EinsteinHamilton

In the Hamiltonian formulation of general relativity, the Einstein-Hamilton equations are:

$$\dot{q}_{ij} = \frac{\partial H}{\partial p^{ij}}$$

$$\quad \dot{p}^{ij} = -\frac{\partial H}{\partial q_{ij}} + \text{constraint terms}$$

The Hamiltonian $H$ is:

$$H = \int ( N \mathcal{H} + N^i \mathcal{H}_i ) d^3 x$$

Lapse Function ( $N$ )

The lapse function is a scalar field that measures the rate at which coordinate time advances relative to proper time along a timelike trajectory in the manifold. Mathematically, if $t^\mu$ is the tangent vector to the trajectory and $n^\mu$ is the unit normal to the spatial slice, then

$$N = - g_{\mu\nu} n^\mu t^\nu$$

Shift Vector ( $N^i$ )

The shift vector is a vector field on the spatial slice that represents the rate at which the spatial coordinates of an observer change as they move from one spatial slice to a neighboring one in the foliation. Formally, if $x^i$ are the spatial coordinates, $t$ is the coordinate time, and $t^\mu$ is the tangent vector to the observer's worldline, then

$$N^i = \frac{g_{\mu i} t^\mu}{N}$$

Spatial Slice

A spatial slice is a three-dimensional, spacelike hypersurface $\Sigma$ embedded in the four-dimensional Lorentzian manifold $M$ representing spacetime. This hypersurface is defined by a smooth embedding function $f: \Sigma \rightarrow M$. The hypersurface is characterized by a unit normal vector field $n^\mu$ that is everywhere timelike, and its induced metric $q_{ij}$ is obtained from the spacetime metric $g_{\mu\nu}$ via the pullback of $f$.