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J. Frauendiener, Center of Mathematics for Applications, University of Oslo
Geometric discretisation of GRICIAM '07, Zürich July 19th
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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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
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A
B
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
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A
B
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
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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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
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
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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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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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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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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