Discrete geometric structures in General Relativity
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1 Discrete geometric structures in General Relativity Jörg Frauendiener Department of Mathematics and Statistics University of Otago and Centre of Mathematics for Applications University of Oslo Jena, August 26, 2010 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
2 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, / 30
3 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, / 30
4 Motivation Motivation GR is a geometric theory invariance under arbitrary diffeomorphisms any two points can be interchanged be a diffeomorphism individual points do not have any meaning Only relations between several points can carry information J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
5 Motivation Motivation Geometry from relations between points Distance between points B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
6 Motivation Motivation Geometry from relations between points Parallel transport from A to B along a curve B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
7 Motivation Motivation Geometry from relations between points Parallel transport from A to B along a curve B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
8 Motivation Motivation Geometry from relations between points Parallel transport from A to B along a curve B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
9 Motivation Motivation Geometry from relations between points Parallel transport from A to B along a curve B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
10 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
11 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
12 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
13 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
14 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
15 Motivation Motivation Geometry from relations between points Holonomy around a closed path A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
16 Motivation Motivation Geometry 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, / 30
17 Motivation Motivation Consequence 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, / 30
18 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, / 30
19 Example: Electromagnetism Electromagnetism Electric field 1-form: electric field E = E x dx + E y dy + E z dz line (1-dim) L B A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
20 Example: Electromagnetism Electromagnetism Electric field 1-form: electric field E = E x dx + E y dy + E z dz line (1-dim) L B A E : A L B E =: W L J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
21 Example: Electromagnetism Electromagnetism Electric field 1-form: electric field E = E x dx + E y dy + E z dz line (1-dim) L B A E : A L B E =: W L Work done on unit charge along L (voltage between A and B) J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
22 Example: Electromagnetism Electromagnetism Magnetic induction 2-form: magnetic induction surface (2-dim) B = B xy dxdy + B yz dydz + B zx dzdx J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
23 Example: Electromagnetism Electromagnetism Magnetic induction 2-form: magnetic induction surface (2-dim) B = B xy dxdy + B yz dydz + B zx dzdx B : A B =: Φ J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
24 Example: Electromagnetism Electromagnetism Magnetic induction 2-form: magnetic induction surface (2-dim) B = B xy dxdy + B yz dydz + B zx dzdx B : A B =: Φ Magnetic flux through loop C = A J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
25 Example: Electromagnetism Electromagnetism Maxwell s equation d Ḃ = rot E B = dt A d Ḋ = rot H D = dt A A E A H J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
26 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d B = E dt Ω Ω J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
27 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
28 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
29 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B, e 3 = 2 1 E J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
30 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B, e 3 = 2 1 E, e 1 = 3 2 E J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
31 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B, e 3 = 2 1 E, e 1 = 3 2 E, e 2 = 1 3 E J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
32 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B, e 3 = 2 1 E, e 1 = 3 2 E, e 2 = 1 3 E ḃ = e 1 + e 2 + e 3 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
33 Example: Electromagnetism Discretisations of Ampere s law Ḃ = rot E Ω d dt i i B = i i E b = B, e 3 = 2 1 E, e 1 = 3 2 E, e 2 = 1 3 E very clear cut, elegant and efficient ḃ = e 1 + e 2 + e 3 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
34 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, / 30
35 Discrete differential forms Discrete differential forms continuous discrete p-dimensional submanifold S p : (0) point, (1) curve, (2) surface p-form: ω : S p ω R S p exterior derivative d: dω = ω S p S p p-simplices S p : (0) node, (1) edge, (2) face discrete p-form: Definition: ω : S p ω[s p ] R dω[s p ] = ω[ S p ] 3 Example: 1 Stokes theorem dω 123 = ω 12 + ω 23 + ω 31 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30 2
36 Discrete differential forms Discrete differential forms continuous Grassmann (wedge) product: ( α, p q β) p+q α β, graded algebra α β = ( 1) pq β α, derivation: d(α β) = dα β + ( 1) p α dβ. discrete discrete Grassmann product: Example: ( 1 α 1 β) 123 = ( α, p q β) p+q α β [α 12β 13 + α 23 β 21 + α 31 β 32 β 12 α 13 β 23 α 21 β 31 α 32 ] discrete d is derivation 3 2 derham cohomology singular cohomology J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
37 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, / 30
38 GR as a differential ideal GR as a differential ideal Basic variables tetrad (θ 0, θ 1, θ 2, θ 3 ) connection ω i k = ω k i J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
39 GR as a differential ideal GR as a differential ideal Basic variables tetrad θ i connection ω i k e e l[e] 2 = η ik θ i [e]θ k [e] exp(ω): holonomy along e J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
40 GR as a differential ideal GR as a differential ideal Basic variables tetrad θ i connection ω i k e e l[e] 2 = η ik θ i [e]θ k [e] exp(ω): holonomy along e related by no-torsion condition dθ i + ω i k θ k = 0 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
41 GR as a differential ideal GR as a differential ideal Einstein s equation To formulate the Einstein s equation one defines 2-forms: L i = 1 2 ɛ ijklη kn ω j n θ l J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
42 GR as a differential ideal GR as a differential ideal Einstein s equation To formulate the Einstein s equation one defines 2-forms: 3-forms: S i = 1 2 ɛ ijkl L i = 1 2 ɛ ijklω jk θ l (ω jk ω l m θ m ω j m ω mk θ l) vacuum equations dl i = S i J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
43 GR as a differential ideal Intermezzo The moment of rotation curvature 2-form: Ω i k = R i klmθ l θ m J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
44 GR as a differential ideal Intermezzo The moment of rotation curvature 2-form: Ω i k = R i klmθ l θ m Warner Miller: moment of rotation Ω [jk θ l] ɛ ijkl Ω jk θ l G ab J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
45 GR as a differential ideal Intermezzo The moment of rotation curvature 2-form: Ω i k = R i klmθ l θ m Warner Miller: moment of rotation Ω [jk θ l] ɛ ijkl Ω jk θ l G ab Identity: dl i = S i ɛ ijklω jk θ l }{{} =:T i J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
46 GR as a differential ideal Intermezzo The moment of rotation curvature 2-form: Ω i k = R i klmθ l θ m Warner Miller: moment of rotation Identity: Ω [jk θ l] ɛ ijkl Ω jk θ l G ab dl i = S i ɛ ijklω jk θ l }{{} =:T i Linearisation of T i on Minkowski space: 4-dim analogue of Snorre s curl T curl operator J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
47 GR as a differential ideal GR as a differential ideal Properties This formulation has close relations to energy balance Einstein: energy-momentum pseudotensor S i Møller: 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, / 30
48 GR as a differential ideal GR as a differential ideal Properties The variables θ i and ω i k are subject to gauge freedom: θ i s i k(x)θ k ω i k s i l (x)ω l ms m k(x) + s i m (x)ds m k for arbitrary function with values on O(1, 3), local Lorentz invariance. J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
49 GR as a differential ideal GR as a differential ideal Properties The variables θ i and ω i k are subject to gauge freedom: θ i s i k(x)θ k ω i k s i l (x)ω l ms m k(x) + s i m (x)ds m 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, / 30
50 GR as a differential ideal GR as a differential ideal Local structure in a normal neighbourhood Fix a normal coordinate system (x i ) around a point O θ i = ( δl i + βi lmx m + (β i lpq 1 ) 2 η jnβ ji pβ n lq)x p x q dx l 1 }{{} 6 Ri plqx p x q dx l + O(x 3 ) exp(β(x)) J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
51 GR as a differential ideal GR as a differential ideal Local structure in a normal neighbourhood Fix a normal coordinate system (x i ) around a point O θ i = dx i 1 6 Ri plqx p x q dx l + O(x 3 ) ω i k = 1 2 Ri kpqx p dx q + O(x 2 ) J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
52 GR as a differential ideal GR as a differential ideal Local structure in a normal neighbourhood Fix a normal coordinate system (x i ) around a point O θ i = dx i 1 6 Ri plqx p x q dx l + O(x 3 ) ω i k = 1 2 Ri kpqx p dx q + O(x 2 ) L i = 1 4 ɛ ijklr jk pqx p dx q dx l + O(x 2 ) S i = O(x 2 ) J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
53 GR as a differential ideal GR as a differential ideal Local structure in a normal neighbourhood Fix a normal coordinate system (x i ) around a point O θ i = dx i 1 6 Ri plqx p x q dx l + O(x 3 ) ω i k = 1 2 Ri kpqx p dx q + O(x 2 ) L i = 1 4 ɛ ijklr jk pqx p dx q dx l + O(x 2 ) S i = O(x 2 ) dl i = S i = O(x 2 ) To second order L i are the conserved fluxes of relativistic energy-momentum J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
54 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, / 30
55 Implementation Implementation with R. Richter, et al Theoretical formulation Geometric discretisation by local domains of dependence (light-like coordinates) non-linear algebraic system of equations solved by Newton s method found order of convergence J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
56 Implementation Implementation with R. Richter, et al Theoretical formulation Geometric discretisation by local domains of dependence (light-like coordinates) non-linear algebraic system of equations solved by Newton s method found order of convergence Application to simple (1 + 1) systems of GR spherically symmetric space-times: Minkowski, Schwarzschild, Kruskal plane waves Gowdy J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
57 Implementation Implementation with R. Richter, et al Theoretical formulation Geometric discretisation by local domains of dependence (light-like coordinates) non-linear algebraic system of equations solved by Newton s method found order of convergence Application to simple (1 + 1) systems of GR spherically symmetric space-times: Minkowski, Schwarzschild, Kruskal plane waves Gowdy verification of the order of convergence J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
58 Implementation Implementation with R. Richter, et al Theoretical formulation Geometric discretisation by local domains of dependence (light-like coordinates) non-linear algebraic system of equations solved by Newton s method found order of convergence Application to simple (1 + 1) systems of GR spherically symmetric space-times: Minkowski, Schwarzschild, Kruskal plane waves Gowdy verification of the order of convergence reproduction of exact solutions J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
59 Implementation Implementation Insights coordinate-free treatment of Einsteins equations purely geometric discretisation convergent method J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
60 Implementation Implementation Insights coordinate-free treatment of Einsteins equations purely geometric discretisation convergent method Questions and problems: no complete understanding of the algebraic structure of the equations J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
61 Implementation Implementation Insights 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 J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
62 Implementation Implementation Insights 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-linearities but: associator vanishes to higher order (known to topologists in the 70s) J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
63 Implementation Implementation Insights 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-linearities but: 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, / 30
64 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, / 30
65 Outlook Outlook General idea: put together small pieces of 4-d space-time i.e., where the discrete equations hold J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
66 Outlook Outlook General idea: put together small pieces of 4-d space-time i.e., where the discrete equations hold glue together across hypersurfaces Σ by gauge transformations J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
67 Outlook Outlook General idea: put together small pieces of 4-d space-time i.e., where the discrete equations hold glue together across hypersurfaces Σ by gauge transformations natural junction conditions (Israel 1966) L i L i = 0 for all 2-dim submanifolds of Σ S J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
68 Outlook Outlook However: not been able to convert this idea into a decent algorithm J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
69 Outlook Outlook However: not been able to convert this idea into a decent algorithm again problem with the gauge transformation J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
70 Outlook Outlook However: not been able to convert this idea into a decent algorithm again problem with the gauge transformation J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
71 Outlook Outlook However: not been able to convert this idea into a decent algorithm again problem with the gauge transformation Way out (?): J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
72 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 scalars How unique is Regge calculus? J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
73 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 scalars How unique is Regge calculus? discrete form ω i k is a hybrid creature use the holonomies directly, edge map into the Lorentz group J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
74 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 scalars How unique is Regge calculus? discrete form ω i k 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, / 30
75 Outlook THANK YOU J. Frauendiener (University of Otago) Discrete geometry Jena, August 26, / 30
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