Consider a time like unit four-vector , and the tensor P whose components are given by Pv = v + UUv. (a) Show that P

Consider a time like unit four-vector , and the tensor P whose components are given by
Pμv = ημv + UμUv.
(a) Show that P is a projection operator that projects an arbitrary vector into one orthogonal to , That is, show that the vector Š¥ whose components are

Is
(i) Orthogonal to ,
And
(ii) Unaffected by P:
VaŠ¥Š¥ : = pa(V(Š¥ = VaŠ¥.
(b) Show that for an arbitrary non-null vector , the tensor that projects orthogonally to it has components
ημν ˆ’ qμqν / (qαqα).
How does this fail for null vectors? How does this relate to the definition of P?
(c) Show that P defined above is the metric tensor for vectors perpendicular to :

= PV = (n"8 + U" Up)V = (n"B +UUp)V V