discrete geometric structures in general...
TRANSCRIPT
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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
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Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 2 / 30
![Page 3: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/3.jpg)
Motivation
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 3 / 30
![Page 4: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/4.jpg)
Motivation
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 4 / 30
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Motivation
MotivationGeometry from relations between points
„Distance“ between points
A
B
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 5 / 30
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Motivation
MotivationGeometry from relations between points
Parallel transport from A to B along a curve
A
B
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 6 / 30
![Page 7: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/7.jpg)
Motivation
MotivationGeometry from relations between points
Parallel transport from A to B along a curve
A
B
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 6 / 30
![Page 8: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/8.jpg)
Motivation
MotivationGeometry from relations between points
Parallel transport from A to B along a curve
A
B
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 6 / 30
![Page 9: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/9.jpg)
Motivation
MotivationGeometry from relations between points
Parallel transport from A to B along a curve
A
B
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 6 / 30
![Page 10: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/10.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
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Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 12: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/12.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 13: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/13.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 14: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/14.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 15: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/15.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 16: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/16.jpg)
Motivation
MotivationGeometry from relations between points
Holonomy around a closed path
A
=⇒ Gauss curvature of a surface
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 7 / 30
![Page 17: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/17.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 8 / 30
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Example: Electromagnetism
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 9 / 30
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Example: Electromagnetism
ElectromagnetismElectric field
1-form: electric field
E = Ex dx + Ey dy + Ez dz
line (1-dim)
A
B
L
E :A
B
L −→∫L
E =: W
Work done on unit charge along L(voltage between A and B)
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 10 / 30
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Example: Electromagnetism
ElectromagnetismElectric field
1-form: electric field
E = Ex dx + Ey dy + Ez dz
line (1-dim)
A
B
L
E :A
B
L −→∫L
E =: W
Work done on unit charge along L(voltage between A and B)
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 10 / 30
![Page 21: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/21.jpg)
Example: Electromagnetism
ElectromagnetismElectric field
1-form: electric field
E = Ex dx + Ey dy + Ez dz
line (1-dim)
A
B
L
E :A
B
L −→∫L
E =: W
Work done on unit charge along L(voltage between A and B)
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 10 / 30
![Page 22: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/22.jpg)
Example: Electromagnetism
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 11 / 30
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Example: Electromagnetism
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 11 / 30
![Page 24: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/24.jpg)
Example: Electromagnetism
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 11 / 30
![Page 25: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/25.jpg)
Example: Electromagnetism
ElectromagnetismMaxwell’s equation
B = rot E
D = −rot H
⇐⇒
ddt
∫A
B =∫
∂AE
ddt
∫A
D = −∫
∂AH
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 12 / 30
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Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt
∫Ω
B =∫
∂ΩE
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
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Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 28: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/28.jpg)
Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 29: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/29.jpg)
Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B , e3 =
∫ 2
1E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
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Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B , e3 =
∫ 2
1E , e1 =
∫ 3
2E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 31: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/31.jpg)
Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B , e3 =
∫ 2
1E , e1 =
∫ 3
2E , e2 =
∫ 1
3E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 32: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/32.jpg)
Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B , e3 =
∫ 2
1E , e1 =
∫ 3
2E , e2 =
∫ 1
3E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 33: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/33.jpg)
Example: Electromagnetism
Discretisations of Ampere’s lawB = rot E
Ωddt ∑
i
∫∆i
B = ∑i
∫∂∆i
E
1 2
3
b =∫
∆B , e3 =
∫ 2
1E , e1 =
∫ 3
2E , e2 =
∫ 1
3E
b = e1 + e2 + e3
very clear cut, elegant and efficient
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 13 / 30
![Page 34: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/34.jpg)
Discrete differential forms
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 14 / 30
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Discrete differential forms
Discrete differential formscontinuous
p-dimensional submanifold Sp:(0) point, (1) curve, (2) surface
p-form:
ω : Sp 7→∫
Sp
ω ∈ R
exterior derivative d:∫Sp
dω =∫
∂Sp
ω
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
3
2
dω123 = ω12 + ω23 + ω31J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 15 / 30
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Discrete differential forms
Discrete differential forms
continuous
Grassmann (wedge) product:
(pα,
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:
(pα,
qβ) 7→
p+qα ∧ β
Example:
1
3
2(1α ∧
1β)123 =12 [α12β13 + α23β21 + α31β32
− β12α13 − β23α21 − β31α32]
discrete d is derivation
singular cohomology
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 16 / 30
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GR as a differential ideal
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 17 / 30
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GR as a differential ideal
GR as a differential idealBasic variables
tetrad
(θ0, θ1, θ2, θ3)
connection
ωik = −ωk
i
e
l[e]2 = ηikθi[e]θk[e]
e
exp(ω): holonomy along e
related by no-torsion condition
dθi + ωik ∧ θk = 0
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 18 / 30
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GR as a differential ideal
GR as a differential idealBasic variables
tetrad
θi
connection
ωik
e
l[e]2 = ηikθi[e]θk[e]
e
exp(ω): holonomy along e
related by no-torsion condition
dθi + ωik ∧ θk = 0
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 18 / 30
![Page 40: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/40.jpg)
GR as a differential ideal
GR as a differential idealBasic variables
tetrad
θi
connection
ωik
e
l[e]2 = ηikθi[e]θk[e]
e
exp(ω): holonomy along e
related by no-torsion condition
dθi + ωik ∧ θk = 0
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 18 / 30
![Page 41: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/41.jpg)
GR as a differential ideal
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 19 / 30
![Page 42: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/42.jpg)
GR as a differential ideal
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 19 / 30
![Page 43: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/43.jpg)
GR as a differential ideal
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︸ ︷︷ ︸
=:Ti
Linearisation of Ti on Minkowski space:4-dim analogue of Snorre’s curl T curl operator
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 20 / 30
![Page 44: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/44.jpg)
GR as a differential ideal
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︸ ︷︷ ︸
=:Ti
Linearisation of Ti on Minkowski space:4-dim analogue of Snorre’s curl T curl operator
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 20 / 30
![Page 45: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/45.jpg)
GR as a differential ideal
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︸ ︷︷ ︸
=:Ti
Linearisation of Ti on Minkowski space:4-dim analogue of Snorre’s curl T curl operator
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 20 / 30
![Page 46: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/46.jpg)
GR as a differential ideal
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︸ ︷︷ ︸
=:Ti
Linearisation of Ti on Minkowski space:4-dim analogue of Snorre’s curl T curl operator
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 20 / 30
![Page 47: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/47.jpg)
GR as a differential ideal
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 21 / 30
![Page 48: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/48.jpg)
GR as a differential ideal
GR as a differential idealProperties
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
k
for arbitrary function with values on O(1, 3), local Lorentz invariance.
trade-in of diffeomorphism freedom for local Lorentz invariance
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 22 / 30
![Page 49: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/49.jpg)
GR as a differential ideal
GR as a differential idealProperties
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
k
for arbitrary function with values on O(1, 3), local Lorentz invariance.
trade-in of diffeomorphism freedom for local Lorentz invariance
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 22 / 30
![Page 50: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/50.jpg)
GR as a differential ideal
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 −
12
ηjnβjipβn
lq)xpxq︸ ︷︷ ︸∼exp(β(x))
)dxl − 1
6Ri
plqxpxqdxl +O(x3)
ωik =
12
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 23 / 30
![Page 51: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/51.jpg)
GR as a differential ideal
GR as a differential idealLocal structure in a normal neighbourhood
Fix a normal coordinate system (xi) around a point O
θi = dxi − 16
Riplqxpxqdxl +O(x3)
ωik =
12
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 23 / 30
![Page 52: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/52.jpg)
GR as a differential ideal
GR as a differential idealLocal structure in a normal neighbourhood
Fix a normal coordinate system (xi) around a point O
θi = dxi − 16
Riplqxpxqdxl +O(x3)
ωik =
12
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 23 / 30
![Page 53: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/53.jpg)
GR as a differential ideal
GR as a differential idealLocal structure in a normal neighbourhood
Fix a normal coordinate system (xi) around a point O
θi = dxi − 16
Riplqxpxqdxl +O(x3)
ωik =
12
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 23 / 30
![Page 54: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/54.jpg)
Implementation
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 24 / 30
![Page 55: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/55.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 25 / 30
![Page 56: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/56.jpg)
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 GRspherically symmetric space-times:Minkowski, Schwarzschild, Kruskalplane wavesGowdy
verification of the order of convergence
reproduction of exact solutions
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 25 / 30
![Page 57: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/57.jpg)
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 GRspherically symmetric space-times:Minkowski, Schwarzschild, Kruskalplane wavesGowdy
verification of the order of convergence
reproduction of exact solutions
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 25 / 30
![Page 58: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/58.jpg)
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 GRspherically symmetric space-times:Minkowski, Schwarzschild, Kruskalplane wavesGowdy
verification of the order of convergence
reproduction of exact solutions
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 25 / 30
![Page 59: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/59.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 26 / 30
![Page 60: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/60.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 26 / 30
![Page 61: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/61.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 26 / 30
![Page 62: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/62.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 26 / 30
![Page 63: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/63.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 26 / 30
![Page 64: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/64.jpg)
Outlook
Outline
1 Motivation
2 Example: Electromagnetism
3 Discrete differential forms
4 GR as a differential ideal
5 Implementation
6 Outlook
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 27 / 30
![Page 65: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/65.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 28 / 30
![Page 66: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/66.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 28 / 30
![Page 67: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/67.jpg)
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 28 / 30
![Page 68: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/68.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 69: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/69.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 70: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/70.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 71: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/71.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 72: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/72.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 73: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/73.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 74: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/74.jpg)
Outlook
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
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 29 / 30
![Page 75: Discrete geometric structures in General Relativitycse.mathe.uni-jena.de/wumds/talks/Frauendiener-WUMDS-2010.pdf · Discrete differential forms Outline 1 Motivation 2 Example: Electromagnetism](https://reader034.vdocuments.mx/reader034/viewer/2022050105/5f436ba785abdf595c2b51fe/html5/thumbnails/75.jpg)
Outlook
THANK YOU
J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, 2010 30 / 30