Needs["xAct`xPand`"]

DefManifold[M,4,{α,β,γ,μ,ν,λ,σ}]
DefMetric[-1,g[-α,-β],CD,{";","(OverBar[∇])"},PrintAs->"(OverBar[g])"];

SetSlicing[g,u,h,cd,{"|","(OverBar[D])"},"FLFlat"]

DefMetricFields[g,dg,h]
DefMatterFields[um,dum,h]

order=2;
VisualizeTensor[dg[LI[1],-μ,-ν]/.SplitMetric[g,dg,h,"NewtonGauge"],h]
VisualizeTensor[dg[LI[2],-μ,-ν]/.SplitMetric[g,dg,h,"NewtonGauge"],h]

MyToxPand[expr_,gauge_,order_]:=ToxPand[expr,dg,um,dum,h, gauge,order]
FourVelSplit=MyToxPand[u[ν],"NewtonGauge",order] 
FourVelMatter=MyToxPand[um[ν],"NewtonGauge",order]