4.2 The ADM formulation
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 (4.29 – 4.32) 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.