Energy-momentum of fields
Conservation laws for particles follow from invariance of the action integral to transformations of spacetime coordinates. This chapter shows how conservation laws for fields follow from the same type of reasoning. It establishes Noether's theorem, one of the central tools of field theory. The action integral associated with a set of physical fields is an integral of the Lagrangian density over spacetime. For particles, invariance of the action under infinitesimal Lorentz transformations (LTs) implies conservation of angular momentum along worldlines. The chapter examines angular momentum as a consequence of Noether's theorem. Electromagnetic fields, represented by the field tensor are generated by the four-current through the Maxwell equation. Invariance under spacetime translations leads to energy-momentum conservation, and invariance under LTs leads to conservation of angular momentum. Conservation of energy-momentum is expressed in terms of continuity equations of the rows of the energy-momentum tensor. The tensor must be symmetric to conserve angular momentum.