Consider the following two situations: One can define a stress energy for AdS which matches with the expectation value for the CFT stress tensor. Consider bulk metric perturbations of the form: $$g_{munu} = g^{AdS}_{mu nu} + h_{mu nu}$$ The boundary value of $h_{munu}$ sources the CFT stress tensor. However the gravity Hamiltonian can be written in the following form (see this): $$H = lim_{rho to pi/2} (cosrho)^{2-d}int d^{d-1}Omega dfrac{h_{00}}{16pi G_N}$$ where $rho to pi/2$ denotes the boundary. Here are my questions: Is the gravity Hamiltonian the $00$th component of the AdS stress energy as defined in the paper above? 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?

Relation between bulk Hamiltonian in AdS and stress energy of CFT

Physics Asked by Michael Williams on February 11, 2021

Consider the following two situations:

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

Here are my questions:

Is the gravity Hamiltonian the $00$th component of the AdS stress energy as defined in the paper above?
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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