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Recent updated results of quasar spectra suggested a 3.9$\sigma$ significanceof spatial variation of the fine-structure constant. Theoretically, it isimportant to examine whether the fine-structure constant, as a fundamentalconstant in quantum theory, is possible varying with space and time. In thispaper, we explore the possibility that spatial variation of the fine-structureconstant could be compatible with Einstein's general relativity. Namely, thespatially dependent fine-structure constant in the Universe could be originatedin different values of the speed of light in separate local frames that are faraway from us, since we have known that light rays must be bending in thepresent of gravity or non-inertial motions. In addition, to learn more aboutthe anisotropic coordinate of FLRW space-time, we also study luminositydistance-redshift relation. It is found that there is a dipole structure inhigh redshift regime, while in low redshift regime, there is not such dipole.

Spatial variation of the fine-structure constant and Hubble's law in anisotropic coordinate of Friedmann-Lemaitre-Robertson-Walker space-time