Numerical General Relativity |

**In General**

* __Motivation__: Make realistic
astrophysical predictions; Use to look for chaos (the first positive claims were
wrong because the constraints were not preserved, and error propagation simulated
negative energy density).

* __History, status__: 1960s,
First attempts at solving a binary black hole spacetime by Hahn and Lindquist;
1976, B DeWitt coins the expression "numerical relativity"; 1989,
Some 3D problems, like collapse and gravitational wave production, can be tackled;
1992, First qualitatively new solution found by numerical methods, Choptuik's
critical collapse; 1994-1999, Binary-black-hole grand challenge; Still relatively
few 3D problems done; Also, better understanding of the convergence of Regge
calculus, theoretically, but in practice method for Regge calculus not as
developed (choosing initial data involves solving elliptical differential
equations); 2005, F Pretorius breakthrough and stable simulation of
black-hole inspiral and merger; 2009, About 11 groups worldwide can now do
full merger simulations; New results have been obtained (gravitational recoil
"kicks", black-hole triplets, gravitational-wave production); Gravity
is in the process of becoming data-driven.

* __Data__: One way of handling the
fact that the region is finite is to give data on a finite spacelike region,
and then free data on the outgoing light front from its boundary.

__Related topics__:
see issues and methods; models
in numerical relativity [collapse, binaries, cosmology, astrophysics].

**Gauge and Coordinate Choices**
> s.a. coordinates; gauge choices.

* __Idea__: It looks like the best gauge choices are dynamical ones.

@ __Choices and effects__: Alcubierre & Massó PRD(98)gq/97 [gauge problems];
Garfinkle & Gundlach CQG(99)gq [approximate Killing vector field];
Garfinkle PRD(02)gq/01 [harmonic coordinates];
Reimann et al PRD(05)gq/04,
Alcubierre CQG(05)gq [gauge shocks].

@ __BCT gauge (minimal strain equations)__: Brady et al;
Gonçalves PRD(00)gq/99;
Garfinkle et al CQG(00)gq.

@ __Special cases__: Gentle et al PRD(01)gq/00 [constant *K* and black holes].

**Constraints** > s.a. Symplectic Integrators.

* __Idea__: Due to finite precision
errors, constraints in numerical relativity are never exactly satisfied, so
one can solve them initially and then simply monitor them as a check on the
evolution (unconstrained evolution), or somehow enforce them as part of the
evolution; 2008, Recent simulations use initial data generated by constraint
solvers that differ by the amount of gravitational radiation they include in
the initial configuration.

@ __General references__: Detweiler PRD(87);
Cook LRR(00)gq;
Tiglio gq/03 [control];
Fiske PRD(04)gq/03 [as attractors];
Gentle et al CQG(04)gq/03 [as evolution equations];
Baumgarte PRD(12)-a1202 [Hamiltonian constraint, alternative approach];
Okawa IJMPA(13)-a1308-ln [elliptic differential equations].

@ __And boundary conditions__: Calabrese et al PRD(02)gq/01;
Calabrese & Sarbach JMP(03) [ill-posed];
Sarbach & Tiglio JHDE(05)gq/04;
Kidder et al PRD(05)gq/04;
Rinne et al CQG(07)-a0704 [comparison of methods];
> s.a. methods in numerical relativity.

@ __Enforcement and violations__: Siebel & Hübner PRD(01)gq [effects of enforcement];
Lindblom & Scheel PRD(02)gq [violations and stability];
Berger GRG(06)gq/04-fs;
Matzner PRD(05)gq/04 [hyperbolicity and constrained evolution];
Marronetti CQG(05)gq [Hamiltonian relaxation],
CQG(06)gq/05,
gq/06-MGXI [constraint relaxation];
Paschalidis et al PRD(07) [well-posed evolution].

**References**

@ __Books and collections of papers__: Centrella ed-86;
Evans et al ed-88;
d'Inverno 92;
Hehl et al ed-96;
issue CQG(06)#16,
CQG(07)#12,
CQG(09)#11;
Alcubierre 08;
Bona et al 09 [and relativistic astrophysics];
Baumgarte & Shapiro 10;
issue CQG(10)#11;
Shibata 16.

@ __Reviews__: Lehner CQG(01)gq,
gq/02-GR16;
van Putten gq/02-conf;
Rezzolla in(14)-a1303-proc;
Cardoso et al LRR(15)-a1409
[fully non-linear evolutions and perturbative approaches, applications to new physics];
Garfinkle RPP(16)-a1606 [applications beyond astrophysics];
Tichy RPP(17)-a1610 [initial-value problem].

@ __Other theories of gravity__: Torsello et al a1904 [bimetric gravity, covariant BSSN formulation].

@ __Other general references__: Hobill & Smarr in(89);
Choptuik et al CQG(92) [spherical, scalar + gravity, 2 codes];
Anninos et al PW(96) [II, black holes];
Alcubierre gq/04-GR17;
Shapiro PTPS(06)gq/05-proc [rev];
Andersson CQG(06)gq [and mathematical relativity];
Babiuc et AppleswithApples CQG(08)-a0709 [standard testbeds];
Sekiguchi CQG(10)-a1009 [taking microphysics into account];
Cardoso et al CQG(12)-a1201 [NR/HEP Workshop summary];
Zilhão a1301-PhD
[extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory].

@ __Computational aspects__: Suen gq/99-rp [and TeraFlop machines];
Löffler et al CQG(12)-a1111,
Zilhão & Löffler IJMPA(13)-a1305-ln [Einstein Toolkit, based on Cactus].

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