Research
Interests
—Juan Antonio Valiente Kroon—
My research interests lie in the area of Mathematical Classical General
Relativity, that is, the qualitative study of the properties of solutions of
the equations governing the gravitational field. In particular, my research
focuses on the study of asymptotic and conformal properties of the Einstein
field equations.
In
recent years I have been analysing the structure of spatial infinity and null
infinity for asymptotically flat spacetimes. This is a long-term project geared
towards providing a mathematically rigorous framework to R. Penrose's proposal
of characterising the gravitational field of isolated systems in General
Relativity by means of the notion of asymptotic simplicity [4]. This project will provide an
assessment of the physical relevance and limitation of Penrose's ideas.
Extending H. Friedrich's ideas —see e.g. [3] on the conformal field equations I
have been able to show that the standard smoothness assumptions on the
behaviour of the gravitational field at the conformal boundary of spacetime
—null infinity— are more restrictive than originally thought
—see [LOP12, LOP15, LOP16, LOP17]. More precisely, I have gathered
evidence for a rigidity conjecture for the notion of asymptotically simplicity.
This conjecture states that if a given asymptotically flat spacetime happens to
have as smooth extension to null infinity (both future and past), then any
Cauchy hypersurface from which the spacetime could have arisen is necessarily
stationary near infinity. The evidence for this conjecture shows that the
developments of the initial data sets for spacetimes containing dynamical black
holes —like the Brill-Lindquist, Misner and Bowen-York— do not have
a smooth null infinity —see [LOP28].
Currently,
I am particularly interested in the effects that the non-smoothness of the
gravitational field at infinity has on objects of physical interest for which
the notion of asymptotically simplicity provides a suitable framework. These
include reductions of the asymptotic symmetry group (the BMS group), the Bondi
mass, angular momentum and the Newman-Penrose constants. On a short term I will
be working on a proof for the rigidity conjecture for asymptotically simple
spacetimes mentioned in the previous paragraph —starting with the case of
time symmetric, conformally flat initial data, and then move to more general
classes of data. Once obtained, such a proof should exhibit a very delicate
interplay between conformal structures of the spacetime, PDE-theory aspects of
the propagation equations and group theoretical properties of the Einstein
field equations.
Initial
data sets of the type aring in the rigidity conjecture have been abstractly
constructed by means of some gluing constructions by Corvino & Schoen and
Chrusciel & Delay —see e.g. [2,1]. Recently, I have started a
collaboration with J. L. Jaramillo
(Institute of Astophysics of Andalucia) to implement these gluing
constructions numerically. In this way one will obtain initial data sets for
numerical simulations with a smooth null infinity. A natural progression of
these calculations is the numerical evolution of such initial data sets.
My
work on the asymptotics of the gravitational field has led to a working
interest on the properties of the Einstein constraint equations and the
construction of initial data sets. In particular, using some ideas from
asymptotics I have been able to provide proofs for some conjectures concerning
the existence of particular hypersurfaces for certain spacetimes —like
conformally flat slices in the Kerr spacetime [LOP13, LOP14]. In the future I plan to work on
some aspects of the construction and characterisation of initial data sets with
a particular emphasis on finding ways of encoding physics in the solutions to
the Einstein constraints —so that the resulting initial data sets are
more realistic.
On
a related issue, I have a collaboration with A. García-Parrado (University
of Linköping, Sweden) in order to develop a 3+1 version of the
equivalence problem for spacetime manifolds —see [LOP18, LOP20]. The aim of this project is to
combine analytical techniques with ideas of the study of exact solutions to the
Einstein field equations to obtain invariant characterisations of initial data
sets of interesting and physically relevant solutions —like Schwarzschild
and Kerr spacetimes. These characterisations have a natural application in
numerical simulations.
Note: LOPX refers to publication number X in the attached list of
publications.
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