The Lapse, Shift, UnitNormalVector and TimeVector are related quantities that, essentially, represent the 4-coordinate degrees of freedom that exist in the representation of a gravitational field (the Lapse and Shift), and identify a particular choice of 3D spacelike hypersurface in the decomposition of spacetime into 3 + 1, its space and time parts (UnitNormalVector and TimeVector). This decomposition is relevant in the context of numerical relativity, and in a Hamiltonian description of gravity.

Regarding the Lapse, Shift, you can set their values using Setup and its lapseandshift keyword; there are three possible values: standard, arbitrary, or a list of algebraic expressions representing them. The value chosen determines the values of the components of all the ThreePlusOne tensors (e.g. ExtrinsicCurvature) and so of tensorial expressions involving them (e.g. the ADMEquations). Those components are always computed first in terms of the Lapse and Shift and the space part $g_{i,j}$ of the 4D metric, then in a second step, if lapseandshift = standard, the Lapse and the Shift are replaced by their expressions in terms of the $g_{0,\mathrm{\mu }}$ part of the 4D metric, according to ${\mathbf{\alpha }}^2={\left(-g_{}^{0,0}\right)}^{-1}$ and ${\mathbf{\beta }}_j=g_{0,j}$, or if lapseandshift was set passing a list to Setup then in terms of the values indicated in that list (these values are not used to set the $g_{0,\mathrm{\mu }}$ part of the 4D metric). When lapseandshift = arbitrary, the second step is not performed and the Lapse and Shift evaluate to themselves, representing an arbitrary value for them. In the three cases, standard, arbitrary, or a list of algebraic expressions, the components of the ThreePlusOne tensors are computed in terms of a metric with line element ${\mathrm{ds}}^2=-{\mathit{\alpha }}^2{\mathrm{dt}}^2+{\mathit{\gamma }}_{i,j}\left({\mathrm{dx}}^i+{\mathit{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}\mathrm{dt}\right)\left({\mathrm{dx}}^j+{\mathit{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\mathrm{dt}\right)$, where $\mathbf{\alpha }$ and ${\mathbf{\beta }}_{}^i$ have the Lapse and Shift values mentioned, and ${\mathbf{\gamma }}_{i,j}=g_{i,j}$ is the ThreePlusOne:-gamma3_ metric of the 3D hypersurface. This design permits working with any 4D metric set and, without changing its value, experimenting with different values of the Lapse and Shift (different values of $g_{0,\mathrm{\mu }}$) for the 3+1 decomposition of Einstein's equations. See also LapseAndShiftConditions.

The presentation in the rest of this page assumes the signature (+ + + -), automatically set when ThreePlusOne is loaded and the denominations and definitions are the standard ones of the ADM formalism (ref [5]). The signature (+ + + -) is physically equivalent to (- + + +), but has the computational advantage of having space indices running from 1 to 3 (instead of from 2 to 4) and the value 0 of a spacetime index pointing to position 4 (instead of position 1). With any of these two signatures, the space components (i, j) of the 4D gamma3_${}_{\mathrm{\mu },\mathrm{\nu }}$ are related to the space components (i, j) of the 4D spacetime metric $g_{\mathrm{\mu },\mathrm{\nu }}$ by

where, as is the case of all the tensors of ThreePlusOne, the 3D metric is displayed in black instead of blue.

From a computational point of view, the Lapse is a scalar, and the Shift is a purely spatial 4D vector, that is, its contravariant timelike component is equal to 0. The UnitNormalVector is a timelike vector (all its covariant space components are zero), perpendicular to the hypersurface determined by the Lapse and Shift, and the TimeVector is neither spacelike nor timelike, but in the adapted coordinates used its contravariant space components are zero. All of them can be defined in terms of a global time function $\mathrm{{\rm T}}\left(x^i\,t\right)=\mathrm{constant}$. In a generic system of references, where a 3+1 split formulates spacetime as a 3D spacelike hypersurface parameterized by $\mathrm{{\rm T}}$ that evolves in time, the Lapse, assumed to be positive, is the lapse of proper time between two 3D neighboring hypersurfaces, measured by observers moving along the direction normal to the hypersurfaces (normal, or Eulerian observers). In the adapted coordinates used in the implementation in Physics, $\mathrm{{\rm T}}\left(x^i\,t\right)=t$. The Lapse is thus defined as

where $▿$ is the 4D covariant derivative operator D_. The UnitNormalVector is a timelike unit vector, normal to the 3D hypersurface, i.e. it is the 4-velocity of a normal observer whose worldline is normal to the 3D hypersurface. The UnitNormalVector is defined in terms of $\mathrm{{\rm T}}\left(x^i\,t\right)$ as

from where $\mathrm{n\_\_\μ}{\mathit{n}}_{}^{\mathrm{\mu }}=\mathrm{-1}$. The 3D metric gamma3_, that contains information on the space part of the 4D metric g_ and that projects into this 3D spacelike hypersurface determined by $\mathrm{{\rm T}}\left(x^i\,t\right)=\mathrm{constant}$, is also expressed in terms of g_ and the UnitNormalVector as

The TimeVector is tangent to the time lines (lines of constant spatial coordinates), so it can also be expressed in terms of the global function $\mathrm{{\rm T}}\left(x^i\,t\right)$ identifying the 3D hypersurface as

from where ${\mathit{t}}_{}^{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\mu }}=-\mathbf{\alpha }$. The Shift can be defined as the projection of the TimeVector onto the 3D hypersurface, i.e.

from where it follows the relationship between these three vectors

From a physical point of view, choosing $\mathrm{{\rm T}}\left(x^i\,t\right)=t$ as the time coordinate, the Lapse and Shift arise naturally when defining simultaneity in the general theory of relativity as the possibility of synchronizing clocks located at different points in space (ref [1], sec. (84)). For that synchronization to be possible, all the components $g_{0,j}$ of the spacetime metric must be equal to zero, and that condition can always be achieved, in any gravitational field, by an appropriate choice of coordinates (ref [1], sec. (84) and (97)). If, in addition, $g_{0,0}=1$, the reference system is synchronous. In this referential the time lines (the TimeVector is tangent to them) are geodesics in the 4D spacetime and normal to the hypersurface $t=\mathrm{constant}$ (because from $g_{0,j}=0$ follows that ${\mathbf{\beta }}_{}^{\mathrm{\mu }}=0$ and so  ${\mathit{t}}_{}^{\mathrm{\mu }}$ and  ${\mathit{n}}_{}^{\mathrm{\mu }}$ are parallel), and the 4D infinitesimal interval is written as.

Under a time rescaling $t\text{'}=G\left(t\right)$ and a general transformation of the space coordinates ${\left(x\text{'}\right)}^i=F^i\left(x^j\,t\right)$, and taking into account ${\mathbf{\gamma }}_{i,j}=g_{i,j}$, this infinitesimal interval changes into

where now ${\mathbf{\beta }}_{}^{\mathrm{\mu }}\ne 0$ and so the new time lines $t=\mathrm{constant}$ (and hence ${\mathit{t}}_{}^{\mathrm{\mu }}$) are no longer orthogonal to the 3D hypersurface whose geometry is described by $\mathrm{\γ\_\_i,j}$. In this decomposition of the infinitesimal interval, the 4-coordinate degrees of freedom are represented by the freedom in choosing the Lapse and the Shift (corresponding to different choices of the system of references; conversely, one can depart from this general form of the infinitesimal interval and construct a transformation to a synchronous reference system, where $g_{0,0}=1$ and $g_{0,j}=0$, by solving a related Hamilton-Jacobi equation see [1] sec.(97)). In this system of references (so-called adapted coordinates system of references), the Lapse and Shift are expressed in terms of the contravariant spatial components of the 4D spacetime metric g_ as

Besides being indexed with a spacetime or a space index, the Shift, TimeVector and UnitNormalVector accept the following keywords as an index:

definition: when passed alone, gamma3_ returns its 4D definition in terms of the UnitNormalVector and the spacetime metric g_. When passed with space indices, it returns its expression in terms of the spatial components of g_.

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in Shift[]) returns an Array that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of Shift[mu]. If this keyword is passed together with indices, that can be covariant or contravariant, for example Shift[~mu, matrix], the resulting array takes into account the character of the indices.

nonzero: returns a set of equations, with the left-hand side as a sequence of two positive numbers identifying the element of $\mathrm{\γ\_\_i,j}$ and the corresponding value on the right-hand side. Note that this set is actually the output of the ArrayElems command when passing to it the Matrix obtained with the keyword matrix.

You can change the value of the 3D metric gamma3_ by changing the value of the 4D metric g_, using Setup or g_. Regarding the geometry in this 3D hypersurface, a 3D covariant derivative, related Christoffel symbols, and the Ricci and Riemann tensors are represented respectively by the D3_, Christoffel3, Ricci3 and Riemann3 commands.

The %Lapse, %Shift, %TimeVector and %UnitNormalVector command are the inert forms of the corresponding active commands, that is represent the same object but entering them does not result in performing any computation. To perform the related computations, e.g. as if %Lapse were Lapse, use value.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}\=\mathrm{true}\,\mathrm{coordinates}\=\mathrm{cartesian}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{with}\left(\mathrm{ThreePlusOne}\right)\:$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{Defined\; as\; 4D\; spacetime\; tensors}\left(\mathrm{see\; ?Physics,ThreePlusOne}\right),{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }},{\mathit{▿}}_{\mathrm{\mu }},{\mathbf{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }},{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }},{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }},{\mathbf{\beta }}_{\mathrm{\mu }},{\mathit{n}}_{\mathrm{\mu }},{\mathit{t}}_{\mathrm{\mu }},{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Changing\; the\; signature\; of\; spacetime\; from}\left(\mathrm{-\; -\; -\; +}\right)\mathrm{to}\left(\mathrm{+\; +\; +\; -}\right)\mathrm{in\; order\; to\; match\; the\; signature\; customarily\; used\; in\; the\; ADM\; formalism}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

${\mathrm{g\_}}_{\[\]}$

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]\right)$

$\mathrm{Setup}\left(\mathrm{spacetimeindices}\,\mathrm{spaceindices}\right)$

$\left[\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\,\mathrm{spacetimeindices}=\mathrm{greek}\right]$

${\mathrm{gamma3\_}}_{\mathrm{definition}}$

${\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}={\mathit{n}}_{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\nu }}+g_{\mathrm{\mu },\mathrm{\nu }}$

${\mathrm{gamma3\_}}_{i\,j\,\mathrm{definition}}$

${\mathbf{\gamma }}_{i,j}=g_{i,j}$

${\mathrm{UnitNormalVector}}_{\mathrm{definition}}$

${\mathit{n}}_{\mathrm{\mu }}=-\mathbf{\alpha }{▿}_{\mathrm{\mu }}\left(t\right)$

${\mathrm{TimeVector}}_{\mathrm{definition}}$

${\mathit{t}}_{\mathrm{\mu }}=\mathbf{\alpha }{\mathit{n}}_{\mathrm{\mu }}+{\mathbf{\beta }}_{\mathrm{\mu }}$

${\mathrm{ExtrinsicCurvature}}_{\mathrm{definition}}$

${\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}=-\frac{{ℒ}_{\mathit{n}}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)}{2}$

${\mathrm{g\_}}_{\mathrm{sc}}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{+\; +\; +\; -}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}-\frac{r}{2m-r}& 0& 0& 0\\ 0& r^2& 0& 0\\ 0& 0& r^2{\sin \left(\mathrm{\θ}\right)}^2& 0\\ 0& 0& 0& \frac{2m-r}{r}\end{array}\right]\right)$

${\mathrm{Shift}}_{\[\]}$

${\mathrm{Shift}}_{\mathrm{\μ}}\=\left(\left[\begin{array}{rrrr}0& 0& 0& 0\end{array}\right]\right)$

${\mathrm{UnitNormalVector}}_{\[\]}$

${\mathrm{UnitNormalVector}}_{\mathrm{\μ}}\=\left(\left[\begin{array}{cccc}0& 0& 0& -\frac{1}{\sqrt{-\frac{r}{2m-r}}}\end{array}\right]\right)$

${\mathrm{TimeVector}}_{\[\]}$