Vectors under Galilean Transformations

Vectors under Galilean Transformations PhL 7.1.12 This subject is not really mentioned in tensor doc. I use the phrase "Galilean transformation" incorrectly here but won't bother to fix it. Think of it as just a local name. 1. The Galilean Transformation with Q and b not functions of time. Consider X = (x,t) X' =(x',t) x' = Q(x)x + b // these two lines describe X' = F(X) . t' = t Here Q is an N-dimensional rotation matrix, but our space is N+1 dimensions. How must a "vector" transform with respect to this underlying transformation? Well, we need to learn something about the linearized transformation at some point X. We expect to get some dX' = R dX Assume for the moment that Q is Q(x), non local, but that b is a constant. Then x' = Q(x)x + b dx' = Q(x) dx dt' = dt So the object R is then a matrix of the form R = diag(Q,1). A vector would be something like dX = (dx,dt). Then "velocity" is V = (v,0) and V' = RV. So in this case, the 4-velocity V = (v,0) does indeed transform as a contravariant vector with respect to our underlying transformation shown above. We have R = Q 1 and things are pretty simple. 2. The Galilean Transformation with Q and b functions of time. Now x' = Q(x,t)x + b(t) t' = t dx' = Q(x,t)dx + [∂tQ(x,t)]x dt + [∂tb]dt dt' = dt How would a vector transform in this case? dX = (dx,dt) dX' = (dx',dt') = ( Qdx + [(∂tQ)x + (∂tb)]dt, dt) We seek to find a matrix R such that dX' = R dX ( Qdx + [(∂tQ)x + (∂tb)]dt, dt) = R (dx,dt) = R Let's write this matrix for the case N = 2: = and this we have R = = R(x,t) = So the triplet dX transforms as a vector here, and anything that transforms like the above would be a vector. But what an ugly way for a vector to transform. 3. The Rotation Transformation with Q a function of time. Just set b = 0 in the above and we get = dX' = R(X) dX If you want something to transform as a contravariant vector under this transformation, then it must transform as shown above, which is a mess. Nevertheless, "velocity" would transform as a vector (just divide both sides by dt) = V' = R(X) V but this thing is NOT a rotation matrix, nor is it R = Q 1 . This just doesn't seem very useful to me.