14.1 On the Bartnik mass and Hawking energy

Although in the literature the notions mass and energy are used almost synonymously, in the present review we have made a distinction between them. By energy we mean the time component of the energy-momentum four-vector, i.e., a reference-frame–dependent quantity, while by mass we mean the length of the energy-momentum, i.e., an invariant. In fact, these two have different properties. The quasi-local energy (both for matter fields and for gravity according to the Dougan–Mason definition) is vanishing precisely for the ‘ground state’ of the theory (i.e., for the vanishing energy-momentum tensor in the domain of dependence D (Σ ) and the flatness of D (Σ ), see Sections 2.2.5 and 8.2.3, respectively). In particular, for configurations describing pure radiation (purely radiative matter fields and pp-waves, respectively) the energy is positive. On the other hand, the vanishing of the quasi-local mass does not characterize the ‘ground state’, rather that is equivalent only to these purely radiative configurations.

The Bartnik mass is a natural quasi-localization of the ADM mass, and its monotonicity and positivity makes it a potentially very useful tool in proving various statements on the spacetime, because it fully characterizes the nontriviality of the finite Cauchy data by a single scalar. However, our personal opinion is that, by its strict positivity requirement for nonflat three-dimensional domains, it overestimates the ‘physical’ quasi-local mass. In fact, if (Σ, hab,χab) is a finite data set for a pp-wave geometry (i.e., a compact subset of the data set for a pp-wave metric), then it probably has an asymptotically flat extension (Σˆ, ˆhab, ˆχab) satisfying the dominant energy condition with bounded ADM energy and no apparent horizon between ∂Σ and infinity. Thus, while the Dougan–Mason mass of ∂ Σ is zero, the Bartnik mass mB (Σ) is strictly positive, unless (Σ, hab,χab) is trivial. Thus, this example shows that it is the procedure of taking the asymptotically flat extension that gives strictly positive mass. Indeed, one possible proof of the rigidity part of the positive energy theorem [38] (see also [488]) is to prove first that the vanishing of the ADM mass implies, through the Witten equation, that the spacetime admits a constant spinor field, i.e., it is a pp-wave spacetime, and then that the only asymptotically flat spacetime that admits a constant null vector field is the Minkowski spacetime. Therefore, it is only the global condition of the asymptotic flatness that rules out the possibility of nontrivial spacetimes with zero ADM mass. Hence, it would be instructive to calculate the Bartnik mass for a compact part of a pp-wave data set. It might also be interesting to calculate its small surface limit to see its connection with the local fields (energy-momentum tensor and probably the Bel–Robinson tensor).

The other very useful definition is the Hawking energy (and its slightly modified version, the Geroch energy). Its advantage is its simplicity, calculability, and monotonicity for special families of two-surfaces, and it has turned out to be a very effective tool in practice in proving for example the Penrose inequality. The small sphere limit calculation shows that the Hawking energy is, in fact, energy rather than mass, so, in principle, one should be able to complete this by a linear momentum to an energy-momentum four-vector. One possibility is Eq. (6.2View Equation), but, as far as we are aware, its properties have not been investigated. Unfortunately, although the energy can be defined for two-surfaces with nonzero genus, it is not clear how the four-momentum could be extended for such surfaces. Although Hawking energy is a well-defined two-surface observable, it has not been linked to any systematic (Lagrangian or Hamiltonian) scenario. Perhaps it does not have any such interpretation, and it is simply a natural (but, in general spacetimes for quite general two-surfaces, not quite viable) generalization of the standard round sphere expression (4.8View Equation). This view appears to be supported by the fact that Hawking energy has strange properties for nonspherical surfaces, e.g., for two-surfaces in Minkowski spacetime, which are not metric spheres.

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