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Overview of the Tensor package

 

Description

Commands for the algebraic manipulation of tensors

Commands for tensor differentiation

Commands for calculating curvature tensors  

Infinitesimal transformation groups

Commands for calculating holonomy

Commands for calculating special tensor fields

Commands for working with Killing tensors

Commands for the 2-component spinor formalism

Commands for the Newman-Penrose formalism

Commands for the algebraic classification of spacetimes

Commands for field theory

Alphabetical listing of all Tensor commands

Description

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The DifferentialGeometry:-Tensor package provides an extensive suite of commands for computations with tensors on the tangent bundle of any manifold or with tensors on any vector bundle.  

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The Tensor package contains commands for the standard algebraic operations on tensors as well as commands for covariant differentiation and curvature calculations (for metric connections, general affine connections, or connections on vector bundles).

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The Tensor package also includes a full implementation of the 2 component spinor and Newman-Penrose formalisms for space-time computations (pseudo-Riemannian manifolds with metric signature [+1, -1, -1, -1] ).  Petrov and Segre classifications of spacetimes can be calculated as well as complete sets of curvature invariants.

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All tensor computations can be done in an arbitrary frame or co-frame on the manifold.  In particular, all curvature computations for a (pseudo-)Riemannian metric can be performed with respect to an orthonormal frame.

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The Tensor package, working in conjunction with other Differential Geometry commands, provides great flexibility for mapping tensors between manifolds.  For example, if G is a Lie group acting on a manifold M, then the PushPullTensor command can be used to push forward the G invariant tensors on M to tensor fields on the quotient manifold M/G.

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Commands are available for calculating the Laplace-Beltrami operator on differential forms and for the Schouten and Frolicher-Nijenhuis brackets of tensor fields.  These bracket operations are important in complex geometry and in Poisson geometry.

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Infinitesimal transformation groups such as the Killing vectors of a metric can be calculated.

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Infinitesimal holonomy of a metric or a connection can be calculated.

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There are commands for computing special tensor fields such as Killing-Yano tensors.  

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The Tensor package is fully integrated with the LieAlgebras and LieAlgebraRepresentations packages which allows for the computation of, for example, the invariant tensors on a Lie algebra.

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The  Differential Geometry Lessons (Lessons 9 and 10) provide a systematic introduction to the commands in the Tensor package.

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Each command in the Tensor package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

Commands for the algebraic manipulation of tensors

CanonicalTensors

ContractIndices

Convert/DGspinor

Convert/DGtensor

FormInnerProduct

DGGramSchmidt

GenerateSymmetricTensors

GenerateTensors

HodgeStar

InverseMetric

KroneckerDelta

MetricDensity

MultiVector

PermutationSymbol

PlebanskiTensor

PushPullTensor

QuadraticFormSignature

RaiseLowerIndices

RearrangeIndices

SymmetrizeIndices

TensorInnerProduct

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CanonicalTensors: create various standard tensors.

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ContractIndices: contract the indices of a tensor.

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Convert/DGspinor: convert a tensor to a spinor.

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Convert/DGtensor: convert an array, vector, p-form, spinor ... to a  tensor.

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DGGramSchmidt: construct an orthonormal basis of vector, forms, tensors with respect to a metric.

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FormInnerProduct: compute the inner product of two forms with respect to a given metric tensor.

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GenerateSymmetricTensors: generate a list of symmetric tensors from a list of tensors.

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GenerateTensors: generate a list of tensors from a list of lists of tensors.

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HodgeStar: apply the Hodge star operator to a differential form.

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InverseMetric: find the inverse of a metric tensor.

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KroneckerDelta: find the Kronecker delta tensor of rank r.

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MetricDensity: use a metric tensor to create a scalar density of a given weight.

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MultiVector: compute the alternating sum of the tensor product of a list of vector fields.

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PermutationSymbol: create a permutation symbol.

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PlebanskiTensor: calculate the Plebanski tensor from a trace-free rank 2 symmetric tensor.

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PushPullTensor: transform a tensor from one manifold or coordinate system to another.

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QuadraticFormSignature : find the signature of a covariant, symmetric, rank 2 tensor.

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RaiseLowerIndices: raise or lower a list of indices of a tensor.

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RearrangeIndices: rearrange the argument/indices of a tensor.

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SymmetrizeIndices: symmetrize or skew-symmetrize a list of tensor indices.

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TensorInnerProduct: compute the inner product of two vectors, forms or tensors with respect to a given metric tensor.

Commands for tensor differentiation

Christoffel

Connection

CovariantDerivative

DirectionalCovariantDerivative

GeodesicEquations

Laplacian

ParallelTransportEquations

TensorBrackets

TorsionTensor

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Christoffel: find the Christoffel symbols of the first or second kind for a metric tensor.

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Connection: define a linear connection on the tangent bundle or on a vector bundle.

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CovariantDerivative: calculate the covariant derivative of a tensor field with respect to a connection.

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DirectionalCovariantDerivative: calculate the covariant derivative of a tensor field in the direction of a vector field and with respect to a given connection.

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GeodesicEquations: calculate the geodesic equations for a symmetric linear connection on the tangent bundle.

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Laplacian: find the Laplacian of a differential form with respect to a metric.

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ParallelTransportEquations: calculate the parallel transport equations for a linear connection on the tangent bundle or a linear connection on a vector bundle.

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TensorBrackets: calculate the Schouten bracket and Frolicher-Nijenhuis brackets of tensor fields.

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TorsionTensor: calculate the torsion tensor for a linear connection on the tangent bundle.

Commands for calculating curvature tensors  

BachTensor

CottonTensor

CurvatureTensor

EinsteinTensor

ProjectiveCurvatureTensor

RicciScalar

RicciTensor

RiemannInvariants

SchoutenTensor

SectionalCurvature

TraceFreeRicciTensor

WeylTensor

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BachTensor: calculate the Bach tensor of a metric.

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CottonTensor: calculate the Cotton tensor for a metric.

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CurvatureTensor: calculate the curvature tensor of a linear connection on the tangent bundle or on a vector bundle.

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EinsteinTensor: calculate the Einstein tensor for a metric.

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ProjectiveCurvatureTensor: calculate the Weyl projective curvature tensor of a connection on the tangent bundle.

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RicciScalar: calculate the Ricci scalar for a metric.

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RicciTensor: calculate the Ricci tensor of a linear connection on the tangent bundle.

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RiemannInvariants: calculate a complete set of scalar curvature invariants in 4 dimensions.

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SectionalCurvature: calculate the sectional curvature for a metric.

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SchoutenTensor: calculate the Schouten tensor of a metric.  

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TraceFreeRicciTensor: calculate the trace-free Ricci tensor for a metric.   

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WeylTensor: calculate the Weyl curvature tensor of a metric.

Infinitesimal transformation groups

ConformalKillingVectors

HomothetyVectors

KillingVectors

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ConformalKillingVectors: calculate the conformal Killing vectors for a given metric.

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HomothetyVectors: calculate the homothety vectors for a given metric.

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KillingVectors: calculate the Killing vectors or infinitesimal isometries for a given metric.

Commands for calculating holonomy

InfinitesimalHolonomy

InvariantTensorsAtAPoint

 

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InfinitesimalHolonomy: find the matrix Lie algebra giving the infinitesimal holonomy of a metric or a connection on the tangent bundle or on a general vector bundle.

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InvariantTensorsAtAPoint: find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices.

Commands for calculating special tensor fields

CovariantlyConstantTensors

KillingSpinors

KillingTensors

KillingYanoTensors

RecurrentTensors

 

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CovariantlyConstantTensors: calculate the covariantly constant tensors with respect to a given metric or connection.

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KillingSpinors: calculate the Killing spinors for a given spacetime.

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KillingYanoTensors:  calculate the Killing tensors of a specified rank for a given metric or connection.

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KillingTensors: calculate the Killing-Yano tensors for a given connection or a given metric.

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RecurrentTensors: calculate the recurrent tensors with respect to a given metric or connection.

Commands for working with Killing tensors

CheckKillingTensor

IndependentKillingTensors

KillingBracket

SymmetricProductsOfKillingTensors

 

 

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CheckKillingTensor: check that a tensor is the Killing tensor for a metric.

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IndependentKillingTensors: create a list of linearly independent Killing tensors.

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KillingBracket: a covariant form of the Schouten bracket for symmetric tensors.

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SymmetricProductsOfKillingTensors: form all possible symmetric tensors of a given rank.

Commands for the 2-component spinor formalism

AdaptedSpinorDyad

BivectorSolderForm

ConjugateSpinor

EpsilonSpinor

FactorWeylSpinor

KroneckerDeltaSpinor

RaiseLowerSpinorIndices

RicciSpinor

SolderForm

SpinConnection

SpinorInnerProduct

WeylSpinor

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AdaptedSpinorDyad: find a spinor dyad which transforms the Weyl spinor to normal form.

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BivectorSolderForm: calculate the rank 4 spin-tensor which maps bivectors to symmetric rank 2 spinors.

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ConjugateSpinor: calculate the complex conjugate of a spinor.

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EpsilonSpinor: calculate the epsilon spinor in the 2 component spinor formalism.

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FactorWeylSpinor: factorize a rank 4 symmetric spinor.

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KroneckerDeltaSpinor: calculate the Kronecker delta spinor in the 2 component spinor formalism.

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RaiseLowerSpinorIndices: raise/lower the indices of a spinor or spin-tensor using the epsilon spinors.

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RicciSpinor: calculate the rank 4 Ricci spinor corresponding to the trace-free Ricci tensor.

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SolderForm: calculate the solder form (or Infeld-van der Waerden symbols) from an orthonormal tetrad.

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SpinConnection: calculate the unique spin connection defined by a solder form.

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SpinorInnerProduct:  contract all spinor indices of a pair of 2-component spin-tensors using the epsilon spinors.

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WeylSpinor: calculate the rank 4 Weyl spinor corresponding to the Weyl tensor.

Commands for the Newman-Penrose formalism

AdaptedNullTetrad

GRQuery

NPBianchiIdentities

NPCurvatureScalars

NPDirectionalDerivatives

NPRicciIdentities

NPSpinCoefficients

NullTetrad

NullTetradTransformation

NullVector

OrthonormalTetrad

PrincipalNullDirections

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AdaptedNullTetrad: find a null tetrad which transforms the Newman-Penrose Weyl scalars to a standard form.

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GRQuery: verify various properties of spacetimes.

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NPBianchiIdentities: calculate the Bianchi identities in the Newman-Penrose formalism.

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NPCurvatureScalars: calculate the Ricci scalars and the Weyl scalars in the Newman-Penrose formalism.

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NPDirectionalDerivatives: define the directional derivative operators used in the Newman-Penrose formalism.

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NPRicciIdentities: calculate the Ricci identities in the Newman-Penrose formalism.

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NPSpinCoefficients: calculate the Newman-Penrose spin coefficients.

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NullTetrad: calculate a null tetrad from an orthonormal tetrad.

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NullTetradTransformation: apply a Lorentz transformation to a null tetrad.

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NullVector: construct a null vector from a solder form and a rank 1 spinor.

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OrthonormalTetrad: calculate an orthonormal tetrad from a null tetrad.

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PrincipalNullDirections: find the principal null directions of a 4-dimensional spacetime.  

Commands for the algebraic classification of spacetimes

CongruenceProperties

IsotropyType

PetrovType

SubspaceType

SegreType

 

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CongruenceProperties: calculate the geometry properties of a line congruence.

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IsotropyType: determine the isotropy type of the isotropy subalgebra of infinitesimal isometries.

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PetrovType: determine the Petrov type of a spacetime from the Weyl tensor.

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SegreType: determine the Plebanski-Petrov type and the Segre type of a trace-free, rank 2 symmetric tensor.

Commands for field theory

BelRobinson

DivergenceIdentities

EnergyMomentumTensor

MatterFieldEquations

RainichConditions

RainichElectromagneticField

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BelRobinson: calculate the rank 4 Bel-Robinson tensor for a metric.

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DivergenceIdentities: check the divergence identity for various energy-momentum tensors.

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EnergyMomentumTensor: calculate the energy-momentum tensor for various fields (scalar, electromagnetic, dust, ...).

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MatterFieldEquations: calculate the field equations for various field theories (scalar, electromagnetic, dust, ...).

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RainichConditions: check that a metric tensor satisfies the Rainich conditions.

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RainichElectromagneticField : from a given metric satisfying the Rainich conditions, calculate an electromagnetic field which solves the Einstein-Maxwell equations.

Alphabetical listing of all Tensor commands

AdaptedNullTetrad

AdaptedSpinorDyad

BachTensor

BelRobinson

BivectorSolderForm

CanonicalTensors

CheckKillingTensor

Christoffel

ConformalKillingVectors

CongruenceProperties

ConjugateSpinor

Connection

ContractIndices

CottonTensor

CovariantDerivative

CovariantlyConstantTensors

CurvatureTensor

DGGramSchmidt

DirectionalCovariantDerivative

DivergenceIdentities

EinsteinTensor

EnergyMomentumTensor

EpsilonSpinor

FactorWeylSpinor

FormInnerProduct

GRQuery

GenerateSymmetricTensors

GenerateTensors

GeodesicEquations

HodgeStar

HomothetyVectors

IndependentKillingTensors

InfinitesimalHolonomy

InvariantTensorsAtAPoint

InverseMetric

IsotropyType

KillingBracket

KillingSpinors

KillingTensors

KillingVectors

KillingYanoTensors

KroneckerDelta

KroneckerDeltaSpinor

Laplacian

MatterFieldEquations

MetricDensity

MultiVector

NPBianchiIdentities

NPCurvatureScalars

NPDirectionalDerivatives

NPRicciIdentities

NPSpinCoefficients

NullTetrad

NullTetradTransformation

NullVector

OrthonormalTetrad

ParallelTransportEquations

PermutationSymbol

PetrovType

PlebanskiTensor

PrincipalNullDirections

ProjectiveCurvatureTensor

PushPullTensor

QuadraticFormSignature

RainichConditions

RainichElectromagneticField

RaiseLowerIndices

RaiseLowerSpinorIndices

RearrangeIndices

RecurrentTensors

RicciScalar

RicciSpinor

RicciTensor

RiemannInvariants

SchoutenTensor

SectionalCurvature

SegreType

SolderForm

SpinConnection

SpinorInnerProduct

SubspaceType

SymmetricProductsOfKillingTensors

SymmetrizeIndices

TensorBrackets

TensorInnerProduct

TorsionTensor

TraceFreeRicciTensor

WeylSpinor

WeylTensor

 

See Also

DifferentialGeometry

GroupActions

JetCalculus

Library

LieAlgebras

Tools

Physics[D_]

Physics[d_]

Physics[Einstein]

Physics[g_]

Physics[LeviCivita]

Physics[Ricci]

Physics[Riemann]

Physics[Weyl]