  Various properties of the Misner-Sharp spherically symmetric gravitational
energy E are established or reviewed. In the Newtonian limit of a perfect
fluid, E yields the Newtonian mass to leading order and the Newtonian kinetic
and potential energy to the next order. For test particles, the corresponding
Hajicek energy is conserved and has the behaviour appropriate to energy in the
Newtonian and special-relativistic limits. In the small-sphere limit, the
leading term in E is the product of volume and the energy density of the
matter. In vacuo, E reduces to the Schwarzschild energy. At null and spatial
infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies
respectively. The conserved Kodama current has charge E. A sphere is trapped if
E>r/2, marginal if E=r/2 and untrapped if E<r/2, where r is the areal radius. A
central singularity is spatial and trapped if E>0, and temporal and untrapped
if E<0. On an untrapped sphere, E is non-decreasing in any outgoing spatial or
null direction, assuming the dominant energy condition. It follows that E>=0 on
an untrapped spatial hypersurface with regular centre, and E>=r_0/2 on an
untrapped spatial hypersurface bounded at the inward end by a marginal sphere
of radius r_0. All these inequalities extend to the asymptotic energies,
recovering the Bondi-Sachs energy loss and the positivity of the asymptotic
energies, as well as proving the conjectured Penrose inequality for black or
white holes. Implications for the cosmic censorship hypothesis and for general
definitions of gravitational energy are discussed.
