In this paper, we suggest that there are two different individual two-dimensional conformal field theories (CFTs) holographically dual to the Kerr-Newman black hole, coming from the corresponding two possible limits-the Kerr/CFT and Reissner-Nordström/CFT correspondences, namely, there exist the Kerr-Newman/CFTs dualities. A probe scalar field at low frequencies turns out can exhibit two different two-dimensional conformal symmetries (named by J and Q pictures, respectively) in its equation of motion when the associated parameters are suitably specified. These twofold dualities are supported by the matchings of entropies, absorption cross sections, and real-time correlators computed from both the gravity and the CFT sides. Our results lead to a fascinating ''microscopic hair conjecture''-for each macroscopic hair parameter, in addition to the mass of a black hole in the Einstein-Maxwell theory, there should exist an associated holographic CFT 2 description.

It is shown that the hidden conformal symmetry, namely SO(2, 2) ∼ SL(2, R) L × SL(2, R) R symmetry, of the non-extremal dyonic Reissner-Nordström black hole can be probed by a charged massless scalar field at low frequencies. The existence of such hidden conformal symmetry suggests that the field theory holographically dual to the 4D Reissner-Nordström black hole indeed should be a 2D CFT. Although the associated AdS 3 structure does not explicitly appear in the near horizon geometry, the primary parameters of the dual CFT 2 can be exactly obtained without the necessity of embedding the 4D Reissner-Nordström black hole into 5D spacetime. The duality is further supported by comparing the absorption cross sections and real-time correlators obtained from both the CFT and the gravity sides.

We have proposed a program for determining the reference for the quasi-local energy defined in the covariant Hamiltonian formalism. Our program has been tested by applying it to the spherically symmetric spacetimes. With respect to different observers we found that the quasi-local energy can be positive, zero, or even negative. The observer measuring the maximum energy was identified; the associated energy is positive for both the Schwarzschild and the Friedmann-Lemaître-RobertsonWalker spacetimes.PACS numbers: 04.20. Cv, 04.20.Fy, 98.80.Jk Introduction. An outstanding fundamental problem in general relativity is that there is no proper definition for the energy density of gravitating systems. (This can be understood as a consequence of the equivalence principle.) The modern concept is that gravitational energy should be non-local, more precisely quasi-local, i.e., it should be associated with a closed two-surface (for a comprehensive review see [1]). Here we consider one proposal based on the covariant Hamiltonian formalism [2] wherein the quasi-local energy is determined by the Hamiltonian boundary term. For a specific spacetime displacement vector field on the boundary of a region (which can be associated with the observer), the associated quasi-local energy depends not only on the dynamical values of the fields on the boundary but also on the choice of reference values for these fields. Thus a principal issue in this formalism is the proper choice of reference spacetime for a given observer.

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.

For a given timelike displacement vector the covariant Hamiltonian quasi-local energy expression requires a proper choice of reference spacetime. We propose a program for determining the reference by embedding a neighborhood of the two-sphere boundary in the dynamic spacetime into a Minkowski reference, so that the two sphere is embedded isometrically, and then extremizing the energy to determine the embedding variables. Applying this idea to Schwarzschild spacetime, we found that for each given future timelike displacement vector our program gives a unique energy value. The static observer measures the maximal energy. Applied to the Friedmann-Lemaître-Robertson-Walker spacetime, we find that the maximum energy value is nonnegative; the associated displacement vector is the unit dual mean curvature vector, and the expansion of the two-sphere boundary matches that of its reference image. For these spherically symmetric cases the reference determined by our program is equivalent to isometrically matching the geometry at the two-sphere boundary and taking the displacement vector to be orthogonal to the spacelike constant coordinate time hypersurface, like the timelike Killing vector of the Minkowski reference.