UNB Technical Report 95-01
Mon Aug 7 18:08:49 ADT 1995
For numerical quantum gravity to be useful, methods must be found to make the calculations meaningful by introducing an appropriate definition of quantum observables. A recent breakthrough has been made in the formulation of black hole and wormhole dynamics. This breakthrough provides a possible solution for finding the desired observables, and is being used for ongoing work into the fully quantum modelling of gravitational collapse on computers. Such models may help answer some of the crucial questions in quantum gravity, and may provide us with new questions to ponder.
It has long been a hope of researchers in quantum gravity to model the full quantum dynamics of black holes and wormholes. Even in semiclassical theory, these systems display strange properties, from Hawking-Unruh radiation [1], with its potential destruction of our ability to make predictions in quantum theory, to mass inflation [2,3], and the corresponding wormhole stability problem. It would be very useful if we could formulate numerical models of the full quantum dynamics of black holes, if we did it in a way that truly corresponded to the construction of a quantum theory. A quantum theory, however, needs more than just a wavefunction: it needs observables. Observables are operators that describe the statements we can make from the quantum theory, without which any result is useless. Therefore, any valid numerical approach to quantum gravity via numerical work must construct these observables. Unfortunately, this has not been done, to date, in any of the numerical studies of quantum gravity. A fully developed numerical solution would thus represent a true breakthrough in quantum gravity research.
There is a problem, however, in that
classical and quantum geometrodynamics (CGD and QGD) are haunted, and made
more interesting, by the problem of time. This is simply
the problem that general relativity, through its general covariance,
does not possess a special, external, time parameter, and thus it is
difficult to understand the theory in terms of dynamics. We can
see this in the quantum theory by looking at the
Arnowitt, Deser and Misner (ADM)
[4] formulation of geometrodynamics. In this formalism, the spacetime 4-metric
is decomposed in terms of the familiar 3-metric 
and the
corresponding lapse N, and shift 
variables, all introduced on
a specific foliation (prescribed by N and 
), labeled by a time
parameter t, and a spatial hypersurface position 
. The
action for the vacuum part of the
theory can then be written in the manifestly canonical form
with the overdot representing partial differentiation with respect to t,
and 
being the momenta conjugate to 
. The
and 
represent vanishing constraint
functions. This allows us
to view the theory as describing the evolving 3-geometry (and
corresponding momenta) on a hypersurface, but with a Hamiltonian
that is constrained to vanish [4,5],
The corresponding quantum equations describing the system are then obtained using the usual canonical quantization rules for replacing phase-space variables with operators,
operating on some wavefunction 
. The vanishing of
the Hamiltonian then means that the wavefunction is constant, the
observables are constants of the motion [5],
and we seem to have lost quantum dynamics! How can we hope to make any
predictions in quantum gravity if we have no dynamics? The Author hopes
to show how this is possible, in a way that may be implemented on computers.
It is possible to construct an alternative formulation of geometrodynamics that both produces classical and quantum dynamics, and tells us the correct quantum observables, at least for model theories. This is the so-called hypertime approach, pioneered by Isham and Kuchar [6,7,8] and investigated by the Author [5,9,10]. In this approach, we view the Hamiltonian given by Equation (2) as representing evolution with respect to other variables, constructed out of the gravitational and matter phase-space variables themselves, instead of with respect to the simple label t. These latter are called embedding variables and they can be considered as representing the location of a given spacelike hypersurface in an embedding into a surrounding spacetime.
The method starts by finding a canonical transformation in which we can divide the standard local gravitational phase-space variables into three classes [5,6]
where 
ranges from one to four and
.
and 
represent
the `true' dynamical variables of the theory discussed
above (which can easily be extended to include matter degrees of freedom
without changing our results [5]). 
represent the corresponding
internal embedding coordinates. The latter describe the location of the
hypersurface in spacetime, commonly dubbed the hypertime. 
are then
the corresponding momenta
(i.e. the energy-momentum densities).
The approach is then to simply write down a solution, for
, to the constraints
where 
represents an index over position, tensor indices,
and potential
boundary terms [5].
Functional differentiation of this gives us a functional form
for the classical dynamics of gravity, relative to hypertime,
where 
denotes a normal Poisson bracket.
These equations can then be quantized to give the
hypertime functional Schrödinger equation
These two equations represent the basic information to solve the problem
of time, and allow us to
describe quantum gravity on a computer. The first equation
tells us that we can represent any set of physical variables at hypertime
by their `initial' values at hypertime 
,
simply by evolving the quantum objects classically back in
time. This
then represents the dynamics in terms of constants of the motion, thus
explaining why the original ADM formalism seems frozen in time. These
are our observables. The second
equation tells us how to get the wavefunction at hypertime 
,
which can then be interpreted using the observables we've constructed.
The first order form of everything makes computer implementation
reasonably straightforward.
How can this be realized in practice? The Author has developed a hypertime formulation of spherically symmetric black holes and wormholes [5] that expresses the hypertime evolution in terms of the effective mass measured by an observer at a varying radius from the hole. In the case of collapsing shells of matter, important for black hole formation, and also important in inflationary models of the universe, it turns out that Equations (6) and (7) are very simple: The classical evolution corresponds to only two ordinary differential equations, and the quantum one is a simple partial differential equation in two variables, first order in the hypertime variables. These equations can be solved on the computer to tell us the wavefunction of these simple models, as well as the observables for the complete quantum formalism. This will be the first fully quantum modelling of black hole collapse with actual calculation of the observables. We can hope that this will allow us to understand some crucial matters in quantum gravity, and also produce some new results. We can, if we wish, actually perform computer-based `experiments' on quantum gravity. For instance, how does the recently discovered universal behaviour of gravitational collapse [11] affect the quantum theory, and if mass is crucial in the hypertime formalism, how does mass inflation come in? The Author is hoping to investigate all these issues, and more.
