Abstract: The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes whose orientable double cover is globally hyperbolic. These spacetimes have generalized Cauchy surfaces on which smooth initial data sets yield unique solutions. A more difficult problem is to characterize the class of spacetimes with closed timelike curves that admit a well-posed initial value problem. Examples of spacetimes with closed timelike curves are given for which initial data at past null infinity has been recently shown to yield unique solutions. Other examples show that confining closed timelike curves to compact regions is not sufficient to guarantee uniqueness. An approach to the characterization problem is suggested by the behavior of congruences of null rays. Interacting fields have not yet been studied, but particle models suggest that uniqueness (and possibly existence) is likely to be lost as the strength of the interaction increases.

Title of lecture 2: Open problems in general relativistic astrophysics

Abstract: The talk reviews a set of unsolved problems that arise from studies of relativistic stars and black holes. Stated in a way that reflects the author's bias on what is likely to be true, the list is:

• I. Symmetries of perfect-fluid equilibria.
  • A. Prove that all static perfect fluid equilibria are spherical.
  • B. Prove that there exist stationary nonaxisymmetric perfect-fluid equilibria, relativistic analogues of the Dedekind ellipsoids of Newtonian gravity.
• II. Stationary non-axisymmetric configurations exist as solutions to the time-independent Einstein-Euler equations, linearized about stationary axisymmetric equilibria. One way to prove IB is to prove linearization stability of the time-independent Einstein-Euler equations.
• III. Prove the existence of an analogue in general relativity of the Newtonian binary systems that are stationary in a rotating frame: Show that there is a class of radiative solutions describing systems of binary stars and binary black holes and having a helical Killing vector. These spacetimes correspond to the Schild solutions, stationary spacetimes describing two opposite charges with equal amounts of ingoing and outgoing radiation.
• IV. Black hole stability
  • A. Show that Kerr black holes are stable (Whiting's proof shows that there are no unstable outgoing modes, but the outgoing modes are not complete).
  • B. Show that the Price fall-off describes the late-time behavior of perturbations of Schwarzschild black holes.

	Streaming audio, video and slides	Streaming audio and slides	Download video (Real Player, ~ 15 Mo)	Download audio (Real Player, ~ 8 Mo)	Download video (WMV file, ~ 150 Mo)	Download video (MPEG file, ~ 200 Mo)	Download slides (pdf)	Download slides (ps)	Download slides (dvi)	Download slides (tex)
First lecture									-	-
Second lecture									-	-
Notes lecture one	-	-	-	-	-	-				-

The slide times are completely random this stage.

The pdf file with the slides might be very slow to start in the video window (50Mo for the first lecture, 15Mo for the second).

Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and J.Friedman