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Compatibility complex for black hole spacetimes. (English) Zbl 07356546
Summary: 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.
##### MSC:
 70Sxx Classical field theories 83Cxx General relativity 58Jxx Partial differential equations on manifolds; differential operators
Janet
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##### References:
 [1] Aksteiner, S.; Andersson, L.; Bäckdahl, T., New identities for linearized gravity on the Kerr spacetime, Phys. Rev. D, 99, 044043 (2019) [2] Aksteiner, S., Bäckdahl, T.: All local gauge invariants for perturbations of the Kerr spacetime. Phys. Rev. Lett. 121, 051104 (2018). arXiv:1803.05341 · Zbl 1437.83006 [3] Andersson, L.; Bäckdahl, T.; Blue, P., Second order symmetry operators, Class. Quantum Gravity, 31, 135015 (2014) · Zbl 1295.35022 [4] Andersson, L., Bäckdahl, T., Blue, P., Ma, S.: Stability for linearized gravity on the Kerr spacetime (2019). arXiv:1903.03859 [math.AP] [5] Bäckdahl, T., A formalism for the calculus of variations with spinors, J. Math. Phys., 57, 022502 (2016) · Zbl 1336.83032 [6] Baer, C., Ginoux, N., Pfaeffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. ESI lectures in mathematics and physics, Vol. 2. European Mathematical Society (2007). arXiv:0806.1036 [7] Barack, L., Ori, A.: Gravitational selfforce and gauge transformations. Phys. Rev. D64, 124003 (2001). arXiv:gr-qc/0107056 [gr-qc] [8] Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The MAPLE package Janet: I. polynomial systems. II. Linear partial differential equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing, Passau (Germany), pp. 31-54. Institut für Informatik, Technische Universität München, Garching (2003) [9] Calderbank, DMJ; Diemer, T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., 537, 67-103 (2001) · Zbl 0985.58002 [10] Canepa, G., Dappiaggi, C., Khavkine, I.: IDEAL characterization of isometry classes of FLRW and inflationary spacetimes. Class. Quantum Gravity 35, 035013 (2018). arXiv:1704.05542 · Zbl 1382.83127 [11] Čap, A.; Slovák, J.; Souček, V., Bernstein-Gelfand-Gelfand sequences, Ann. Math., 154, 97-113 (2001) · Zbl 1159.58309 [12] Coll, B.; Ferrando, JJ, Thermodynamic perfect fluid Its Rainich theory, J. Math. Phys., 30, 2918-2922 (1989) · Zbl 0691.76134 [13] Douglis, A.; Nirenberg, L., Interior estimates for elliptic systems of partial differential equations, Commun. Pure Appl. Math., 8, 503-538 (1955) · Zbl 0066.08002 [14] Ferrando, JJ; Sáez, JA, An intrinsic characterization of the Schwarzschild metric, Class. Quantum Gravity, 15, 1323-1330 (1998) · Zbl 0937.83006 [15] Ferrando, JJ; Sáez, JA, An intrinsic characterization of the Kerr metric, Class. Quantum Gravity, 26, 075013 (2009) · Zbl 1161.83363 [16] Ferrando, JJ; Sáez, JA, An intrinsic characterization of spherically symmetric spacetimes, Class. Quantum Gravity, 27, 205024 (2010) · Zbl 1202.83023 [17] Fröb, M.B., Hack, T.-P., Higuchi, A.: Compactly supported linearised observables in single-field inflation. J. Cosmol. Astropart. Phys. 2017, 043 (2017). arXiv:1703.01158 [18] Fröb, M.B., Hack, T.-P., Khavkine, I.: Approaches to linear local gauge-invariant observables in inflationary cosmologies. Class. Quantum Gravity 35, 115002 (2018). arXiv:1801.02632 · Zbl 1393.83034 [19] Goldschmidt, H., Existence theorems for analytic linear partial differential equations, Ann. Math., 86, 246-270 (1967) · Zbl 0154.35103 [20] Jezierski, J.: Energy an angular momentum of the weak gravitational waves on the Schwarzschild backround-quasilocal gauge-invariant formulation. General Relat. Gravitat. 31, 1855 (1999). arXiv:gr-qc/9801068 · Zbl 0952.83020 [21] Khavkine, I., Cohomology with causally restricted supports, Ann. Henri Poincaré, 17, 3577-3603 (2016) · Zbl 1354.81042 [22] Khavkine, I., The Calabi complex and Killing sheaf cohomology, J. Geom. Phys., 113, 131-169 (2017) · Zbl 1364.53066 [23] Khavkine, I.: Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation. Class. Quantum Gravity 36, 185012 (2019). arXiv:1805.03751 [24] Khavkine, I.: IDEAL characterization of higher dimensional spherically symmetric black holes. Class. Quantum Gravity 36, 045001 (2019). arXiv:1807.09699 [25] Kruglikov, BS; Lychagin, VV; Krupka, D.; Saunders, D., Geometry of differential equations, Handbook of Global Analysis, 725-771 (2008), Amsterdam: Elsevier, Amsterdam · Zbl 1236.58039 [26] Martel, K.; Poisson, E., Gravitational perturbations of the Schwarzschild spacetime: a practical covariant and gauge-invariant formalism, Phys. Rev. D, 71, 104003 (2005) [27] Merlin, C., Ori, A., Barack, L., Pound, A., van de Meent, M.: Completion of metric reconstruction for a particle orbiting a Kerr black hole. Phys. Rev. D 94, 104066 (2016). arXiv:1609.01227 [28] Penrose, R.; Rindler, W., Spinors and Space-time I & II. Cambridge Monographs on Mathematical Physics (1986), Cambridge: Cambridge University Press, Cambridge [29] Pommaret, J.-F.: Minkowski, Schwarzschild and Kerr metrics revisited. J. Mod. Phys. 9, 1970-2007 (2018). arXiv:1805.11958 [physics.gen-ph] [30] Pommaret, J.-F.: Generating compatibility conditions and general relativity. J. Mod. Phys. 10, 371-401 (2019). arXiv:1811.12186 [math.DG] [31] Pommaret, J.F.: “A mathematical comparison of the Schwarzschild and Kerr metrics (2020). arXiv:2010.07001 [32] Pound, A.; Merlin, C.; Barack, L., Gravitational self-force from radiation-gauge metric perturbations, Phys. Rev. D, 89, 024009 (2014) [33] Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra, Algorithms and Computation in Mathematics, vol. 24. Springer (2010) · Zbl 1205.35003 [34] Shah, A.G., Whiting, B.F., Aksteiner, S., Andersson, L., Bäckdahl, T. : Gauge-invariant perturbations of Schwarzschild spacetime (2016), arXiv:1611.08291 [35] Spencer, DC, Overdetermined systems of linear partial differential equations, Bull. Am. Math. Soc., 75, 179-240 (1969) · Zbl 0185.33801 [36] Stewart, JM; Walker, M., Perturbations of space-times in general relativity, Proc. R. Soc. Lond. A Math. Phys. Sci., 341, 49-74 (1974) [37] Tarkhanov, NN, Complexes of Differential Operators, Mathematics and Its Applications (1995), Dordrecht: Kluwer, Dordrecht [38] Thompson, JE; Chen, H.; Whiting, BF, Gauge invariant perturbations of the Schwarzschild spacetime, Class. Quantum Gravity, 34, 174001 (2017) · Zbl 1372.83051 [39] Thompson, JE; Wardell, B.; Whiting, BF, Gravitational self-force regularization in the Regge-Wheeler and easy gauges, Phys. Rev. D, 99, 124046 (2019) [40] Walker, M.; Penrose, R., On quadratic first integrals of the geodesic equations for type $$\{2,2\}$$ spacetimes, Commun. Math. Phys., 18, 265-274 (1970) · Zbl 0197.26404
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