In the usual 3+1 decomposition of Einstein’s field equations (see, for example, [214], for a through discussion of it) one evolves the three metric and the extrinsic curvature (the first and second fundamental forms) relative to a foliation of spacetime by spacelike hypersurfaces. The motivation for this formulation stems from the Hamiltonian description of general relativity (see, for instance, Appendix E in [429]) where the “” variables are the three metric and the associated canonical momenta (the “” variables) are related to the extrinsic curvature according to
where denotes the determinant of the three-metric and the trace of the extrinsic curvature.

In York’s formulation [444] of the 3+1 decomposed Einstein equations, the evolution equations are

Here, the operator is defined as with and denoting lapse and shift, respectively. It is equal to the Lie derivative along the future-directed unit normal to the time slices when acting on covariant tensor fields orthogonal to . Next, and are the Ricci tensor and covariant derivative operator belonging to the three metric , and and are the energy density and the stress tensor as measured by observers moving along the future-directed unit normal to the time slices. Finally, denotes the trace of the stress tensor. The evolution system (4.20, 4.21) is subject to the Hamiltonian and momentum constraints,
where is the flux density.

#### 4.2.1 Algebraic gauge conditions

One issue with the evolution equations (4.20, 4.21) is the principle part of the Ricci tensor belonging to the three-metric,

which does not define a positive-definite operator. This is due to the fact that the linearized Ricci tensor is invariant with respect to infinitesimal coordinate transformations generated by a vector field . This has the following implications for the evolution equations (4.20, 4.21), assuming for the moment that lapse and shift are fixed, a priori specified functions, in which case the system is equivalent to the second-order system for the three metric. Linearizing and localizing as described in Section 3 one obtains a linear, constant coefficient problem of the form (3.56), which can be brought into first-order form via the reduction in Fourier space described in Section 3.1.5. The resulting first-order system has the form of Eq. (3.58) with the symbol
where is, up to a factor , the principal symbol of the Ricci operator,
Here, , and refer to the frozen lapse, shift and three-metric, respectively. According to Theorem 2, the problem is well posed if and only there is a uniformly positive and bounded symmetrizer such that is symmetric and uniformly positive for . Although is diagonalizable and its eigenvalues are not negative, some of them are zero since for of the form with an arbitrary one-form , so cannot be positive.

These arguments were used in [308] to show that the evolution system (4.20, 4.21) with fixed lapse and shift is weakly but not strongly hyperbolic. The results in [308] also analyze modifications of the equations for which the lapse is densitized and the Hamiltonian constraint is used to modify the trace of Eq. (4.21). The conclusion is that such changes cannot make the evolution equations (4.20, 4.21) strongly hyperbolic. Therefore, these equations, with given shift and densitized lapse, are not suited for numerical evolutions.

#### 4.2.2 Dynamical gauge conditions leading to a well-posed formulation

The results obtained so far often lead to the popular statement “The ADM equations are not strongly hyperbolic.” However, consider the possibility of determining the lapse and shift through evolution equations. A natural choice, motivated by the discussion in Section 4.1, is to impose the harmonic gauge constraint (4.3). Assuming that the background metric is Minkowski in Cartesian coordinates for simplicity, this yields the following equations for the 3+1 decomposed variables,

with a constant, which is equal to one for the harmonic time coordinate . Let us analyze the hyperbolicity of the evolution system (4.27, 4.28, 4.20, 4.21) for the fields , where for generality and later use, we do not necessarily assume in Eq. (4.27). Since this is a mixed first/second-order system, we base our analysis on the first-order pseudodifferential reduction discussed in Section 3.1.5. After linearizing and localizing, we obtain the constant coefficient linear problem
where , and refer to the quantities corresponding to , , of the background metric when frozen at a given point. In order to rewrite this in first-order form, we perform a Fourier transformation in space and introduce the variables with
where and the hatted quantities refer to their Fourier transform. With this, we obtain the first-order system where the symbol has the form with
where , , , and . In order to determine the eigenfields such that is diagonal, we decompose
into pieces parallel and orthogonal to , similar to Example 15. Then, the problem decouples into a tensor sector, involving , into a vector sector, involving and a scalar sector involving . In the tensor sector, we have
which has the eigenvalues with corresponding eigenfields . In the vector sector, we have
which is also diagonalizable with eigenvalues , and corresponding eigenfields and . Finally, in the scalar sector we have
It turns out is diagonalizable with purely imaginary values if and only if and . In this case, the eigenvalues and corresponding eigenfields are , , and , , , respectively. A symmetrizer for , which is smooth in , , and , can be constructed from the eigenfields as in Example 15.

Remarks:

• If instead of imposing the dynamical shift condition (4.28), is a priori specified, then the resulting evolution system, consisting of Eqs. (4.27, 4.20, 4.21), is weakly hyperbolic for any choice of . Indeed, in that case the symbol (4.36) in the vector sector reduces to the Jordan block
which cannot be diagonalized.
• When linearized about Minkowski spacetime, it is possible to classify the characteristic fields into physical, constraint-violating and gauge fields; see [106]. For the system (4.294.32) the physical fields are the ones in the tensor sector, , the constraint-violating ones are and , and the gauge fields are the remaining characteristic variables. Observe that the constraint-violating fields are governed by a strongly-hyperbolic system (see also Section 4.2.4 below), and that in this particular formulation of the ADM equations the gauge fields are coupled to the constraint-violating ones. This coupling is one of the properties that make it possible to cast the system as a strongly hyperbolic one.

We conclude that the evolution system (4.27, 4.28, 4.20, 4.21) is strongly hyperbolic if and only if and . Although the full harmonic gauge condition (4.3) is excluded from these restrictions, there is still a large family of evolution equations for the lapse and shift that give rise to a strongly hyperbolic problem together with the standard evolution equations (4.20, 4.21) from the 3+1 decomposition.

#### 4.2.3 Elliptic gauge conditions leading to a well-posed formulation

Rather than fixing the lapse and shift algebraically or dynamically, an alternative, which has been considered in the literature, is to fix them according to elliptic equations. A natural restriction on the extrinsic geometry of the time slices is to require that their mean curvature, , vanishes or is constant [391]. Taking the trace of Eq. (4.21) and using the Hamiltonian constraint to eliminate the trace of yields the following equation for the lapse,

which is a second-order linear elliptic equation. The operator inside the square parenthesis is formally positive if the strong energy condition, , holds, and so it is invertible when defined on appropriate function spaces. See also [203] for generalizations of this condition. Concerning the shift, one choice, which is motivated by eliminating the “bad” terms in the expression for the Ricci tensor, Eq. (4.24), is the spatial harmonic gauge [25]. In terms of a fixed (possibly time-dependent) background metric on , this gauge is defined as (cf. Eq. (4.3))
where is the Levi-Civita connection with respect to and denote the corresponding Christoffel symbols. The main importance of this gauge is that it permits one to rewrite the Ricci tensor belonging to the three metric in the form
where denotes the covariant derivative with respect to the background metric and where the lower-order terms “l.o.” depend only on and its first derivatives . When the operator on the right-hand side is second-order quasilinear elliptic, and with this, the evolution system (4.20, 4.21) has the form of a nonlinear wave equation for the three-metric . However, the coefficients and source terms in this equation still depend on the lapse and shift. For constant mean curvature slices the lapse satisfies the elliptic scalar equation (4.39), and with the spatial harmonic gauge the shift is determined by the requirement that Eq. (4.40) is preserved throughout evolution, which yields an elliptic vector equation for it. In [25] it was shown that the coupled hyperbolic-elliptic system consisting of the evolution equations (4.20, 4.21) with the Ricci tensor rewritten in elliptic form using the condition , the constant mean curvature condition (4.39), and this elliptic equation for , gives rise to a well-posed Cauchy problem in vacuum. Besides eliminating the “bad” terms in the Ricci tensor, the spatial harmonic gauge also has other nice properties, which were exploited in the well-posed formulation of [25]. For example, the covariant Laplacian of a function is
which does not contain any derivatives of the three metric if . For applications of the hyperbolic-elliptic formulation in [25] to the global existence of expanding vacuum cosmologies; see [26, 27].

Other methods for specifying the shift have been proposed in [391], with the idea of minimizing a functional of the type

where is the strain tensor. Therefore, the functional minimizes time changes in the three metric in an averaged sense. In particular, attains its absolute minimum (zero) if is a Killing vector field. Therefore, one expects the resulting gauge condition to minimize the time dependence of the coordinate components of the three metric. An alternative is to replace the strain by its trace-free part on the right-hand side of Eq. (4.43), giving rise to the minimal distortion gauge. Both conditions yield a second-order elliptic equation for the shift vector, which has unique solutions provided suitable boundary conditions are specified. For generalizations and further results on these type of gauge conditions; see [73, 203, 204]. However, it seems to be currently unknown whether or not these elliptic shift conditions, together with the evolution system (4.20, 4.21) and an appropriate condition on the lapse, lead to a well-posed Cauchy problem.

#### 4.2.4 Constraint propagation

The evolution equations (4.20, 4.21) are equivalent to the components of the Einstein equations corresponding to the spatial part of the Ricci tensor,

and in order to obtain a solution of the full Einstein equations one also needs to solve the constraints and . As in Section 4.2.3, the constraint propagation system can be obtained from the twice contracted Bianchi identities, which, in the 3+1 decomposition, read
The condition of the stress-energy tensor being divergence-free leads to similar evolution equations for and . Therefore, the equations (4.44) lead to the following symmetric hyperbolic system [190, 445] for the constraint variables and ,
As has also been observed in [190], the constraint propagation system associated with the standard ADM equations, where Eq. (4.44) is replaced by its trace-reversed version is
which is only weakly hyperbolic. Therefore, it is much more difficult to control the constraint fields in the standard ADM case than in York’s formulation of the 3+1 equations.