ADM 3+1 Formalism

The story goes, Arnowitt, Deser and Misner weren't necessarily looking for a numerical relativity method when they developed their Hamiltonian version of the Einstein-Hilbert field equations. No, they were looking for a way to develop a quantum theory of gravity.

But, their clever method of treating 3-dimensional foliations of space within the 4-dimensional space-time manifold allowed those interested in numerical relativity to format the general relativity field equations as an initial value problem.

Foliation here is meant to be the breaking up 4-dimensional space-time into non-intersection 3-dimensional spatial hypersurfaces. (Think of the 2-dimensional leaves of a tree forming separate surfaces in 3-dimensional space.) The drawing below shows two nearby hypersurfaces a "distance" Δt away from each other in time. The lower one is denoted as Σ_Δt and the upper one as Σ_t+Δt

Note that the vector that represents the time dimension t above in the drawing (the vector from point 'O' to point 'B') may actually not be parallel to the normal (the vector from point 'O' to point 'A') to the surface Σ_Δt. This means that in some increment of time, t , the distance between the two hypersurfaces along the normal to Σ_Δt  is actually α times Δ t instead of t. The vector between where the normal lands on Σ_t+Δt  and Σ_t+Δt  is  β.

Another name for α is the lapse function (as in a lapse of time) and another name for the offset vector β between point 'A' and point 'B' is the shift function. Collectively, α and β are also known as the gauge variables.