Discrete geometric structures in General...

Discrete geometric structures in General Relativity

Jörg Frauendiener

Department of Mathematics and StatisticsUniversity of Otago

andCentre of Mathematics for Applications

University of Oslo

Jena, August 26, 2010

J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 1 / 30

Outline

1 Motivation

2 Example: Electromagnetism

3 Discrete differential forms

4 GR as a differential ideal

5 Implementation

6 Outlook

Motivation

Outline

1 Motivation

5 Implementation

6 Outlook

Motivation

GR is a geometric theoryinvariance under arbitrary diffeomorphisms

any two points can be interchanged be a diffeomorphismindividual points do not have any meaning

Only relations between several points can carry information

Motivation

MotivationGeometry from relations between points

„Distance“ between points

Motivation

Parallel transport from A to B along a curve

Motivation

Holonomy around a closed path

=⇒ Gauss curvature of a surface

Motivation

MotivationConsequence

Objects should not be localised entirely on points

cp. Finite Difference Methods:all tensor components are represented as grid functions

type of object is relevant for localisation on line, surface, volume

Example: Electromagnetism

Outline

1 Motivation

5 Implementation

6 Outlook

ElectromagnetismElectric field

1-form: electric field

E = Ex dx + Ey dy + Ez dz

line (1-dim)

L −→∫L

E =: W

Work done on unit charge along L(voltage between A and B)

line (1-dim)

L −→∫L

E =: W

line (1-dim)

L −→∫L

E =: W

ElectromagnetismMagnetic induction

2-form: magnetic induction

B = Bxy dxdy + Byz dydz + Bzx dzdx

surface (2-dim)

B : −→∫

AB =: Φ

Magnetic flux through loop C = ∂A

surface (2-dim)

B : −→∫

AB =: Φ

surface (2-dim)

B : −→∫

AB =: Φ

ElectromagnetismMaxwell’s equation

B = rot E

D = −rot H

⇐⇒

B =∫

D = −∫

Discretisations of Ampere’s lawB = rot E

B =∫

∂ΩE

b = e1 + e2 + e3

very clear cut, elegant and efficient

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

∆B , e3 =

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

∆B , e3 =

1E , e1 =

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

∆B , e3 =

1E , e1 =

2E , e2 =

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

∆B , e3 =

1E , e1 =

2E , e2 =

b = e1 + e2 + e3

Ωddt ∑

∫∆i

B = ∑i

∫∂∆i

b =∫

∆B , e3 =

1E , e1 =

2E , e2 =

b = e1 + e2 + e3

Discrete differential forms

Outline

1 Motivation

5 Implementation

6 Outlook

Discrete differential formscontinuous

p-dimensional submanifold Sp:(0) point, (1) curve, (2) surface

p-form:

ω : Sp 7→∫

ω ∈ R

exterior derivative d:∫Sp

dω =∫

Stokes’ theorem

discrete

p-simplices Sp:(0) node, (1) edge, (2) face

discrete p-form:

ω : Sp 7→ ω[Sp] ∈ R

Definition:

dω[Sp] = ω[∂Sp]

Example: 1

dω123 = ω12 + ω23 + ω31J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 15 / 30

continuous

Grassmann (wedge) product:

qβ) 7→

p+qα ∧ β,

graded algebra

α ∧ β = (−1)pq β ∧ α,

derivation:

d(α ∧ β) = dα ∧ β + (−1)pα ∧ dβ.

deRham cohomology

discrete

discrete Grassmann product:

qβ) 7→

p+qα ∧ β

Example:

2(1α ∧

1β)123 =12 [α12β13 + α23β21 + α31β32

− β12α13 − β23α21 − β31α32]

discrete d is derivation

singular cohomology

GR as a differential ideal

Outline

1 Motivation

5 Implementation

6 Outlook

GR as a differential idealBasic variables

tetrad

(θ0, θ1, θ2, θ3)

connection

ωik = −ωk

l[e]2 = ηikθi[e]θk[e]

exp(ω): holonomy along e

related by no-torsion condition

dθi + ωik ∧ θk = 0

tetrad

connection

tetrad

connection

GR as a differential idealEinstein’s equation

To formulate the Einstein’s equation one defines

2-forms: Li =12

εijklηknωj

n ∧ θl

3-forms: Si =12

εijkl

(ωjk ∧ωl

m ∧ θm −ωjm ∧ωmk ∧ θl

)vacuum equations

dLi = Si

GR as a differential idealEinstein’s equation

To formulate the Einstein’s equation one defines

2-forms: Li =12

εijklωjk ∧ θl

3-forms: Si =12

εijkl

(ωjk ∧ωl

m ∧ θm −ωjm ∧ωmk ∧ θl

)vacuum equations

dLi = Si

IntermezzoThe moment of rotation

curvature 2-form:Ωi

k = Riklmθl ∧ θm

Warner Miller: moment of rotation

Ω[jk ∧ θl] ∼ εijklΩjk ∧ θl ∝ Gab

Identity:

dLi = Si +12

εijklΩjk ∧ θl︸︷︷︸

Linearisation of Ti on Minkowski space:4-dim analogue of Snorre’s curl T curl operator

Identity:

dLi = Si +12

Identity:

dLi = Si +12

Identity:

dLi = Si +12

GR as a differential idealProperties

This formulation has close relations toenergy balance

Einstein: energy-momentum pseudotensor SiMøller: energy-balance in tetrad form

Bondi-Sachs mass loss formula

focussing of light rays due to gravity

The variables θi and ωik are subject to gauge freedom:

θi 7→ sik(x)θk

ωik 7→ sl

i(x)ωlmsm

k(x) + smi(x)dsm

for arbitrary function with values on O(1, 3), local Lorentz invariance.

trade-in of diffeomorphism freedom for local Lorentz invariance

The variables θi and ωik are subject to gauge freedom:

θi 7→ sik(x)θk

ωik 7→ sl

i(x)ωlmsm

k(x) + smi(x)dsm

for arbitrary function with values on O(1, 3), local Lorentz invariance.

trade-in of diffeomorphism freedom for local Lorentz invariance

GR as a differential idealLocal structure in a normal neighbourhood

Fix a normal coordinate system (xi) around a point O

θi =(

δil + βi

lmxm + (βilpq −

ηjnβjipβn

lq)xpxq︸︷︷︸∼exp(β(x))

)dxl − 1

plqxpxqdxl +O(x3)

ωik =

Rikpqxpdxq +O(x2)

Li =14

εijklRjkpqxpdxqdxl +O(x2)

Si = O(x2)

dLi = Si = O(x2)

To second order Li are the conserved fluxes of relativistic energy-momentum

θi = dxi − 16

Riplqxpxqdxl +O(x3)

ωik =

Rikpqxpdxq +O(x2)

Li =14

Si = O(x2)

dLi = Si = O(x2)

θi = dxi − 16

Riplqxpxqdxl +O(x3)

ωik =

Rikpqxpdxq +O(x2)

Li =14

Si = O(x2)

dLi = Si = O(x2)

θi = dxi − 16

Riplqxpxqdxl +O(x3)

ωik =

Rikpqxpdxq +O(x2)

Li =14

Si = O(x2)

dLi = Si = O(x2)

Implementation

Outline

1 Motivation

5 Implementation

6 Outlook

Implementation

Implementationwith R. Richter, et al

Theoretical formulationGeometric discretisation by local domains of dependence(light-like coordinates)non-linear algebraic system of equationssolved by Newton’s methodfound order of convergence

Application to simple (1 + 1) systems of GR

spherically symmetric space-times:Minkowski, Schwarzschild, Kruskalplane wavesGowdy

verification of the order of convergence

reproduction of exact solutions

Implementation

Application to simple (1 + 1) systems of GRspherically symmetric space-times:Minkowski, Schwarzschild, Kruskalplane wavesGowdy

Implementation

ImplementationInsights

coordinate-free treatment of Einsteins equations

purely geometric discretisation

convergent method

Questions and problems:

no complete understanding of the algebraic structure of the equations

no adapted solution algorithm

discrete form algebra is not associative: affects non-linearitiesbut: associator vanishes to higher order(known to topologists in the ’70s)

no understanding of the gauge freedom in the discrete setting

Implementation

convergent method

Implementation

convergent method

Implementation

convergent method

Implementation

convergent method

Outlook

Outline

1 Motivation

5 Implementation

6 Outlook

Outlook

General idea:

put together small pieces of 4-d space-timei.e., where the discrete equations hold

glue together across hypersurfaces Σ by gauge transformations

natural junction conditions (Israel 1966)∫S

Li − Li = 0 for all 2-dim submanifolds of Σ

Outlook

General idea:

Outlook

General idea:

Outlook

However:

not been able to convert this idea into a decent algorithm

again problem with the gauge transformation

Way out (?):

find a formulation of GR entirely in terms of scalarsHow unique is Regge calculus?

discrete form ωik is a hybrid creature

use the holonomies directly, edge map into the Lorentz group

many (yet unknown) problems to solve

Outlook

However:

Way out (?):

Outlook

However:

Way out (?):

Outlook

However:

Way out (?):

Outlook

However:

Way out (?):

Outlook

However:

Way out (?):

Outlook

However:

Way out (?):

Outlook

THANK YOU

Discrete geometric structures in General...

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