New second derivative theories of gravity for spherically symmetric spacetimes

Abstract

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 $g_{θθ}$ 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 $g_{θθ}$ 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 $g_{θθ}$ 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.