c = 1;
\[Omega] = 1;
kz = \[Omega]/c;
vx = 0.01 t1 c; (*for the sake of the animation the velocity grows with time, but we are neglecting any acceleration*)
v = {vx, 0, 0};
dir = FullSimplify[v/Sqrt[v[[1]]^2 + v[[2]]^2 + v[[3]]^2]];
\[Gamma] = 1/Sqrt[1 - Norm[v]^2/c^2];
e = {Cos[kz z - \[Omega] t], Sin[kz z - \[Omega] t], 0}; (*electric field in the "still" frame of reference. In this case a circularly polarized plane wave propagating along z*)
\[CurlyPhi] = 0;
A = FullSimplify[-Integrate[e, t]]; (*Find potentials in the Weyl Gauge*)
(*test for Lorenz gauge!*)Div[A, {x, y, z}] == -1/c^2 D[\[CurlyPhi], t]
lorentzcoord = Simplify[{x1, y1, z1} + (\[Gamma] - 1) {x1, y1, z1}.dir*dir + \[Gamma] t1 v];
\[CurlyPhi]1 = (\[Gamma] (\[CurlyPhi] - v.A) /. {t -> \[Gamma] (t1 + v.{x1, y1, z1}/c^2), x -> lorentzcoord[[1]], y -> lorentzcoord[[2]], z -> lorentzcoord[[3]]});
A1 = (\[Gamma] (A.dir*dir - v/c^2 \[CurlyPhi]) + (A - A.dir*dir)) /. {t -> \[Gamma] (t1 + v.{x1, y1, z1}/c^2), x -> lorentzcoord[[1]], y -> lorentzcoord[[2]], z -> lorentzcoord[[3]]};
e1 = FullSimplify[-Grad[\[CurlyPhi]1, {x1, y1, z1}] - D[A1, t1] , Assumptions -> {t1 \[Element] Reals}] (*Electric field in the "moving" frame of reference*)

p0 = Table[
   Graphics[{Arrowheads[0.02],
     Table[ Arrow[{{x1, y1}, {x1, y1} + 0.8 e1[[1 ;; 2]] /. {t1 -> \[Tau], z1 -> 0}}], {x1, -10, 10, 2}, {y1, -10, 10, 2}],
     Orange, Table[Line[Table[{x1, y1} + 0.8 e1[[1 ;; 2]] /. {t1 -> \[Tau]}, {z1, 0, 10, 0.1}] ], {x1, -10, 10, 2}, {y1, -10, 10, 2}],
     Purple, Thick, Arrow[{{0, -11.5}, {0, -11.5} + 5 v[[1 ;; 2]] /. {t1 -> \[Tau]}}],
     Text[ Style[StringForm["v=``c", NumberForm[Norm[v /. {t1 -> \[Tau]}], {3, 2}]], Purple, Bold], {-2, -11.5}]
     }, PlotRange -> {{-12, 12}, {-12, 12}}]
   , {\[Tau], 0, 50, 0.5}];
ListAnimate[p0]