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The Arnowitt-Deser-Misner energy is the Hamiltonian boundary charge generating an asymptotic time translation in the ADM formalism. For a spatial metric tensor $h_{ij}$ in asymptotically Cartesian coordinates, use geometrized units $G=c=1$:Here $n^i$ is the outward Euclidean unit normal vector. Standard differentiable asymptotic flatness gives $h_{ij}={\delta }_{ij}+O(r^{-1})$ and ${\partial }_k{h}_{ij}=O(r^{-2})$; suitable additional decay and integrability conditions ensure that the limit exists. Together with ADM momentum, this charge forms the asymptotic energy-momentum four-vector.^[ref]

For a time-symmetric Schwarzschild spacetime slice in isotropic coordinates, $h_{ij}={\psi }^4{\delta }_{ij}$ with $\psi =1+M/(2r)$. Its surface integrand is $-2{\partial }_r{\psi }^4=4M/r^2+O(r^{-3})$. Integrating over $S_r$ therefore gives $E_{ADM}=M$. The corresponding slice of Minkowski spacetime has constant $h_{ij}={\delta }_{ij}$ and zero ADM energy. In a moving asymptotic frame, the energy component need not equal the invariant ADM mass.

Charles W. Misner co-developed the canonical formulation of general relativity with Richard Arnowitt and Stanley Deser. Their account separates spatial geometry from its time evolution, expressing the gravitational system through dynamical variables and constraint equations.^[ref]

This turns spacetime evolution into an initial-value problem. Initial geometric data cannot be freely assigned component by component; they must satisfy the gravitational constraints.

Richard Arnowitt co-developed a canonical treatment of general relativity with Stanley Deser and Charles W. Misner. Their framework organizes spatial geometry, conjugate momenta and constraints, and supplies an asymptotic definition of gravitational energy.^[ref]

The Arnowitt-Deser-Misner energy is associated with suitable asymptotic boundary conditions; it is not a universal pointwise energy-density formula for an arbitrary spacetime.

Stanley Deser, with Richard Arnowitt and Charles W. Misner, developed the canonical dynamics of general relativity. The formulation separates physical evolution from coordinate freedom and makes the constraint structure explicit.^[ref]

The distinction matters when counting degrees of freedom: the number of written metric components is larger than the number of independently propagating gravitational modes.