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The harmonic formulation of Einstein's field equations is considered, wherethe gauge conditions are introduced as dynamical constraints. The differencebetween the fully constrained approach (used in analytical approximations) andthe 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 complementary3+1 approach is also considered, where constraints are related instead withenergy and momentum first integrals and the gauge must be provided separately.The relationship between the two formalisms is discussed in a more generalframework (Z4 formalism). Different strategies in black hole simulations followwhen introducing singularity avoidance as a requirement. More flexiblequasi-stationary gauge conditions are proposed in this context, which can beseen as generalizations of the current 'freezing shift' prescriptions.

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