  We construct an action for four-dimensional extended string Newton-Cartan
gravity which is an extension of the string Newton-Cartan gravity that
underlies nonrelativistic string theory. The action can be obtained as a
nonrelativistic limit of the Einstein-Hilbert action in General Relativity
augmented with a term that contains an auxiliary two-form and one-form gauge
field that both have zero flux on-shell. The four-dimensional extended string
Newton-Cartan gravity is based on a central extension of the algebra that
underlies string Newton-Cartan gravity.
  The construction is similar to the earlier construction of a
three-dimensional Chern-Simons action for extended Newton-Cartan gravity, which
is based on a central extension of the algebra that underlies Newton-Cartan
gravity. We show that this three-dimensional action is naturally obtained from
the four-dimensional action by a reduction over the spatial isometry direction
longitudinal to the string followed by a truncation of the extended string
Newton-Cartan gravity fields. Our construction can be seen as a special case of
the construction of an action for extended p-brane Newton-Cartan gravity in p+3
dimensions.
