Compatibility complex for black hole spacetimes

Abstract

The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (S.A, T.B.) in [2] is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from [2] with the first compatibility operator from that complex.

Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

“…As in both cases, the central operator is the Spencer operator but not the exterior derivative, contrary to what is done in ([6] [7] [8]) and the corresponding references, in particular ( [6], Ref. [8]), we do not agree on the effectivity of their definition of "involutivity" ( [6], pp. 1608-1609).…”

“…R J T ⊂ can be achieved as a simple exercise of formal integrability, with no extra physical technical tool, contrary to ( [6] [7] [8]). As the computations will be explicitly done, the numbers of CC obtained will bring serious doubts about the validity of the results obtained in the above references, later confirmed with the K metric.…”

“…We have just a "jump" in the order and, for the details, refer the reader to the quite delicate Example 3.14 of ( [9], p 119-125) in which it is already difficult to discover how many new second order CC should be introduced though the initial system is trivially FI with coefficients in  . Such a result could not even be imagined while using the methods of ( [6] [7] [8]).…”