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 (Aksteiner and Bäckdahl in Phys Rev Lett 121:051104, 2018) 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 Aksteiner and Bäckdahl (Phys Rev Lett 121:051104, 2018) with the first compatibility operator from that complex.

A Mathematical Comparison of the Schwarzschild and Kerr Metrics

“…= 0 but not strictly exact because R 2 is quite far from being FI as we have even R It follows from these examples and the many others presented in ( [20]) that we cannot agree with ( [1][2][3][4]). Indeed, it is clear that one can use successive prolongations in order to look for CC of order 1, 2, 3, ... and so on, selecting each time the new generating ones and knowing that Noetherian arguments will stop such a procedure ... after a while !.…”

“…Nevertheless, we obtain the following unexpected formal linearized result that will be used in a crucial intrinsic way for finding out the generating second order and third order CC: THEOREM 4.2: The rank of the previous system with respect to the four jet coodinates (ξ 1 3 , ξ 2 0 , ξ 1 0 , ξ 2 3 ) is equal to 2, for both the S and K metrics. We obtain in particular the two striking identities:…”

“…Proof: In the case of he S metric with a = 0, only the framed terms may not vanish and, denoting by " ∼ " a linear proportionality, we have already obtained mod(j 2 (Ω)): R 01,03 ∼ ξ 1 3 , R 03,23 ∼ ξ 0 2 , R 02,03 = 0, R 03,13 = 0 Hence, the rank of the system with respect to the 4 parametric jets (ξ 1 3 , ξ 2 0 , ξ 1 0 , ξ 2 3 ) just drops to 2 and this fact confirms the existence of the 5 additional first order equations obtained, as we saw, after two prolongations.…”

“…, that is 1 2 the coefficient of dtdφ in the metric ds 2 which is indeed 2ω tφ dtdφ. We may obtain therefore successively the Killing equations for the Kerr type metric, using sections of jet bundles and writing simply ξ∂ω = ξ r ∂ r ω = ξ 1 ∂ 1 ω + ξ 2 ∂ 2 ω while framing the principal derivative ξ j i of Ω ij :…”

“…Keeping in mind the study of the S-metric and the fact that ρ 01,03 = 0, ρ 03,13 = 0, ρ 02,03 = 0, ρ 03,13 = 0 while framing the leading terms not vanishing when a = 0, we get: R 01,03 ≡ ρ 01,01 ξ 1 3 + ρ 03,03 ξ 3 1 + (ρ 01,23 + ρ 03,21 )ξ 2 0 + ρ 01,02 ξ 2 3 + ρ 01,13 ξ 1 0 = 0 Then, taking into account the fact that ρ 01,12 = 0, ρ 02,12 = 0, ρ 12,13 = 0, we obtain similarly: R 01,12 ≡ (ρ 01,32 + ρ 03,12 )ξ 3 1 + ρ 12,21 ξ 2 0 + ρ 01,10 ξ 0 2 + +ρ 01,13 ξ 3 2 + ρ 01,02 ξ 0 1 = 0 The leading determinant does not vanish when a = 0 because, in this case, all terms are vanishing and we are left with the two linearly independent framed terms, a result amounting to ξ 1 3 = 0 ⇔ ξ 3 1 = 0 and ξ 0 2 = 0 ⇔ ξ 2 0 = 0 in the case of the S-metric in ( [19]). In the case of the K-metric, we may use the relations already framed in order to keep only the four parametric jets (ξ 1 3 , ξ 2 0 , ξ 1 0 , ξ 2 3 ) on the right side. We may also rewrite them as follows:…”