gST family. Relating these invariants to the linearization of the IDEAL characterization tensors, as well as vice versa, can accomplish two goals: give a geometric interpretation to the invariants of [15] and to prove the completeness of the linearized invariants that can be obtained from the present work.…”

“…For future reference, we also compute some algebraic combinations among these tensors (see [15,Sec.3.3] for more intermediate steps of the calculations):…”

Generating Compatibility Conditions and General Relativity

“…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:…”

Linear Yang–Mills Theory as a Homotopy AQFT

“…[SDH14, BDS14, BDHS14, FL16, Ben16, BSS16], but there also exist similar developments for e.g. linearized gravity [FH13,BDM14,Kha16,Kha18] and linearized supergravity [HS13]. In addition to such non-interacting models, examples of perturbatively interacting quantum gauge theories were constructed in [Hol08, FR12, FR13, TZ18] by means of an appropriate adaption of the BRST/BV formalism to AQFT.…”