Consider the following two situations:

  1. One can define a stress energy for AdS which matches with the expectation value for the CFT stress tensor.

  2. Consider bulk metric perturbations of the form: $$g_{\mu\nu} = g^{AdS}_{\mu \nu} + h_{\mu \nu}$$ The boundary value of $h_{\mu\nu}$ sources the CFT stress tensor. However the gravity Hamiltonian can be written in the following form (see this): $$H = \lim_{\rho \to \pi/2} (cos\rho)^{2-d}\int d^{d-1}\Omega \dfrac{h_{00}}{16\pi G_N}$$ where $\rho \to \pi/2$ denotes the boundary.

Here are my questions:

  1. Is the gravity Hamiltonian the $00$th component of the AdS stress energy as defined in the paper above?
  2. If that is so, isn't it contradictory that $00$th component of the AdS stress energy sources the CFT stress energy, resulting in the matching of the AdS stress energy with the expectation value of the CFT stress energy?

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