The
Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum
P. Conservation of four-momentum is given by the
continuity equation: \partial_\nu T^{\mu\nu} = 0\,, where T^{\mu\nu} \, is the
stress–energy tensor, and ∂ are
partial derivatives that make up the
four-gradient (in non-Cartesian coordinates this must be replaced by the
covariant derivative). Integrating over space: \int d^3 x T^{\mu 0}\left(\vec{x}, t\right) = P^\mu gives the
four-momentum vector at time
t. The Noether current for a rotation about the point
y is given by a tensor of 3rd order, denoted M^{\alpha\beta\mu}_y. Because of the
Lie algebra relations M^{\alpha\beta\mu}_y(x) = M^{\alpha\beta\mu}_0(x) + y^\alpha T^{\beta\mu}(x) - y^\beta T^{\alpha\mu}(x)\,, where the 0 subscript indicates the
origin (unlike momentum, angular momentum depends on the origin), the integral: \int d^3 x M^{\mu\nu}_0(\vec{x},t) gives the
angular momentum tensor M^{\mu\nu} \, at time
t. ==Definition==