The Weyl Tensor - Universität zu Köln · I Algebraic equations for the traces of the Riemann...

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Transcript of The Weyl Tensor - Universität zu Köln · I Algebraic equations for the traces of the Riemann...

  • The Weyl TensorIntroduction, properties and applications

    David Wichmann

    Dec 06, 2016

    1 / 24

  • Overview

    1. Motivation

    2. Derivation and properties of the Weyl Tensor

    3. Where the Weyl Tensor appears

    4. Application: Classification of Spacetimes using the WeylTensor (Petrov Classification)

    2 / 24

  • Motivation

    G = R 1

    2Rg = T

    I Algebraic equations for the traces of the Riemann Tensor

    I Determine 10 components of the Riemann Tensor

    I No direct visibility of curvature propagation

    Traceless part of R is the Weyl tensor, C .

    3 / 24

  • Decomposition of Curvature Tensor

    DefinitionFor A symmetric, define:

    A = Ag + Ag Ag Ag

    A fulfills:

    1. A = A2. A = A

    3. A + A + A

    = 0

    Possible AnsatzR = C + AR

    + BRg

    , A,B R.

    4 / 24

  • The Weyl Tensor

    C = R 1

    n 2(Rg + Rg Rg Rg)

    +1

    (n 1)(n 2)R(gg gg)

    See for example: H.-J. Schmidt, Gravitoelectromagnetism and other decompositions of the Riemann tensor (2004,arXiv:gr-qc/04070531v1)

    Get a propagation equation:

    C =n 3n 2

    (T T

    1

    n 1(Tg Tg)

    )See for example: Matthias Blau, Lecture Notes on General Relativity(ttp://www.blau.itp.unibe.ch/Lecturenotes.html)

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  • Weyl Tensor Properties

    1. Same algebraic symmetries as Riemann Tensor

    2. Traceless: gC = 0

    3. Conformally invariant:I That means:

    g = 2(x)g C = C

    6

    I C = 0 is sufficient condition for g = 2 in n 4

    4. Vanishes identically in n < 4

    5. In vacuum it is equal to the Riemann tensor.

    6 / 24

  • Applications of the Weyl tensor

    I Vacuum GR: Geodesic deviation, gravitational waves, ...I Cosmology: Weyl Curvature Hypothesis and isotropic

    singularities 1

    I Weyl Tensor is small at initial singularities 2

    I g = 2g, g regular at t = 0:

    limt0

    C

  • The Petrov Classification

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  • The Petrov Classification: Mathematical Preliminaries I

    Definition (Bivector Space B(p))Space of second order skew symmetric contravariant tensors X

    at point p M:

    X B(p) X = X .

    B(p) is a 6-dimensional space.

    Notation (Bivector indices)

    X : = {01, 02, 03, 23, 31, 12}XA : A = {1, 2, 3, 4, 5, 6}

    AB =(

    0 I33I33 0

    )9 / 24

  • The Petrov Classification: Mathematical Preliminaries II

    1. Weyl Tensor as linear map:

    C : B(p) B(p)XB 7 CABXB

    2. Eigenvalue equation:

    C (X ) = X , C

    3. In coordinates(CAB GAB)XB = 0

    GAB G = = diag(1,1,1, 1, 1, 1)

    10 / 24

  • The Petrov Classification: Mathematical Preliminaries III

    Definition (Hodge Duality Transformation)

    ? : B(p) B(p)XA 7 ?XA = ABXB

    Definition (Hodge dual of Weyl Tensor)

    Weyl tensor as double-bivector define left and right dual:

    ?CAB := D

    A CDB

    C ?AB := D

    B CAD

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  • The Petrov Classification: Magic of the Weyl Tensor I

    Can show:

    I Minkowski space: ??X = X = X ??

    I Duality property: C (?X ) = C ?(X )

    I Tracelessnes of the Weyl Tensor: ?C = C ?

    Definition ((Anti-)self-dual Bivector)

    I X B(p) is self-dual, X S+(p), if ?X = iXI X B(p) is anti-self-dual, X S(p), if ?X = iX

    It follows for X B(p):1. X+ := X + i?X S+(p)2. X := X i?X S(p)

    3. X S+(p) X =(AiA

    )12 / 24

  • The Petrov Classification: Magic of the Weyl Tensor II

    The restricted Weyl tensor C |S+ is an automorphism on S+(p).C+ = C + i?C maps S(p) to zero:

    X S+(p) C+(X ) = [C + i?C ](X ) = 2C (X )X S(p) C+(X ) = [C + i?C ](X ) = 0

    For C ,X real, C(X) = C+(X+). That is, it is enough tostudy the Eigenvalue equation of C+, as they have tooccur in complex conjugate pairs!

    13 / 24

  • Petrov Classification: Constructing the Weyl Tensor

    Use:

    1. Symmetry CAB = CBA

    2. Tracelessnes C = 0

    3. Algebraic Bianchi Identity C + C + C = 0

    One finds:

    C =

    (M NN M

    ) C+ =

    (M iN N + iMN + iM M + iN

    ):=

    (Q iQiQ Q

    )M,N real traceless symmetric 3 3 matrices.For X S+(p) X = (A, iA)T for 3-dim vector A. We get theEigenvalue equation QA = A.

    14 / 24

  • Finding eigenvalues

    I Complex symmetric 3 3 matrix Q.I Similarity transformations, Q P1QP. Same eigenvalues,

    rank, characteristic and minimal polynomials

    I In general, Q not diagonalizable (only if real)

    I Best we can do is construct the Jordan normal form

    I Cayley-Hamilton Theorem: Every matrix fulfills its owncharacteristic equation

    15 / 24

  • Some linear algebra: Example 1

    P1QP =

    a 0 00 a 00 0 b

    (a )2(b ) = 0

    Cayley-Hamilton minimal polynomial:

    (Q aI)(Q bI) = 0

    16 / 24

  • Some linear algebra: Example 2

    P1QP =

    a 1 00 a 00 0 b

    (a )2(b ) = 0

    Cayley-Hamilton minimal polynomial:

    (Q aI)2(Q bI) = 0

    Dimension of Eigenvectors of = a (geometric multiplicity) is 1,less than the algebraic multiplicity 2!

    17 / 24

  • The Petrov TypesBased on the Eigenvalues of the Matrix Q:From tracelessnes of the matrix Q:

    1 + 2 + 3 = 0

    I Type I: All eigenvalues different dimension of eigenspacesis 3

    I Type D: One double eigenvalue with eigenspace-dimension 2

    I Type II: One double eigenvalue with eigenspace-dimension 1

    I Type III: All eigenvalues equal to zero. 1-dimensionaleigenspace

    I Type N: All eigenvalues equal to zero. 2-dimensionaleigenspaces.

    I Type O: All eigenvalues equal to zero. 3-dimensionaleigenspaces, i.e. C=0

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  • Penrose Diagram for Petrov Types

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  • The Petrov Types (other formulation)

    Based on the Eigenvalues of the Matrix Q:

    Use Caley-Hamilton theorem: Every similar matrix fulfills the sameminimal polynomial.

    Type Eigenvalues Minimal PolynomialI {1, 2, 3} (Q 1I )(Q 2I )(Q 3I ) = 0D {1, 1, 2 = 21} (Q 1I )(Q 2I ) = 0II {1, 1, 2 = 21} (Q 1I )2(Q 2I ) = 0III {1 = 2 = 3 = 0} Q3 = 0N {1 = 2 = 3 = 0} Q2 = 0O {1 = 2 = 3 = 0} Q = 0

    20 / 24

  • Some examples from physics

    Spacetime Petrov TypeSpherically Symmetric spacetimes D

    Kerr D

    FLRW O

    Transversal gravitational waves N

    21 / 24

  • More physics...

    Vacuum curvature tensor of an isolated matter distribution obeysunder certain assumptions:

    R =N

    r+

    IIIr2

    +Dr3

    + . . .

    (See for example H. Stephani: Relativity, 2004)

    22 / 24

  • Literature

    I G. S. Hall: Symmetries and Curvature Structure in GeneralRelativity. World Scientific Publishing Company (April 29,2004)

    I Joan Mart inez i Portillo: Classification of Weyl and RicciTensors (Bachelors Thesis, 2015/16) 4

    I A. Papapetrou: Lectures on general relativity. SpringerScience & Business Media, 2012.

    I H. Stephani: Relativity: An Introduction to Special andGeneral Relativity, Cambridge University Press (March 29,2004)

    4https://upcommons.upc.edu/bitstream/handle/2117/87155/memoria.pdf?sequence=1&isAllowed=y

    23 / 24

  • Thats it.

    24 / 24