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The Non-integrable Mass and the Scalar Charge

The non-integrable mass is studied explicitly in this paper. We study Einstein-scalar gravities with weakened boundary conditions, and calculate the mass with the Hamiltonian formula and Wald's formula respectively. We find the masses calculated by these two formulas are non-integrable. One way to solve this non-integrability problem is to impose boundary conditions; however, we find the mass calculated in this way has many other problems. This implies the macroscopic thermodynamic properties of the scalar hairy black holes should be described by one more charge beside the mass, which we call a scalar charge. In fact, the non-integrability of mass will always arise when the matter fields have charges which is not associate to any diffeomorphisms of spacetime. We find the mass becomes non-integrable just because Wald's formula is used in a wrong way. Based on Wald's formula and the existence of the scalar charge, we propose a new definition for mass, with the modification that we require the variation of the Hamiltonian to have no contribution from the variation of the other charges. This new definition is also valid for much more general gravities coupled to matter fields with other charges.