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Piotr T. Chrusciel - Research
Those are the research areas to which I have contributed:
- Global structure of solutions of Einstein equations:
global properties, cosmic censorship, asymptotics
- Global Lorentzian geometry: the structure of horizons and the area
theorem
- The general relativistic constraint equations
- The mass in general relativity
- Classification of black hole space-times
- Nonlinear partial differential equations
Here is a short description of some of my papers - it
illustrates fairly well my research interests.
- Global structure of solutions of Einstein equations:
global structure of Robinson-Trautman space-times
In the papers [14,15] I prove that all
Robinson-Trautman space-times which describe asymptotically flat
(in light-like directions) smooth metrics form black holes. This
is the only existing result of this kind in the literature without
any symmetry hypotheses (all the remaining results on black hole
formation assume spherical symmetry). The result is obtained by
proving global existence to the future of solutions of the
Calabi-Robinson-Trautman equation [14], and by proving an
asymptotic expansion for its solutions as time goes to infinity
[15].
- Global structure of solutions of Einstein equations:
cosmic censorship
In the paper [34] I prove, in collaboration with A.
Rendall, that strong cosmic censorship holds in the class of
locally spatially homogeneous metrics. This problem turns out to
be rather subtle, because of the complicated, presumably chaotic,
dynamical behavior of some metrics in this class ( e.g. the
so called Bianchi IX metrics).
In the paper [7], written in collaboration with B. Berger
and V. Moncrief, I prove that strong cosmic censorship holds in
the class of asymptotically flat space-times with cylindrical
symmetry. This is the only existing result of this kind in which a
non-linear problem is considered and in which no smallness
hypotheses are made on the initial data.
In [13] I prove that the existence of a curvature
singularity is stable under non-linear
perturbations of a class of Kasner metrics.
Until [45] (where my results are generalised)
this was the only existing result on non-linear stability of a
curvature singularity.
In [30] we give an exhaustive description of Cauchy
horizons in Gowdy space-times; this result, together
with [45], gives the first proof of SCC in this
class of space-times.
- Global structure of solutions of Einstein equations:
asymptotics
In the paper [12] I classify the asymptotic structures
of Ashtekar-Hansen [4] type at spatial infinity
under a geodesic regularity condition, and shown that the
non-uniqueness of those structures does not lead to ambiguities in
the definition of global quantities such as energy-momentum, etc.
In the papers [3,2] (cf. also
[1]), written in collaboration with L. Andersson and
H. Friedrich, we establish existence of a large class of solutions
of vacuum Einstein equations, the asymptotic properties of which
correspond those conjectured by Bondi [8], Sachs
[46] and Penrose [42] many years ago.
Until the work in [3,2] the existence of such
space-times was the main open problem in the understanding of the
gravitational field in the radiating regime. A fundamental
contribution to this problem has been made by Friedrich in
[38], who shifted the problem to that of the
construction of appropriate initial data ``of hyperboloidal
type''. The construction of appropriate initial data has been
performed in [3,2].
In the paper [32], written in collaboration with M.
McCallum and D. Singleton, I show that the Bondi-Sachs analysis
of the asymptotic behavior of the gravitational field in
light-like directions can be generalized to include logarithmic
expansions in , in a manner consistent with the Einstein vacuum
equations. We find families of new constants of motion in the
radiation regime.
- Global Lorentzian geometry: the structure of horizons and the area
theorem
A classical theorem of Hawking and Ellis asserts that the
area of sections of piecewise smooth horizons is non-increasing
towards the future (under appropriate energy conditions and global
restrictions on the space-times into consideration). Now in the
paper [24], written in collaboration with G.
Galloway, Cauchy and event horizons are constructed which are not
differentiable on a dense set. Those examples make it clear that
the hypothesis that the horizon is piecewise smooth made by
Hawking and Ellis does not have any justification. In the
paper [23], written in collaboration with E. Delay, G.
Galloway, and R. Howard, we show that the area theorem holds
without any supplementary hypotheses of differentiability of the
event horizon. This requires overcoming various severe technical
hurdles related to the low potential differentiability of the
horizon under consideration. The above quoted papers also prove
several new results concerning the structure of horizons.
- The general relativistic constraint equations
The initial data for
Einstein equations are not aribrary, they have to satisfy a set of
constraint equations. In a series of papers in collaboration with
R. Beig, E. Delay, J. Isenberg, R. Mazzeo, D. Pollack and R.
Schoen we show how to obtain new solutions of those equations by
gluing old ones. This allows the construction of initial data sets
with interesting properties: For example, those techniques can be
used to prove existence of asymptotically simple vacuum
space-times [22,21]; the existence of such
space-times has been conjectured by Penrose in the 1960's, but was
open until our work. We prove that one can glue together generic
initial data sets [27,6], leading sometimes to
black hole space-times with non-connected
horizons [33]. The gluing can be used to construct
spatially compact, vacuum, maximal globally hyperbolic space-times
which do not contain any compact hypersurfaces with constant mean
curvature [26]; this is very annoying, as CMC
hypersurfaces provide a very convenient time function, whenever
they exist.
- The mass in general relativity
In the paper [10] I prove that the ADM mass is a
geometric invariant of an initial data set. While this result is
often attributed to Bartnik [5] in the literature,
the proof in [10] has been obtained independently, about
at the same time, and is much simpler than Bartnik's proof. (The
hypotheses of [10] are marginally stronger than those of
[5].) In the paper [11] I show that
the ADM mass is a geometric invariant of a boost-type domain in a
space-time. This generalizes the results of
[5,10] to a space-time setting.
In the paper [28] I prove, in collaboration with J.
Jezierski and Malcolm McCallum, that the Trautman-Bondi mass is (up
to a multiplicative constant) the unique functional, in a large
class of natural functionals, which is monotonous in time for all
vacuum solutions of Einstein equations for which it is well defined.
This is the first uniqueness result for the mass of the
gravitational field in the radiating regime.
In a series of classical
papers [44,43], Penrose and
collaborators have suggested proofs of the positive energy
theorem using purely Lorentzian techniques. In [25] and
in [20] we show that those methods can indeed be used
to prove the result, assuming that the metrics are uniformly
Schwarzschildian near infinity.
- Classification of black hole space-times
In the paper [18] I have finished the classification of
static vacuum black holes which contain an asymptotically flat
spacelike hypersurface with compact interior with a ``black hole
inner boundary''. Recall that this problem has a long history,
starting with the pioneering work of Israel [40].
The most complete result existing in the literature before my paper
was that of Bunting and Masood-ul-Alam [9] who
show, roughly speaking, that all appropriately regular such black
holes which do not contain degenerate horizons belong to the
Schwarzschild family. In the paper [18] I remove the
condition of non-degeneracy of the event horizon, and show that the
Schwarzschild black holes exhaust the family of all appropriately
regular black hole space-times. (The paper [19]
contains an improvement of similar previous results concerning the
electro-vacuum black holes, under the restrictive condition that
all degenerate components of the black hole carry charges of the
same sign.)
In the paper [17] I show, by constructing a
counterexample, that one of the key theorems of the classification
theory of black holes, due to Hawking (the so-called ``strong
rigidity theorem''), was wrong. In [16] I give a
corrected statement of the theorem, together with a proof.
In the paper [36], written in collaboration with R.
Wald, it is shown that the space-time exterior to a stationary black
hole is simply connected, and that sections of stationary black holes
have spherical topologies. This result fills a gap in a similar result
by Hawking [39].
- Nonlinear partial differential equations of hyperbolic type
In the paper [35], written in collaboration with J.
Shatah, global solvability of the Cauchy problem for the Yang-Mills
equations on an arbitrary globally hyperbolic four dimensional
Lorentzian manifold is proved. This was a classical open problem
since the fundamental work of Eardley and Moncrief
[37] (cf. also
[41]), who have proved such a statement on
four dimensional Minkowski space-time.
In [31], written in collaboration with my PhD
student O. Lengard, we prove propagation of polyhomogeneous
singularities for a class of semi-linear wave equations.
- Miscellaneous
In the paper [29] an invariant definition of Yang Mills
charges at spatial infinity is given.
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Piotr Chrusciel
2004-10-28