= Exponentiated dilaton wave equation
{title2=$(\Box_g+2C)e^{-2\varphi}=0$}
For the <dilaton equation from contracted Bianchi identity>, put $F=e^{-2\varphi}$. The chain rule gives $\Box_gF=F(-2\Box_g\varphi+4|\nabla\varphi|^2)=-2CF$. Thus the exponential obeys a linear <Klein-Gordon equation> on the given background, although the original <dilaton> equation is nonlinear and the <string-frame metric> remains coupled to the <dilaton>. With the opposite named-field convention $\varphi=-\Phi$, this variable is $F=e^{2\Phi}$.
Back to article page
