In Lattice QCD space-time is approximated by a grid.
To me this doesn't seem to handle relativity well. Due to
(1) A Lorentz transformation of the grid will distort the hyper-cube volumes into parallel-prisms.
(2) Imagine modelling two protons heading towards each other at 99% the speed of light. They could be length contracted so much that their width is smaller than the grid itself.
(3) If a Lorentz transformation changes the cells, but the cell size is related to renormalisation scale, this seems to suggest the effective theory at any scale would not be Lorentz invariant (to a required approximation).
This leads me to think that a Lattice QCD model of a proton could miss out certain relativistic effects if some of the quarks were moving close to the speed of light. (It shouldn't affect the gluons too much as things at light speed aren't affected by Lorentz transformations.)
Hence, it seems like Lattice QCD could only be well approximated if either we assume quarks move at the speed of light or alternatively that they move at non-relativistic speeds.
The problem seems to be that although a Euclidean space can be divided into equal volume cells. Minkowski space-time, such tilings don't make sense due to the signature.
My question is, what (if any) relativistic effects might Lattice QCD not model well. And secondly, are there any potential fixes to this? (Perhaps dynamically changing the size of the cells depending on the speed of the quarks. Such a thing is common in fluid dynamics, for example using a smaller cell size where the fluid gets turbulent.)