Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
ADMDecomposition
Represent a canonical decomposition of the metric for a Riemannian or pseudo-Riemannian manifold via the ADM formalism
Contributed by:
Jonathan Gorard
ResourceFunction["ADMDecomposition"][MetricTensor[…],t,α,βi] represents an ADM decomposition with the given (initial) spatial MetricTensor, distinguished "time" coordinate t, lapse function α and shift vector (field) βi. | |
ResourceFunction["ADMDecomposition"]["name",…] represents a named ADM decomposition "name". | |
ResourceFunction["ADMDecomposition"][{"name",…},…] represents a named parameterized ADM decomposition "name", with additional parameter(s) specified within a list. | |
ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],t] transforms a specified ResourceFunction["ADMDecomposition"] into one with new distinguished "time" coordinate t. | |
ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],α,βi] transforms a specified ResourceFunction["ADMDecomposition"] into one with new lapse function α and new shift vector (field) βi. | |
ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],t,α,βi] transforms a specified ResourceFunction["ADMDecomposition"] into one with new distinguished "time" coordinate t, new lapse function α and new shift vector (field) βi. |
Details
Formally, the ADM decomposition (due to Arnowitt, Deser and Misner) represents a foliation of a Riemannian or pseudo-Riemannian manifold into a sequence of submanifolds, i.e. it designates an equivalence relation on the manifold whose equivalence classes are injectively-immersed submanifolds of uniform dimension. Typically, these submanifolds are known as the "leaves", "slices" or "hypersurfaces" of the foliation.
Typically, the ADM formalism is applied in the case of general relativity, in which the ambient manifold is a Lorentzian spacetime and the submanifolds are usually spacelike hypersurfaces of codimension-1, with the equivalence classes being the level surfaces of a smooth universal "time function" whose gradient is everywhere timelike. Such a decomposition thus allows one to formulate the Einstein field equations as an explicit time evolution/initial value problem.
The dynamical variables of the ADM formalism are the components of the spatial metric tensor (i.e. the metric tensor field associated to each submanifold/spacelike hypersurface), along with their respective conjugate momenta, otherwise known as the components of the extrinsic curvature tensor (i.e. the rank-2 tensor field characterizing how the submanifolds/spacelike hypersurfaces are embedded into the ambient manifold/spacetime). Together, these variables allow one to define a Hamiltonian, thus permitting one to write the equations of motion for general relativity as a special case of Hamilton's equations, describing how the components of the spatial metric tensor and the extrinsic curvature tensor "evolve" from one submanifold/spacelike hypersurface to the next.
There is, in addition, a family of Lagrange multipliers associated with the ADM formalism, namely the lapse function α and the shift vector (field) βi; these correspond to the degrees of gauge/coordinate freedom inherent to the ADM decomposition. Geometrically, the lapse function α represents the "timelike" coordinate distance between neighboring submanifolds/spacelike hypersurfaces, and therefore determines the "slicing" of the ambient manifold/spacetime, while the shift vector (field) βi represents how the spatial coordinates are relabeled between neighboring submanifolds/spacelike hypersurfaces.
If the manifold/spacetime is globally hyperbolic, then there exists a foliation (and therefore a gauge choice) in which the submanifolds/spacelike hypersurfaces do not intersect, and thus in which the initial value problem formulation of the Einstein field equations is guaranteed to be well-posed.
The Einstein field equations lead to the evolution equations/equations of motion for the spatial metric tensor and the extrinsic curvature tensor, while the (contracted) Bianchi identities lead to the so-called Hamiltonian constraint and momentum constraint equations. The evolution equations are typically hyperbolic in character, while the Hamiltonian and momentum constraint equations are typically elliptic in character (albeit with many notable exceptions, such as harmonic and generalized harmonic gauge, in which the constraint equations are intentionally formulated so as to be hyperbolic). The resulting system can then be closed by specifying appropriate governing equations for the lapse and shift gauge variables.
ResourceFunction["ADMDecomposition"] comes equipped with several in-built slicing conditions (i.e. governing equations for the lapse function α) as well as coordinate conditions (i.e. governing equations for the shift vector field βi), including the geodesic, maximal, harmonic and 1+log slicing conditions for the lapse, and the normal, harmonic, minimal distortion (also known as the generalized Smarr-York conditions) and pseudo-minimal distortion coordinate conditions for the shift.
Furthermore, any valid ADM decomposition must additionally satisfy the Gauss equations (relating projections of the Riemann curvature tensor over the ambient manifold/spacetime to components of the Riemann curvature tensor over the submanifolds/spacelike hypersurfaces) and the Codazzi-Mainardi equations (relating projections of the Ricci curvature tensor over the ambient manifold/spacetime to covariant derivatives of the extrinsic curvature tensor over the submanifolds/spacelike hypersurfaces), which specify the compatibility conditions necessary for the submanifolds/spacelike hypersurfaces in the decomposition to "embed" correctly into the ambient manifold/spacetime.
By default, ResourceFunction["ADMDecomposition"][MetricTensor[…],t,α,βi] represents an ADM decomposition by a given (initial) spatial metric tensor, a distinguished "time" coordinate t, a lapse function α and a shift vector (field) βi, although it can be applied to construct decompositions of arbitrary Riemannian and pseudo-Riemannian manifolds that do not necessarily have distinguished "time" and "space" directions in the physical sense. In this way, ResourceFunction["ADMDecomposition"] can be used to set up a Cauchy problem for the Einstein field equations in general relativity.
ResourceFunction["ADMDecomposition"] does not assume any particular number of dimensions, nor any particular metric convention, for the ambient manifold (although some properties, such as "LorentzianConditions", implicitly assume a (-,+,+,+,…) signature). ResourceFunction["ADMDecomposition"] also does not require the ambient metric to be strictly symmetric (i.e. spin and torsion connections are also supported), although the computation of the ADM decomposition implicitly assumes the torsion-free Levi-Civita connection by default.
ResourceFunction["ADMDecomposition"] implicitly keeps track of all known algebraic equivalences between tensor expressions, and can apply them for simplification purposes where necessary. Requesting any property with "Reduced" in its name has the effect of applying all known tensor equivalences and simplifying (note that this can have the effect of increasing computation time significantly).
By default, ResourceFunction["ADMDecomposition"] evaluates all partial derivatives of the metric tensor(s) and gauge variables automatically. In certain cases, however, these partial derivatives may be difficult or even impossible to compute, in which case the evaluation may not terminate in a reasonable time. Requesting any property with "Symbolic" in its name has the effect of leaving all partial derivative operators unevaluated instead (note that this can have the effect of increasing expression length significantly).
In ResourceFunction["ADMDecomposition"]["name",…] or ResourceFunction["ADMDecomposition"][{"name",…},…], the following named (in-built) ADM decompositions are supported:
| "Minkowski" | ADM decomposition of the metric for 4-dimensional (or 1+3-dimensional) flat/Minkowski spacetime in Cartesian coordinates |
| {"Minkowski",d} | ADM decomposition of the metric for d-dimensional (or 1+(d-1)-dimensional) flat/Minkowski spacetime in Cartesian coordinates |
| "Schwarzschild" | ADM decomposition of the metric describing the exterior spacetime geometry surrounding an uncharged, spherically-symmetric, non-rotating mass distribution (e.g. an uncharged, non-rotating black hole) in Schwarzschild/spherical polar coordinates, with purely symbolic mass "M" |
| {"Schwarzschild",M} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding an uncharged, spherically-symmetric, non-rotating mass distribution (e.g. an uncharged, non-rotating black hole) in Schwarzschild/spherical polar coordinates, with numerical mass M |
| "Kerr" | ADM decomposition of the metric describing the exterior spacetime geometry surrounding an uncharged, axially-symmetric, rotating mass distribution (e.g. an uncharged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with purely symbolic mass "M" and purely symbolic angular momentum "J" |
| {"Kerr",M} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding an uncharged, axially-symmetric, rotating mass distribution (e.g. an uncharged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with numerical mass M and purely symbolic angular momentum "J" |
| {"Kerr",M,J} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding an uncharged, axially-symmetric, rotating mass distribution (e.g. an uncharged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with numerical mass M and numerical angular momentum J |
| "ReissnerNordstrom" | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, spherically-symmetric, non-rotating mass distribution (e.g. a charged, non-rotating black hole) in Schwarzschild/spherical polar coordinates, with purely symbolic (total) mass "M" and purely symbolic electric charge "Q" |
| {"ReissnerNordstrom",M} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, spherically-symmetric, non-rotating mass distribution (e.g. a charged, non-rotating black hole) in Schwarzschild/spherical polar coordinates, with numerical (total) mass M and purely symbolic electric charge "Q" |
| {"ReissnerNordstrom",M,Q} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, spherically-symmetric, non-rotating mass distribution (e.g. a charged, non-rotating black hole) in Schwarzschild/spherical polar coordinates, with numerical (total) mass M and numerical electric charge Q |
| "KerrNewman" | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, axially-symmetric, rotating mass distribution (e.g. a charged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with purely symbolic (total) mass "M", purely symbolic angular momentum "J" and purely symbolic electric charge "Q" |
| {"KerrNewman",M} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, axially-symmetric, rotating mass distribution (e.g. a charged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with numerical (total) mass M, purely symbolic angular momentum "J" and purely symbolic electric charge "Q" |
| {"KerrNewman",M,J} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, axially-symmetric, rotating mass distribution (e.g. a charged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with numerical (total) mass M, numerical angular momentum J and purely symbolic electric charge "Q" |
| {"KerrNewman",M,J,Q} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a charged, axially-symmetric, rotating mass distribution (e.g. a charged, spinning black hole) in Boyer-Lindquist/oblate spheroidal coordinates, with numerical (total) mass M, numerical angular momentum J and numerical electric charge Q |
| "BrillLindquist" | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a pair of uncharged, spherically-symmetric, non-rotating mass distributions (e.g. an uncharged, non-rotating binary black hole system) in Schwarzschild/spherical polar coordinates, with purely symbolic mass "M" and purely symbolic separation distance "z0" |
| {"BrillLindquist",M} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a pair of uncharged, spherically-symmetric, non-rotating mass distributions (e.g. an uncharged, non-rotating binary black hole system) in Schwarzschild/spherical polar coordinates, with numerical mass M and purely symbolic separation distance "z0" |
| {"BrillLindquist",M,z0} | ADM decomposition of the metric describing the exterior spacetime geometry surrounding a pair of uncharged, spherically-symmetric, non-rotating mass distributions (e.g. an uncharged, non-rotating binary black hole system) in Schwarzschild/spherical polar coordinates, with numerical mass M and numerical separation distance z0 |
| "FLRW" | ADM decomposition of the metric describing the spacetime geometry of a homogeneous, isotropic universe which is either contracting or expanding in spherical polar coordinates, with purely symbolic global curvature "k" and purely symbolic scale factor "a" (treated as a function of coordinate time) |
| {"FLRW",k} | ADM decomposition of the metric describing the spacetime geometry of a homogeneous, isotropic universe which is either contracting or expanding in spherical polar coordinates, with numerical global curvature k and purely symbolic scale factor "a" (treated as a function of coordinate time) |
| {"FLRW",k,a} | ADM decomposition of the metric describing the spacetime geometry of a homogeneous, isotropic universe which is either contracting or expanding in spherical polar coordinates, with numerical global curvature k and numerical scale factor a (treated as a function of coordinate time) |
Calling ResourceFunction["ADMDecomposition"][] returns a list of all named (built-in) ADM decompositions.
If the function succeeds in performing the specified ADM decomposition, it will return an ResourceFunction["ADMDecomposition"] expression.
Calling ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],t] has the effect of transforming the ADM decomposition (as well as its underlying spatial metric tensor) to use the new distinguished "time" coordinate t. Calling ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],α,βi] has the effect of transforming the ADM decomposition to use the new lapse function α and shift vector (field) βi. Calling ResourceFunction["ADMDecomposition"][ResourceFunction["ADMDecomposition"][…],t,α,βi] has the effect of performing both transformations simultaneously.
Based on the eigenvalues of the matrix representation of the ambient metric tensor in covariant form, the ambient manifold/spacetime will be classified by ResourceFunction["ADMDecomposition"] as either Riemannian (all eigenvalues positive or all eigenvalues negative), pseudo-Riemannian (some eigenvalues positive and some eigenvalues negative), Lorentzian (all eigenvalues positive except for one negative, or all eigenvalues negative except for one positive) or Indeterminate.
The property "RiemannianConditions" returns the conditions necessary to guarantee that the eigenvalues of the matrix representation of the ambient metric tensor are strictly positive; "PseudoRiemannianConditions" returns the conditions necessary to guarantee that the eigenvalues of the matrix representation of the ambient metric tensor are all non-zero; "LorentzianConditions" returns the conditions necessary to guarantee that the eigenvalue corresponding to the {1,0,0,…} eigenvector of the ambient metric tensor (if it exists) if negative, with all other eigenvalues being positive. Note that this is strictly less general than the behavior of properties such as "RiemannianQ" (described in the point above), since it assumes certain features of the metric signature.
In ResourceFunction["ADMDecomposition"], the following properties are supported:
| "SpatialMetricTensor" | underlying (spatial) metric tensor on each submanifold/spacelike hypersurface of the ADM decomposition |
| "SpacetimeMetricTensor" | metric tensor on the resulting ambient manifold/spacetime of the ADM decomposition |
| "NormalVector" | (future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the ADM decomposition |
| "ReducedNormalVector" | (future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the ADM decomposition, modulo all tensor equivalences |
| "SymbolicNormalVector" | (future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the ADM decomposition, with purely symbolic partial derivative operators |
| "TimeVector" | (future-pointing, timelike) "time vector" connecting corresponding points on neighboring submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "SymbolicTimeVector" | (future-pointing, timelike) "time vector" connecting corresponding points on neighboring submanifolds/spacelike hypersurfaces of the ADM decomposition, with purely symbolic partial derivative operators |
| "TimeCoordinate" | distinguished "time" coordinate symbol for the ADM decomposition |
| "SpatialCoordinates" | list of distinguished "spatial" coordinate symbols for the ADM decomposition |
| "CoordinateOneForms" | list of differential 1-form symbols for the ambient/spacetime coordinates of the ADM decomposition |
| "LapseFunction" | lapse function determining "timelike" coordinate distance between neighboring submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "ShiftVector" | shift vector (field) determining how "spatial" coordinates are relabeled between neighboring submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "GaussEquations" | list of Gauss equations relating projections of the Riemann curvature tensor over the ambient manifold/spacetime to components of the Riemann curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "SymbolicGaussEquations" | list of Gauss equations relating projections of the Riemann curvature tensor over the ambient manifold/spacetime to components of the Riemann curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition, with purely symbolic partial derivative operators |
| "CodazziMainardiEquations" | list of Codazzi-Mainardi equations relating projections of the Ricci curvature tensor over the ambient manifold/spacetime to covariant derivatives of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "SymbolicCodazziMainardiEquations" | list of Codazzi-Mainardi equations relating projections of the Ricci curvature tensor over the ambient manifold/spacetime to covariant derivatives of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition, with purely symbolic partial derivative operators |
| "GeodesicSlicingCondition" | condition on the lapse function required to guarantee geodesic slicing (i.e. unit lapse over the entire ambient manifold/spacetime) for the ADM decomposition |
| "ReducedGeodesicSlicingCondition" | condition on the lapse function required to guarantee geodesic slicing (i.e. unit lapse over the entire ambient manifold/spacetime) for the ADM decomposition, modulo all tensor equivalences |
| "MaximalSlicingCondition" | condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition |
| "ReducedMaximalSlicingCondition" | condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicMaximalSlicingCondition" | condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition, with purely symbolic partial derivative operators |
| "HarmonicSlicingCondition" | condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the "time" coordinate) for the ADM decomposition |
| "ReducedHarmonicSlicingCondition" | condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the "time" coordinate) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicHarmonicSlicingCondition" | condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the "time" coordinate) for the ADM decomposition, with purely symbolic partial derivative operators |
| "OnePlusLogSlicingCondition" | condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition |
| "ReducedOnePlusLogSlicingCondition" | condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicOnePlusLogSlicingCondition" | condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition, with purely symbolic partial derivative operators |
| "NormalCoordinateConditions" | list of conditions on the shift vector (field) required to guarantee normal coordinates (i.e. vanishing shift over the entire ambient manifold/spacetime) for the ADM decomposition |
| "ReducedNormalCoordinateConditions" | list of conditions on the shift vector (field) required to guarantee normal coordinates (i.e. vanishing shift over the entire ambient manifold/spacetime) for the ADM decomposition, modulo all tensor equivalences |
| "HarmonicCoordinateConditions" | list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the "spatial" coordinates) for the ADM decomposition |
| "ReducedHarmonicCoordinateConditions" | list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the "spatial" coordinates) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicHarmonicCoordinateConditions" | list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the "spatial" coordinates) for the ADM decomposition, with purely symbolic partial derivative operators |
| "MinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition |
| "ReducedMinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicMinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition, with purely symbolic partial derivative operators |
| "PseudoMinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition |
| "ReducedPseudoMinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition, modulo all tensor equivalences |
| "SymbolicPseudoMinimalDistortionConditions" | list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition, with purely symbolic partial derivative operators |
| "Dimensions" | number of dimensions of the ambient manifold/spacetime described by the ADM decomposition |
| "Signature" | list of +1s and -1s designating the signature of the ambient manifold/spacetime described by the ADM decomposition (+1 for each positive eigenvalue of the metric, -1 for each negative eigenvalue of the metric) |
| "RiemannianQ" | whether the ambient manifold/spacetime described by the ADM decomposition is Riemannian (i.e. all eigenvalues of the metric have the same sign) |
| "PseudoRiemannianQ" | whether the ambient manifold/spacetime described by the ADM decomposition is pseudo-Riemannian (i.e. all eigenvalues of the metric are non-zero, but not all have the same sign) |
| "LorentzianQ" | whether the ambient manifold/spacetime described by the ADM decomposition is Lorentzian (i.e. all eigenvalues of the metric have the same sign, except for one eigenvalue which has the opposite sign) |
| "RiemannianConditions" | list of conditions required to guarantee that the ambient manifold/spacetime described by the ADM decomposition is Riemannian (i.e. all eigenvalues of the metric are positive) |
| "PseudoRiemannianConditions" | list of conditions required to guarantee that the ambient manifold/spacetime described by the ADM decomposition is pseudo-Riemannian (i.e. all eigenvalues of the metric are non-zero) |
| "LorentzianConditions" | list of conditions required to guarantee that the ambient manifold/spacetime described by the ADM decomposition is Lorentzian (i.e. the "time" eigenvalue of the metric is negative and all other eigenvalues are positive) |
| "EvolutionEquations" | list of equations characterizing the "time" evolution of components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition |
| "SymbolicEvolutionEquations" | list of equations characterizing the "time" evolution of components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition, with purely symbolic partial derivative operators |
| "HamiltonianConstraint" | value of the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the ADM decomposition |
| "SymbolicHamiltonianConstraint" | value of the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the ADM decomposition, with purely symbolic partial derivative operators |
| "MomentumConstraints" | list of values of the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the ADM decomposition |
| "SymbolicMomentumConstraints" | list of values of the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the ADM decomposition, with purely symbolic partial derivative operators |
| "HamiltonianConstraintEquation" | condition required to guarantee that the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the ADM decomposition vanishes identically |
| "SymbolicHamiltonianConstraintEquation" | condition required to guarantee that the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the ADM decomposition vanishes identically, with purely symbolic partial derivative operators |
| "MomentumConstraintEquations" | list of conditions required to guarantee that the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the ADM decomposition vanish identically |
| "SymbolicMomentumConstraintEquations" | list of conditions required to guarantee that the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the ADM decomposition vanish identically, with purely symbolic partial derivative operators |
| "Properties" | list of properties |
Examples
Basic Examples (4)
Perform an ADM decomposition for the Schwarzschild metric (e.g. for an uncharged, non-rotating black hole with symbolic mass "M") in standard spherical polar coordinates, using the most general/maximally-unconstrained choice of gauge:
| In[1]:= | ![adm = ResourceFunction["ADMDecomposition"]["Schwarzschild"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/54619d01a050fb3d.png){kind=small} |
| Out[1]= |  |
Show the spatial metric tensor for the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[2]:= | ![adm["SpatialMetricTensor"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/679122925944857a.png){kind=small} |
| Out[2]= |  |
| In[3]:= | ![%["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/19349130a832f3d0.png){kind=small} |
| Out[3]= |  |
Show the spacetime metric tensor for the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[4]:= | ![adm["SpacetimeMetricTensor"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6254bbd6561365ac.png){kind=small} |
| Out[4]= |  |
| In[5]:= | ![%["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/732645a14f36469f.png){kind=small} |
| Out[5]= |  |
Show the lapse function for the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[6]:= | ![adm["LapseFunction"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/75e7a7976b5e680b.png){kind=small} |
| Out[6]= |  |
Show the shift vector for the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[7]:= | ![adm["ShiftVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5efdbd36c3f614f9.png){kind=small} |
| Out[7]= |  |
Show the future-pointing, timelike unit vector normal to spacelike hypersurfaces in the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[8]:= | ![adm["NormalVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/763a7d461c35822e.png){kind=small} |
| Out[8]= |  |
Show the condition on the lapse function that must hold in order to achieve geodesic slicing in the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[9]:= | ![adm["GeodesicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2840f5a63490041a.png){kind=small} |
| Out[9]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve normal coordinates in the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[10]:= | ![adm["NormalCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/369ff024bf8043b5.png){kind=small} |
| Out[10]= |  |
Show the condition on the lapse function that must hold in order to achieve 1+log slicing in the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric:
| In[11]:= | ![adm["OnePlusLogSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6c81b93486b5fd79.png){kind=small} |
| Out[11]= |  |
Show the condition on the lapse function that must hold in order to achieve 1+log slicing in the (maximally-unconstrained) ADM decomposition of the Schwarzschild metric, with all algebraic equivalences imposed:
| In[12]:= | ![adm["ReducedOnePlusLogSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1e325e8633cbe045.png){kind=small} |
| Out[12]= |  |
Perform the same ADM decomposition for the Schwarzschild metric in spherical polar coordinates, but with numerical mass 1, time coordinate symbol t, spatial coordinate symbols r, a1 and a2, using a partially-constrained choice of gauge (defined in terms of scalar functions a and b):
| In[13]:= | ![adm2 = ResourceFunction["ADMDecomposition"][{"Schwarzschild", 1}, t, {r, a1, a2}, a[t, r, a1, a2], {b[t, r, a1, a2], 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6ee7480fff9b945b.png){kind=small} |
| Out[13]= |  |
Show the spacetime metric tensor:
| In[14]:= | ![adm2["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1ce33b60d4e778c5.png){kind=small} |
| Out[14]= |  |
Show the list of equations of motion describing the evolution of the extrinsic curvature tensor with respect to coordinate time t for the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[15]:= | ![adm2["EvolutionEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2502cc39c7317440.png){kind=small} |
| Out[15]= |  |
Show that the evolution equations for the (partially-constrained) ADM decomposition of the Schwarzschild metric all hold identically:
| In[16]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/003bf3b40547b3be.png){kind=small} |
| Out[16]= |  |
Show the list of Gauss equations relating projections of the spacetime Riemann curvature tensor to components of the Riemann curvature tensor over spacelike hypersurfaces for the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[17]:= | ![adm2["GaussEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7302374d09e15bca.png){kind=small} |
| Out[17]= |  |
Show that the Gauss equations for the (partially-constrained) ADM decomposition of the Schwarzschild metric all hold identically:
| In[18]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/279f7f61fd79d990.png){kind=small} |
| Out[18]= |  |
Show the Codazzi-Mainardi equations relating projections of the spacetime Ricci curvature tensor to covariant derivatives of the extrinsic curvature tensor over spacelike hypersurfaces for the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[19]:= | ![Short[adm2["CodazziMainardiEquations"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/02965bde604f3828.png){kind=small} |
| Out[19]= |  |
Show that the Codazzi-Mainardi equations for the (partially-constrained) ADM decomposition of the Schwarzschild metric all hold identically:
| In[20]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1729411d2641202c.png){kind=small} |
| Out[20]= |  |
Show the equation requiring that the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) vanishes identically for the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[21]:= | ![adm2["HamiltonianConstraintEquation"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/644b1d9e287ce625.png){kind=small} |
| Out[21]= |  |
Show that the Hamiltonian constraint for the (partially-constrained) ADM decomposition of the Schwarzschild metric vanishes identically:
| In[22]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/475030b7fafe4057.png){kind=small} |
| Out[22]= |  |
Show the list of equations requiring that the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) vanish identically for the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[23]:= | ![adm2["MomentumConstraintEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/423cd021613e2f3f.png){kind=small} |
| Out[23]= |  |
Show that the momentum constraints for the (partially-constrained) ADM decomposition of the Schwarzschild metric vanish identically:
| In[24]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/290900e95ab8e4c0.png){kind=small} |
| Out[24]= |  |
Show the condition on the lapse function that must hold in order to achieve maximal slicing in the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[25]:= | ![adm2["MaximalSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/744c80b515e9139a.png){kind=small} |
| Out[25]= |  |
Show that condition on the lapse function that must hold in order to achieve maximal slicing in the (partially-constrained) ADM decomposition of the Schwarzschild metric, with all algebraic equivalences imposed:
| In[26]:= | ![adm2["ReducedMaximalSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/370891ee2fb49ac7.png){kind=small} |
| Out[26]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve harmonic coordinates in the (partially-constrained) ADM decomposition of the Schwarzschild metric:
| In[27]:= | ![Short[adm2["HarmonicCoordinateConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/32941d62ef17c705.png){kind=small} |
| Out[27]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve harmonic coordinates in the (partially-constrained) ADM decomposition of the Schwarzschild metric, with all algebraic equivalences imposed:
| In[28]:= | ![adm2["ReducedHarmonicCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/4f7660365d6157a2.png){kind=small} |
| Out[28]= |  |
Perform an ADM decomposition for the Kerr metric (e.g. for an uncharged, spinning black hole with symbolic mass "M" and symbolic angular momentum "J") in Boyer-Lindquist/oblate spheroidal coordinates, using a partially-constrained choice of gauge (defined in terms of scalar functions a and b):
| In[29]:= | ![adm = ResourceFunction["ADMDecomposition"]["Kerr", t, {r, a1, a2}, a[t, r, a1, a2], {0, 0, b[t, r, a1, a2]}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0a45638be97303e5.png){kind=small} |
| Out[29]= |  |
| In[30]:= | ![adm["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/16cd507ab1add6cf.png){kind=small} |
| Out[30]= |  |
Show the list of evolution equations for the (partially-constrained) ADM decomposition of the Kerr metric:
| In[31]:= | ![adm["EvolutionEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3b282b75028985c2.png){kind=small} |
| Out[31]= |  |
Show that the evolution equations for the (partially-constrained) ADM decomposition of the Kerr metric all hold identically:
| In[32]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2b9b4f2247053647.png){kind=small} |
| Out[32]= |  |
Compute the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (partially-constrained) ADM decomposition of the Kerr metric:
| In[33]:= | ![adm["HamiltonianConstraint"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0d2f701250b74d1b.png){kind=small} |
| Out[33]= |  |
Show that the Hamiltonian constraint for the (partially-constrained) ADM decomposition of the Kerr metric vanishes identically:
| In[34]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/429d2465cd309db2.png){kind=small} |
| Out[34]= |  |
Compute the list of momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (partially-constrained) ADM decomposition of the Kerr metric:
| In[35]:= | ![adm["MomentumConstraints"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/43b252c79a0e4677.png){kind=small} |
| Out[35]= |  |
Show that the momentum constraints for the (partially-constrained) ADM decomposition of the Kerr metric vanish identically:
| In[36]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/27d78bd41b76b718.png){kind=small} |
| Out[36]= |  |
Show the condition on the lapse function that must hold in order to achieve harmonic slicing in the (partially-constrained) ADM decomposition of the Kerr metric:
| In[37]:= | ![Short[adm["HarmonicSlicingCondition"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/35cad2a39fa2a708.png){kind=small} |
| Out[37]= |  |
Show the condition on the lapse function that must hold in order to achieve harmonic slicing in the (partially-constrained) ADM decomposition of the Kerr metric, with all algebraic equivalences imposed:
| In[38]:= | ![adm["ReducedHarmonicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0d14480d7c4960ad.png){kind=small} |
| Out[38]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve minimal distortion coordinates in the (partially-constrained) ADM decomposition of the Kerr metric:
| In[39]:= | ![Short[adm["MinimalDistortionConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3619694ce3e56fc9.png){kind=small} |
| Out[39]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve minimal distortion coordinates in the (partially-constrained) ADM decomposition of the Kerr metric, with all algebraic equivalences imposed:
| In[40]:= | ![adm["ReducedMinimalDistortionConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/002b766d5a6386d0.png){kind=small} |
| Out[40]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve pseudo-minimal distortion coordinates in the (partially-constrained) ADM decomposition of the Kerr metric:
| In[41]:= | ![Short[adm["PseudoMinimalDistortionConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/796c2a8d11e9fb49.png){kind=small} |
| Out[41]= |  |
Show the list of conditions on the shift vector that must hold in order to achieve pseudo-minimal distortion coordinates in the (partially-constrained) ADM decomposition of the Kerr metric, with all algebraic equivalences imposed:
| In[42]:= | ![adm["ReducedPseudoMinimalDistortionConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/35b78321c183abbf.png){kind=small} |
| Out[42]= |  |
Perform an ADM decomposition from an (initial) spatial metric tensor directly, using the most general/maximally-unconstrained choice of gauge:
| In[43]:= | ![metric = ResourceFunction["MetricTensor"]["Euclidean"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/172946292d56b97f.png){kind=small} |
| Out[43]= |  |
| In[44]:= | ![adm = ResourceFunction["ADMDecomposition"][metric]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/70e7157859f52ecb.png){kind=small} |
| Out[44]= |  |
| In[45]:= | ![adm["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/55bdc62bfd4f04fe.png){kind=small} |
| Out[45]= |  |
Show the list of gauge conditions that must hold for the resulting ambient/spacetime metric to be Riemannian:
| In[46]:= | ![adm["RiemannianConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/592cf8e8bd3625bf.png){kind=small} |
| Out[46]= |  |
Show the list of gauge conditions that must hold for the resulting ambient/spacetime metric to be pseudo-Riemannian:
| In[47]:= | ![adm["PseudoRiemannianConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5915681f388e4fb4.png){kind=small} |
| Out[47]= |  |
Perform an ADM decomposition for the Brill-Lindquist metric (e.g. for an uncharged, non-rotating binary black hole system with symbolic mass "M" and initial separation distance "z0") in standard spherical polar coordinates, using a partially constrained choice of gauge (defined in terms of a single scalar function a):
| In[48]:= | ![adm = ResourceFunction["ADMDecomposition"]["BrillLindquist", t, {r, a1, a2}, a[t, r, a1, a2], {0, 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3c3c17adb7898f03.png){kind=small} |
| Out[48]= |  |
Show the list of evolution equations for the (partially-constrained) ADM decomposition of the Brill-Lindquist metric:
| In[49]:= | ![adm["EvolutionEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/37c2744845b4a0c7.png){kind=small} |
| Out[49]= |  |
Show that the evolution equations for the (partially-constrained) ADM decomposition of the Brill-Lindquist metric all hold identically:
| In[50]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7d3623b7e6b8970f.png){kind=small} |
| Out[50]= |  |
Show the Hamiltonian constraint equation for the (partially-constrained) ADM decomposition of the Brill-Lindquist metric:
| In[51]:= | ![adm["HamiltonianConstraintEquation"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7a42833e2a318a02.png){kind=small} |
| Out[51]= |  |
Show that the Hamiltonian constraint equation for the (partially-constrained) ADM decomposition of the Brill-Lindquist metric holds identically:
| In[52]:= | ![FullSimplify[%]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2faf6abc1dc98a02.png){kind=small} |
| Out[52]= |  |
Scope (5)
ADM decompositions can be performed directly from an (initial) spatial MetricTensor expression:
| In[53]:= | ![adm = ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"]["Euclidean"]]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/08d710c623a1286b.png){kind=small} |
| Out[53]= |  |
| In[54]:= | ![adm["SpacetimeMetricTensor"]["LineElement"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/724696c568c296b8.png){kind=small} |
| Out[54]= |  |
Additional arguments can be used to specify the distinguished "time" coordinate symbol (otherwise the default symbol "t" will be chosen automatically):
| In[55]:= | ![adm2 = ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"]["Euclidean"], T]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7d7b2c09dda49209.png){kind=small} |
| Out[55]= |  |
| In[56]:= | ![adm2["SpacetimeMetricTensor"]["LineElement"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/4ed99d2a961c7a4b.png){kind=small} |
| Out[56]= |  |
Or the lapse/shift gauge conditions:
| In[57]:= | ![adm3 = ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"]["Euclidean"], 1, {0, 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/13ee8ce81bda9f74.png){kind=small} |
| Out[57]= |  |
| In[58]:= | ![adm3["SpacetimeMetricTensor"]["LineElement"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2621493df321c277.png){kind=small} |
| Out[58]= |  |
Or both simultaneously:
| In[59]:= | ![adm4 = ResourceFunction["ADMDecomposition"][
ResourceFunction["MetricTensor"]["Euclidean"], T, 1, {0, 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6fa834d939286ff5.png){kind=small} |
| Out[59]= |  |
| In[60]:= | ![adm4["SpacetimeMetricTensor"]["LineElement"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5ac9af2ed8488acf.png){kind=small} |
| Out[60]= |  |
Common ADM decompositions can also be performed using an in-built name:
| In[61]:= | ![adm = ResourceFunction["ADMDecomposition"]["ReissnerNordstrom"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2983e785938beb0f.png){kind=small} |
| Out[61]= |  |
| In[62]:= | ![adm["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/740a1395bf518111.png){kind=small} |
| Out[62]= |  |
When an in-built/named ADM decomposition has one or more parameters, those parameters can be left unspecified (in which case they are filled with purely symbolic defaults, such as "M" in the above), or can be specified explicitly in list form:
| In[63]:= | ![adm2 = ResourceFunction[
"ADMDecomposition"][{"ReissnerNordstrom", 1, 1/3}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/52f53d20f186dda7.png){kind=small} |
| Out[63]= |  |
| In[64]:= | ![adm2["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/077ee66b8c47bbcb.png){kind=small} |
| Out[64]= |  |
If only some parameters are explicitly specified, then the remainder are filled with symbolic defaults (e.g. if one specifies only a numerical mass for the Reissner-Nordström metric, then ADMDecomposition will use a purely symbolic electric charge, namely "Q"):
| In[65]:= | ![adm3 = ResourceFunction["ADMDecomposition"][{"ReissnerNordstrom", 1}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0d2a76a215e99c39.png){kind=small} |
| Out[65]= |  |
| In[66]:= | ![adm3["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/29f95583348f5a05.png){kind=small} |
| Out[66]= |  |
Coordinate and lapse/shift gauge information can also be specified for in-built/named ADM decompositions:
| In[67]:= | ![adm4 = ResourceFunction["ADMDecomposition"][{"ReissnerNordstrom", 1}, t, {r, a1, a2}, a[t, r, a1, a2], {b[t, r, a1, a2], 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3ff6dfc1315fda6a.png){kind=small} |
| Out[67]= |  |
| In[68]:= | ![adm4["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/54abf423ac10faae.png){kind=small} |
| Out[68]= |  |
A new distinguished "time" coordinate symbol can be specified for any ADM decomposition:
| In[69]:= | ![adm = ResourceFunction["ADMDecomposition"]["FLRW"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1ad07eba9781a429.png){kind=small} |
| Out[69]= |  |
| In[70]:= | ![adm["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/746458fc7d206d87.png){kind=small} |
| Out[70]= |  |
| In[71]:= | ![adm2 = ResourceFunction["ADMDecomposition"][adm, T]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/221969274619e6ee.png){kind=small} |
| Out[71]= |  |
| In[72]:= | ![adm2["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/573ec6c176f8f0e0.png){kind=small} |
| Out[72]= |  |
New lapse/shift gauge conditions can also be specified for any ADM decomposition:
| In[73]:= | ![adm3 = ResourceFunction["ADMDecomposition"][adm, 1, {0, 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2c7a2872e00916a8.png){kind=small} |
| Out[73]= |  |
| In[74]:= | ![adm3["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7ce7c4a1be9bc49d.png){kind=small} |
| Out[74]= |  |
New coordinate symbols and new lapse/shift gauge conditions can also be specified simultaneously:
| In[75]:= | ![adm4 = ResourceFunction["ADMDecomposition"][adm, T, 1, {0, 0, 0}]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0938de09d8a8a4d2.png){kind=small} |
| Out[75]= |  |
| In[76]:= | ![adm4["SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7546d5119f4babad.png){kind=small} |
| Out[76]= |  |
Show the list of all in-built/named ADM decompositions:
| In[77]:= | ![ResourceFunction["ADMDecomposition"][]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7f1a502667897bcf.png){kind=small} |
| Out[77]= |  |
Perform an ADM decomposition for the 4-dimensional (i.e. 1+3-dimensional) Minkowski metric (default) in Cartesian coordinates:
| In[78]:= | ![ResourceFunction["ADMDecomposition"]["Minkowski"][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7a047faf44f1ac5f.png){kind=small} |
| Out[78]= |  |
Perform an ADM decomposition for the 7-dimensional (i.e. 1+6-dimensional) Minkowski metric in Cartesian coordinates:
| In[79]:= | ![ResourceFunction["ADMDecomposition"][{"Minkowski", 7}][
"SpacetimeMetricTensor"]["MatrixRepresentation"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/314f8a49a18131b9.png){kind=small} |
| Out[79]= |  |
Perform an ADM decomposition for the Schwarzschild metric, e.g. the exterior spacetime of an uncharged, non-rotating black hole, with symbolic mass "M" (default) in spherical polar coordinates:
| In[80]:= | ![ResourceFunction["ADMDecomposition"]["Schwarzschild"][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/57e5105b3261df57.png){kind=small} |
| Out[80]= |  |
Perform an ADM decomposition for the Schwarzschild metric with numerical mass 1 in spherical polar coordinates:
| In[81]:= | ![ResourceFunction["ADMDecomposition"][{"Schwarzschild", 1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0fb0fe6ac8be8bb5.png){kind=small} |
| Out[81]= |  |
Perform an ADM decomposition for the Kerr metric, e.g. the exterior spacetime of an uncharged, spinning black hole, with symbolic mass "M" and symbolic angular momentum "J" (default) in Boyer-Lindquist/oblate spheroidal coordinates:
| In[82]:= | ![ResourceFunction["ADMDecomposition"]["Kerr"]["SpacetimeMetricTensor"][
"MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/268a974c95c19194.png){kind=small} |
| Out[82]= |  |
Perform an ADM decomposition for the Kerr metric with numerical mass 1 and symbolic angular momentum "J" in Boyer-Lindquist/oblate spheroidal coordinates:
| In[83]:= | ![ResourceFunction["ADMDecomposition"][{"Kerr", 1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/704ebcdc9410b6cb.png){kind=small} |
| Out[83]= |  |
Perform an ADM decomposition for the Kerr metric with numerical mass 1 and numerical angular momentum 1/2 in Boyer-Lindquist/oblate spheroidal coordinates:
| In[84]:= | ![ResourceFunction["ADMDecomposition"][{"Kerr", 1, 1/2}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5867b9b84385abb1.png){kind=small} |
| Out[84]= |  |
Perform an ADM decomposition for the Reissner-Nordström metric, e.g. the exterior spacetime of a charged, non-rotating black hole, with symbolic mass "M" and symbolic electric charge "Q" (default) in spherical polar coordinates:
| In[85]:= | ![ResourceFunction["ADMDecomposition"]["ReissnerNordstrom"][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/77b2abc7f3ac4fc9.png){kind=small} |
| Out[85]= |  |
Perform an ADM decomposition for the Reissner-Nordström metric with numerical mass 1 and symbolic electric charge "Q" in spherical polar coordinates:
| In[86]:= | ![ResourceFunction["ADMDecomposition"][{"ReissnerNordstrom", 1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/047bbe3bd0f371a4.png){kind=small} |
| Out[86]= |  |
Perform an ADM decomposition for the Reissner-Nordström metric with numerical mass 1 and numerical electric charge 1/3 in spherical polar coordinates:
| In[87]:= | ![ResourceFunction["ADMDecomposition"][{"ReissnerNordstrom", 1, 1/3}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2a71fe659bf8a85d.png){kind=small} |
| Out[87]= |  |
Perform an ADM decomposition for the Kerr-Newman metric, e.g. the exterior spacetime of a charged, spinning black hole, with symbolic mass "M", symbolic angular momentum "J" and symbolic electric charge "Q" (default) in Boyer-Lindquist/oblate spheroidal coordinates:
| In[88]:= | ![ResourceFunction["ADMDecomposition"]["KerrNewman"][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/68312332eac606d9.png){kind=small} |
| Out[88]= |  |
Perform an ADM decomposition for the Kerr-Newman metric with numerical mass 1, symbolic angular momentum "J" and symbolic electric charge "Q" in Boyer-Lindquist/oblate spheroidal coordinates:
| In[89]:= | ![ResourceFunction["ADMDecomposition"][{"KerrNewman", 1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/66ecee3a2811911b.png){kind=small} |
| Out[89]= |  |
Perform an ADM decomposition for the Kerr-Newman metric with numerical mass 1, numerical angular momentum 1/2 and symbolic electric charge "Q" in Boyer-Lindquist/oblate spheroidal coordinates:
| In[90]:= | ![ResourceFunction["ADMDecomposition"][{"KerrNewman", 1, 1/2}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5bc801af9ace66d5.png){kind=small} |
| Out[90]= |  |
Perform an ADM decomposition for the Kerr-Newman metric with numerical mass 1, numerical angular momentum 1/2 and numerical electric charge 1/3 in Boyer-Lindquist/oblate spheroidal coordinates:
| In[91]:= | ![ResourceFunction["ADMDecomposition"][{"KerrNewman", 1, 1/2, 1/3}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5d8fad082add77e1.png){kind=small} |
| Out[91]= |  |
Perform an ADM decomposition for the Brill-Lindquist metric, e.g. the exterior spacetime of a pair of uncharged, non-rotating black holes, with symbolic mass "M" and symbolic separation distance "z0" (default) in spherical polar coordinates:
| In[92]:= | ![ResourceFunction["ADMDecomposition"]["BrillLindquist"][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/75a646b0df664613.png){kind=small} |
| Out[92]= |  |
Perform an ADM decomposition for the Brill-Lindquist metric with numerical mass 1 and symbolic separation distance "z0" in spherical polar coordinates:
| In[93]:= | ![ResourceFunction["ADMDecomposition"][{"BrillLindquist", 1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3536596a3e3a41ca.png){kind=small} |
| Out[93]= |  |
Perform an ADM decomposition for the Brill-Lindquist metric with numerical mass 1 and numerical separation distance 2 in spherical polar coordinates:
| In[94]:= | ![ResourceFunction["ADMDecomposition"][{"BrillLindquist", 1, 2}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0716f931367132ff.png){kind=small} |
| Out[94]= |  |
Perform an ADM decomposition for the Friedmann-Lemaître-Robertson-Walker/FLRW metric, i.e. the metric for a homogeneous, isotropic and uniformly expanding/contracting universe, with symbolic global curvature "k" and symbolic scale function "a" (default) in spherical polar coordinates:
| In[95]:= | ![ResourceFunction["ADMDecomposition"]["FLRW"]["SpacetimeMetricTensor"][
"MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/32446eb3cf1d7df4.png){kind=small} |
| Out[95]= |  |
Perform an ADM decomposition for the FLRW metric with numerical global curvature -1 and symbolic scale function "a" in spherical polar coordinates:
| In[96]:= | ![ResourceFunction["ADMDecomposition"][{"FLRW", -1}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/02080277d898d7ca.png){kind=small} |
| Out[96]= |  |
Perform an ADM decomposition for the FLRW metric with numerical global curvature -1 and numerical scale function (#*3)& in spherical polar coordinates:
| In[97]:= | ![ResourceFunction["ADMDecomposition"][{"FLRW", -1, (#*3) &}][
"SpacetimeMetricTensor"]["MatrixRepresentation"] // MatrixForm](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/535111a7e5c4596a.png){kind=small} |
| Out[97]= |  |
Perform an ADM decomposition for the Schwarzschild metric, with symbolic mass "M":
| In[98]:= | ![adm = ResourceFunction["ADMDecomposition"]["Schwarzschild"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5fca6d0824fb6895.png){kind=small} |
| Out[98]= |  |
Show the list of properties:
| In[99]:= | ![adm["Properties"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/666bcc55a0e64155.png){kind=small} |
| Out[99]= |  |
Show the spatial metric tensor (i.e. the metric tensor on submanifolds/spacelike hypersurfaces) for the ADM decomposition:
| In[100]:= | ![adm["SpatialMetricTensor"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/643170fde0e5d76e.png){kind=small} |
| Out[100]= |  |
Show the spacetime metric tensor (i.e. the metric tensor on the ambient manifold/spacetime) for the ADM decomposition:
| In[101]:= | ![adm["SpacetimeMetricTensor"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/30587657fe38e0e7.png){kind=small} |
| Out[101]= |  |
Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces for the ADM decomposition:
| In[102]:= | ![adm["NormalVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/580ef0a8f3d6bb06.png){kind=small} |
| Out[102]= |  |
Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces for the ADM decomposition, with all algebraic equivalences imposed:
| In[103]:= | ![adm["ReducedNormalVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/568bbbd104329cf5.png){kind=small} |
| Out[103]= |  |
Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[104]:= | ![adm["SymbolicNormalVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2f586d408e33d97f.png){kind=small} |
| Out[104]= |  |
Show the future-pointing, timelike "time vector" for the ADM decomposition:
| In[105]:= | ![adm["TimeVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0572d86263543875.png){kind=small} |
| Out[105]= |  |
Show the future-pointing, timelike "time vector" for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[106]:= | ![Short[adm["SymbolicTimeVector"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/08bc090f6c9d70b7.png){kind=small} |
| Out[106]= |  |
Show the distinguished time coordinate symbol for the ADM decomposition:
| In[107]:= | ![adm["TimeCoordinate"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0c4d802c4bb56d0c.png){kind=small} |
| Out[107]= |  |
Show the list of distinguished spatial coordinate symbols for the ADM decomposition:
| In[108]:= | ![adm["SpatialCoordinates"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/4f3e0ef8423ff641.png){kind=small} |
| Out[108]= |  |
Show the list of differential 1-form symbols for the (ambient/spacetime) coordinates of the ADM decomposition:
| In[109]:= | ![adm["CoordinateOneForms"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3a5ce45378cc3f63.png){kind=small} |
| Out[109]= |  |
Show the lapse function for the ADM decomposition:
| In[110]:= | ![adm["LapseFunction"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/75028568a9085d2b.png){kind=small} |
| Out[110]= |  |
Show the shift vector (field) for the ADM decomposition:
| In[111]:= | ![adm["ShiftVector"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1fdc7fc93cb2628b.png){kind=small} |
| Out[111]= |  |
Show the list of Gauss equations relating projections of the Riemann curvature tensor over the ambient manifold/spacetime to components of the Riemann curvature tensor over the submanifolds/spacelike hypersurfaces for the ADM decomposition:
| In[112]:= | ![adm["GaussEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2745ee6cbef1e733.png){kind=small} |
| Out[112]= |  |
Show the list of Gauss equations relating projections of the Riemann curvature tensor over the ambient manifold/spacetime to components of the Riemann curvature tensor over the submanifolds/spacelike hypersurfaces for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[113]:= | ![adm["SymbolicGaussEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2f836cb1b3c6d68c.png){kind=small} |
| Out[113]= |  |
Show the list of Codazzi-Mainardi equations relating projections of the Ricci curvature tensor over the ambient manifold/spacetime to covariant derivatives of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition:
| In[114]:= | ![adm["CodazziMainardiEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0b21f602566231cc.png){kind=small} |
| Out[114]= |  |
Show the list of Codazzi-Mainardi equations relating projections of the Ricci curvature tensor over the ambient manifold/spacetime to covariant derivatives of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[115]:= | ![adm["SymbolicCodazziMainardiEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2af7239e8c667658.png){kind=small} |
| Out[115]= |  |
Show the condition on the lapse function required to guarantee geodesic slicing (i.e. unit lapse over the entire ambient manifold/spacetime) for the ADM decomposition:
| In[116]:= | ![adm["GeodesicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1f24f995bb5d31e0.png){kind=small} |
| Out[116]= |  |
Show the condition on the lapse function required to guarantee geodesic slicing (i.e. unit lapse over the entire ambient manifold/spacetime) for the ADM decomposition, with all algebraic equivalences imposed:
| In[117]:= | ![adm["ReducedGeodesicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/68fc6df153bee8e9.png){kind=small} |
| Out[117]= |  |
Show the condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition:
| In[118]:= | ![Short[adm["MaximalSlicingCondition"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/08925969be936a04.png){kind=small} |
| Out[118]= |  |
Show the condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition, with all algebraic equivalences imposed:
| In[119]:= | ![adm["ReducedMaximalSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3ce4944e4faf1e3c.png){kind=small} |
| Out[119]= |  |
Show the condition on the lapse function required to guarantee maximal slicing (i.e. maximum spatial volume of each submanifold/spacelike hypersurface) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[120]:= | ![Short[adm["SymbolicMaximalSlicingCondition"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/71f5c63746fd2265.png){kind=small} |
| Out[120]= |  |
Show the condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the time coordinate) for the ADM decomposition:
| In[121]:= | ![Short[adm["HarmonicSlicingCondition"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/397e642ec90cd5f2.png){kind=small} |
| Out[121]= |  |
Show the condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the time coordinate) for the ADM decomposition, with all algebraic equivalences imposed:
| In[122]:= | ![adm["ReducedHarmonicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5ee5c5d63d9bfa83.png){kind=small} |
| Out[122]= |  |
Show the condition on the lapse function required to guarantee harmonic slicing (i.e. vanishing of the curved spacetime d'Alembertian of the time coordinate) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[123]:= | ![adm["SymbolicHarmonicSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/13ff1f538d5a82a0.png){kind=small} |
| Out[123]= |  |
Show the condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition:
| In[124]:= | ![adm["OnePlusLogSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/15b065c01ae09162.png){kind=small} |
| Out[124]= |  |
Show the condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition, with all algebraic equivalences imposed:
| In[125]:= | ![adm["ReducedOnePlusLogSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6016892a469cf107.png){kind=small} |
| Out[125]= |  |
Show the condition on the lapse function required to guarantee 1+log slicing (i.e. generalized harmonic slicing) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[126]:= | ![adm["SymbolicOnePlusLogSlicingCondition"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/1817b4f0324157e1.png){kind=small} |
| Out[126]= |  |
Show the list of conditions on the shift vector (field) required to guarantee normal coordinates (i.e. vanishing shift over the entire ambient manifold/spacetime) for the ADM decomposition:
| In[127]:= | ![adm["NormalCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/10c0481be5621c0b.png){kind=small} |
| Out[127]= |  |
Show the list of conditions on the shift vector (field) required to guarantee normal coordinates (i.e. vanishing shift over the entire ambient manifold/spacetime) for the ADM decomposition, with all algebraic equivalences imposed:
| In[128]:= | ![adm["ReducedNormalCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/779d1bb21f6c582f.png){kind=small} |
| Out[128]= |  |
Show the list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the spatial coordinates) for the ADM decomposition:
| In[129]:= | ![Short[adm["HarmonicCoordinateConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/51303bfdc9a35d20.png){kind=small} |
| Out[129]= |  |
Show the list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the spatial coordinates) for the ADM decomposition, with all algebraic equivalences imposed:
| In[130]:= | ![adm["ReducedHarmonicCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/61c676accd5a6597.png){kind=small} |
| Out[130]= |  |
Show the list of conditions on the shift vector (field) required to guarantee harmonic coordinates (i.e. vanishing of the curved spacetime d'Alembertian of the spatial coordinates) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[131]:= | ![adm["SymbolicHarmonicCoordinateConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3ac5c4da149d1bf7.png){kind=small} |
| Out[131]= |  |
Show the list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition:
| In[132]:= | ![Short[adm["MinimalDistortionConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6f99a78cac6031d2.png){kind=small} |
| Out[132]= |  |
Show the list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition, with all algebraic equivalences imposed:
| In[133]:= | ![adm["ReducedMinimalDistortionConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7a15bd48ff093af3.png){kind=small} |
| Out[133]= |  |
Show the list of conditions on the shift vector (field) required to guarantee minimal distortion coordinates (i.e. minimum strain on each submanifold/spacelike hypersurface) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[134]:= | ![adm["SymbolicMinimalDistortionConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/4db81594e3ac6b1d.png){kind=small} |
| Out[134]= |  |
Show the list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition:
| In[135]:= | ![Short[adm["PseudoMinimalDistortionConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/2fd1fbf75743f975.png){kind=small} |
| Out[135]= |  |
Show the list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition, with all algebraic equivalences imposed:
| In[136]:= | ![adm["ReducedPseudoMinimalDistortionConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3a184b63fcd1d1d2.png){kind=small} |
| Out[136]= |  |
Show the list of conditions on the shift vector (field) required to guarantee pseudo-minimal distortion coordinates (i.e. minimal distortion coordinates, but with all covariant derivatives replaced with partial derivatives) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[137]:= | ![Short[adm["SymbolicPseudoMinimalDistortionConditions"], 10]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7d9bc50010c30604.png){kind=small} |
| Out[137]= |  |
Show the number of dimensions of the ambient manifold/spacetime represented by the ADM decomposition:
| In[138]:= | ![adm["Dimensions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/6a6193aaa728939f.png){kind=small} |
| Out[138]= |  |
Show the signature of the ambient manifold/spacetime represented by the ADM decomposition (with +1s representing positive eigenvalues and -1s representing negative eigenvalues of the metric tensor):
| In[139]:= | ![adm["Signature"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7117bd77ec05fa24.png){kind=small} |
| Out[139]= |  |
Determine whether the ambient manifold/spacetime represented by the ADM decomposition is Riemannian (i.e. all eigenvalues of the metric tensor have the same sign):
| In[140]:= | ![adm["RiemannianQ"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/51925c3ab9439faa.png){kind=small} |
| Out[140]= |  |
Determine whether the ambient manifold/spacetime represented by the ADM decomposition is pseudo-Riemannian (i.e. all eigenvalues are non-zero, but not all have the same sign):
| In[141]:= | ![adm["PseudoRiemannianQ"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/35a4efa90cdde723.png){kind=small} |
| Out[141]= |  |
Determine whether the ambient manifold/spacetime represented by the ADM decomposition is Lorentzian (i.e. all eigenvalues of the metric tensor have the same sign, except for one eigenvalue which has the opposite sign):
| In[142]:= | ![adm["LorentzianQ"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/120e73ff9f305d5b.png){kind=small} |
| Out[142]= |  |
Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the ADM decomposition is Riemannian (i.e. all eigenvalues of the metric tensor are positive):
| In[143]:= | ![Short[adm["RiemannianConditions"], 20]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0c619661b1434194.png){kind=small} |
| Out[143]= |  |
Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the ADM decomposition is pseudo-Riemannian (i.e. all eigenvalues of the metric tensor are non-zero):
| In[144]:= | ![Short[adm["PseudoRiemannianConditions"], 20]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/0aef247404ee62fb.png){kind=small} |
| Out[144]= |  |
Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the ADM decomposition is Lorentzian (i.e. the "time" eigenvalue is negative, and all other eigenvalues are positive):
| In[145]:= | ![adm["LorentzianConditions"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/591f6c4760502f21.png){kind=small} |
| Out[145]= |  |
Show the list of equations characterizing the time evolution of components of the extrinsic curvature tensor over submanifold/spacelike hypersurfaces for the ADM decomposition:
| In[146]:= | ![adm["EvolutionEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7df5edbffed14144.png){kind=small} |
| Out[146]= |  |
Show the list of equations characterizing the time evolution of components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[147]:= | ![adm["SymbolicEvolutionEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3a82d449da3a8662.png){kind=small} |
| Out[147]= |  |
Show the value of the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the ADM decomposition:
| In[148]:= | ![adm["HamiltonianConstraint"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/60f55e61bbfa6594.png){kind=small} |
| Out[148]= |  |
Show the value of the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[149]:= | ![adm["SymbolicHamiltonianConstraint"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/173111f3fe80dcbb.png){kind=small} |
| Out[149]= |  |
Show the list of values of the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the ADM decomposition:
| In[150]:= | ![adm["MomentumConstraints"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/3483997fea818c18.png){kind=small} |
| Out[150]= |  |
Show the list of values of the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the ADM decomposition, with all partial derivative operators left purely symbolic:
| In[151]:= | ![adm["SymbolicMomentumConstraints"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/75012b935414cf1a.png){kind=small} |
| Out[151]= |  |
Show the condition required to guarantee that the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the ADM decomposition vanishes identically:
| In[152]:= | ![adm["HamiltonianConstraintEquation"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/7cf2c9c6af08cf9f.png){kind=small} |
| Out[152]= |  |
Show the condition required to guarantee that the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the ADM decomposition vanishes identically, with all partial derivative operators left purely symbolic:
| In[153]:= | ![adm["SymbolicHamiltonianConstraintEquation"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5259568da73a064e.png){kind=small} |
| Out[153]= |  |
Show the list of conditions required to guarantee that the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the ADM decomposition vanish identically:
| In[154]:= | ![adm["MomentumConstraintEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/5bd4bfa0f50e6fe7.png){kind=small} |
| Out[154]= |  |
Show the list of conditions required to guarantee that the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the ADM decomposition vanish identically, with all partial derivative operators left purely symbolic:
| In[155]:= | ![adm["SymbolicMomentumConstraintEquations"]](https://www.wolframcloud.com/obj/resourcesystem/images/a45/a45fe35b-02b8-487e-860c-55871a038944/75bae33d4a6af0c6.png){kind=small} |
| Out[155]= |  |
Publisher
Related Links
- Einstein Field Equations - Wolfram MathWorld
- The Wolfram Physics Project
- Stephen Wolfram's A New Kind of Science | Online
- Multiway System–Wolfram MathWorld
Version History
- 1.0.0 – 22 March 2023
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This work is licensed under a Creative Commons Attribution 4.0 International License