Abstract A typical geometry extracted from the path integral of a quantum theory gravity may be quite complicated in UV region. Even if single configuration is not physical, its properties interest to understand details nature, since some universal features can important for physics model. If formalism describing coordinate independent, which case model studied here, such understanding facilita...

In this Letter, we introduce the nonlinear partial differential equation ${\ensuremath{\partial}}_{\ensuremath{\tau}}^{2}\ensuremath{\pi}\ensuremath{\propto}(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\pi}{)}^{2}$ showing a new type of instability. Such equations appear in effective field theory (EFT) dark energy for $k$-essence model as well many other theories based o...