  This paper explores the use of geometric algebra to study the Galilean
spacetime and its physical implications. The authors introduce the concept of
geometric algebra and its advantages over tensor algebra for describing
physical phenomena. They define the Galilean-spacetime algebra (GSTA) as a
geometric algebra generated by a four-dimensional vector space with a
degenerate metric. They show how the GSTA can be used to represent Galilean
transformations, rotations, translations, and boosts. The authors also derive
the general form of Galilean transformations in the GSTA and show how they
preserve the scalar product and the pseudoscalar. They develop a tensor
formulation of Galilean electromagnetism using the GSTA and show how it reduces
to the usual Maxwell equations in the non-relativistic limit. They introduce
the concept of Galilean spinors as elements of the minimal left ideals of the
GSTA and show how the Galilean spinors can be used to construct the
Levy-Leblond equation for a free electron and its matrix representation. They
provide a suitable matrix representation for the Galilean gamma matrices and
the Galilean pseudoscalar. They relate the GSTA to the four component dual
numbers introduced by Majernik to express Galilean transformations and show how
the dual numbers can be used to develop a Newton-Cartan theory of gravity. The
paper concludes by summarizing the main results and suggesting some possible
applications and extensions of the GSTA.
