2.6 The constraint and evolution equations 41
The Hamiltonian constraint (2.90) and the momentum constraint (2.96)arethedirect
equivalent of the constraints (2.5) and (2.6) in electrodynamics. They involve only the
spatial metric, the extrinsic curvature, and their spatial derivatives. They are the conditions
that allow a 3-dimensional slice with data (γ
ab
, K
ab
) to be embedded in a 4-dimensional
manifold M with data (g
ab
). Field data (γ
ab
, K
ab
) that are being imposed on a timeslice
have to satisfy the two constraint equations. We will discuss strategies for solving the
constraint equations and finding initial data that represent a snapshot of the gravitational
fields at a certain instant of time in Chapter 3.
The evolution equations that evolve the data (γ
ab
, K
ab
) forward in time can be found
from (2.53), which can be considered as the definition of the extrinsic curvature, and the
Ricci equation (2.82). However, the Lie derivative along n
a
, L
n
, is not a natural time
derivative since n
a
is not dual to the surface 1-form
a
, i.e., their dot product is not unity
but rather
n
a
a
=−αg
ab
∇
a
t∇
b
t = α
−1
. (2.97)
Instead, consider the vector
t
a
= αn
a
+ β
a
, (2.98)
which is dual to
a
for any spatial shift vector β
a
,
t
a
a
= αn
a
a
+ β
a
a
= 1. (2.99)
It will prove useful to choose t
a
to be the congruence along which we propagate the spatial
coordinate grid from one time slice to the next slice. In other words, t
a
will connect points
with the same spatial coordinates on neighboring time slices. Then the shift vector β
a
will measure the amount by which the spatial coordinates are shifted within a slice with
respect to the normal vector, as illustrated in Figure 2.4. As we have noted before, the lapse
function α measures how much proper time elapses between neighboring time slices along
the normal vector. The lapse and the shift therefore determine how the coordinates evolve
in time. The choice of α and β
a
is quite arbitrary, and we will postpone a discussion of
some common choices to Chapter 4. The freedom to choose these four gauge functions
α and β
a
completely arbitrarily embodies the four-fold coordinate degrees of freedom
inherent in general relativity.
11
Specifically, the lapse function reflects the freedom to
choose the sequence of time slices, pushing them forward by different amounts of proper
time at different spatial points on a slice and thus exploiting “the many-fingered nature
of time”.
12
The shift vector reflects the freedom to relabel spatial coordinates on each
slice in an arbitrary way. Observers who are “at rest” relative to the slices follow the
normal congruence n
a
and are called either normal or Eulerian observers, while observers
11
Recall that β
a
is spatial and therefore subject to the constraint that n
a
β
a
= 0, hence only three of its components may
be freely specified.
12
See, e.g., Misner et al. (1973), p. 527.