Maximal Slicing of Schwarzschild or 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/ and probably 2 or 3 further papers…

Deriving the Maximal Slices 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 throat,r  3M/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

Maximal Slices in Carter-Penrose Diagrams odd “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 g is 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 (Smarr & York, 1978, Beig & O’Murchadha, 1998) They found   1.82M numerically!

Slice Stretching for “zgp” BCs and zero Shift Infinity “Lefthand” EH Throat “Righthand” EH Puncture Slices penetrate EH to approach r = 3M/2 asymptoticallyCollapse of the lapse, “outward moving shoulder” Growing peak in the radial part of the metricThroat and “righthand” EH move to infinite x 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 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! The “zgp” lapse at late times has  new = ½!

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.

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 Picking up twice a factor ln[  ] at the throat! If   0 then 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… (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! Alcubierre et al., 2003.

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 Remarks Very 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. 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.