Addition of deformations: numerical tests

On the basis of the discussion in the previous section we *define*
normal deformation as those that are orthogonal to all special deformations,
in the sense of Eq. (4.5). Obviously there are `good' normal
deformations for which the WNA is an excellent approximation
(P1 and W8 in Fig. 3.4, for example), and there are
`bad' normal deformations for which the WNA
is not a very good approximation (FR in Fig. 3.4,
and the normal component in Fig. 4.8b).
In this section we present numerical evidence that verifies
the theoretical results of the previous section,
and investigate how `bad' a normal deformation has to be for them
to break down.

From what we have claimed it follows that if
and
are orthogonal
normal deformations, then
.
We could as well write

because by assumption the three correlation functions are approximately flat. We demonstrate this addition rule in the case of two `good' deformations which are orthogonal in Fig. 4.2. We found that small `pistons' (P2 is significant on only of the perimeter) were needed to achieve addition of the accuracy (a few %) shown. However, the restriction on the `wiggle' type of deformation was somewhat more lenient (WG is times wider than P2 yet obeys the WNA better than P2 does).

In general we observe that the quality of the addition
rule is limited by the
deviation from the WNA of the *better* of the two deformations.
In Fig. 4.3 we see that
if both
and
are bad, then also
the addition rule (4.8) becomes quite bad.
Fig. 4.4 shows that the addition rule (4.8)
is reasonably well satisfied also
if *either*
or
is a `good' normal deformation.
We have chosen
as WG (good),
and
as SX which is almost completely
dominated by the special x-translation deformation.
The addition rule (4.8) is obeyed at all .
This proves that our assertions Eq.(4.4) about the vanishing
of
is indeed correct.
It holds here as a non-trivial statement
(
is general and `bad').

Finally, we consider the case where
is
general and
is special.
This is
illustrated in Fig. 4.5.
The addition rule (4.8) becomes exact in the limit
of small frequency corresponding to the vanishing of
as implied by
Eq.(4.3). In particular this implies that
.
Note that there is *no* condition on the orthogonality of
and
).
This will be the key to for improving over the WNA, which
we are going to discuss in the next section.