7.6.2.d. Spin
A plane wave solution to eq(7.119) is
hk  x     eik x
(7.124)
where   is a symmetric polarization tensor of at most
independent components.
1
 4   4  1  10
2
In the harmonic gauge, (7.119) reduces to Wh  0 so
that k satisfies k k   0 . The gauge condition (7.123) itself becomes
1
1


  h    h   i  k   i  k   e  ik x  0
2
2



1
2
  k     k 
(7.125)
which provides 4 constraints on   , thus reducing the number of independent
components of the latter to at most 6.
Setting q  q  0 in (7.122), we see that
further gauge transformations that preserve the harmonic gauge can be constructed
using   (x) that satisfy
W   x   0
Thus, we can write
   x   i  eikx
(7.126)
where k k   0 and   is an arbitrary real constant 4-vector.
Under such a
transformation, (7.121) becomes
h   x   h   x    k    k     e  i k x
(a)
With the help of (7.124) and writing
hk  x     eik x
eq(a) becomes
      k    k  
(7.127)
The 4 arbitrary constants in   further reduce the number of independent components
of   to 2. Since the graviton is massless, these 2 polarizations must be the states
of opposite helicities. The determination of the values of these helocities is
straightforward but tedious. We shall merely quote the result: h  2 . Thus, the
graviton, if exists, is a massless spin 2 particle. As with the case of the spin 1 photon,
helicities states with h  0, 1 correspond to purely gauge degrees of freedom and
have no physical significance.
To date, no graviton has yet been detected experimentally. However, indirect
evidence for the existence of gravitational waves is provided by the peculiar behavior
of a binary pulsar discovered by R.A.Hulse and J.H.Taylor in the 1970s.