lorentz transformations

• January 23rd 2008, 03:11 PM
ppyvabw
lorentz transformations
in coordinates $(t,x_1,x_2,x_3)$ the infinitesimal isometries of minkowski space are 3 boosts and 3 rotations (and 4 displacements).

I.e

$(t',x'_1,x'_2,x'_3)^\mu=(t,x_1,x_2,x_3)^\mu+\epsil on\xi^\mu$

where $\xi$ are killing vectors given by $(t,x_1,0,0)$, $(x_2,0,t,0)$, $(x_3,0,0,t)$, $(0,-x_2,x_1,0)$, $(0,-x_3,0,x_1)$ and $(0,0,-x_3,x_2)$

I am probably being really thick, but what are the killing vectors that preserve the surfaces of $t-x_3=constant$ I am totally confused, or atleast how do I work them out? I expect there should be three such vectors, and are linear combinations of the above vectors.

If these where constant t surfaces, then this would be easy, it's just 3 rotations, but i need to know $t-x_3=constant$, where the constant could be zero I suppose if that makes it easier.