View source for Arnowitt-Deser-Misner formalism

==Introduction==
General relativity is unlike any previous physical theory. It abolishes the framework of an a priori given space-time,
replacing it by a dynamically determined one, within the framework of (pseudo-) Riemannian spaces. The space-time 
metric is the fundamental field here, and the conceptual difficulty in coming from special relativity, say, is that the metric
not only contains physical information -which spacetime it describes- but also coordinate information: what do the <math>(t,x,y,z)</math>
label and how? They are like generalized coordinates in the usual sense, but  there are no "really true" old-fashioned Cartesian 
reference coordinates. Furthermore, the metric, which obeys the second differential order Einstein equations, also contains
constraints as well as dynamical and coordinate or gauge components. In this respect it resembles the gauge potentials of 
Maxwell, or better, Yang-Mills theories. 
Disentangling these different aspects of the field variables was long a most confusing part of understanding even the basic
physics described by the theory, from what the Black Hole radius means to whether there are gravitational waves, and if so, how
is their invariant content described when any and all coordinate frames are equally valid and their nature is purely implicit? How
even to understand what differentiates the weak fields of the Newtonian limit from gauge artifacts? One basic symptom is that 
the Hamiltonian of the theory is identically zero by any standard definition, a problem that plagued general relativity for many years.

In the present work, a precise determination of the independent dynamical modes of the
gravitational field is arrived at by casting  general relativity into canonical form. [Actually, the method is universally applicable to any modified covariant model of gravity, or indeed any gauge theory, all of which share some of the above paradoxical aspects.]
This reveals the minimal number of variables specifying the state of the system,
all the relevant information about the field's behavior being exhibited in familiar, "<math>L=p\mathrm{d}q/\mathrm{d}t-H(p,q)</math>",  language. The canonical formulation, involving only the minimal set of variables (which will turn out to be two degrees of freedom per space point), is also essential to the quantization program, since it yields directly simple Poisson bracket (P.B.) relations among these conjugate, unconstrained, variables. Two essential aspects of canonical form are: (1) that the field equations are of first order in the time derivatives; and (2) that time has been singled out by expressing the theory in 3+1 dimensional form, without of course losing general covariance by this "gauge choice". 
These two features are characteristic of Hamilton (or P.B.) equations of motion, in contrast to the Lagrange equations. The
first requirement may be achieved in general relativity, since its Lagrangian may be written in a form linear in the time derivatives
('Palatini' form). The type of variable fulfilling the second requirement will be seen also to possess a natural geometrical interpretation.

The use of the Palatini Lagrangian and of 3+1 dimensional notation
does not, of course, impair the general covariance of the theory
under arbitrary coordinate transformations. In possessing this
covariance, general relativity is precisely analogous to the
parameterized form of mechanics in which the Hamiltonian and the
time are introduced as a conjugate pair of variables of a new
degree of freedom. When in parameterized form, a theory is
invariant under an arbitrary re-parameterization, just as general
relativity is invariant under an arbitrary change of coordinates.
The action of general relativity will thus be seen to be in
"already parameterized" form. The well-known relations between
the usual canonical form and the parameter description will thus
provide the key for deriving the desired canonical form for the
gravitational field. We will therefore begin with
a brief review of parameterized particle mechanics. Then,
the Lagrangian of general relativity will be cast into Palatini
and 3+1 dimensional form, and the geometrical significance of the
variables will be discussed. We will see then that relativity has
a form identical to parameterized mechanics. To complete
the analysis, we obtain the canonical variables and their
relations as well as the P.B. equations of motion. Once canonical form 
is reached, the physical interpretation of
quantities involved follows directly. Thus, the canonical variables 
themselves represent the
independent excitations of the field (and hence provide the basis
for defining gravitational radiation in a coordinate-independent
way). Further, the numerical value of the Hamiltonian for a
particular state of the system provides the primary definition of
total energy. It is also possible to set up the
analysis of gravitational radiation in a fashion closely analogous
to electrodynamics by introducing a suitable definition of the
wave zone. In this region, gravitational waves propagate as free
radiation, independent of the strong field interior sources. The
waves obey ordinary (flat-space) wave equations and consequently
satisfy superposition. The Poynting vector may also be defined
invariantly in the wave zone. In contrast, the Newtonian-like
parts of the metric cannot be determined within the wave zone;
they depend strongly on the interior non-linearities.