286 research outputs found
The hyperboloidal foliation method
The Hyperboloidal Foliation Method presented in this monograph is based on a
(3+1)-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It
allows us to establish global-in-time existence results for systems of
nonlinear wave equations posed on a curved spacetime and to derive uniform
energy bounds and optimal rates of decay in time. We are also able to encompass
the wave equation and the Klein-Gordon equation in a unified framework and to
establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a
large class of nonlinear interactions. The hyperboloidal foliation of Minkowski
spacetime we rely upon in this book has the advantage of being geometric in
nature and, especially, invariant under Lorentz transformations. As stated, our
theory applies to many systems arising in mathematical physics and involving a
massive scalar field, such as the Dirac-Klein-Gordon system. As it provides
uniform energy bounds and optimal rates of decay in time, our method appears to
be very robust and should extend to even more general systems.Comment: 160 page
The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields
We study the initial value problem for two fundamental theories of gravity,
that is, Einstein's field equations of general relativity and the
(fourth-order) field equations of f(R) modified gravity. For both of these
physical theories, we investigate the global dynamics of a self-gravitating
massive matter field when an initial data set is prescribed on an
asymptotically flat and spacelike hypersurface, provided these data are
sufficiently close to data in Minkowski spacetime. Under such conditions, we
thus establish the global nonlinear stability of Minkowski spacetime in
presence of massive matter. In addition, we provide a rigorous mathematical
validation of the f(R) theory based on analyzing a singular limit problem, when
the function f(R) arising in the generalized Hilbert-Einstein functional
approaches the scalar curvature function R of the standard Hilbert-Einstein
functional. In this limit we prove that f(R) Cauchy developments converge to
Einstein's Cauchy developments in the regime close to Minkowski space. Our
proofs rely on a new strategy, introduced here and referred to as the
Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of
the Hyperboloidal Foliation Method (HFM) which we used earlier for the
Einstein-massive field system but for a restricted class of initial data. Here,
the data are solely assumed to satisfy an asymptotic flatness condition and be
small in a weighted energy norm. These results for matter spacetimes provide a
significant extension to the existing stability theory for vacuum spacetimes,
developed by Christodoulou and Klainerman and revisited by Lindblad and
Rodnianski.Comment: 127 pages. Selected chapters from a boo
Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem
We study gravitational perturbations of Schwarzschild spacetime by solving a
hyperboloidal initial value problem for the Bardeen-Press equation.
Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us
to have access to the gravitational waveform at null infinity in a general
setup. We argue that this hyperboloidal approach leads to a more accurate and
efficient calculation of the radiation signal than the common approach where a
timelike outer boundary is introduced. The method can be generalized to study
perturbations of Kerr spacetime using the Teukolsky equation.Comment: 14 pages, 9 figure
Stability of a coupled wave-Klein-Gordon system with quadratic nonlinearities
Using the hyperboloidal foliation method, we establish stability results for
a coupled wave-Klein-Gordon system with quadratic nonlinearities. In
particular, we investigate quadratic wave-Klein-Gordon interactions in which
there are no derivatives on the massless wave component. By combining
hyperboloidal energy estimates with appropriate transformations of our fields,
we are able to show global existence of solutions for sufficiently small
initial data.Comment: Published version. Both authors are grateful to the anonymous
referees, whose comments improve a lot on the presentation and the precision
of the articl
Binary black hole coalescence in the large-mass-ratio limit: the hyperboloidal layer method and waveforms at null infinity
We compute and analyze the gravitational waveform emitted to future null
infinity by a system of two black holes in the large mass ratio limit. We
consider the transition from the quasi-adiabatic inspiral to plunge, merger,
and ringdown. The relative dynamics is driven by a leading order in the mass
ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To
compute the waveforms we solve the Regge-Wheeler-Zerilli equations in the
time-domain on a spacelike foliation which coincides with the standard
Schwarzschild foliation in the region including the motion of the small black
hole, and is globally hyperboloidal, allowing us to include future null
infinity in the computational domain by compactification. This method is called
the hyperboloidal layer method, and is discussed here for the first time in a
study of the gravitational radiation emitted by black hole binaries. We
consider binaries characterized by five mass ratios, ,
that are primary targets of space-based or third-generation gravitational wave
detectors. We show significative phase differences between finite-radius and
null-infinity waveforms. We test, in our context, the reliability of the
extrapolation procedure routinely applied to numerical relativity waveforms. We
present an updated calculation of the gravitational recoil imparted to the
merger remnant by the gravitational wave emission. As a self consistency test
of the method, we show an excellent fractional agreement (even during the
plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the
gravitational wave angular momentum flux computed at null infinity. New results
concerning the radiation emitted from unstable circular orbits are also
presented.Comment: 22 pages, 18 figures. Typos corrected. To appear in Phys. Rev.
Hyperboloidal evolution of test fields in three spatial dimensions
We present the numerical implementation of a clean solution to the outer
boundary and radiation extraction problems within the 3+1 formalism for
hyperbolic partial differential equations on a given background. Our approach
is based on compactification at null infinity in hyperboloidal scri fixing
coordinates. We report numerical tests for the particular example of a scalar
wave equation on Minkowski and Schwarzschild backgrounds. We address issues
related to the implementation of the hyperboloidal approach for the Einstein
equations, such as nonlinear source functions, matching, and evaluation of
formally singular terms at null infinity.Comment: 10 pages, 8 figure
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