  A density-dependent conformal killing vector (CKV) field is attained from a
conformally transformed action composed of a unique constraint and a
Klein-Gordon field. The CKV is re-expressed into an information identity and
studied in its integro-differential form for both null and time-like geodesics.
It is conjectured that the identity corresponds to a generalized second law of
thermodynamics which holographically relates the covariant entropy contained
within a volumetric $n$- and $(n-1)$-form, starting from an $(n-2)$-spatial
area. The time-like geodesics inherit an effective `geometric spin' while the
null geodesics are suggested to obey the generalized covariant entropy bound so
long as they conform to Einstein's equation of state. To then comply with the
equation of state, a metriplectic system is introduced, whereby a newly defined
energy functional is derived for the entropy. Such an entropy functional
mediates the Casimir invariants of the Hamiltonian and therefore preserves the
symplectic form of quantum mechanics. For null geodesics, the Poisson bracket
of the entropy functional with the Hamiltonian is shown to elegantly result in
Einstein's energy-mass relation.
