States of Negative Energy and AdS₅ times S₅/Z_k

We develop a careful definition of energy for nonsupersymmetric warped product asymptotically $AdS_d \times M_q$ solutions which include a nonzero p-form. In the case of an electric p-form extending along all the AdS directions, and in particular in the case of self-dual fields like those used in the Freund-Rubin construction, the Hamiltonian is well defined only if a particular asymptotic gauge for the p-form is used. Rather surprisingly, asymptotically this gauge is time dependent, despite the fact the field and metric are not. We then consider a freely orbifolded $AdS_5 \times S_5$ and demonstrate that the standard boundary conditions allow states of arbitrarily negative energy. The states consist of time symmetric initial data describing bubbles that are regular up to singularities due to smeared D3-branes. We discuss the evolution of this data and point out that if the usual boundary conditions are enforced such bubbles may never reach infinity.