## Hidden symmetries and decay for the wave equation on the Kerr spacetime.(English)Zbl 1373.35307

In this paper the authors consider the covariant wave equation $$\nabla^{a}\nabla_{a}\psi =0$$ in the exterior region of the slowly rotating Kerr spacetime. They prove boundedness and integrated energy decay for solutions of the above stated equation. Introduce $$r_{+}=M+\sqrt{M^2-a^2}$$, $$\Sigma = r^2+a^2\cos^2\theta$$; $$M$$ is the mass. Then in Boyer-Lindquist coordinates $$(x^{\alpha }) = (t, r, \theta , \phi )$$, the exterior region is given by the manifold $$\mathbb{R}\times (r_{+},\infty ) \times S^2$$ with the Lorentz metric $$g_{\mu\nu }$$. It is known that for $$0\leq |a| \leq M$$, the Kerr family of metrics describes an asymptotically flat, stationary and axi-symmetric solution of the vacuum Einstein equation, containing a rotating black hole with mass $$M$$ and angular momentum $$M a$$, and with the horizon located at $$r = r_{+}$$. The Schwarzschild spacetime is the subcase with $$a = 0$$. Here $$M>0$$ is fixed and the authors study the slowly rotating case, that is, $$|a|\ll M$$. The exterior region is globally hyperbolic, with the surfaces $$\Sigma_t$$ of constant $$t$$ as Cauchy surfaces. Then the wave equation is well posed in the exterior region even though the Kerr spacetime can be extended. Here it is considered some initial data on the hypersurface $$\Sigma_0$$.
The authors establish new original methods to prove two known and at the same time very important results. The first is the following statement known as Uniformly bounded positive energy, that is: “Given $$M>0$$, then there are positive constants $$C$$ and $$\bar{a}$$ such that if $$|a|\leq \bar{a}$$ and $$\psi : \mathbb{R}\times (r_{+},\infty )\times S^2\to \mathbb{R}$$ is a solution of the wave equation $$\nabla^{a}\nabla_{a}\psi =0$$, then for all $$t$$, $$E_{\text{model,3}}(\Sigma_t)\leq C E_{\text{model,3}}(\Sigma_0)$$”. Here $$E_{\text{model,3}}(\Sigma_t)$$ is the model energy.
The second result is known as Morawetz estimate, that is: “Given $$M > 0$$, there are positive constants $$\bar{a}$$, $$\bar{r}$$, $$C$$ and a function that is identically one for $$|r - 3 M | > \bar{r}$$ and zero otherwise, such that for all $$|a|\leq\bar{a}$$ and all smooth $$\psi$$ solving the above stated wave equation, there exists an integral inequality in which takes place the angular gradient in Boyer-Lindquist coordinates in the measure $$d^3\mu dt$$ and the model energy”.

### MSC:

 35Q76 Einstein equations 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C57 Black holes 83F05 Relativistic cosmology 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
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### References:

 [1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz, M. and Stegun, I. A., Eds., New York: Dover Publications, 1992. [2] S. Alexakis, A. D. Ionescu, and S. Klainerman, ”Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces,” Comm. Math. Phys., vol. 299, iss. 1, pp. 89-127, 2010. · Zbl 1200.83059 [3] L. Andersson and P. Blue, Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, 2013. · Zbl 1338.83094 [4] L. Andersson, P. Blue, and J. Nicolas, ”A decay estimate for a wave equation with trapping and a complex potential,” Int. Math. Res. Not., vol. 2013, iss. 3, pp. 548-561, 2013. · Zbl 1326.35038 [5] J. M. Bardeen, ”Timelike and null geodesics in the Kerr metric,” in Black Holes/Les Astres Occlus, DeWitt, C. and DeWitt, B. S., Eds., New York: Gordon and Breach Science Publishers, 1973, pp. 215-239. [6] P. Blue, ”Decay of the Maxwell field on the Schwarzschild manifold,” J. Hyperbolic Differ. Equ., vol. 5, iss. 4, pp. 807-856, 2008. · Zbl 1169.35057 [7] P. Blue and A. Soffer, ”A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold,” Lett. Math. Phys., vol. 81, iss. 3, pp. 227-238, 2007. · Zbl 1137.58011 [8] P. Blue and A. Soffer, ”Phase space analysis on some black hole manifolds,” J. Funct. Anal., vol. 256, iss. 1, pp. 1-90, 2009. · Zbl 1158.83007 [9] P. Blue and J. Sterbenz, ”Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space,” Comm. Math. Phys., vol. 268, iss. 2, pp. 481-504, 2006. · Zbl 1123.58018 [10] B. Carter, ”Global structure of the Kerr family of gravitational fields,” Phys. Rev., vol. 174, pp. 1559-1571, 1968. · Zbl 0167.56301 [11] B. Carter, ”Killing tensor quantum numbers and conserved currents in curved space,” Phys. Rev. D, vol. 16, iss. 12, pp. 3395-3414, 1977. [12] G. Caviglia, ”Conformal Killing tensors of order 2 for the Schwarzschild metric,” Meccanica, vol. 18, pp. 131-135, 1983. · Zbl 0543.53025 [13] C. Chanu, L. Degiovanni, and R. G. McLenaghan, ”Geometrical classification of Killing tensors on bidimensional flat manifolds,” J. Math. Phys., vol. 47, iss. 7, p. 073506, 2006. · Zbl 1112.53008 [14] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton, NJ: Princeton Univ. Press, 1993. · Zbl 0827.53055 [15] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $$|a| < M$$, 2014. · Zbl 1347.83002 [16] M. Dafermos and I. Rodnianski, ”A proof of Price’s law for the collapse of a self-gravitating scalar field,” Invent. Math., vol. 162, iss. 2, pp. 381-457, 2005. · Zbl 1088.83008 [17] M. Dafermos and I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, 2007. [18] M. Dafermos and I. Rodnianski, ”The red-shift effect and radiation decay on black hole spacetimes,” Comm. Pure Appl. Math., vol. 62, iss. 7, pp. 859-919, 2009. · Zbl 1169.83008 [19] M. Dafermos and I. Rodnianski, ”A new physical-space approach to decay for the wave equation with applications to black hole spacetimes,” in XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 421-432. · Zbl 1211.83019 [20] M. Dafermos and I. Rodnianski, ”A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds,” Invent. Math., vol. 185, iss. 3, pp. 467-559, 2011. · Zbl 1226.83029 [21] M. Dafermos and I. Rodnianski, ”Lectures on black holes and linear waves,” in Evolution Equations, Providence, RI: Amer. Math. Soc., 2013, vol. 17, pp. 97-205. · Zbl 1300.83004 [22] R. Donninger, W. Schlag, and A. Soffer, ”On pointwise decay of linear waves on a Schwarzschild black hole background,” Comm. Math. Phys., vol. 309, iss. 1, pp. 51-86, 2012. · Zbl 1242.83054 [23] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, New York: McGraw-Hill Book Company, 1953. · Zbl 0051.30303 [24] F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, ”Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry,” Comm. Math. Phys., vol. 230, iss. 2, pp. 201-244, 2002. · Zbl 1026.83029 [25] F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, ”An integral spectral representation of the propagator for the wave equation in the Kerr geometry,” Comm. Math. Phys., vol. 260, iss. 2, pp. 257-298, 2005. · Zbl 1089.83017 [26] H. Friedrich, ”Cauchy problems for the conformal vacuum field equations in general relativity,” Comm. Math. Phys., vol. 91, iss. 4, pp. 445-472, 1983. · Zbl 0555.35116 [27] V. P. Frolov and I. D. Novikov, Black Hole Physics, Dordrecht: Kluwer Academic Publishers Group, 1998, vol. 96. · Zbl 0978.83001 [28] D. Häfner, ”Sur la théorie de la diffusion pour l’équation de Klein-Gordon dans la métrique de Kerr,” Dissertationes Math. $$($$Rozprawy Mat.$$)$$, vol. 421, p. 102, 2003. · Zbl 1075.35093 [29] D. Häfner and J. Nicolas, ”Scattering of massless Dirac fields by a Kerr black hole,” Rev. Math. Phys., vol. 16, iss. 1, pp. 29-123, 2004. · Zbl 1064.83036 [30] G. Holzegel, Ultimately Schwarzschildean spacetimes and the black hole stability problem, 2010. [31] S. Klainerman, ”Uniform decay estimates and the Lorentz invariance of the classical wave equation,” Comm. Pure Appl. Math., vol. 38, iss. 3, pp. 321-332, 1985. · Zbl 0635.35059 [32] T. H. Koornwinder, ”Book Review: Symmetry and separation of variables,” Bull. Amer. Math. Soc., vol. 1, iss. 6, pp. 1014-1019, 1979. [33] J. Kronthaler, Decay rates for spherical scalar waves in the Schwarzschild geometry, 2007. [34] I. Łaba and A. Soffer, ”Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds,” Helv. Phys. Acta, vol. 72, iss. 4, pp. 274-294, 1999. · Zbl 0976.58019 [35] H. Lindblad and I. Rodnianski, ”Global existence for the Einstein vacuum equations in wave coordinates,” Comm. Math. Phys., vol. 256, iss. 1, pp. 43-110, 2005. · Zbl 1081.83003 [36] J. Luk, ”A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole,” Anal. PDE, vol. 5, iss. 3, pp. 553-625, 2012. · Zbl 1267.83065 [37] J. Marzuola, J. Metcalfe, D. Tataru, and M. Tohaneanu, ”Strichartz estimates on Schwarzschild black hole backgrounds,” Comm. Math. Phys., vol. 293, iss. 1, pp. 37-83, 2010. · Zbl 1202.35327 [38] W. Miller Jr., Symmetry and Separation of Variables, Reading, MA: Addison-Wesley Publishing Co., 1977, vol. 4. · Zbl 0368.35002 [39] C. S. Morawetz, ”The decay of solutions of the exterior initial-boundary value problem for the wave equation,” Comm. Pure Appl. Math., vol. 14, pp. 561-568, 1961. · Zbl 0101.07701 [40] P. J. Olver, Applications of Lie Groups to Differential Equations, Second ed., New York: Springer-Verlag, 1993, vol. 107. · Zbl 0785.58003 [41] B. O’Neill, The Geometry of Kerr Black Holes, Wellesley, MA: A K Peters Ltd., 1995. · Zbl 0828.53078 [42] R. H. Price, ”Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields,” Phys. Rev. D, vol. 5, pp. 2439-2454, 1972. [43] R.  H. Price, ”Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields,” Phys. Rev. D, vol. 5, pp. 2439-2454, 1972. [44] J. V. Ralston, ”Solutions of the wave equation with localized energy,” Comm. Pure Appl. Math., vol. 22, pp. 807-823, 1969. · Zbl 0209.40402 [45] J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds - with applications to black hole spacetimes, 2013. · Zbl 1343.35229 [46] Y. Shlapentokh-Rothman, ”Quantitative mode stability for the wave equation on the Kerr spacetime,” Ann. Henri Poincaré, vol. 16, iss. 1, pp. 289-345, 2015. · Zbl 1308.83104 [47] A.  A. Starobinskiv i, ”Amplification of waves during reflection from a rotating “black hole”,” Soviet J. Experimental Theor. Phys., vol. 37, p. 28, 1973. [48] D. Tataru, ”Local decay of waves on asymptotically flat stationary space-times,” Amer. J. Math., vol. 135, iss. 2, pp. 361-401, 2013. · Zbl 1266.83033 [49] D. Tataru and M. Tohaneanu, ”A local energy estimate on Kerr black hole backgrounds,” Int. Math. Res. Not., vol. 2011, iss. 2, pp. 248-292, 2011. · Zbl 1209.83028 [50] E. Teo, ”Spherical photon orbits around a Kerr black hole,” Gen. Relativity Gravitation, vol. 35, iss. 11, pp. 1909-1926, 2003. · Zbl 1047.83024 [51] S. Teukolsky, ”Perturbations of a rotating black hole I: fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations,” Astrophys. J., vol. 185, pp. 635-647, 1973. [52] M. Tohaneanu, ”Strichartz estimates on Kerr black hole backgrounds,” Trans. Amer. Math. Soc., vol. 364, iss. 2, pp. 689-702, 2012. · Zbl 1234.35275 [53] R. M. Wald, General Relativity, Chicago, IL: University of Chicago Press, 1984. · Zbl 0549.53001 [54] M. Walker and R. Penrose, ”On quadratic first integrals of the geodesic equations for type $$\{22\}$$ spacetimes,” Comm. Math. Phys., vol. 18, pp. 265-274, 1970. · Zbl 0197.26404 [55] B. F. Whiting, ”Mode stability of the Kerr black hole,” J. Math. Phys., vol. 30, iss. 6, pp. 1301-1305, 1989. · Zbl 0689.53041
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