28O RELATIVITY, THE GENERAL THEORY
in a coordinate system for which the 'gauge condition'holds true (Eq. 15.14 is sometimes called the Hilbert condition since Hilbert was
the first to prove in general that the coordinate condition Eq. 15.14 can always be
satisfied to the first order in h^ [H5]).
Einstein noted not only that in the weak-field approximation there exist grav-
itational waves which propagate with light velocity but also that only two of the
ten h',,, have independent physical significance, or, as we now say, that there are
only two helicity states. He also pointed out that the existence of radiationless
stable interatomic orbits is equally mysterious from the electromagnetic as from
the gravitational point of view! 'It seems that the quantum theory will have to
modify not only Maxwell's electrodynamics but also the new gravitational theory.'
Perhaps this renewed concern with quantum physics spurred him, a few months
later, to make one of his great contributions to quantum electrodynamics: in the
fall of 1916 he introduced the concepts of spontaneous and induced transitions and
gave a new derivation of Planck's radiation law [E21].
In the same June 1916 paper, Einstein also attempted to calculate the amount
of gravitational radiation emitted by an excited isolated mechanical system with
linear dimensions R. He introduced two further approximations: (1) only wave-
lengths A for which X/R » 1 are considered and (2) all internal velocities of
the mechanical system are « c. At that time he mistakenly believed that a
permanently spherically symmetric mechanical system can emit gravitational
radiation. There the matter lay until he corrected this error in 1918 and presented
the quadrupole formula [E22]: the energy loss of the mechanical system is given
by*
where
is the mass quadrupole moment and p the mass density of the source.
After 1918 Einstein returned one more time to gravitational waves. In 1937 he
and Rosen studied cylindrical wave solutions of the exact gravitational equations
[E23], which were analyzed further in [W15].
*Einstein's result was off by a factor of 2. This factor is corrected in Eq. 15.15, which has also been
written in modernized form. Dots denote time derivatives. Equation 15.15 represents, of course, the
leading term in a gravitational multipole expansion. For a review of this expansion, see [T4].