A Basic Field Equations

In the BSSN-puncture formalism, the typical variables to be evolved are

&tidle;γij = det(γij)−1∕3γij, (131 ) −1∕6 −1∕3 W = det (γij) ((orχ = det (γij)) ), (132 ) &tidle; −1∕3 1- Aij = det(γij) Kij − 3γijK , (133 ) K = tr(K ), (134 ) i ijij &tidle;Γ = − ∂j&tidle;γ . (135 )
The typical basic equations are
l &tidle; k k 2- k (∂t − β ∂l)&tidle;γij = − 2 αAij + &tidle;γikβ ,j + &tidle;γjkβ ,i − 3&tidle;γijβ ,k, (136 ) [ ( 1 ) ( 1 ) ] (∂t − βl∂l)A&tidle;ij = W 2 α Rij − -γijRkk − DiDj α − -γijΔ α 3 3 k k k 2-k + α (K &tidle;Aij − 2 &tidle;Aik &tidle;Aj ) + β ,iA&tidle;kj + β ,jA &tidle;ki − 3β ,kA&tidle;ij ( 1 ) − 8π αW 2 Sij − -γijSkk , (137 ) ( 3) [ ( )] l W-- k l 2χ- k (∂t − β ∂l)W = 3 αK − β ,k or (∂t − β ∂l)χ = 3 αK − β ,k , (138 ) [ 1 ] (∂t − βl∂l)K = α &tidle;Aij &tidle;Aij +-K2 − Δ α + 4π α(ρH + Skk), (139 ) 3 [ ] l &tidle;i &tidle;ij &tidle;i &tidle;jk 2- ij ik W,j- &tidle;ij (∂t − β ∂l)Γ = − 2A ∂jα + 2α Γ jkA − 3&tidle;γ K,j − 8πγ&tidle; jk − 3 W A (140 ) − &tidle;Γ j∂jβi + 2-&tidle;Γ i∂jβj + 1&tidle;γikβj + &tidle;γjkβi,jk, (141 ) 3 3 ,jk
where &tidle;Γ ijk is the Christoffel symbol associated with &tidle;γij, and
ρH = Tμνn μnν, (142 ) μ ν ji = − Tμνn γi, (143 ) Sij = T μνγμiγνj. (144 )
The gauges in the puncture formulation are
(∂ − βi∂ )α = − 2αK, (145 ) t i (∂ − βj∂ )βi = 3-Bi, (146 ) t j 4 (∂t − βj∂j)Bi = (∂t − βj∂j)&tidle;Γ i − ηBBi, (147 )
where Bi is an auxiliary function and ηB is a constant chosen to be of order 1∕m0 with m0 being the total mass of the system.

A point in the BSSN formalism is to rewrite the Ricci tensor, e.g., as

W Rij = &tidle;Rij + Rij , (148 )
where &tidle;R ij is the Ricci tensor with respect to &tidle;γ ij and
1 ( 1 2 ) RWij = --D&tidle;i &tidle;DjW + &tidle;γij ---&tidle;Dk &tidle;DkW − --2D&tidle;kW &tidle;DkW , (149 ) W W W
Here, D&tidle;i is the covariant derivative with respect to &tidle;γij. Then, &tidle;Rij is written using &tidle;Γ i, e.g., as
( ) &tidle;Rij = − 1&tidle;γkl&tidle;γij,kl + 1- &tidle;γki∂j&tidle;Γ k + &tidle;γkj∂i&tidle;Γ k (150 ) 2( 2 ) 1- kl kl &tidle;l &tidle;l &tidle;k − 2 &tidle;γil,k&tidle;γ ,j + γ&tidle;jl,k&tidle;γ ,i − Γ &tidle;γij,l − ΓikΓjl. (151 )
With this prescription, R&tidle;ij ∼ − Δ &tidle;γij∕2 in the weak field limit, and thus, Equations (136View Equation) and (137View Equation essentially constitute a Klein–Gordon-type wave equations for &tidle;γij.

In the generalized harmonic GH formalism, the typical variables to be evolved are spacetime metric gμν, and

Qμν = − nα ∂αgμν, (152 ) Diμν = ∂igμν. (153 )
The evolution equations in the GH formalism are written in first-order form. In contrast to the BSSN formalism, there are several options for the basic equations, in particular, in the choice of the constraint damping terms. The details of the formalism employed in the CCCW and LBPLI groups are described in [129] and [6Jump To The Next Citation Point], respectively.

The GH gauge condition is explicitly written as

(∂ − βk∂ )α = − α (H − βkH + αK ), (154 ) t k t k (∂t − βk∂k)βi = α γij(αHj + αγklΓ j,kl − ∂jαK ), (155 )
where j Γkl is the Christoffel symbol associated with gμν. A suitable choice for H μ is still under development, e.g., the CCCW group employs a rather phenomenological function [58], and LBPLI employs H μ = 0 [6].


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