(^54) The Quantum Structure of Space and Tame
The work of Christodoulou on spherically symmetric massless scalar collapse
[40] [41] [42] [43] shows that Cosmic Censorship in the strict sense fails, because of
transient singularities associated with a very special choice of initial data.
These singularities are associated with a discrete self-similar behaviour, referred
to as critical behaviour, first uncovered numerically by Choptuik [44].
It is rather doubtful that this behaviour will survive quantum corrections.
The strategy of Christodoulou is essentially to reduce the problem to a l+l
dimensional non-linear wave problem.
This technique has been exploited recently by Dafermos [35] [36] [34] [33] [32] who
is able to treat charged gravitational collapse and establish a form of cosmic censor-
ship in the presence of Cauchy horizons assuming the existence of closed trapped
surfaces and also to justify the assumption that Z+ is complete.
Dafermos’s methods also extend to higher dimensions [46] where, following nu-merical numerical work by Bizon, Chmaj and Schmidt in 4fl [30] and 8+1 [31]
Dafermos and Holzegel [29] were recently able to extend some of these results vac-
uum gravitational collapse in 4+ 1.
There has been a great deal of work on homogeneous solutions of Einstein’s
equations, particularly near singularities. This can be partially extended to inho-
mogeneous solutions provided one assumes spatial derivatives are small (velocity
dominated approximation).
On this basis, Belinsky, Lifshitz and Khalatnikov proposed that generically,
singularities are of chaotic, oscillatory, Mixmaster type, first seen in Bianchi IX
models.
Recent mathematically rigorous work tends to confirm that this may happen,
at least for an open set of Cauchy data.
The story in higher dimensions will be the subject of a report by T Damour.
3.1.4 Classical Boundary Conditions and Stability
The basic problem raised by spacetime by singularities is what boundary conditions
are to be posed in their presence? Cosmic Censorship is an attempt to evade that
problem as long as one is outside the event horizon. Even if it is true, what happens
inside the horizon?.
The choice of boundary conditions may affect questions of genericity or stability.
In some cases, despite singularities the choice of boundary condition may be
~nique.~
Typically in gravitational situations however, this is not the case and choices
must be made.
For example, in a recent paper, Gibbons, Hartnoll and Ishibashi [5] showed that
there is a choice of boundary conditions such that even negative mass Schwarzschild
’for example the unique self-adjoint extension for the Hydrogen Atom