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Constructing a well-posed variational principle is a non-trivial issue ingeneral relativity. For spacelike and timelike boundaries, one knows that theaddition of the Gibbons-Hawking-York (GHY) counter-term will make thevariational principle well-defined. This result, however, does not directlygeneralize to null boundaries on which the 3-metric becomes degenerate. In thiswork, we address the following question: What is the counter-term that may beadded on a null boundary to make the variational principle well-defined? Wepropose the boundary integral of $2 \sqrt{-g} \left( \Theta+\kappa \right)$ asan appropriate counter-term for a null boundary. We also conduct a preliminaryanalysis of the variations of the metric on the null boundary and conclude thatisolating the degrees of freedom that may be fixed for a well-posed variationalprinciple requires a deeper investigation.

A Boundary Term for the Gravitational Action with Null Boundaries