# Constraint equation for Einstein-Hilbert action in Light-cone gauge

In Light cone coordinate system $(+,-,i)$, where $i=1,2$, the light cone coordinates are defined as $x^{\pm}=\frac{x^0 \pm x^3}{\sqrt{2}}$, if we consider the $+$ coordinate to be our "timelike" coordinate, constraint equations are those which appear independent of the "timelike derivative", $\partial_+$, in the equations of motion. For the Einstein-Hilbert Lagrangian, the first constraint equation has the form $R_{--}=0$. When the metric, $g_{\mu\nu}$ is parametrized as $g_{+-}=-e^{\phi}$, $g_{ij}=e^{\psi}\gamma_{ij}$, where $\phi, \psi$ are real, and $\gamma_{ij}$ is a real, symmetric matrix; and we have $g_{--}=g_{-i}=0$ as the gauge choice, the constraint equation is supposed to turn out as: $2\partial_-\phi\partial_-\psi-2\partial_-^2\psi-(\partial_-\psi)^2+\frac{1}{2}\partial_-\gamma_{ij}\partial_-\gamma^{ij}=0$.

I have been trying to work this out, but I am getting certain extra terms, of the form $\gamma^{ij}\partial_-\gamma_{ij}$, $\gamma^{ij}\partial_-^2\gamma_{ij}$. Is there any way I can show that these terms vanish, or is there something that I am missing while solving for $R_{--}=0$, such as extra terms, which might cancel these extra terms? The relevant formulae can be obtained from the appendix of this paper. Light cone gravity in AdS_4

• Minor comment to the post (v2): In the future please link to abstract pages rather than pdf files. – Qmechanic Aug 30 '18 at 7:24