wrote:
In Carroll's text Einestein's equations are obtained from varying the
Hilbert action with respect to the full degrees of freedom of the
metric, i.e. the metric variation is not restricted to preserving
constant signature.

Maintaining constant signature is absolutely essential. But note the
variational technique is a CONTINUOUS technique, and both the manifold
and the metric are continuous as well. For any given metric, there is a
neighborhood in which the signature remains unchanged, and the variation
must occur within that neighborhood -- this is implicit in using such
variational techniques. IOW: the technique is valid only for SMALL
variations, and the definition of "small" must include not changing the
metric signature.


Particularly, if I try to explore what happens if the matter term in
the
action is ignored and vary with the full (inappropriate) degrees of
freedom of
the metric, I obtain solutions that violate the Bianchi identity.

I'm not sure what you mean. If one ignores the matter terms, then the
action is just \integral R sqrt(-g) d^4x, and varying that gives the
(vacuum) Einstein field equation, which clearly satisfies the Bianchi
identities, identically. I suspect you made a mistake.


Imagine a
boundary condition on G_mu_nu such that it is non-zero in some regions
of a
spacelike hypersurface then the equation resulting from this
variation: G_mu_nu=0 outside the boundary would mean the G is not
conserved.

You cannot just dictate such a condition on a boundary. There are
constraint equations which must be satisfied on any Cauchy surface or
boundary.


This violation seems to me more a result of varying with
respect to inappropriate degrees of freedom of metric than of ignoring
the matter term in the action.
Is this correct?

I doubt it. I suspect you tried to use unphysical initial or boundary
conditions (i.e. ones which do not satisfy the relevant constraints).


Tom Roberts