calculate properties of a congruence of curves

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
- テンソル解析
- パッケージ
- ファンクションアドバイザ
- ベキ級数
- ベクトル解析
- 不活性関数
- 代数
- 因数分解と方程式解法
- 基本数学
- 基本的な注意事項
- 変分法
- 変換
- 幾何
- 微分代数方程式
- 微分幾何学
  - Group Actions
  - JetCalculus
  - LieAlgebras
  - Tensor
    - 概要
    - AdaptedNullTetrad
    - AdaptedSpinorDyad
    - CanonicalTensors
    - CheckKillingTensor
    - Christoffel
    - ConformalKillingVectors
    - CongruenceProperties
    - ContractIndices
    - CottonTensor
    - CovariantDerivative
    - CovariantlyConstantTensors
    - CurvatureTensor
    - DirectionalCovariantDerivative
    - EinsteinTensor
    - FactorWeylSpinor
    - GenerateSymmetricTensors
    - GenerateTensors
    - GeodesicEquations
    - GRQuery
    - HodgeStar
    - HomothetyVectors
    - IndependentKillingTensors
    - InverseMetric
    - KillingBracket
    - KillingSpinors
    - KillingTensors
    - KillingVectors
    - KillingYanoTensors
    - KroneckerDelta
    - MetricDensity
    - MultiVector
    - NPCurvatureScalars
    - NullVector
    - ParallelTransportEquations
    - PermutationSymbol
    - PetrovType
    - PlebanskiTensor
    - PrincipalNullDirections
    - PushPullTensor
    - QuadraticFormSignature
    - RainichConditions
    - RainichElectromagneticField
    - RaiseLowerIndices
    - RearrangeIndices
    - RecurrentTensors
    - RicciScalar
    - RicciTensor
    - SegreType
    - SymmetricProductsOfKillingTensors
    - SymmetrizeIndices
    - TensorBrackets
    - TensorInnerProduct
    - TorsionTensor
    - TraceFreeRicciTensor
    - WeylSpinor
    - WeylTensor
    - コネクション
    - ラプラシアン
  - ツール
  - ライブラリ
  - レッスンおよびチュートリアル
  - 概要
  - &wedge
  - Annihilator
  - ApplyTransformation
  - ChangeFrame
  - ComplementaryBasis
  - ComposeTransformations
  - DeRhamHomotopy
  - DGbasis
  - DGDualBasis
  - DGsetup
  - DGzip
  - evalDG
  - ExteriorDerivative
  - Flow
  - FrameData
  - GetComponents
  - Hook
  - InfinitesimalTransformation
  - IntegrateForm
  - IntersectSubspaces
  - InverseTransformation
  - LieBracket
  - LieDerivative
  - Pullback
  - PullbackVector
  - Pushforward
  - RemoveFrame
  - Transformation
  - 参考文献
  - 変換
  - 環境設定
- 微分方程式
- 微積分
- 数
- 数値計算
- 数学関数
- 数論
- 最適化
- 特殊関数
- 統計
- 線形代数
- 群論
- 評価
- 論理
- 過去のパッケージ
- 金融
- 離散数学
- piecewise examples
- グラフ電卓
- 区分関数
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
エラーメッセージガイド
マニュアル
数学アプリ

Tensor[CongruenceProperties] - calculate properties of a congruence of curves

Calling Sequences

CongruenceProperties()

Parameters

g - a metric tensor

U - a unit vector

K,L - normalized null vectors, the vector defines an affinely parameterized, geodesic null congruence.

NT - a list of 4 vectors, defining a null tetrad, the first vector in the tetrad defines the geodesic null congruence.

Description

•	The command CongruenceProperties returns a table of properties associated to a line congruence defined by a unit (time-like or space-like) vector field or a null vector field .

•	Let , set . The following scalar and tensor fields are calculated by the first calling sequence.

- Acceleration:

- Expansion: Θ = .

- Rotation Tensor : 1/2 (

- Shear Tensor: 1/2 (

•	The left-hand side of the Raychaudhuri equation valid when the congruence is geodesic (0), where is the Ricci tensor and is also calculated.

•	The first calling sequence returns a table with indices "Acceleration", "Expansion", "RotationTensor", "ShearTensor", "Raychaudhuri".

•	The remaining three calling sequences apply only to an affinely parameterized, geodesic null congruence , that is, and

•	The second calling sequence requires where Set and

- Expansion: Θ = .

- Rotation Tensor:

- Rotation Scalar: omega[] = (alpha[])/(2)`ε`^(abcd ) L[a]K[b] (∇)[c] K[d] .

- Complex expansion: .

- Shear Tensor:

The Raychaudhuri equation is as above but using these definitions of and and with

•	The second calling sequence returns a table with 8 indices "Expansion", "RotationNormSquared" "ShearNormSquared", "RotationTensor", "RotationScalar", "ShearTensor" , "ComplexExpansion" and "Raychaudhuri".

•

The third calling sequence calculates: Expansion: Θ = Rotation norm squared = and Shear norm squared = The definitions are as in the second calling sequence but, as these scalars do not in fact depend upon the choice of L, only the vector K is needed as input. The third calling sequence returns a table with indices "Expansion", "RotationNormSquared", "ShearNormSquared" and "Raychaudhuri".

•	Finally, from the 4th calling sequence we set and and calculate, in addition to the 8 quantities calculated for the second calling sequence , Newman-Penrose Spin Coefficients.

Examples

Example 1.

For our first example we use the standard metric on the sphere.

(2.1)

(2.2)

Define a unit vector field .

M >

(2.3)

We see that the congruence is geodesic on the equator () but is accelerating elsewhere. It is shearing, rotating and non-expanding.

M >

(2.4)

Example 2.

For the next example we consider a class of Robinson-Trautman metrics. These are of Petrov type II and admit a null congruence which is shear-free.

M >

(2.5)

RT >

_DG([["tensor", RT, [["cov_bas", "cov_bas"], []]], [[[1, 1], -2*H(zeta, zetab, r, u)], [[1, 2], -1], [[2, 1], -1], [[3, 4], r^2/P(zeta, zetab, u)^2], [[4, 3], r^2/P(zeta, zetab, u)^2]]])

(2.6)

Here is a null tetrad for this metric.

RT >

(2.7)

The null congruence is very simple:

RT >

(2.8)

First calling sequence:

RT >

(2.9)

Third calling sequence:

RT >

table( [( "RotationTensor" ) = _DG([["tensor", RT, [["cov_bas", "cov_bas"], []]], [[[1, 1], 0]]]), ( "Expansion" ) = 2/r, ( "RotationNormSquared" ) = 0, ( "RotationScalar" ) = 0, ( "Raychaudhuri" ) = 0, ( "ShearNormSquared" ) = 0, ( "ShearTensor" ) = _DG([["tensor", RT, [["cov_bas", "cov_bas"], []]], [[[1, 1], 0]]]) ] )

(2.10)

Fourth calling sequence

RT >

(2.11)

Example 3.

Here is an example of a Newman-Tamburino metric of Petrov type I and which admits a null geodesic congruence with non-vanishing shear.

RT >

(2.12)

M >

g := evalDG(r^2*`&t`(dx, dx)+x^2*`&t`(dy, dy)-2*r*`&s`(du, dx)/x-`&s`(du, dr)+(c+ln(r^2*x^4))*`&t`(du, du)/x^2)

_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], (c+ln(r^2*x^4))/x^2], [[1, 2], -1], [[1, 3], -2*r/x], [[2, 1], -1], [[3, 1], -2*r/x], [[3, 3], r^2], [[4, 4], x^2]]])

(2.13)

Here is a null tetrad for this metric.

M >

NT := [D_r, D_u+(c+ln(r^2*x^4))*D_r/(2*x^2), -sqrt(2)*D_r/x+sqrt(2)*D_x/(2*r)+(I*(1/2))*sqrt(2)*D_y/x, -sqrt(2)*D_r/x+sqrt(2)*D_x/(2*r)-(I*(1/2))*sqrt(2)*D_y/x]

(2.14)

Again we consider the first leg of this tetrad.

M >

(2.15)

First calling sequence:

RT >

(2.16)

Third calling sequence:

RT >

table( [( "RotationTensor" ) = _DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 0]]]), ( "Expansion" ) = 1/r, ( "RotationNormSquared" ) = 0, ( "RotationScalar" ) = 0, ( "Raychaudhuri" ) = 0, ( "ShearNormSquared" ) = (1/2)/r^2, ( "ShearTensor" ) = _DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[3, 3], (1/2)*r], [[4, 4], -(1/2)*x^2/r]]]) ] )