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 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 [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. (6.2), 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 (4.8
).
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.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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