Spatial variation of the fine-structure constant and Hubble's law in
  anisotropic coordinate of Friedmann-Lemaitre-Robertson-Walker space-time
  Recent updated results of quasar spectra suggested a 3.9$\sigma$ significance
of spatial variation of the fine-structure constant. Theoretically, it is
important to examine whether the fine-structure constant, as a fundamental
constant in quantum theory, is possible varying with space and time. In this
paper, we explore the possibility that spatial variation of the fine-structure
constant could be compatible with Einstein's general relativity. Namely, the
spatially dependent fine-structure constant in the Universe could be originated
in different values of the speed of light in separate local frames that are far
away from us, since we have known that light rays must be bending in the
present of gravity or non-inertial motions. In addition, to learn more about
the anisotropic coordinate of FLRW space-time, we also study luminosity
distance-redshift relation. It is found that there is a dipole structure in
high redshift regime, while in low redshift regime, there is not such dipole.
