numerical relativity is still relativity

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Numerical Relativity is still Relativity ERE Salamanca 2008 Palma Group Alic, Dana · Bona, Carles · Bona-Casas, Carles

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Numerical Relativity is still Relativity. ERE Salamanca 2008 Palma Group Alic, Dana · Bona, Carles · Bona-Casas, Carles. Most recent successful stories in BH simulations. Long term evolutions: Harmonic (4D spacetime, excision, harmonic gauge source functions) - PowerPoint PPT Presentation

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Page 1: Numerical Relativity is still Relativity

Numerical Relativity is still Relativity

ERE Salamanca 2008 Palma Group

Alic, Dana · Bona, Carles · Bona-Casas, Carles

Page 2: Numerical Relativity is still Relativity

• Long term evolutions:– Harmonic (4D spacetime, excision, harmonic gauge source

functions)– BSSN (3+1 decomposition, punctures/excision, 1+log and

gamma freezing)

• Isn’t the gauge choice too limited? Shouldn’t numerical relativity be relativity?

Most recent successful stories in BH simulations

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Page 3: Numerical Relativity is still Relativity

Do we have any choice?

• Reported experiences:– No long term simulations with normal coordinates

(zero shift).– Generalised harmonic slicing but strictly harmonic

shift.– BSSN normal coordinates (zero shift) and 1+log

slicing crashes at 30-40M (gr-qc/0206072).– Gaugewave test: gauge imposed is harmonic, so

harmonic code succeeds, but BSSN crashes.Para ver esta película, debedisponer de QuickTime™ y de

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Page 4: Numerical Relativity is still Relativity

Looking for a gauge polyvalent code

• Z4 formalism

• MoL with 3rd order SSP Runge-Kutta.• Powerful 3rd order FD algorithm (submitted to

JCP). See a variant in http://arxiv.org/abs/0711.4685 (ERE 2007)

• Scalar field stuffing.• Cactus. Single grid calculation. Logarithmic grid

for long runs.

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Page 5: Numerical Relativity is still Relativity

Gaugewave Test

• Minkowski spacetime:

• Harmonic coordinates x,y,z,t.

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Page 6: Numerical Relativity is still Relativity

t=1000; Amplitude 0.1

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Page 7: Numerical Relativity is still Relativity

BSSN Comparison

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t=1000

t=30

Page 8: Numerical Relativity is still Relativity

t=1000; Amplitude 0.5

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Page 9: Numerical Relativity is still Relativity

Single BH Test

• Singularity avoidant conditions (Bona-Massó)

Q = f (trK-2)

• 1+log (f=2/) slicing with normal coordinates (zero shift) up to 1000M and more! Never done before (BSSN reported to crash at 30-40M without shift).

• Unigrid simulation. Logcoords =1.5.

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R = λ sinh r /λ( )

Page 10: Numerical Relativity is still Relativity

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Lapse function at t=1000M

Page 11: Numerical Relativity is still Relativity

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R/M=20r/M=463000

Page 12: Numerical Relativity is still Relativity

More gauges (zero shift)

• Isotropic coords. Boundaries at 20M.

• Logcoords f=1/ 150M.

Slicing (f) 2/ 1+1/ 1/2+1/ 1/ 1/4+3/4 1/2+1/2

Vol. Elem. left

37% 25% 20% 14% 10% 6%

Time lasting (0.2 / 0.1 resol)

50M

/

50M

50M

/

50M

50M

/

50M

6M

/

50M

6M

/

20M

5M/12M

2

fα∂tα =∂t ln γ( )

Page 13: Numerical Relativity is still Relativity

Shift

• 1st order conditions.

• Vectorial.– Harmonic? xi = 0.

1

α γ∂0

γ

αβ i

⎣ ⎢

⎦ ⎥−

1

α γ∂k

γ

αβ iβ k

⎣ ⎢

⎦ ⎥= −

1

α γ∂k α γ γ ki[ ]

∂0

β i

α

⎣ ⎢

⎦ ⎥−β k∂k

β i

α

⎣ ⎢

⎦ ⎥= trK β i −α A i −Di + 2V i

( )1st order version

Page 14: Numerical Relativity is still Relativity

Advection terms

• Lie derivative “advection/damping”

• Covariant advection term

1

γ(∂0 − Lβ )

γ

αβ i

⎣ ⎢

⎦ ⎥=

1

α∂tβ

i + (Q− trK)β i

(∂0 −β k∇ k )β i

α

⎣ ⎢

⎦ ⎥=∂tβ

i −β kBki − Γ jk

i β jβ k +α Qβ i

Page 15: Numerical Relativity is still Relativity

1st order vector ingredients

• Time-independent coordinate transformations.

Ai =∂i lnα

Di −Di t= 0=∂i ln γ /γ 0

E i − E i t= 0= D ji

j −D jij

0

M

Zi

Page 16: Numerical Relativity is still Relativity

∂tβi =

3

4α 2 (E − E0)i −

(D−D0)i

3+ Z i

⎝ ⎜

⎠ ⎟

+β kBki + Γ jk

i β jβ k −α Qβ i

∂tβi =

3

4α 2 (E − E0)i − (D−D0)i + Z i( )

+β kBki + Γ jk

i β jβ k −α Qβ i

Page 17: Numerical Relativity is still Relativity

∂tβi =

1

2α 2A i + β kBk

i + Γ jki β jβ k −α Qβ i

∂tβi =

1

2α 2A i −α (Q− (trK − 2θ))β i