A Field evolution equations

In the BSSN evolution system, we define the following variables in terms of the standard ADM 4-metric g ij, 3-metric γ ij, and extrinsic curvature K ij:

1 ϕ ≡ ---log [det γij] = logψ, (35 ) 12 &tidle;γij ≡ e −4ϕγij, (36 ) K ≡ gijK , (37 ) ij[ ] &tidle; −4ϕ 1- Aij ≡ e Kij − 3gijK, (38 ) &tidle;i jk &tidle;i Γ ≡ γ&tidle; Γ jk. (39 )
The evolution system consists of 15 equations for the various field terms,
( ) ij ij 1 2 ( ij ) ∂0K = − γ &tidle;DiD&tidle;j α + α &tidle;A &tidle;Aij + 3K + 4 π E + γ Sij , (40 ) ∂ ϕ = − 1-(αK − ∂ βk ), (41 ) 0 6 k k 2- k ∂0&tidle;γij = − 2αA&tidle;ij + 2γ&tidle;k(i∂j)β − 3&tidle;γij∂kβ , (42 ) [ ]TF ∂0A&tidle;ij = e−4ϕ αR&tidle;ij + αR ϕij − D&tidle;i &tidle;Dj α + αK A&tidle; − 2α &tidle;A A&tidle;k + 2 &tidle;A ∂ βk − 2-&tidle;A ∂ βk − 8π αe−4ϕST F, (43 ) ij [ ik j k(i j) 3 ij k] ij i ij i kl ij 2 ij ∂0&tidle;Γ = − 2A&tidle; ∂jα + 2α Γ&tidle;klA &tidle; + 6A&tidle; ∂jϕ − -&tidle;γ K,j 3 − &tidle;Γ j∂ βi + 2-&tidle;Γ i∂ βj + 1&tidle;γikβj + &tidle;γjkβi − 16 παγ&tidle;ikS , (44 ) j 3 j 3 ,jk ,jk k
where the matter source terms contain various projections of the stress-energy tensor, defined through the relations
μ ν -1-( i i j ) E ≡ n n Tμν = α2 T00 − 2β T0i + β β Tij , (45 ) 1 ( ) Si ≡ − nμγνi T μν = −-- T0i − βjTij , (46 ) μ ν α Sij ≡ γi γj Tμν. (47 )
We have introduced the notation j ∂0 = ∂t − β ∂j. All quantities with a tilde involve the conformal 3-metric &tidle;γij, which is used to raise and lower indices. In particular, &tidle; Di and &tidle; k Γ ij refer to the covariant derivative and the Christoffel symbols with respect to &tidle;γij. Parentheses indicate symmetrization of indices, and the expression [⋅⋅⋅]TF denotes the trace-free part of the expression inside the brackets. In the BSSN approach, the Ricci tensor is typically split into two pieces, whose respective contributions are given by
R&tidle; = − 1&tidle;γkl∂ ∂ &tidle;γ + &tidle;γ ∂ &tidle;Γ k − ∂ &tidle;γ ∂ &tidle;γkl + 1&tidle;Γ l&tidle;γ − &tidle;Γ l&tidle;Γ k , (48 ) ij 2 k l ij k(ij) k l(ij) 2 ij,l ik jl R ϕ = − 2D&tidle; D&tidle; ϕ − 2&tidle;γ D&tidle;k D&tidle; ϕ + 4D&tidle; ϕD&tidle; ϕ − 4&tidle;γ &tidle;Dk ϕ &tidle;D ϕ. (49 ) ij i j ij k i j ij k

These equations must be supplemented with gauge conditions that determine the evolution of the lapse function α and shift vector i β. Noting that some groups introduce slight variants of these, the moving puncture gauge conditions that have become popular for all GR merger calculations involving BHs and NSs typically take the form

∂0α = − 2 αK, (50 ) 3 ∂tβi = -Bi, (51 ) 4 ∂tBi = ∂tΓ&tidle;i − ηBi, (52 )
where i B is an intermediate quantity used to convert the second-order “Gamma-driver” shift condition into a pair of first-order equations, and η is a user-prescribed term used to control dissipation in the simulation. Note that it is possible to replace the three instances of ∂t in the shift evolution equations 51View Equation and 52View Equation by the shift-advected time derivative (∂t − βj∂j) without changing the stability or hyperbolicity properties of the evolution scheme; in both cases moving punctures translate smoothly across a grid over long time periods [322] and both systems are strongly hyperbolic so long as the shift vector does not grow too large within the simulation domain [129].

The generalized harmonic formulation involves recasting the Einstein field equations, Eq. 9View Equation in the form

( ) R μν = 8π T μν − 1gμνT (53 ) 2
and, after some tensor algebra, rewriting the Ricci tensor in the form
− 2R μν = g γδg αβ,γδ + gγδ,βgαδ,γ + gγ,δαgβδ,γ + 2H (α,β) − 2H δΓ δαβ + 2Γ γδβΓ δγα. (54 )
The Christoffel coefficients are calculated from the full 4-metric,
Γ α ≡ 1gαδ [g + g − g ], (55 ) βγ 2 βδ,γ γδ,β βγ,δ
and the gauge source terms H μ are defined in Eq. 25View Equation.

Given well-posed initial data for the metric and its first time derivative (since the system is second-order in time according to Eq. 54View Equation), the evolution of the system may be treated by a first-order reduction that specifies the evolution of the four functions H μ along with the spacetime metric gμν, its projected time derivatives Π μν = − nα∂αg μν, and its spatial derivatives

Φiμν = ∂ig μν, (56 )
subject to a constraint specifying that the derivative terms Φi μν remain consistent with the metric gμν in time. In practice, one typically introduces a constraint for the source functions, defining
C μ = H μ − □x μ (57 )
and then modifies the evolution equation by appending a constraint damping term to the RHS of the stress energy-term (following [130, 231Jump To The Next Citation Point]
δ − 8π (2T μν − gμνT) ⇒ − 8π(2Tμν − gμνT ) − κ (nμC ν + n νCμ − gμνn C δ), (58 )
where nμ is the unit normal vector to the hypersurface (see Eq. 15View Equation). The gauge conditions used in the first successful simulations of merging BH binaries [231] consisted of the set
α − 1 ν □Ht = − ξ1-α-η--+ ξ2Ht,νn ; Hi = 0, (59 )
with ξ1,ξ2,η constant parameters used to tune the evolution. The first one drives the coordinates toward the ADM form and the latter provides dissipation. The binary NS-NS merger work of [6] chose harmonic coordinates with H μ = 0.
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