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J. Frauendiener, Center of Mathematics for Applications, University of Oslo Geometric discretisation of GR ICIAM '07, Zürich July 19 th

Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

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Page 1: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Geometric discretisation of GRICIAM '07, Zürich July 19th

Page 2: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Motivation

Numerical simulation of general relativistic space-times(cosmology, gravitational waves, ...)

GR is a geometric theory

invariance under arbitrary diffeomorphisms

any two points can be interchanged be a diffeo

individual points do not have any meaning

Only relations between several points carry information

metric provides distance between two points

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Page 3: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Motivation

geometry is determined by relations between points and higher dimensional objects

'distance' between two points

3

A

B

Page 4: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Motivation

geometry is determined by relations between points and higher dimensional objects

parallel transport from A to B along a curve

4

A

B

Page 5: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Motivation

geometry is determined by relations between points and higher dimensional objects

holonomy around a closed path

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➡ Gauss curvature of a surface

Page 6: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Consequence

localise geometric objects not only on points

FD: tensor components as grid functions

but also on lines, surfaces, volumes according to their meaning

use discrete exterior calculus

requires understanding of

principal fibre bundles over discrete domains

discrete vector valued differential forms

so far: naive procedure

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Page 7: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

GR with differential forms

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tetrad connection

!ik

!!0,!1,!2,!3"

related by the torsion free condition

d!i + "ik ! !k = 0

l2 = !ik!i[e]!k[e]

e

: holonomy along e exp(!)

e

four 2-form equations

Page 8: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

GR with differential forms

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To formulate the Einstein equations one defines

2-forms: Li =12!ijkl!

jk ! "l

3-forms: Si =12!ijkl

!!kl ! !j

m ! "m " !km ! !ml ! "j"

dLi = SiEinstein equationin vacuum

!ijkl = ![ijkl]

!0123 = 1

four 3-form equations

Page 9: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Meaning of the equations

energy balance

Einstein: energy-momentum pseudotensor ~ Si

Møller: energy-balance in tetrad form

Bondi-Sachs mass loss formula

in terms of spinors at the basis of Witten PET proof

focussing of light rays due to gravity

Penrose inequality (?)

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Page 10: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Theoretical formulation algorithm for discrete equations 1+1 dimensions second order convergence discrete wedge product non-abelian gauge freedom timestep through local domains of dependence

(lightlike coordinates)

Results

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Page 11: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Results

Application to simple systems of GR

spherically symmetric space-times

Minkowski, Schwarzschild, Kruskal

colliding plane waves

cosmological (Gowdy) space-times

verification of the order of convergence

reproduction of exact solutions

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Page 12: Geometric discretisation of GR - Institute for Mathematics ...arnold/dgcd-talks/Frauendiener.pdf · Geometric discretisation of GR ICIAM '07, Zürich July 19th. J. Frauendiener,

J. Frauendiener, Center of Mathematics for Applications, University of Oslo

Results

✦ coordinate-free treatment of Einsteins equations✦ purely geometric discretisation

Questions and problems what should/can be taken over from continuum? do we have the right variables? how is gauge freedom represented? treatment vector valued differential forms? treatment of discrete principle fibre bundles? synthetic differential geometry? groupoid definition of PFB?

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