Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

Abstract

Abstract. In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t −2λ , where λ = 1, 2, · · · is the angular momentum. Our technique is to use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t −2λ . For the second set, in general, the solutions tend to some explicit profile at the rate t −2λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green's functions for time independent Schrödinger equations associated with these wave equations.

Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case $$|a|\ll M$$

“…1.3.9.) For the cases and see [ 106 ]. See [ 49 ] for the related massive Dirac equation not covered by ( 2 ).…”

Peeling of Dirac field on Kerr spacetime

“…On the other hand, Ma in [24] establishes the decays by transforming the Dirac equation to the Teukolsky spin wave equations and then use the vector field method which consists Morawertz estimates and r p -theory to obtain the decays in both time and spatial directions for these equations in Schwarzschild spacetime (see also [4] for the linearized gravity fields). The decay results obtained for the Dirac field in [24] (in detail the decay rates of the components are v −α τ −β ) is improved the one in [46]. Moreover, the time decay for the Dirac field on Kerr spacetime can be found in the works Finster et al [10,11], where the the decay rate is t −5/6 .…”

“…The inequality (24) guarantees H τ is a Cauchy hypersurface (for more details see the proof of [4, Lemma 2.1]). There are some results about the pointwise decays (also called Price's law) of the Dirac fields on the Schwarzschild and Kerr spacetimes [10,11,46,24]. In particular, in Schwarzschild spacetime Smoller and Xie [46] use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations, then show that the wave decays as t −2λ , where λ = 1, 2... is the angular momentum.…”

“…There are some results about the pointwise decays (also called Price's law) of the Dirac fields on the Schwarzschild and Kerr spacetimes [10,11,46,24]. In particular, in Schwarzschild spacetime Smoller and Xie [46] use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations, then show that the wave decays as t −2λ , where λ = 1, 2... is the angular momentum. On the other hand, Ma in [24] establishes the decays by transforming the Dirac equation to the Teukolsky spin wave equations and then use the vector field method which consists Morawertz estimates and r p -theory to obtain the decays in both time and spatial directions for these equations in Schwarzschild spacetime (see also [4] for the linearized gravity fields).…”

“…The pointwise decays (also called Price's law) of the Dirac and the generalized spin-n/2 zero rest-mass fields in the Minkowski spacetime were established by Andersson et al in [3] by analyzing Hertz potentials. The pointwise decays for the Dirac field on the Schwarzschild spacetime were studied by Smoller and Xie [46] and these results are improved in the recent work of Ma [24]. Another interesting aspect of the massless spin fields is the local integral formula which were establish initially in the Minkowski spacetime by Penrose [36].…”