  We present new second derivative, generally covariant theories of gravity for
spherically symmetric spacetimes (general covariance is in the $t-r$ plane)
belonging to the class where the spherically symmetric Einstein-Hilbert theory
is modified by the presence of $g_{\theta\theta}$ dependent functions. In $3+1$
dimensional vacuum spacetimes there is three-fold infinity of freedom in
constructing such theories as revealed by the presence of three arbitrary
$g_{\theta\theta}$ dependent functions in the Hamiltonian (matter Hamiltonian
also has the corresponding freedom). This result is not a contradiction to the
theorem of Hojman et. al. [1] which is applicable to the full theory whereas
the above conclusion is for symmetry reduced sector of the theory (which has a
much reduced phase space). In the full theory where there are no special
symmetries, the result of Hojman et. al. will continue to hold. In the process
we also show that theories where the constraint algebra is deformed by the
presence of $g_{\theta\theta}$ dependent functions - as is the case in the
presence of inverse triad corrections in loop quantum gravity - can always be
brought to the form where they obey the standard (undeformed) constraint
algebra by performing a suitable canonical transformation. We prove that
theories obtained after performing canonical transformation are inequivalent to
the symmetry reduced general relativity and that the resulting theories fall
within the purview of the theories mentioned above.

General Relativity and Quantum Cosmology;High Energy Physics - Theory