Gauge and constraint degrees of freedom: from analytical to numerical approximations in General Relativity

The harmonic formulation of Einstein’s field equations is considered, where the gauge conditions are introduced as dynamical constraints. The difference between the fully constrained approach (used in analytical approximations) and the free evolution one (used in most numerical approximations) is pointed out. As a generalization, quasi-stationary gauge conditions are also discussed, including numerical experiments with the gauge-waves testbed. The complementary 3+1 approach is also considered, where constraints are related instead with energy and momentum first integrals and the gauge must be provided separately. The relationship between the two formalisms is discussed in a more general framework (Z4 formalism). Different strategies in black hole simulations follow when introducing singularity avoidance as a requirement. More flexible quasi-stationary gauge conditions are proposed in this context, which can be seen as generalizations of the current ’freezing shift’ prescriptions. 1. Harmonic formulation Einstein’s field equations Rab = 8 π (Tab − 1 2 T gab), (1) are usually expressed as a system of partial differential equations for the space-time metric gab. Soon after Einstein’s 1915 paper, their mathematical structure was closely investigated, leading to a very convenient formulation (De Donder 1921, 1927), namely 1 2 g∂ cd gab+∂(aHb) = ΓcabH +2 gg [∂egac ∂fgbd−Γace Γbdf ]−8 π (Tab− T 2 gab) , (2) where indices inside round brackets are symmetrized and we have noted H ≡ −g Γabc = 1/ √ g ∂b( √ g g). (3) One can now take advantage of the general covariance of the theory. Let us define spacetime coordinates by a set of four independent harmonic functions, namely x = 1/ √ g ∂b( √ g g) = 0 , (4) where the box stands for the general-covariant wave operator acting on functions. In this harmonic coordinate system, the field equations (2) get the simpler form gab = · · · − 16 π (Tab − T 2 gab) , (5) Gauge and constraint degrees of freedom 2 where the dots stand for terms quadratic in the metric first derivatives. If we look at the principal part (the second derivatives terms), we see just a set of independent wave equations, one for every metric component. The coupling comes only through the right-hand-side terms. System (5) is very convenient in analytical approximations. Let us assume for instance that the metric admits a development of the form gab = ηab + h (1) ab + h (2) ab + · · · , (6) where h is the nth-order perturbation. Then, one can express (5) in a recursive way: η∂ cd h (n+1) ab = Fab(h , ∂h) r, s ≤ n , (7) which can by integrated just by inverting the standard (flat-space) wave operator. System (5), however, is not equivalent to the original field equations (2). The coordinate conditions (4) can be interpreted as first-order constraints to be imposed on the solutions of the ’relaxed’ system (5). The hard point in proving the wellposedness of the Cauchy problem for Einstein’s equations was precisely to prove that the harmonic constraints (4) where actually first integrals of the relaxed system (Choquet-Bruhat 1952). In the analytical perturbation framework, this translates into the fact that fulfilling the harmonic constraints at the nth-level does not imply the same thing at the next level. Obtaining a true solution of the Einstein equations implies adjusting the integration constants in such a way that H a ≡ 0 , (8) and this must be done at every order in the perturbation development. 1.1. Numerical Relativity applications The relaxed system (5) is also very useful in numerical approximations. The usual practice is using explicit time-discretization algorithms. This means that the metric coefficients are computed at a given time slice, assuming that one knows their values at the previous ones. But again fulfilling the harmonic constraints (4) is not granted. Moreover, in numerical approximations one has no adjustable integration constants. This means that numerical errors make the contracted Christoffel symbols obtained from the relaxed system to depart from their assumed harmonic (zero) value: Γ 6= 0 . (9) In this ’free evolution’ approach, one can use the non-zero values (9) to monitor the quality of the simulation. This can be done by introducing a ’zero’ four-vector Z as the difference between the relaxed and the harmonic (zero) contracted Christoffel symbols, namely Γ − Γ = −2Z , (10) so that true Einstein’s solutions would correspond to Z = 0, which amounts to fulfilling the harmonic constraints. On the contrary, allowing for (10), solutions of the relaxed system would verify a generalized version of (2), in which (3) must be replaced by H ≡ −g Γabc − 2Z . (11) Gauge and constraint degrees of freedom 3 The vector Z provides a new degree of freedom which arises quite naturally in numerical simulations. A fully covariant description is provided by the ’Z4 system’ (Bona et al. 2003) Rab + Z a; b + Z b; a = 8π (Tab − T/2 gab) , (12) where the semicolon stands for the covariant derivative. We can easily see that the true Einstein’s solutions are recovered when Z = 0. Moreover, allowing for the contracted Bianchi identities, the stress-energy tensor conservation implies gbcZa;bc +R a b Z b = 0 . (13) It is clear that the vanishing of Z, leading to true Einstein’s solutions, is a first integral of the ’subsidiary system’ (13), which follows from the field equations just assuming the stress-energy tensor conservation. 1.2. Testing coordinate conditions We have seen how harmonic coordinates can be preferred for the sake of simplifying the (approximate) integration of the field equations. But one would like to use instead the General Relativity coordinate freedom for simplifying the physical interpretation of the results. On the contrary, choosing a convoluted coordinate system can complicate even the simplest physical situations. A good example of these unphysical gauge complications is the ’gauge waves’ testbed (Alcubierre et al 2004). The Minkowsy (flat) metric can be written in some non-trivial harmonic coordinate system as: ds = F (x− t)(−dt + dx) + dy + dz , (14) where F is an arbitrary function of its argument. One could naively interpret this as the propagation of an arbitrary wave profile with unit speed. But it is a pure gauge effect, because (14) is nothing but the Minkowsky metric in disguise. A more natural coordinate system should be adapted to the fact that flat spacetime is stationary, and this is not granted by the harmonic condition, as (14) shows dramatically. The problem of finding ’quasi-stationary coordinates’ (as stationary as possible) in a generic spacetime has been addressed recently (Bona et al 2005a). The idea is to find ’almost-Killing’ vector fields ξ by means of a standard variational principle δS = 0 , S ≡ ∫ L √ g dx , (15) where the Lagrangian density L is given by L = ξ(a ; b)ξ (a ; b) − k 2 (ξc; c) 2 (16) (k being an arbitrary parameter), and the variations of the vector field ξ are considered in a fixed spacetime. The resulting Euler-Lagrange equations get the form [ ξ a; b + ξ b; a − k ξc; c g ]; b = 0 . (17) (’almost-Killing’ equation), or the equivalent one gξa;bc +Rab ξ b + (1 − k) ∂a(ξc; c) = 0 . (18) We will consider here the particular ’harmonic’ choice k = 1. Note that, in this case, the subsidiary system (13) is nothing but condition (18) for the Z vector. We can then interpret that the combination Z(a ; b) in the Z4 system (12) gets minimized, so Gauge and constraint degrees of freedom 4