In "Gravitational Dynamics From Entanglement "Thermodynamics"" by Lashkari/McDermott/Van Raamsdonk, the authors derive the linearised Einstein equations from ADS/CFT. At page 6 they use $$t_{\mu{\nu}}(x)=\frac{d}{16{\pi}G_{N}}H_{\mu\nu}\tag{1}$$ where $t_{\mu{\nu}}(x)$ is the stress-energy tensor on the boundary and $H_{\mu\nu}$ comes from the perturbed ADS metric given by $$\mathrm{d}s^2=\frac{1}{z^{2}}(\mathrm{d}z^{2}+\mathrm{d}x_{\mu}\mathrm{d}x^{\mu}+z^{d}H_{\mu\nu}(x,z)\mathrm{d}x^{\mu}\mathrm{d}x^{\nu})$$

In "Stress Tensors and Casimir Energies in the AdS/CFT Correspondence" by Myers, the following procedure to obtain $(1)$ is suggested:

One needs to follow an $\text{ADS}_{d+1}/\text{CFT}_{d}$ prescription: $$\mathrm{e}^{-I_{Ads}[\phi]}=\langle\int{dx^{d}\phi_{0}(x)O(x)}\rangle_\text{CFT}$$ Yet in this case instead of $O(x)$ we consider $t^{\mu\nu}$ and instead of $\phi_{0}(x)$ we consider the graviton field/metric perturbation$h_{\mu\nu}$.

At this point I have a problem. From what I know, the ADS action in this prescription should be evaluated on-shell. In the case of the graviton field we will have $h_{\mu\nu}$ satisfying the linearised Einstein equations in the bulk with some boundary conditions. But we can't use these equations because we are trying to prove them in the article. How to resolve this contradiction?

If there is another way to obtain $(1)$ I will be glad to see it.


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