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I was reading The Weiss Variation of the Gravitational Action by Feng and Matzner, where the authors take the variations of the gravitational action with respect to the bulk metric $g$, the induced metric $\gamma$, and the displacement $\delta x$ of the boundary. It seems that to do so one requires that $\delta \gamma_{\mu \nu} |_{\partial \mathcal{M}} \neq 0$. I have a hard time understanding the implications of such variations on the equations of motion of the theory (equation $(4.34)$). In particular, while the bulk variation gives the well known Einstein Field Equations in the vacuum, the variation with respect to the induced metric seems to give another equation, \begin{equation} K_{\mu \nu}-K \gamma_{\mu \nu}=0, \tag{1} \end{equation} where $K_{\mu \nu}$ is the extrinsic curvature and $K$ its trace. Does this equation determine the induced metric, just like EFE determines the bulk metric? If so, why isn't the induced metric just $g_{\mu \nu}-\epsilon n_{\mu} n_\nu$? Maybe $(1)$ determines $n_\mu$, but then aren't $(1)$ and the equation of motion associated with the variation with respect to $\delta x$ redundant?

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After sending an email to J. Feng, the principal author of the article I linked, it simply appeared that I've misunderstood the purpose of doing such variational. Indeed one doesn't want to find a specific equation of motion but to identify conjugated quantities to the variationals. So the LHS of equation (1) is simply the generalized momentum associated with the induced metric, and for $\delta x$ one identifies the canonical stress-energy-momentum tensor.

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