  In this paper we explore the correspondence between four-dimensional
Born-Infeld theory and five-dimensional classical gravity. The Born-Infeld
theory side corresponds to the low energy effective theory for open strings
ending on coincident D3-branes in a (slowly varying) background Kalb-Ramond
field, including all higher-derivative corrections. On the gravity side one has
the gravitational (closed string) description of D3-branes in the same
background Kalb-Ramond field and the correspondence is thus a consequence of
the open/closed string duality. According to the correspondence the gravity
side provides a description of the strong coupling limit of Born-Infeld theory.
This is a correspondence between effective theories in a similar sense as in
the fluid/gravity correspondence. We match the Born-Infeld and gravity sides up
to, and including, two-derivative corrections. To this end, we find a new
gravity solution for D3-branes with flat embedding in the background of an
arbitrary constant background Kalb-Ramond field and show that there are no
two-derivative corrections to this for a slowly varying Kalb-Ramond field.
