Defining Mass in General Relativity: Concepts and Obstacles
Generalizing this definition to general relativity, however, is problematic; in fact, it turns out to be impossible to find a general definition for a system's total mass (or energy). The main reason for this is that "gravitational field energy" is not a part of the energy-momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situation it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the Stress-energy-momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.
How, then, does one define a concept as a system's total mass – which is easily defined in classical mechanics? As it turns out, at least for spacetimes which are asymptotically flat (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space), the ADM 3+1 split leads to a solution: as in the usual Hamiltonian formalism, the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the ADM mass (or, equivalently, ADM energy). Alternatively, there is a possibility to define mass for a spacetime that is stationary, in other words, one that has a time-like Killing vector field (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called Komar mass Although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes. The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity. In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity. Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of positivity: if there were no lower limit, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that Minkowski space (for which both are zero) is indeed stable. While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.
The disadvantage of all the definitions mentioned so far is that they are defined only at (null or spatial) infinity; since the 1970s, physicists and mathematicians have worked on the more ambitious endeavor of defining suitable quasi-local quantities, such as the mass of an isolated system defined using only quantities defined within a finite region of space containing that system. However, while there is a variety of proposed definitions such as the Hawking energy, the Geroch energy or Penrose's quasi-local energy-momentum based on twistor methods, the field is still in flux. Eventually, the hope is to use a suitable defined quasi-local mass to give a more precise formulation of the hoop conjecture, prove the so-called Penrose inequality for black holes (relating the black hole's mass to the horizon area) and find a quasi-local version of the laws of black hole mechanics.
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