Equations of motion of the metric in Classical Dilaton Gravity

Question

In these lecture notes by Strominger section 3.3 we derive the equations of motion of the Classical Dilaton Gravity action $$ S = \frac{1}{2\pi}\int{d^2x}{\sqrt{-g}e^{-2\phi}\left(R +4 (\nabla\phi)^2+4\lambda^2\right)}\tag{3.6} $$

The equations of motion for the metric are $$ 2 e^{-2\phi}(\nabla_\mu\nabla_\nu\phi+g_{\mu\nu}((\nabla\phi)^2-\nabla^2\phi-\lambda^2)=0\tag{3.7} $$

Then by introducing light-cone coordinates and gauge fixing the metric we have $$ g_{+-}=-\frac{1}{2}e^{2\rho}\qquad g_{++}=g_{--}=0\tag{3.12} $$

Finally for the $g_{++}$ and $g_{--}$ the equations of motion in terms of the light-cone coordinates give $$ e^{-2\phi}(4\partial_+\rho\partial_+\phi-2\partial_+^2\phi)=0 \\e^{-2\phi}(4\partial_-\rho\partial_-\phi-2\partial_-^2\phi)=0\tag{3.15} $$

I don't understand where the last 2 equations come from. Are they not supposed to be the components $_{++}$ and $_{--}$ of the equation of motion of the metric found before?

Answer 1

To derive (3.15), you need only (3.7). The $++$ component of (3.7) gives the first one in (3.15); while the $--$ component give the second one.

For example, $++$ component gives $2 e^{-2\phi}(\nabla_+\nabla_+\phi)=0$, where I have used gauge condition $g_{++}=0$; on the other side $\nabla_+\nabla_+\phi$ is $\partial_+\partial_+\phi-2\partial_+\rho\partial_+\phi$, where $2\partial_+\rho$ comes from a component of Christoffel symbol; as a result, we have $2 e^{-2\phi}(\partial_+\partial_+\phi-2\partial_+\rho\partial_+\phi)=0$.