ADM formalism
The Arnowitt–Deser–Misner (ADM) formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity.
[2] The comprehensive review of the formalism that the authors published in 1962[3] has been reprinted in the journal General Relativity and Gravitation,[4] while the original papers can be found in the archives of Physical Review.
[2][5] The formalism supposes that spacetime is foliated into a family of spacelike surfaces
The dynamic variables of this theory are taken to be the metric tensor of three-dimensional spatial slices
The text here uses Einstein notation in which summation over repeated indices is assumed.
The absolute value of the determinant of the matrix of metric tensor coefficients is represented by
It separates the spacetime metric into its spatial and temporal parts, which facilitates the study of the evolution of gravitational fields.
The basic idea is to express the spacetime metric in terms of a lapse function that represents the time evolution between hypersurfaces, and a shift vector that represents spatial coordinate changes between these hypersurfaces) along with a 3D spatial metric.
is the lapse function encoding the proper time evolution,
is the shift vector, encoding how spatial coordinates change between hypersurfaces.
This decomposition allows for a separation of the spacetime evolution equations into constraints (which relate the initial data on a spatial hypersurface) and evolution equations (which describe how the geometry of spacetime changes from one hypersurface to another).
The starting point for the ADM formulation is the Lagrangian which is a product of the square root of the determinant of the four-dimensional metric tensor for the full spacetime and its Ricci scalar.
The desired outcome of the derivation is to define an embedding of three-dimensional spatial slices in the four-dimensional spacetime.
The metric of the three-dimensional slices will be the generalized coordinates for a Hamiltonian formulation.
The conjugate momenta can then be computed as using standard techniques and definitions.
are Christoffel symbols associated with the metric of the full four-dimensional spacetime.
The lapse and the shift vector are the remaining elements of the four-metric tensor.
Having identified the quantities for the formulation, the next step is to rewrite the Lagrangian in terms of these variables.
Although the variables in the Lagrangian represent the metric tensor on three-dimensional spaces embedded in the four-dimensional spacetime, it is possible and desirable to use the usual procedures from Lagrangian mechanics to derive "equations of motion" that describe the time evolution of both the metric
The result and is a non-linear set of partial differential equations.
Taking variations with respect to the lapse and shift provide constraint equations and and the lapse and shift themselves can be freely specified, reflecting the fact that coordinate systems can be freely specified in both space and time.
and the spatial metric functions by linear functional differential operators More precisely, the replacing of classical variables by operators is restricted by commutation relations.
There are relatively few known exact solutions to the Einstein field equations.
The most common approaches start with an initial value problem based on the ADM formalism.
ADM energy is a special way to define the energy in general relativity, which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity – for example a spacetime that asymptotically approaches Minkowski space.
The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form.
In other words, the ADM energy is computed as the strength of the gravitational field at infinity.
If the required asymptotic form is time-independent (such as the Minkowski space itself), then it respects the time-translational symmetry.
Noether's theorem then implies that the ADM energy is conserved.
By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons–Hawking–York boundary term for modified gravity theories "whose Lagrangian is an arbitrary function of the Riemann tensor".
