Energy-momentum Tensor: Noether vs Hilbert

Abstract

We revisit the old problem of the energy-momentum tensor in general relativistic field theories. On the basis of the general covariance we derive a simple equation for the Hilbert and Noether energy-momentum tensors for the scalar and electromagnetic field theories. We see that the two definitions of energy-momentum tensors coincide and identify the Noether current if the space-time has the Killing vector. Relation to the Wald entropy is also briefly discussed.