BeginPackage["Minkowski`", "Normalize`"]

Minkowski::usage = "\n
   Minkowski[x], where x has head List, is a vector in Minkowski space.\n
   Such vectors can be added and multiplied by scalars.\n
   Dot, Cross and Normalize are suitably modified."

Cross::usage = "\n
  Cross[x_,y_], where x and y are vectors in Minkowski space, gives their\n
  cross product."

Begin["`private`"]

Minkowski[x_List]+Minkowski[y_List] ^:= Minkowski[x+y]
x_*Minkowski[y_List] ^:= Minkowski[x*y] (* here x is meant as a scalar *)

Minkowski[{xs__,lastx_}].Minkowski[{ys__,lasty_}] ^:= {xs}.{ys} - lastx lasty

Normalize::isotropic = "Cannot normalize ``"
Normalize[x_Minkowski] := (Message[Normalize::isotropic,x];Null) /; x.x==0
Normalize[x_Minkowski] := x / Sqrt[Abs[x.x]]

Unprotect[Cross]

Cross[Minkowski[x_List],Minkowski[y_List]] := 
 Minkowski[{1,1,-1} RotateLeft[RotateLeft[y] x - RotateLeft[x] y]]

End[]
EndPackage[]