Function Repository Resource:

SolveVacuumADMEquations

Determine whether a given ADM decomposition is a solution to the vacuum ADM equations

Contributed by: Jonathan Gorard

ResourceFunction["SolveVacuumADMEquations"][ResourceFunction["ADMDecomposition"][…],cosmo]

attempts to solve the vacuum ADM evolution equations for the given ADMDecomposition, assuming cosmological constant cosmo.

ResourceFunction["SolveVacuumADMEquations"][ResourceFunction["ADMDecomposition"][…]]

uses the default (symbolic) value "" as the cosmological constant.

Details

The vacuum ADM evolution equations are a formulation of the vacuum Einstein field equations (i.e. the equations of motion of general relativity in the absence of any stress-energy tensor, and thus in the absence of any matter-energy content of spacetime) as a special case of Hamilton's equations, describing how the components of the spatial metric tensor (i.e. the metric tensor field associated to each submanifold/spacelike hypersurface) and 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), "evolve" from one submanifold/spacelike hypersurface to the next.

The vacuum ADM equations are based on the ADM decomposition (due to Arnowitt, Deser and Misner), in which a Riemannian or pseudo-Riemannian manifold is foliated into a sequence of submanifolds; in the case of general relativity, the ambient manifold is a Lorentzian spacetime and the submanifolds are spacelike hypersurfaces of codimension-1, thus allowing one to formulate the Einstein field equations as an explicit time evolution/initial value problem.

In addition to the (typically hyperbolic) vacuum evolution equations/vacuum equations of motion for the spatial metric tensor and the extrinsic curvature tensor, there are also several (typically elliptic, though with many notable exceptions) constraint equations, namely equations for the Hamiltonian and momentum constraints; the vacuum evolution equations are obtained directly from the vacuum Einstein field equations, while the constraint equations are obtained from the timelike and spacelike projections of the (contracted) Bianchi identities.

ResourceFunction["SolveVacuumADMEquations"] assumes a non-zero cosmological constant (represented by the purely symbolic value "") by default, although this can be overridden as part of the second argument of the function.

ResourceFunction["SolveVacuumADMEquations"] classifies all vacuum ADM solutions as either exact (i.e. no additional assumptions required) or non-exact (i.e. some additional assumptions required), depending upon whether any additional field equations must be assumed in order for the given ADMDecomposition to constitute a valid solution to the vacuum ADM evolution equations. Any such field equations (if they exist) can be computed using the property "FieldEquations".

ResourceFunction["SolveVacuumADMEquations"] 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["SolveVacuumADMEquations"] 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 and its corresponding metric and curvature tensors implicitly assumes the torsion-free Levi-Civita connection by default.

ResourceFunction["SolveVacuumADMEquations"] 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["SolveVacuumADMEquations"] 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).

If ResourceFunction["SolveVacuumADMEquations"] succeeds in solving the vacuum ADM evolution equations (or in proving that no such solution can exist), it will return a VacuumADMSolution expression.

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["SolveVacuumADMEquations"] 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) is 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 VacuumADMSolution, the following properties are supported:

"SolutionQ"

whether the ADM decomposition is a valid solution to the vacuum ADM evolution equations

"ExactSolutionQ"

whether the ADM decomposition is a valid exact solution to the vacuum ADM evolution equations (i.e. no additional field equations need to be assumed)

"FieldEquations"

list of any (additional) field equations that must be assumed in order for the ADM decomposition to be a valid solution to the vacuum ADM evolution equations

"EvolutionEquations"

list of vacuum ADM evolution equations characterizing the "time" evolution of the components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the underlying ADM decomposition

"ReducedEvolutionEquations"

"SymbolicEvolutionEquations"

"ADMDecomposition"

underlying ADM decomposition associated to the vacuum ADM solution

"SpatialMetricTensor"

underlying (spatial) metric tensor on each submanifold/spacelike hypersurface of the vacuum ADM solution

"SpacetimeMetricTensor"

metric tensor on the resulting ambient manifold/spacetime of the vacuum ADM solution

"NormalVector"

(future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the vacuum ADM solution

"ReducedNormalVector"

(future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the vacuum ADM solution, modulo all tensor equivalences

"SymbolicNormalVector"

(future-pointing, timelike) unit vector normal to each submanifold/spacelike hypersurface of the vacuum ADM solution, with purely symbolic partial derivative operators

"TimeVector"

(future-pointing, timelike) "time vector" connecting corresponding points on neighboring submanifolds/spacelike hypersurfaces of the vacuum ADM solution

"SymbolicTimeVector"

(future-pointing, timelike) "time vector" connecting corresponding points on neighboring submanifolds/spacelike hypersurfaces of the vacuum ADM solution, with purely symbolic partial derivative operators

"TimeCoordinate"

distinguished "time" coordinate symbol for the vacuum ADM solution

"SpatialCoordinates"

list of distinguished "spatial" coordinate symbols for the vacuum ADM solution

"CoordinateOneForms"

list of differential 1-form symbols for the ambient/spacetime coordinates of the vacuum ADM solution

"LapseFunction"

lapse function determining "timelike" coordinate distance between neighboring submanifolds/spacelike hypersurfaces of the vacuum ADM solution

"ShiftVector"

shift vector (field) determining how "spatial" coordinates are relabeled between neighboring submanifolds/spacelike hypersurfaces of the vacuum ADM solution

"HamiltonianConstraintSatisfiedQ"

whether the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the vacuum ADM solution vanishes identically

"HamiltonianConstraint"

value of the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the vacuum ADM solution

"ReducedHamiltonianConstraint"

value of the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the vacuum ADM solution, modulo all tensor equivalences

"SymbolicHamiltonianConstraint"

value of the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the vacuum ADM solution, 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 vacuum ADM solution vanishes identically

"ReducedHamiltonianConstraintEquation"

condition required to guarantee that the Hamiltonian constraint (derived from the "timelike" component of the contracted Bianchi identities) of the vacuum ADM solution vanishes identically, modulo all tensor equivalences

"SymbolicHamiltonianConstraintEquation"

"MomentumConstraintsSatisfiedQ"

whether the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the vacuum ADM solution vanish identically

"MomentumConstraints"

list of values of the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the vacuum ADM solution

"ReducedMomentumConstraints"

list of values of the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the vacuum ADM solution, modulo all tensor equivalences

"SymbolicMomentumConstraints"

list of values of the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the vacuum ADM solution, 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 vacuum ADM solution vanish identically

"ReducedMomentumConstraintEquations"

list of conditions required to guarantee that the momentum constraints (derived from the "spacelike" components of the contracted Bianchi identities) of the vacuum ADM solution vanish identically, modulo all tensor equivalences

"SymbolicMomentumConstraintEquations"

"CosmologicalConstant"

value of the cosmological constant for the vacuum ADM solution

"Dimensions"

number of dimensions of the ambient manifold/spacetime described by the vacuum ADM solution

"Signature"

list of +1s and -1s designating the signature of the ambient manifold/spacetime described by the vacuum ADM solution (+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 vacuum ADM solution is Riemannian (i.e. all eigenvalues of the metric have the same sign)

"PseudoRiemannianQ"

whether the ambient manifold/spacetime described by the vacuum ADM solution 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 vacuum ADM solution 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 vacuum ADM solution 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 vacuum ADM solution 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 vacuum ADM solution is Lorentzian (i.e. the "time" eigenvalue of the metric is negative and all other eigenvalues are positive)

"Properties"

list of properties

Examples

Basic Examples (3)

Show that the ADM decomposition for the Schwarzschild metric (e.g. for an uncharged, non-rotating black hole with symbolic mass M) in standard spherical polar coordinates with time coordinate symbol t and spatial coordinate symbols r, a1 and a2, using a partially-constrained choice of gauge (defined in terms of scalar functions a and b), is a valid solution to the vacuum ADM evolution equations with zero cosmological constant:

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Show that the (partially-constrained) vacuum ADM Schwarzschild solution is not exact (i.e. additional field equations need to be assumed):

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Show the list of vacuum equations of motion describing the evolution of the extrinsic curvature tensor with respect to coordinate time t for the (partially-constrained) vacuum ADM Schwarzschild solution:

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Show that the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution does not vanish identically:

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Compute the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution:

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Compute the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution, with all algebraic equivalences imposed:

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Show that the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution do not vanish identically:

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Compute the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution:

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Compute the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (partially-constrained) vacuum ADM Schwarzschild solution, with all algebraic equivalences imposed:

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Show that the ADM decomposition for the Schwarzschild metric using a fully-constrained choice of gauge is now an exact solution to the vacuum ADM equations with zero cosmological constant (i.e. no additional field equations need to be assumed):

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Show that the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (maximally-constrained) vacuum ADM Schwarzschild solution vanishes identically:

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Show that the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (maximally-constrained) vacuum ADM Schwarzschild solution vanish identically:

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Show that the ADM decomposition for the Schwarzschild metric using a geodesic slicing condition on the lapse (i.e. the lapse function is identically one) and normal coordinate conditions on the shift (i.e. the shift vector is identically zero) is only a valid solution to the vacuum ADM equations with zero cosmological constant if either the mass of the black hole is equal to zero or the value of the radial coordinate is equal to infinity:

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Find the conditions under which the ADM decomposition for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric (for a homogeneous, isotropic and uniformly-expanding/contracting universe, with symbolic curvature parameter "k" and symbolic scale factor "a") in standard spherical polar coordinates with time coordinate symbol t and spatial coordinate symbols r, a1 and a2, using a partially-constrained choice of gauge (defined in terms of scalar function a), is a valid solution to the vacuum ADM evolution equations with symbolic cosmological constant "":

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Find the condition under which the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (partially-constrained) ADM FLRW solution vanishes identically:

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Find the conditions under which the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (partially-constrained) ADM FLRW solution vanish identically:

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Find the corresponding conditions for the case of zero cosmological constant:

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Show that in neither case is the solution exact (i.e. additional field equations must be assumed):

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Show that the 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 with time coordinate symbol t and spatial coordinate symbols r, a1 and a2, using an appropriate maximally-constrained choice of gauge, is a valid (exact) solution to the vacuum ADM evolution equations with zero cosmological constant:

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$adm = ResourceFunction["ADMDecomposition"][{"Kerr", M, J}, t, {r, a1, a2}, Sqrt[(J^2 + M^2 r (-2 M + r)) (J^2 + 2 M^2 r^2 + J^2 Cos[2 a1])]/ Sqrt[2 (J^2 + M^2 r^2) (M^2 r^2 + J^2 Cos[a1]^2) + 4 J^2 M^3 r Sin[a1]^2], {0, 0, -((2 J M^4 r)/((J^2 + M^2 r^2) (M^2 r^2 + J^2 Cos[a1]^2) + 2 J^2 M^3 r Sin[a1]^2))}]$

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Show the list of vacuum equations of motion describing the evolution of the extrinsic curvature tensor with respect to coordinate time t for the (maximally-constrained) vacuum ADM Kerr solution:

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Show the list of vacuum equations of motion describing the evolution of the extrinsic curvature tensor with respect to coordinate time t for the (maximally-constrained) vacuum ADM Kerr solution, with all algebraic equivalences imposed, and confirm that they hold identically:

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Show that the Hamiltonian constraint (derived from the timelike component of the contracted Bianchi identities) for the (maximally-constrained) vacuum ADM Kerr solution vanishes identically:

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Show that the momentum constraints (derived from the spacelike components of the contracted Bianchi identities) for the (maximally-constrained) vacuum ADM Kerr solution vanish identically:

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Show the lapse function for the (maximally-constrained) vacuum ADM Kerr solution, demonstrating the effects of gravitational time dilation:

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Show the shift vector for the (maximally-constrained) vacuum ADM Kerr solution, demonstrating the effects of frame dragging:

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Scope (2)

By default, SolveVacuumADMEquations will use a symbolic value ("") for the cosmological constant:

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This default choice may be overridden using a second argument:

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Solve the vacuum ADM evolution equations for the ADM decomposition of the FLRW metric, with symbolic curvature parameter "k", symbolic scale factor "a" and a partially-constrained choice of gauge (defined in terms of scalar function a), and assuming symbolic cosmological constant "":

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Show the list of properties:

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Determine whether the ADM decomposition is a valid solution to the vacuum ADM evolution equations:

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Determine whether the ADM decomposition is a valid exact solution to the vacuum ADM evolution equations (i.e. no additional field equations need to be assumed):

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Show the list of (additional) field equations that must be assumed in order for the ADM decomposition to be a valid solution to the vacuum ADM evolution equations:

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Show the list of vacuum ADM evolution equations characterizing the time evolution of components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition:

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Show the list of vacuum ADM evolution equations characterizing the time evolution of components of the extrinsic curvature tensor over submanifolds/spacelike hypersurfaces of the ADM decomposition, with all algebraic equivalences imposed:

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Show the underlying ADM decomposition associated to the vacuum ADM solution:

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Show the spatial metric tensor (i.e. the metric tensor on submanifolds/spacelike hypersurfaces) for the vacuum ADM solution:

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Show the spacetime metric tensor (i.e. the metric tensor on the ambient manifold/spacetime) for the vacuum ADM solution:

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Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces of the vacuum ADM solution:

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Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces of the vacuum ADM solution, with all algebraic equivalences imposed:

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Show the future-pointing, timelike unit vector normal to submanifolds/spacelike hypersurfaces of the vacuum ADM solution, with all partial derivative operators left purely symbolic:

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Show the future-pointing, timelike "time vector" for the vacuum ADM solution:

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Show the future-pointing, timelike "time vector" for the vacuum ADM solution, with all partial derivative operators left purely symbolic:

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Show the distinguished time coordinate symbol for the vacuum ADM solution:

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Show the list of distinguished spatial coordinate symbols for the vacuum ADM solution:

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Show the list of differential 1-form symbols for the (ambient/spacetime) coordinates of the vacuum ADM solution:

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Show the lapse function for the vacuum ADM solution:

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Show the shift vector (field) for the vacuum ADM solution:

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Determine whether the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution vanishes identically:

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Show the value of the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution:

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Show the value of the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution, with all algebraic equivalences imposed:

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Show the value of the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution, with all partial derivative operators left purely symbolic:

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Show the condition required to guarantee that the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution vanishes identically:

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Show the condition required to guarantee that the Hamiltonian constraint (i.e. the timelike component of the contracted Bianchi identities) for the vacuum ADM solution vanishes identically, with all algebraic equivalences imposed:

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Determine whether the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution vanish identically:

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Show the list of values of the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution:

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Show the list of values of the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution, with all algebraic equivalences imposed:

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Show the list of values of the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution, with all partial derivative operators left purely symbolic:

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Show the list of conditions required to guarantee that the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution vanish identically:

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Show the list of conditions required to guarantee that the momentum constraints (i.e. the spacelike components of the contracted Bianchi identities) for the vacuum ADM solution vanish identically, with all algebraic equivalences imposed:

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Show the value of the cosmological constant for the vacuum ADM solution:

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Show the number of dimensions of the ambient manifold/spacetime represented by the vacuum ADM solution:

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Show the signature of the ambient manifold/spacetime represented by the vacuum ADM solution (with +1s representing positive eigenvalues and -1s representing negative eigenvalues of the metric tensor):

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Determine whether the ambient manifold/spacetime represented by the vacuum ADM solution is Riemannian (i.e. all eigenvalues of the metric tensor have the same sign):

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Determine whether the ambient manifold/spacetime represented by the vacuum ADM solution is pseudo-Riemannian (i.e. all eigenvalues are non-zero, but not all have the same sign):

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Determine whether the ambient manifold/spacetime represented by the vacuum ADM solution is Lorentzian (i.e. all eigenvalues of the metric tensor have the same sign, except for one eigenvalue which has the opposite sign):

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Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the vacuum ADM solution is Riemannian (i.e. all eigenvalues of the metric tensor are positive):

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Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the vacuum ADM solution is pseudo-Riemannian (i.e. all eigenvalues of the metric tensor are non-zero):

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Show the list of conditions on the coordinates required to guarantee that the ambient manifold/spacetime represented by the vacuum ADM solution is Lorentzian (i.e. the "time" eigenvalue is negative, and all other eigenvalues are positive):

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Publisher

Jonathan Gorard

Version History

1.0.0 – 21 August 2023

Source Metadata

Citation:
- Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (1973), Gravitation
- Stephen Wolfram (2002), A New Kind of Science
- Stephen Wolfram (2020), "A Class of Models with Potential to Represent Fundamental Physics"
- Jonathan Gorard (2020), "Some Relativistic and Gravitational Properties of the Wolfram Model"
- Jonathan Gorard (2020), "Some Quantum Mechanical Properties of the Wolfram Model"

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

SolveVacuumADMEquations

Details

Examples

Basic Examples (3)

Scope (2)

Publisher

Related Links

Version History

Source Metadata

Related Resources

License Information

SolveVacuumADMEquations

Details

Examples

Basic Examples (3)

Scope (2)

Publisher

Related Links

Version History

Source Metadata

Related Resources

Related Symbols

License Information