on equivalence of definition of the energy-momentum tensor in general relativity and field theory

This work investigates a fundamental pedagogical question concerning the definition of the energy-momentum tensor in theoretical physics. students learn the definition of energy-momentum tensor in the field theory and on the other hand a different definition of energy-momentum tensor in general relativity. why these two definitions are equivalent. we present a proof establishing the equivalence between two seemingly distinct definitions: (1) the energy-momentum tensor is obtained from the einstein-hilbert action in general relativity by varying lagrangian of matter with respect to the metric field, and (2) the canonical energy-momentum tensor derived from noether’s theorem in classical field theory where the lagrangian vary with respect to the field. we use generic coordinate transformation where the fields and metric vary at the same time. using the noether theorem and setting the variation of action to zero, we bridge between these two approaches. our analysis clarifies the conceptual link between geometric and field theory formulations of energy-momentum.