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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

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• 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)

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• 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 .

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• 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.

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• 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.

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• 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)

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• 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

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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!

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• 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.

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• 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

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• 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

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• 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!

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• 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

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• Some examples from physics

Spacetime Petrov TypeSpherically Symmetric spacetimes D

Kerr D

FLRW O

Transversal gravitational waves N

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• 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)

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• 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

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• Thats it.

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