Can a tensorial analogue of gravitational force explain away the
  galactic rotation curves without dark matter?
  The dark matter problem is one of the most pressing problems in modern
physics. As there is no well-established claim from a direct detection
experiment supporting the existence of the illusive dark matter that has been
postulated to explain the flat rotation curves of galaxies, and since the whole
issue of an alternative theory of gravity remains controversial, it may be
worth to reconsider the familiar ground of general relativity (GR) itself for a
possible way out.
  It has recently been discovered that a skew-symmetric rank-three tensor field
- the Lanczos tensor field - that generates the Weyl tensor differentially,
provides a proper relativistic analogue of the Newtonian gravitational force.
By taking account of its conformal invariance, the Lanczos tensor leads to a
modified acceleration law which can explain, within the framework of GR itself,
the flat rotation curves of galaxies without the need for any dark matter
whatsoever.
