  Disformal transformation is a generalisation of the well-known conformal
transformation commonly elaborated in mainstream graduate texts in gravity
(relativity) and modern cosmology. This transformation is one of the most
important mathematical operations in scalar tensor theories attempting to
address pressing problems involving dark energy and dark matter. With this
topic yet to penetrate these texts, we present a pedagogically oriented
derivation of the disformal transformation of the Einstein-Hilbert action.
Along the way of calculation, we encounter apparently problematic terms that
could be construed as leading to equations of motion that go beyond second
order in derivatives, signalling instability. We demonstrate that these terms
can be eliminated and absorbed through the definition of the Riemann curvature
tensor. The transformed Einstein-Hilbert action turns out to be a special case
of the Horndeski action and the equations of motion for the scalar field that
it describes are all up to second order only in derivatives, implying
stability.
