  Nonrelativistic open string theory is defined by a worldsheet theory that
produces a Galilean invariant string spectrum and is described at low energies
by a nonrelativistic Yang-Mills theory. We study T-duality transformations in
the path integral for the sigma model that describes nonrelativistic open
string theory coupled to an arbitrary closed string background, described by a
string Newton-Cartan geometry, Kalb-Ramond, and dilaton field. We prove that
T-duality transformations map nonrelativistic open string theory to
relativistic and noncommutative open string theory in the discrete light cone
quantization (DLCQ), a quantization scheme relevant for Matrix string theory.
We also show how the worldvolume dynamics of nonrelativistic open string theory
described by the Dirac-Born-Infeld type action maps to the Dirac-Born-Infeld
actions describing the worldvolume theories of the DLCQ of open string theory
and noncommutative open string theory.
