  I introduce a covariant four-vector $\mathcal{G}^a[v]$, which can be
interpreted as the momentum density attributed to the spacetime geometry by an
observer with velocity $v^a$, and describe its properties: (a) Demanding that
the total momentum of matter plus geometry is conserved for all observers,
leads to the gravitational field equations. Thus, how matter curves spacetime
is entirely determined by this principle of momentum conservation. (b) The
$\mathcal{G}^a[v]$ can be related to the gravitational Lagrangian in a manner
similar to the usual definition of Hamiltonian in, say, classical mechanics.
(c) Geodesic observers in a spacetime will find that the conserved total
momentum vanishes on-shell. (d) The on-shell, conserved, total energy in a
region of space, as measured by the comoving observers, will be equal to the
total heat energy of the boundary surface. (e) The off-shell gravitational
energy in a region will be the sum of the ADM energy in the bulk plus the
thermal energy of the boundary. These results suggest that $\mathcal{G}^a[v]$
can be a useful physical quantity to probe the gravitational theories.
