On Super-Planckian Fields at Sub-Planckian Energies

Abstract

For a light scalar coupled to gravity, I study the gravitational backreaction associated with large field variations. I show a generic obstruction in sourcing a super-Planckian scalar profile without making the whole experiment collapse into a black hole. In empty space the scalar variation obeys an absolute bound, of order of the Planck scale. A Newtonian analysis suggests that inside its sources the scalar can undergo arbitrarily large variations without causing large gravitational backreactions. However the maximum attainable \Delta \phi increases only logarithmically with the size of the source. The bound straightforwardly generalizes to any number of dimensions, and to moduli space-like cases, where it applies to the invariant length in field space as measured by the kinetic metric.