Maximal Slicing of Schwarzschildor

The One-Body-Problem of Numerical Relativity

Bernd Reimann

Mexico City, 8/12/2003

ICN, UNAM AEI

Outline

Here in this talk I want to

• motivate why to look at a) maximal slicing & b) only one BH

• sketch how the maximal slices are derived

• point out the role of BCs imposed on the lapse

• show the slices in Carter-Penrose diagrams

• present analytical & numerical results for “puncture evolutions”

• derive late time statements which could be used to test codes

• introduce and discuss “indicators for slice stretching” for evolutions with zero shift

• study boundary conditions which make slice stretching effects occur “late” in a numerical simulation

• discuss if (and how) slice stretching can be avoided

Exact solutions are ideal – if only we knew some useful ones…

Why look at Maximal Slicing of the Schwarzschild Spacetime?

Motivation

- maximal slices avoid singularities

- being geometrically motivated,

K=0 simplifies equations and the

slices can be studied analytically

- hyperbolic lapse choices (“1+log”

family) “mimic” maximal slicing

- astrophysical relevance

- “the” spacetime for gauge choices

to be studied analytically and

tested numerically

- understand foliations of one BH

should help in the “two-BH” case

• The analytic solution allows to confirm/disproof numerical statements and to discuss the “One-Body-Problem of NR”:

Can one avoid/control Slice Stretching to evolve a BH forever?

• Topic of Diploma thesis supervised by B. Brügmann & E. Seidel

and subject of gr-qc/0307036 and probably 2 or 3 further papers…

Deriving the Maximal Slices

0

02

KgK

KKKR

aabab

a

abab

)2()(

2)(c

bacababbaabt

ababt

KKKKRKL

KgL

.0; aantrKK

Following Estabrook et al.,1973, one can solve Einstein’s equations

in “3+1” form, i.e. the constraints

together with the evolution equations

while demanding the maximality condition

Analytically as functions of time at infinity and areal radius r one obtains the lapse , the shift r and the 3-metric ij.

Boundary Conditions for the Lapse• For the lapse a second-order ODE

is found, so 2 BCs have to be specified.• To measure proper time the lapse is set to

one at infinity.• Antisymmetry w.r.t. throat yields the static

Schwarzschild metric in iso- tropic coordinates with the odd lapse.

• Demanding symmetry w.r.t. throat one obtains the even lapse.

• Since the ODE is linear, by the super-position principal any other lapse can be constructed as linear combination

R

evennewoddnewnew )1(

throat,r3M/2

“left” EH, r=2M

initially:“later”:

“infinity”, {x,r }

“puncture”, {x=0,r }

throat=EH, x=M/2,r=2M

“right”EH, r=2M

22242)3( )( drdxxds

x

Mx

21)(

Maximal Slices in Carter-Penrose Diagramsodd

“zgp”

even

“infinity”“puncture”

“right” EH“left” EH throat

“Collapse of the Lapse” at times 0,1,10,100M

left EH

throat

right EH

Approach of the Limiting Slice r = 3M/2

(Areal radius obtained as root of angular metric part)

Development of a Peak in the Metric

Note that the peak in gis located in between throat and “right” EH

Fundamental Problem with Black Holes and Singularity Avoiding Slicings

constant time slices wrap up around the singularity: (“Slice Wrapping”)

• evolution essentially “frozen” in the inner region

• evolution marches ahead outside to cover a large fraction of the spacetime

infalling observers (“Slice Sucking”)

=> “Slice Stretching”

Throat

Collapsing Star

Singularity

t=150

t=100

t=50

t=0

Event Horizon

Throat

• The idea here is to introduce ,

to expand all terms in leading order of and discuss the limit

• It turns out that - unless odd boundary conditions are used - is decaying exponentially with time, where a fundamental time-scale is given by

Late Time Analysis• The analytically found 4-metric is valid for all times at infinity, but it

is “difficult” to evaluate these expressions “at late times”• Hence it is useful to perform a “late time analysis” in order to discuss the

behavior of the maximal slices at 5 “markers” being from left to right

puncture - “lefthand” EH – throat - “righthand” EH – infinity

2

3Mrthroat

0

MM 8371.164

3

(Smarr & York, 1978, Beig & O’Murchadha, 1998)

They found 1.82M numerically!

MM

constx 3 ]ln[3189.14

0x

constx

x

MM

constx 3 ]ln[1594.7

r

Mr 2

Mr 2

r

23Mr

Slice Stretching for “zgp” BCs and zero Shift

322

11359.1

M

322

19428.0

M

214714.0

M

322

13967.03248.0

M

1

Infinity

“Lefthand” EH

Throat

“Righthand” EH

MM eeM 8371.16742.36236.0

Puncture Slices penetrate EH to approach r = 3M/2 asymptoticallyCollapse of the lapse, “outward moving shoulder”

1g

constg

1g

34

]ln[3189.14

Mconstg

34

]ln[1594.7

Mconstg

Growing peak in the radial part of the metric34Throat and “righthand” EH move to infinite x3

1

From left to right in terms of

Holds not for the odd but for all other BCs!

Numerically a value of 0.3 has been used to

locate the EH for excision!

Puncture Lapse evaluated at the “markers”“puncture”

“lefthand” EH

throat

“righthand” EH

Slice Stretching evaluated

at the “markers”• “Slice Sucking” occurs like

as measured at throat or EH• “Slice Wrapping” takes place as a peak in the radial

metric component develops like

3/1

,

EHthroatx

3/4

,

EHthroatxxg

right EH

peak in g

throat

left EH

right EH

peak in g

throat

left EH

The peak in g does not grow exponentially!

“Exploit”/”Explore” the Maximal Slices Further?

In quite some detail both analytically and numerically for- odd, even and “zgp” BCs

- evolutions of puncture data

the maximal slices of Schwarzschild have been discussed.

up to now:

but further interesting questions are:

Do BCs exist which are more “favorable” for numerical purposes?

What happens if for puncture evolutions one uses a shift?

How does the behavior of the 3-metric change when starting with

other (e.g. logarithmic) spatial coordinates? Plus shift?

Is it possible to avoid slice stretching? How?

BCs which make Slice Stretching occur “late”• The late time analysis shows that in terms of the results for

“slice stretching indicators” (location of the throat and righthand EH, blow-up of the 3-metric there) do not depend on BCs!

• The idea is hence to make for the lapse

slice stretching occur “late”, i.e. at large values of , by demanding that “approaches zero slowly” as a function of .

• But one obtains that is independent of BCs .

• So in order to make “large”, new has to go to zero and the odd lapse is approached, .

• But this lapse is negative in the “lefthand region”!

• Demanding in addition , i.e. new ½, it turns out that the average of odd and even lapse is the “best-possible” lapse!

oddnew

0new

The “zgp” lapse at late times has

new = ½!

evennewoddnewnew )1(

)(

newnew

d

d

d

d new

Slice Stretching for a 1-parameter family of BCs (I)

• (Preliminary) numerical test have been performed by implementing

with new = const,

i.e demanding

and

.

• At late times

is found.

evennewoddnewnew )1(

120

newxnew

1xnew

]8317.1

exp[M

new

Slice Stretching for a 1-parameter family of BCs (II)

Since the expansions in of “slice stretching indicators” do not depend on the BCs, this implies that both

“Slice Sucking”

and

“Slice Wrapping”

occur “later” in a numerical evolution if the lapse is “closer” to the odd lapse.

The “Slice Stretching Integral”• In order to discuss the slice stretching effects arising for

- arbitrary (!) spatial coordinates y on the maximal slices

- evolutions with any possible (!) shift,starting from

one can write down the coordinate transformation

• Integrating from the “lefthand” to the “righthand” EH yields

214

22 )

)(21(),( dr

r

C

r

MdyyGyy

ry

yy dzz

C

z

MdzzG 2/1

4

2

))(2

1(),(

M

r

y

y

yythroat

rightEH

leftEH

dzz

C

z

MdzzG

2 2/14

2

))(2

1(2),(

Picking up twice a factor ln[] at

the throat!

]ln[ const

constIf 0 then ]exp[

const

THAT’S BAD!

Avoid Slice Stretching I: Excision• The diverging factor , being for “reasonable“ BCs

proportional to time at infinity , is picked up by integrating “over the throat”. It’s essentially like integrating 1/ “over = 0”.

• But if the throat is not part of the slice, i.e. if a piece in between “lefthand” and “righthand” EH containing the throat is excised, one obtains in leading order

and slice stretching is stopped as the evolution “freezes”.

• In the context of several different shifts this is shown e.g. in

Since the throat is excised one should not speak of “Singularity Excision” but instead of

“Throat Excision”!The singularity is not even part of the slices…

]ln[

constdzzGrightEH

boundary

y

y

yy ),(

(Anninos et al., 1995)

Avoid Slice Stretching II: Conformal Factor• The radial metric component appearing in the “slice stretching

integral” can be written as making use of a conformal factor.

• In particular, if has a coordinate singularity, the integral

can diverge while numerically evolved quantities “freeze”.

• For puncture evolutions, with a (finite) shift has to act such that throat and “lefthand” EH move towards y = 0 like

For “1+log lapse” and “gamma-freezing shift” numerical evidence is found in

• But for logarithmic coordinates, with there is no chance to avoid slice stretching – which explains numerical observations made in the PhD Thesis of Bernstein & Daues!

)(),( 4 yyg yy ),( yGyy

constdzzzgrightEH

leftEH

y

y

yy )(),( 2

3/1

,

leftEHthroat

y

yM 21

Alcubierre et al., 2003.]2cosh[2 yM

Test Codes with“Maximal Slicing of Schwarzschild”?

+ Foliation of

a (“the”) fundamental spacetime with

a (“the”) geometrically motivated gauge choice

+ Solution known analytically

- Not every code is capable of solving the (computationally expensive) elliptic equation for the lapse

- The analytic solution is non-trivial, but suitable for a numerical test are

* Timescale describing the exponential decay

* Late time statements for the lapse

(valid, however, only at late times…)

Final RemarksVery similar results for the

- singularity avoiding behavior

- collapse of the lapse

- slice stretching effects

have been observed numerically for “1+log” slicing and

modifications such as .42 Kt

Whereas it is straightforward to generalize (some of) the analysis - including electrical charge to Reissner-Nordström

- setting K = const to constant mean curvature,

it is non-trivial to discuss- maximal slicings of Kerr

- hyperbolic lapse choices for Schwarzschild

analytically.