PHZ 6607 Fall 2006
Homework #3, Due Friday, September 22
1. For any timelike vector Uµthere is a frame in which Uhas only a t-component, and in
this frame Uis invariant under rotations, a three-dimensional subset of all possible Lorentz
transformations. A null vector Vµwith components Vt=Vz=1,Vx=Vy=0alsohasa
three-dimensional subset of Lorentz transformations, called the “little group of V,” that leave
the components of Vunchanged. A pure rotation in the x-yplane is such a transformation.
Find a transformation that is not a pure rotation but is in the little group of V.
2. A Lorentz transformation is the product of a boost with rapidity ζin direction ˆ
n
1,
followed by a boost with rapidity ζin direction ˆ
n
2, followed by a boost with rapidity ζin
direction ˆ
n
3,whereζis the same in each case, and the three directions ˆ
n
1,ˆ
n
2,and ˆ
n
3lie
in the same plane separated by 120◦. What is the resulting transformation? To lowest order
for small ζ, is it a boost or a rotation? At what order does the other (boost or rotation)
enter?
3.(a)Observer
O
at “rest” sees a symmetric tensor Tµν to be diagonal with components
(ρ, p, p, p). What are the components of Tµν ?
(b)Frame
O
0moves with speed vin the +xdirection with respect to
O
. What are the
components of Tµ0ν0in frame
O
0? What are the components of Tµ0ν0? How can the “rest
frame” be identified? Suppose that p=
ρin the original frame
O
;whatisTµ0ν0then?
Make an insightful observation.
4. Tαβ···ρis a tensor in an n-dimensional space.
(a)IfThas rank r(rindices) and no symmetries, how many independent components does
it have?
(b)IfTis antisymmetric in aindices, how many independent components does it have?
(c)IfTis symmetric in sindices, how many independent components does it have?
5. Let Aµν be an antisymmetric tensor, Aµν =A[µν ],andletSµν be a symmetric tensor,
Sµν =S(µν). Show that Aµν Sµν =0.