The Size and Pervasiveness of Ly$\alpha$-UV Spatial Offsets in
  Star-Forming Galaxies at $z\sim6$
  We study the projected spatial offset between the ultraviolet continuum and
Ly$\alpha$ emission for 65 lensed and unlensed galaxies in the Epoch of
Reionization ($5\leq z\leq7$), the first such study at these redshifts, in
order to understand the potential for these offsets to confuse estimates of the
Ly$\alpha$ properties of galaxies observed in slit spectroscopy. While we find
that ~40% of galaxies in our sample show significant projected spatial offsets
($|\Delta_{Ly\alpha-UV}|$), we find a modest average offset of 0.61$\pm$0.08
kpc. A small fraction of our sample, ~10%, exhibits offsets of 2-4 kpc, sizes
that are larger than the effective radii of typical galaxies at these
redshifts. An internal comparison and a comparison to studies at lower redshift
yielded no significant evidence of evolution of $|\Delta_{Ly\alpha-UV}|$ with
redshift. In our own sample, UV-bright galaxies showed offsets a factor of
three greater than their fainter counterparts, 0.89$\pm$0.18 vs. 0.27$\pm$0.05
kpc, respectively. We argue that offsets are likely not the result of merging
processes, but are rather due to internal anisotropic processes resulting from
stellar feedback facilitates Ly$\alpha$ fluorescence and/or backscattering from
nearby or outflowing gas. The reduction in the Ly$\alpha$ flux due to offset
effects for various observational setups was quantified through mock
observations of simple simulations. It was found that the loss of Ly$\alpha$
photons for galaxies with average offsets is not, if corrected for, a limiting
factor for all but the narrowest slit widths (<0.4''). However, for the largest
offsets, if such offsets are mostly perpendicular to the slit major axis, slit
losses were found to be extremely severe in cases where slit widths of
$\leq$1'' were employed, such as those planned for James Webb Space
Telescope/NIRSpec observations. (abridged)
