We present an introduction to mass and angular momentum in General
Relativity. After briefly reviewing energy-momentum for matter fields, first in
the flat Minkowski case (Special Relativity) and then in curved spacetimes with
or without symmetries, we focus on the discussion of energy-momentum for the
gravitational field. We illustrate the difficulties rooted in the Equivalence
Principle for defining a local energy-momentum density for the gravitational
field. This leads to the understanding of gravitational energy-momentum and
angular momentum as non-local observables that make sense, at best, for
extended domains of spacetime. After introducing Komar quantities associated
with spacetime symmetries, it is shown how total energy-momentum can be
unambiguously defined for isolated systems, providing fundamental tests for the
internal consistency of General Relativity as well as setting the conceptual
basis for the understanding of energy loss by gravitational radiation. Finally,
several attempts to formulate quasi-local notions of mass and angular momentum
associated with extended but finite spacetime domains are presented, together
with some illustrations of the relations between total and quasi-local
quantities in the particular context of black hole spacetimes. This article is
not intended to be a rigorous and exhaustive review of the subject, but rather
an invitation to the topic for non-experts. In this sense we follow essentially
the expositions in Szabados 2004, Gourgoulhon 2007, Poisson 2004 and Wald 84,
and refer the reader interested in further developments to the existing
literature, in particular to the excellent and comprehensive review by Szabados
(2004).Comment: 41 pages. Notes based on the lecture given at the C.N.R.S. "School on
Mass" (June 2008) in Orleans, France. To appear as proceedings in the book
"Mass and Motion in General Relativity", eds. L. Blanchet, A. Spallicci and
B. Whiting. Some comments and references added
