3+1 Splitting of the Generalized Harmonic Equations

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "3+1 Splitting of the Generalized Harmonic Equations"

Λυκούργος Διάβολος Αγγελόπουλος
5 χρόνια πριν
Προβολές:

1 3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011

2 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime derivatives into time and space derivatives µ t i

3 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime derivatives into time and space derivatives µ t i Formulations: ġ K ADM BSSN etc

4 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime derivatives into time and space derivatives µ t i Formulations: ġ K ADM BSSN etc Also split tensors g µν α β i g ij

5 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime derivatives into time and space derivatives µ t i Formulations: ġ K ADM BSSN etc Also split tensors g µν α β i g ij Generalized Harmonic formulation can be interpreted as an initial value problem without splitting tensor indices.

6 Einstein Equations R µν = 0 Generalized Harmonic Eqs. R µν = (µ C ν) C µ H µ + Γ µ σρg σρ = 0

7 Einstein Equations R µν = 0 Generalized Harmonic Eqs. R µν = (µ C ν) C µ H µ + Γ µ σρg σρ = 0 Initial Value Problem t g ij = L β g ij 2αK t K ij = L β K ij + αkk ij + H K 2 K ij K ij + R = 0 M i D j K j i D i K = 0 H = M i = 0 preserved in time if H = M i = 0 initially Generate solution by evolving g ij and K ij with α and β i chosen freely

8 Einstein Equations R µν = 0 Initial Value Problem t g ij = L β g ij 2αK t K ij = L β K ij + αkk ij + H K 2 K ij K ij + R = 0 M i D j K j i D i K = 0 H = M i = 0 preserved in time if H = M i = 0 initially Generate solution by evolving g ij and K ij with α and β i chosen freely Generalized Harmonic Eqs. R µν = (µ C ν) C µ H µ + Γ µ σρg σρ = 0 Initial Value Problem g µν = 2 (µ H ν) + C µ = C ν (µ C ν) C µ = 0 preserved in time if C µ = 0 and t C µ = 0 (equiv. to H = M i = 0) initially Generate solution by evolving g µν with H µ chosen freely

9 Goal: Write generalized harmonic (GH) equations with 3+1 splitting of derivatives and tensor indices: t g ij =... t K ij =... t α =... t β i =....

10 Goal: Write generalized harmonic (GH) equations with 3+1 splitting of derivatives and tensor indices: Motivation: t g ij =... t K ij =... t α =... t β i =... GH is expressed in the same language/notation as ġ K, ADM, BSSN, etc. Comparison and insights..

11 Goal: Write generalized harmonic (GH) equations with 3+1 splitting of derivatives and tensor indices: Motivation: t g ij =... t K ij =... t α =... t β i =... GH is expressed in the same language/notation as ġ K, ADM, BSSN, etc. Comparison and insights. Result is nice..

12 3+1 Splitting of GH: technical details Covariant GH equations: where H µ is a vector and R µν = (µ C ν) C µ H µ + Γ µ σρg σρ = 0 Γ µ σρ Γ µ σρ Γ µ σρ = 1 2 g µν ( σ g ρν + ρ g σν ν g σρ )

13 3+1 Splitting of GH: technical details Covariant GH equations: where H µ is a vector and R µν = (µ C ν) C µ H µ + Γ µ σρg σρ = 0 Γ µ σρ Γ µ σρ Γ µ σρ = 1 2 g µν ( σ g ρν + ρ g σν ν g σρ ) Split R µν, Γ µ σρ, Γ µ σρ, H µ, µ C ν Absorb terms F (g µν, g µν, σ g µν ) into H and H i Define K ij ( t g ij L β g ij )/(2α) π ( t α β i D i α)/α 2 + H ρ i ( t β i β k D k β i )/α 2 + D i α/α H i

14 GH Equations in 3+1 form ( t L β ) g ij = 2αK ij [ ] K ij = α R ij 2K ik Kj k + KK ij D i D j α αc K ij αd (i C j) α = α 2 π α 2 H t β i = β j D j β i + α 2 ρ i αd i α + α 2 H i π = αk ij K ij + D i D i α + C i D i α ρ i = αd i π πd i α 2K ij D j α + 2αK jk Γ i jk + g kl D k D l β i C π + K C i ρ i + Γ i jk g jk H K 2 K ij K ij + R M i D j K j i D i K

15 Constraint Evolution C = αkc + αh + C i D i α αd i C i C i = C D i α αd i C 2αM i 2αK ij C j H = 2απH + 2αRC + 2α(K ij Kg ij )D i C j 4M i D i α 2αD i M i M i = HD i α + (Kδ j i K j i )D j(αc ) 1 2 αd ih απm i + D j αd [i C j ] + D i (αd j C j ) 1 2 αr ijc j αd j D j C i

16 Now What? Direct comparison of GH with ġ K, ADM, BSSN, etc Which terms are responsible for stable evolution? Moving puncture coordinates/evolution in GH? New formulations?

17 Symmetric Hyperbolicity Conserved energy: [ ] 1 ε = M ijkl 4 g mn m g ij n g kl + (K ij i β j /α)(k kl k β l /α) [ ] 1 +N ij α 2 g kl k β i l β j + (ρ i i α/α)(ρ j j α/α) [ +C ππ + 1 ] α 2 g ij i α j α where M ijkl, N ij and C are positive definite.

18 GH with Constraint Damping R µν (µ C ν) + κ [ n (µ C ν) g µν n σ C σ /2 ] = 0 = g ij = 2αK ij [ ] K ij = α R ij 2K ik Kj k + KK ij D i D j α αc K ij αd (i C j) 1 2 κg ijc α = α 2 π α 2 H t β i = β j D j β i + α 2 ρ i αd i α + α 2 H i π = αk ij K ij + D i D i α + C i D i α 1 2 καc ρ i = αd i π πd i α 2K ij D j α + 2αK jk Γ i jk +g kl D k D l β i + καc i

19 GH with Constraint Damping C = αkc + αh + C i D i α αd i C i 2καC C i = C D i α αd i C 2αM i 2αK ij C j καc i H = 2απH + 2αRC + 2α(K ij Kg ij )D i C j 4M i D i α 2αD i M i 2καC M i = HD i α + (Kδ j i K j i )D j(αc ) 1 2 αd ih απm i + D j αd [i C j ] + D i (αd j C j ) 1 2 αr ijc j αd j D j C i + κd i (αc )

Σχετικά έγγραφα

Higher Derivative Gravity Theories

Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)