Numerical Relativity

or

Jimmy, You Haven’t Done

Enough Yet!

Why is Numerical Relativity important?

1. Find out new things about Einstein equations

2. Predict wave forms for experiments (LIGO,

LISA . . . )

Examples of (1):

• Critical behavior in black hole formation

(Choptuik 1993)

• Toroidal black holes

(Shapiro and Teukolsky 1994)

Predicting wave forms is urgent!

• To improve sensitivity

• To compare theory and experiment

What can we learn from experiments like LIGO?

We know

Lots of mass in small volume

M > Mmax for neutron star

Deduce black hole

BUT

Are these the black holes of Einstein’s theory?

Event horizon = 1-way membrane

Geometry of spacetime = Schwarzschild metric

etc.

GR tested only in weak-field regime

Solar system expts:v2

c2corrections to Φnewt

Hulse-Taylor binary pulsar: (existence of GW)

“only”v5

c5

Only black holes test the full nonlinear,

strong-field aspects of GR

If we can detect GW signals from binary black

hole coalescence, we may finally have the

opportunity to confront the most dramatic

predictions of the theory with experiment.

Why is it so hard to solve the binary black hole

problem?

1992 — brute force?

The Numerical Problem — Scale

16 – 50 variables, 3-D

Length scale M

1% accuracy FD → resolution M/20

λ for radiation from 6M ∼ 100M

→ 2000 grid points per dimension (uniform)

→ 3 terabytes

2 orbits + plunge ∼ 500M

→ 104 timesteps (Courant limit)

Complicated equations: ∼ 200 flops per variable

per grid pt per timestep

→ 106 teraflops for 1 modest simulation

(Mesh refinement)

The Numerical Problem — Black Hole

Singularities

1. Singularity avoiding time slicing

→ grid stretching

Calculations crash eventually (steep gradients)

2. Black hole excision

Finite differencing with excision:

������

������ ������

������ ��

������

� ��� ������

������

������������

������������

������

������

��� �

!�!"�"

#�#$�$

%�%&�&

'�'(�(

)�)*�*

+�+,�,

-�-.�.

/�/0�0

1�12�2

3�34�4

Can we cut the black hole out of the

computational domain?

Standard formulation:

Can require info from inside

No obvious b.c.

Coordinate freedom → “gauge” modes

New formulations:

Manifestly hyperbolic in strict

mathematical sense

Hyperbolic Formulations of GR

• Advantages

– Simpler equations (wave equations!).

– Well-posed.(Note: does not exclude exponential growth.)

– Characteristic fields ⇒Knowledge of allowed boundary conditions.

Horizon excision well-defined.

Well-defined prescription for outer boundary.

• Disadvantages

– More variables.

– More constraints must be satisfied.

– More complicated non-principal terms.

Example of hyperbolic formulation of GR

Einstein-Christoffel (Anderson & York 1999)

(30 variables):

∂0gij = −2αKij

∂0Kij + α∂kfkij = αX(gij,Kij, fkij)

∂0fkij + α∂kKij = αY (gij,Kij, fkij)

where

fkij ≡ Γ(ij)k + g`m(gkiΓ[`j]m + gkjΓ[`i]m)

3+1:

∂0gij = −2αKij

∂0Kij =α

2gk` (∂k∂`gij + ∂i∂jgk` − ∂i∂`gkj − ∂k∂jgi`)

+ αF (gij, Kij, ∂kgij)

Pseudospectral Collocation Methods

• Basic idea

– Equations, boundary conditions remain exact .

– Solution written as sum of N basis functions.

f(x, t) =N−1∑

k=0

fk(t)φk(x)

– Spectral method:

fk(t) =

∫

f(x, t)φk(x) dx

– Pseudospectral method (Lanczos 1938):

fk(t) =N−1∑

n=0

wnf(xn, t)φk(xn)

– Uses N collocation points {xn} → {f(xn, t)}.– Compute spatial derivatives analytically.

– Compute nonlinear terms at points {xn}.– Evolve f(xn, t) as ODEs (“method of lines”).

(Elliptic equations:

Solve simultaneous algebraic equations.)

• Advantages

– Exponential convergence for smooth solutions.(Solutions of Einstein’s equations are smooth invacuum!)

– No need to discretize boundary conditions.

⇒ Black hole excision is trivial.

• Disadvantages

– More work per coefficient.

– More restrictive Courant condition.

– Supression of high-frequency modes sometimes needed.

– No advantage for non-smooth solutions.

Prediction

In the next 10 years spectral methods will become

the method of choice for numerical simulations of

black holes. (Neutron stars?)

Spherical Symmetry (1D)

(Kidder, Scheel, Teukolsky, Carlson, Cook, PRD 62, 084032 (2000))

• Einstein-Christoffel formulation.

– 6 dynamical variables : grr, gθθ, Krr,Kθθ, frrr, frθθ.

– 6 characteristic fields:

U±r ≡ Krr ±

frrr√grr

, U 0

r ≡ grr,

U±θ ≡ Kθθ ±

frθθ√grr

, U 0

θ ≡ gθθ.

U+U−U+

U−U+

U−Horizon

r

t

• Use Chebyshev polynomials in r as basis.

Schwarzschild BH, Spherical Symmetry (1D)

r = 1.9M

r = 11.9M

• Kerr-Schild coordinates

• Analytic gauge conditions.

• Inner boundary: No BC.

• Outer boundary:

U 0r , U

0

θ , U−r , U−

θ frozen.

U+r , U+

θ untouched.

• 4th-order Runge-Kutta, ∆t = 0.007M .

• Radial resolutions12, 16, 20, 24, 27, 32, 36, 40, 45, 48, 54, 60.

3D Black Hole Evolutions

(L. Kidder, M. Scheel, D. Shoemaker, H. Pfeiffer, L. Lindblom)

• Rectangular domains:

– x,y,z basis: Chebyshev polynomials.

• Spherical shell domains:

– r basis: Chebyshev polynomials.

– Angular basis: Y`m(θ, φ).

– Evolve Cartesian components.

• How do we handle 2 black holes?

Multiple domains:

• Interdomain boundaries:

Match characteristic fields (interpolate for

overlapping domains).

Single Schwarzschild BH, 3D

r = 1.9M

r = 11.9M

`max = 7

• Painleve-Gullstrand coordinates.

• Analytic gauge conditions.

• Inner boundary: No BC

• Outer boundary:

– Ingoing fields frozen.

– Outgoing fields untouched.

• 4th-order Runge-Kutta, ∆t = 0.0075M

• Radial resolutions8, 10, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54.

The Analytic Problem

Einstein’s equations are an overdetermined system

Pick a subset to solve (e.g., evolution equations)

“All” 3-d black hole evolutions unstable

(Important accomplishment of past decade)

How to deal with this . . .

• New formulations

– Partial success in special cases

• Control the instability

– Constraint-attracting system (Brodbeck, Frittelli,Huebner, Reula, 1999)

– Dynamical feedback of parameters (Tiglio 2003)

• Reimpose constraints (fast elliptic solver)

– Projection methods (Holst, Lindblom, Owen, Pfeiffer,Scheel, Kidder 2004)

• Least-squares methods

I =

∫

(

‖evolution equations‖2 + ‖constraints‖2)

dV

• . . .

We’re optimistic, but Jimmy, we sure could use

some help!