Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 011, 57 pages      arXiv:2004.09651      https://doi.org/10.3842/SIGMA.2022.011

Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

Igor Khavkine ab
a) Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
b) Charles University in Prague, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic

Received March 16, 2021, in final form January 22, 2022; Published online February 04, 2022

Abstract
We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations (for spins $s=0,1,2$). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.

Key words: Schwarzschild black hole; linearized gravity; harmonic gauge; Regge-Wheeler equation; rational ODE; computer algebra; rational solution; decoupling.

pdf (786 kb)   tex (76 kb)  

References

  1. Abramov S.A., On the differential and full algebraic complexities of operator matrices transformations, in Computer Algebra in Scientific Computing, Lecture Notes in Comput. Sci., Vol. 9890, Springer, Cham, 2016, 1-14.
  2. Abramov S.A., Ryabenko A.A., Khmelnov D.E., Revealing matrices of linear differential systems of arbitrary order, Program. Comput. Softw. 43 (2017), 67-74.
  3. Aksteiner S., Andersson L., Bäckdahl T., New identities for linearized gravity on the Kerr spacetime, Phys. Rev. D 99 (2019), 044043, 19 pages, arXiv:1601.06084.
  4. Aksteiner S., Bäckdahl T., Symmetries of linearized gravity from adjoint operators, J. Math. Phys. 60 (2019), 082501, 21 pages, arXiv:1609.04584.
  5. Barack L., Lousto C.O., Perturbations of Schwarzschild black holes in the Lorenz gauge: formulation and numerical implementation, Phys. Rev. D 72 (2005), 104026, 25 pages, arXiv:gr-qc/0510019.
  6. Barack L., Ori A., Gravitational self-force and gauge transformations, Phys. Rev. D 64 (2001), 124003, 13 pages, arXiv:gr-qc/0107056.
  7. Barack L., Pound A., Self-force and radiation reaction in general relativity, Rep. Progr. Phys. 82 (2018), 016904, 46 pages, arXiv:1805.10385.
  8. Benini M., Dappiaggi C., Murro S., Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states, J. Math. Phys. 55 (2014), 082301, 28 pages, arXiv:1404.4551.
  9. Berndtson M.V., Harmonic gauge perturbations of the Schwarzschild metric, Ph.D. Thesis, University of Colorado, 2007, arXiv:0904.0033.
  10. Berti E., Cardoso V., Starinets A.O., Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26 (2009), 163001, 108 pages, arXiv:0905.2975.
  11. Bičák J., Katz J., On the uniqueness of harmonic coordinates, Czechoslovak J. Phys. 55 (2005), 105-118, arXiv:gr-qc/0503018.
  12. Brunetti R., Fredenhagen K., Rejzner K., Quantum gravity from the point of view of locally covariant quantum field theory, Comm. Math. Phys. 345 (2016), 741-779, arXiv:1306.1058.
  13. Bussola F., De Donder gauge graviton Green's function in Schwarzschild spacetime with an outlook toward the Feynman propagator, Master's Thesis, University of Trento, 2015.
  14. Chandrasekhar S., The mathematical theory of black holes, International Series of Monographs on Physics, Vol. 69, The Clarendon Press, Oxford University Press, New York, 1983.
  15. Cluzeau T., Quadrat A., Factoring and decomposing a class of linear functional systems, Linear Algebra Appl. 428 (2008), 324-381.
  16. Couch W.E., Newman E.T., Algebraically special perturbations of the Schwarzschild metric, J. Math. Phys. 14 (1973), 285-286.
  17. Dafermos M., Holzegel G., Rodnianski I., The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math. 222 (2019), 1-214, arXiv:1601.06467.
  18. Dafermos M., Rodnianski I., Lectures on black holes and linear waves, in Evolution Equations, Clay Math. Proc., Vol. 17, Amer. Math. Soc., Providence, RI, 2013, 97-205, arXiv:0811.0354.
  19. Dimock J., Kay B.S., Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. I, Ann. Physics 175 (1987), 366-426.
  20. Donninger R., Schlag W., Soffer A., A proof of Price's law on Schwarzschild black hole manifolds for all angular momenta, Adv. Math. 226 (2011), 484-540, arXiv:0908.4292.
  21. Dotti G., Black hole non-modal linear stability: the Schwarzschild (A)dS cases, Classical Quantum Gravity 33 (2016), 205005, 43 pages, arXiv:1307.3340.
  22. Fewster C.J., Hunt D.S., Quantization of linearized gravity in cosmological vacuum spacetimes, Rev. Math. Phys. 25 (2013), 1330003, 44 pages, arXiv:1203.0261.
  23. Frolov V.P., Novikov I.D., Black hole physics: basic concepts and new developments, Fundamental Theories of Physics, Vol. 96, Kluwer Academic Publishers Group, Dordrecht, 1998.
  24. Glampedakis K., Johnson A.D., Kennefick D., Darboux transformation in black hole perturbation theory, Phys. Rev. D 96 (2017), 024036, 14 pages, arXiv:1702.06459.
  25. Gross D.J., Perry M.J., Yaffe L.G., Instability of flat space at finite temperature, Phys. Rev. D 25 (1982), 330-355.
  26. Hintz P., Vasy A., The global non-linear stability of the Kerr-de Sitter family of black holes, Acta Math. 220 (2018), 1-206, arXiv:1606.04014.
  27. Hung P.-K., The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part, arXiv:1803.03881.
  28. Hung P.-K., The linear stability of the Schwarzschild spacetime in the harmonic gauge: even part, arXiv:1909.06733.
  29. Johnson T.W., The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge, Ph.D. Thesis, Imperial College London, 2018, arXiv:1803.04012.
  30. Johnson T.W., The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge, Ann. PDE 5 (2019), 13, 92 pages, arXiv:1810.01337.
  31. Johnson T.W., On the link between the Maxwell and linearised Einstein equations on Schwarzschild, Classical Quantum Gravity 37 (2020), 067001, 10 pages, arXiv:1907.09521.
  32. Khavkine I., Explicit triangular decoupling of the separated vector wave equation on Schwarzschild into scalar Regge-Wheeler equations, J. Phys. Conf. Ser. 968 (2018), 012006, 20 pages, arXiv:1711.00585.
  33. Khavkine I., IDEAL characterization of higher dimensional spherically symmetric black holes, Classical Quantum Gravity 36 (2019), 045001, 19 pages, arXiv:1807.09699.
  34. Khavkine I., Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations, Differential Geom. Appl. 70 (2020), 101632, 14 pages, arXiv:1801.09800.
  35. Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  36. Maassen van den Brink A., Analytic treatment of black-hole gravitational waves at the algebraically special frequency, Phys. Rev. D 62 (2000), 064009, 16 pages, arXiv:gr-qc/0001032.
  37. Martel K., Poisson E., Gravitational perturbations of the Schwarzschild spacetime: a practical covariant and gauge-invariant formalism, Phys. Rev. D 71 (2005), 104003, 13 pages, arXiv:gr-qc/0502028.
  38. Oaku T., Takayama N., Tsai H., Polynomial and rational solutions of holonomic systems, J. Pure Appl. Algebra 164 (2001), 199-220, arXiv:math.AG/0001064.
  39. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
  40. Regge T., Wheeler J.A., Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957), 1063-1069.
  41. Rosa J.G., Dolan S.R., Massive vector fields on the Schwarzschild spacetime: quasinormal modes and bound states, Phys. Rev. D 85 (2012), 044043, 16 pages, arXiv:1110.4494.
  42. Sasaki M., Tagoshi H., Analytic black hole perturbation approach to gravitational radiation, Living Rev. Relativ. 6 (2003), 2003-6, 48 pages, arXiv:gr-qc/0306120.
  43. Seiler W.M., Involution: the formal theory of differential equations and its applications in computer algebra, Algorithms and Computation in Mathematics, Vol. 24, Springer-Verlag, Berlin, 2010.
  44. Shah A.G., Whiting B.F., Raising and lowering operators of spin-weighted spheroidal harmonics, Gen. Relativity Gravitation 48 (2016), 78, 19 pages, arXiv:1503.02618.
  45. Stewart J.M., Hertz-Bromwich-Debye-Whittaker-Penrose potentials in general relativity, Proc. Roy. Soc. London Ser. A 367 (1979), 527-538.
  46. Tataru D., Decay of linear waves on black hole space-times, in Surveys in Differential Geometry 2015. One Hundred Years of General Relativity, Surv. Differ. Geom., Vol. 20, Int. Press, Boston, MA, 2015, 157-182.
  47. Wald R.M., Construction of solutions of gravitational, electromagnetic, or other perturbation equations from solutions of decoupled equations, Phys. Rev. Lett. 41 (1978), 203-206.
  48. Wasow W., Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1965.
  49. Weibel C.A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.

Previous article  Next article  Contents of Volume 18 (2022)