The Z4 has been introduced initially inBona et al.(2003) to get a fully covari-ant formulation of Einstein equations which also provides a consistent way of treating terms proportional to the constraints in the evolution equations. The starting point is to introduce an additional auxiliary field, the 4-vectorZµ, in the Einstein equations, so that they take the form:
Rµν+ 2∇(µZµ)= 8π(Tµν−1
2T gµν). (4.8)
When the conditionZµ= 0is satisfied, solutions to equations (4.8) are the same as solutions of the original equations (2.10). Note that these equations, as well as the Z4 system that results from them, can also be derived from an action principle (Bona et al.,2010).
Applying the 3+1 split to these equations as described in section2.4, results in the following system of equations:
∂⊥γij =−2αKij (4.9a)
∂⊥Kij =−∇i∇jα+α
Rij−2KikKkj+ (K−2Θ)Kij
(4.9b)
−8πα
Sij−1
2γij(S−E)
∂⊥Θ = α 2
R+ 2∇kZk+ (K−2Θ)K−KijKij
−2
αZk∇kα−16πE
(4.9c)
∂⊥Zi =α
∇j(Kij−δijK) +∂iΘ−2KijZj
−1
αΘ∇iα−8πSi
, (4.9d)
whereΘ = nµZµ = αZ0. The evolution equation for the extrinsic curvature almost identical to the original ADM one (2.33), while the evolution equations for the components of theZµvector replace the Hamiltonian and momentum constraint equations. Note that they are indeed evolution equations, not ellip-tic ones. However by imposing the conditionsZµ = ∂tZµ = 0on the initial slice, these two equations reduce to the usual constraint equations (Bona et al., 2003). This means that initial data for the Z4 system can be constructed as usual by solving the constraint equations on the initial slice, plus the initial valueZi = Θ = 0for the auxiliary fields. Evolution with the system (4.9) will preserve these constraint.
Starting from the Z4 system, a number of other formulations of Einstein equations can be recovered, including the BSSNOK one and the KST one (Kid-der et al.,2001). The hyperbolicity of the system has been studied inBona et al.
(2004) for a family of slicing conditions (including generalized versions of the harmonic and 1+log slicing) and zero shift, and was proven to be strongly hy-perbolic. A first order version of the system, first considered inBona et al.
(2004) and then in more detail inAlic et al.(2009) where it was cast in flux-conservative form, was also found to be strongly hyperbolic.
Despite these encouraging results neither the second order version of the Z4 system nor the first order one have been applied to realistic simulations.
The main drawback has been the inability to identify a suitable replacement of the Gamma driver shift conditions with the same horizon freezing properties:
tests involving BH spacetimes and different shift prescription have resulted in a growth of the horizon radius (seeAlic et al.(2009)).
To address this difficulty, conformal formulations of the Z4 system have been developed, in which the usual gauge conditions can be applied unmodi-fied and used to obtain long-term stable evolutions as for the BSSNOK formu-lation. These systems, namely the Z4c one and the CCZ4 one, are the subject of the next two sections.
4.3.1 The Z4c formulation
The Z4c is a non-covariant, conformal formulation of the Einstein equations based on the second order Z4 one, written in such a way to be as close as possible to BSSNOK, initially developed byBernuzzi and Hilditch(2010). Its derivation starts from the following form of the Einstein equations:
Rµν+ 2∇(µZµ)= 8π(Tµν−1
2T gµν) +κ1
2n(µZν)−(1 +κ2)gµνnτZτ
. (4.10)
In equation (4.10) the additional terms on the right hand side, absent in (4.8), are damping terms which drive to zero the growth of constraint violations (κ1
and κ2 being tunable damping constants). In applying the 3+1 approach to these equations,Bernuzzi and Hilditch(2010) chose to discard non-principal, non-damping terms, which breaks the covariance of the system, but allows to write the final form of the equations in a form which is very similar to the BSSNOK one (in the interest of brevity we do not report the full form of the equations here). The resulting system was proven to be strongly hyperbolic for the usual 1+log and Gamma driver gauge choices.
The Z4c formulation was extensively tested, in standard testbeds scenar-ios as well as in realistic BH and NS simulations, including compact binaries (Bernuzzi and Hilditch,2010;Hilditch et al.,2013). In each case it was shown to be superior to BSSNOK, leading to very similar results, but with much lower levels of constraint violations. In the case of compact binaries, the system also allows for the extraction of more accurate waveforms, reducing both the de-phasing and the amplitude variation in simulations performed at different res-olutions. In a further development constraint-preserving outgoing boundary conditions have been found for the Z4c system (Ruiz et al.,2011), while they are not available for BSSNOK. Due to these desirable properties, Z4c has been employed in a host of different simulations of compact binaries, seee.g. Diet-rich et al.(2015);Bernuzzi and Dietrich(2016);Dietrich et al.(2017a).
However successful, the Z4c system suffers from the drawback of being non-covariant. The CCZ4 system was initially developed precisely to address this issue.
4.3.2 The CCZ4 formulation
The CCZ4 (conformal and covariant Z4) system was developed byAlic et al.
(2012), starting from the Z4 form of the Einstein equations with damping terms, i.e. equations (4.10). Casting the equations in the 3+1 form, one obtains the
system
∂⊥γij =−2αKij (4.11a)
∂⊥Kij =−∇i∇jα+α
Rij−2KikKkj+ (K−2Θ)Kij (4.11b)
−κ1(1 +κ2)Θγij]−8πα
Sij−1
2γij(S−E)
∂⊥Θ = α 2
R+ 2∇kZk+ (K−2Θ)K−KijKij
−2
αZk∇kα−2κ1(2 +κ2)Θ−16πE
(4.11c)
∂⊥Zi =α
∇j(Kij−δijK) +∂iΘ−2KijZj
−1
αΘ∇iα−κ1Zi−8πSi
, (4.11d)
analogous to the (4.9) one, but including the constraint-damping terms. In contrast toBernuzzi and Hilditch(2010),Alic et al.(2012) did not discard any terms in equations (4.11), obtaining a fully covariant formulation.
A conformal transformation is then applied to the evolution variables, in a fashion similar but slightly different from the BSSNOK case (4.5):
φ :=γ−16 (4.12a)
K :=γijKij (4.12b)
˜
γij :=φ2γij (4.12c)
A˜ij:=φ2
Kij−1 3Kγij
(4.12d)
Γ˜i := ˜γjkΓ˜ijk (4.12e)
Γˆi := ˜Γi+ 2˜γijZj (4.12f)
Θ :=αZ0. (4.12g)
The differences from the BSSNOK variables are a different choice for the con-formal factorφ; the definition of a new variableΓˆi which depends on theZi
for vector and it is evolved instead ofΓ˜i; and naturally the inclusion of the Z4 variableΘ.
With this choice of variables, the CCZ4 system can finally be written as: where again the 1+log and Gamma driver conditions have been included as well. In equations (4.13) the Ricci tensorRijis once again computed asRij = R˜φij+ ˜Rij, with the two terms defined as in equations (4.7), as for the BSSNOK system. Note that an arbitrary given functionK0has been added to the slicing condition. This will be exploited in the following sections to enforce some particular behaviour of the lapse in a few test cases. Note also theκ3coefficient which has been added in equation (4.13f). This coefficient is not present in the original system (4.11), and has been added to allow for stable evolutions of BH spacetimes as described in the following.
The CCZ4 system has been thoroughly tested in standard testbeds and BH spacetimes (including BBHs) (Alic et al.,2012), as well as in BNS simulations (Alic et al.,2013). In all of the case considered the system allows for better results than the BSSNOK formulation, both concerning the constraint viola-tions and the GW dephasing; and it compares favourably with the Z4c system.
One unexpected caveat that has emerged in the use of the CCZ4 formulation in BH spacetimes is the presence of non-linear couplings between the various terms, such that in order to obtain stable evolutions with the valueκ16= 0, the coefficientκ3 needs to be set to the value0.5, breaking the covariance of the system (seeAlic et al.(2012)). However in (Alic et al.,2013) a different solution
was found, which enables stable evolutions of BH spacetimes while maintain-ing spatial covariance of the system (and full covariance of its non-dampmaintain-ing terms).
The CCZ4 system has been extensively applied to realistic simulations of compact objects (seee.g. Radice et al.(2014b) and the review article byBaiotti and Rezzolla(2017)). To date however no proof of hyperbolicity of the system (in its second order form) is available in the literature.