  In four dimensions complexified General Relativity (GR) can be non-trivially
deformed: There exists an (infinite-parameter) set of modifications all having
the same count of degrees of freedom. It is trivial to impose reality
conditions that give versions of the deformed theories corresponding to
Riemannian and split metric signatures. We revisit the Lorentzian signature
case. To make the problem tractable, we restrict our attention to a
four-parameter set of deformations that are natural extensions of Ashtekar's
Hamiltonian formalism for GR. The Hamiltonian of the later is a linear
combination of $EEE$ and $EEB$. We consider theories for which the Hamiltonian
constraint is a general linear combination of $EEE, EEB, EBB$ and $BBB$. Our
main result is the computation of the evolution equations for the modified
theories as geometrodynamics evolution equations for the 3-metric. We show that
only for GR (and the related theory of Self-Dual Gravity) these equations close
in the sense that they can be written in terms of only the metric and its first
time derivative. Modified theories are therefore seen to be essentially
non-metric in the sense that their dynamics cannot be reduced to
geometrodynamics. We then show this to be related to the problem with
Lorentzian reality conditions: the conditions of reality of the 3-metric and
its time derivative are not acceptable because they are not preserved by the
dynamics. Put differently, their conservation implies extra reality conditions
on higher-order time derivatives, which then leaves no room for degrees of
freedom.
