Chapter 3 Constructing initial data 55
at any time depends on their past history. Constraint equations, on the other hand, tend to
be elliptic in nature and constrain the fields in space at one instant of time, independently of
their past history. It is therefore natural that the constraint equations serve to constrain only
the “longitudinal” parts of the fields, while the “transverse” parts, related to the dynamical
degrees of freedom, remain freely specifiable.
Ideally one would like to separate unambiguously the longitudinal from the transverse
parts of the fields at some initial time, freely specifying the latter and then solving the
constraints for the former. Given the nonlinear nature of general relativity such a rigorous
separation is not possible; instead, all these fields are entangled in the spatial metric and
the extrinsic curvature. We can nevertheless introduce decompositions of γ
ij
and K
ij
that allow for a convenient split of constrained from freely specifiable variables. These
decompositions often amount to an approximate split of transverse from longitudinal
pieces of the field and, in any case, serve to simplify the solution of the resulting constraint
equations.
Typically, then, the solution of Einstein’s initial value equations proceeds along the
following lines. We first decide which field variables we want to determine by solving
constraint equations. This amounts to choosing a particular decomposition of the constraint
equations. We then have to make choices for the remaining, freely specifiable variables.
These choices should reflect the physical or astrophysical situation at hand, but may also be
guided by any resulting simplifications that they induce in the constraint equations. Lastly,
we must solve these equations for the constrained field variables.
We point out that this situation is similar to what we encounter in electrodynamics. As
we have seen in Chapter 2.2, Maxwell’s equations also split into constraint and evolution
equations. The constraint equations (2.5) and (2.6)havetobesatisfiedbyanyelectricand
magnetic field at each instant of time, but they are not sufficient to completely determine
these fields. Consider the equation for the electric field E
i
,
D
i
E
i
= 4πρ. (3.3)
Given an electrical charge density ρ, we can solve this equation for one of the com-
ponents of E
i
, but not all three of them. For example, we could make certain choices
for E
x
and E
y
, and then solve (3.3)forE
z
, even though we might be troubled by the
asymmetry in singling out one particular component in this approach. Alternatively, we
may prefer to write E
i
as some “background” field
¯
E
i
times some overall scaling factor,
say ψ
4
,
E
i
= ψ
4
¯
E
i
. (3.4)
We could now insert (3.4)into(3.3), make certain choices for all three components
of the background field
¯
E
i
, and then solve (3.3) for the scaling factor ψ
4
. Though it
might not be so useful for treating Maxwell’s equations, such an approach leads to a very
convenient and tractable system for Einstein’s equations, as we shall discuss in the following
sections.