  Gravitational waves in cylindrically symmetric Einstein gravity are described
by an effective energy tensor with the same form as that of a massless Klein-
Gordon field, in terms of a gravitational potential generalizing the Newtonian
potential. Energy-momentum vectors for the gravitational waves and matter are
defined with respect to a canonical flow of time. The combined energy-momentum
is covariantly conserved, the corresponding charge being the modified Thorne
energy. Energy conservation is formulated as the first law expressing the
gradient of the energy as work and energy-supply terms, including the energy
flux of the gravitational waves. Projecting this equation along a trapping
horizon yields a first law of black-hole dynamics containing the expected term
involving area and surface gravity, where the dynamic surface gravity is
defined with respect to the canonical flow of time. A first law for dynamic
cosmic strings also follows. The Einstein equation is written as three wave
equations plus the first law, each with sources determined by the combined
energy tensor of the matter and gravitational waves.
