# Question about the Weiss variational of gravitational action and related equations of motion

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, $$$$K_{\mu \nu}-K \gamma_{\mu \nu}=0, \tag{1}$$$$ where $$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?

## 1 Answer

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.