Consider transforming back from Lorentz-transformed coordinates......
Simplify[Inverse[{{Cosh[p],Sinh[p]},{Sinh[p],Cosh[p]}}]]
%.{x,t}
tran[{x_,t_},p_]={xCosh[p]-tSinh[p],tCosh[p]-xSinh[p]}
Position[FromCAState[CAEvolveList[ElementaryRule[90],CenterList[100,{1}],48]],1];
Show[Surround[Graphics[(Point[tran[Reverse[{-1,1}#1],.2]]&)/@%,AspectRatioAutomatic]]];
Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[150]],{ConstantList[100,0],CenterList[100,{1}]},48]],1];
Show[Surround[Graphics[(Point[tran[Reverse[{-1,1}#1],.2]]&)/@%,AspectRatioAutomatic]]];
Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[150]],{ConstantList[300,0],CenterList[300,{1}]},148]],1];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0]]&)/@%14},AspectRatioAutomatic]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],.1]]&)/@%14},AspectRatioAutomatic]]];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],.25]]&)/@%14},AspectRatioAutomatic]]];
Show[Surround[Graphics[{AbsolutePointSize[.3],(Point[tran[Reverse[{-1,1}#1],.3]]&)/@%14},AspectRatioAutomatic]]];
Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[22]],{ConstantList[300,0],CenterList[300,{1}]},148]],1];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0.25]]&)/@%21},AspectRatioAutomatic]]];
Position[FromCAState[CAEvolveList[Reversible[ElementaryRule[110]],{ConstantList[300,0],CenterList[300,{1}]},148]],1];
Show[Surround[Graphics[{AbsolutePointSize[.5],(Point[tran[Reverse[{-1,1}#1],0.25]]&)/@%23},AspectRatioAutomatic]]];
hyppts=FlattenTableTable+,x,{x,-4,4,.2},{c,0,5},1;
2
c
2
x
Show[Surround[Graphics[(Point[tran[Reverse[{-1,1}#1],.2]]&)/@%,AspectRatioAutomatic]]];
Values are a function only of the t^2-x^2 distance from the origin......
Uniform density clearly satisfies this constraint.....