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 non-triviality of the finite Cauchy data by a single scalar. However,
our personal opinion is that, just by its strict positivity for non-flat 3-dimensional domains, it
overestimates the ‘physical’ quasi-local mass. In fact, if 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
satisfying the dominant energy condition with
bounded ADM energy and no apparent horizon between
and infinity. Thus while the
Dougan-Mason mass of
is zero, the Bartnik mass
is strictly positive unless
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 [24] (see also [354]) 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 just the global condition
of the asymptotic flatness that rules out the possibility of non-trivial 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 2-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 it is energy rather than mass, so in principle one should be able to
complete this to an energy-momentum 4-vector. One possibility is Equation (39, 40
), but, as far as we are
aware, its properties have not been investigated. Unfortunately, although the energy can be
defined for 2-surfaces with nonzero genus, it is not clear how the 4-momentum could be extended
for such surfaces. Although the Hawking energy is a well-defined 2-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 2-surfaces not quite viable) generalization of the standard round sphere expression (27
).
This view appears to be supported by the fact that the Hawking energy has strange properties
for non-spherical surfaces, e.g. for 2-surfaces in Minkowski spacetime which are not metric
spheres.
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