# Asymptotic Invariants in General Relativity

I was trying to understand Witten's proof of the Positive Energy Theorem in General Relativity by reading the original argument given by Witten. I am comfortable with the overall argument, but I would like to understand the following statement, made on the second paragraph of page 394 in the previous link:

"The only invariants that can be formed from the $1/r$ term in the metric tensor are the total energy and the total momentum."

These invariants (which I will call "$1/r$-built" for short) refer to an asymptotically flat initial value (spacelike smooth) 3-surface in spacetime.

Is there an obvious reason why this should be true? Looking at the definition of the ADM-energy and momentum, it seems plausible because the only data we have are the first and second fundamental form, and it should be possible to write any invariant as a combination of these, and consequently, any $1/r$-built invariant as a function of ADM energy and momentum. However, this reasoning is too hand-wavy, so I wonder if there is a clear cut explanation.

My interest in this fact is that although it is not really a logical step in the proof, if true it is probably one of the best ways to motivate a spinorial proof (I was thinking of something along the lines of "if we can construct a manifestly non-negative $1/r$-built invariant by means of the asymptotic behavior of spinors, then it should be possible to prove that energy is non-negative by writing this invariant as a function of ADM energy and momentum").

From a purely physical perspective, if it were true that ADM energy and momentum suffice to specify the system (to order $1/r$), then they should be the only independent invariants. However, this suggests that an asymptotic observer who knew the total energy and momentum could reconstruct the metric up to order $1/r$. I was thinking if it is not possible to construct a counterexample by defining an axisymmetric spacetime in which the Killing field of azimuthal symmetry is build up only with $1/r$ terms and showing that it is asymptotically distinguishable from its non-rotating counterpart. In this spirit, I think it is worth asking a broader version of my question:

What kinds of "physically interesting" boundary terms can appear in a Hamiltonian formulation for an asymptotically flat spacetime manifold in General Relativity?

Thanks in advance.

• An azimulthal symmetry cannot be built up only with $1/r$ terms. In fact, for the killing vector field $\phi^\nu$ to be asymptotically a rotation, $g_{\mu\nu}\phi^\mu\phi^\nu$ must be size $r$ near infinity. Incidentally, to define "rotation", whose corresponding invariant would be like an angular momentum, you need to impose more restrictive fall-off conditions on the metric than the ones required to define ADM energe-momenta. See for example this review paper. – Willie Wong Jun 25 '11 at 12:28
• Also, I'd like to pick a bone with the statement of "an asymptotic observer who knew the total energy and momentum". The ADM quantities are defined by integrating over a sphere at infinity, so I don't think it is reasonable to say that an asymptotic observer can know the total energy and momentum. – Willie Wong Jun 25 '11 at 12:32
• Hi Willie, thank you for your comments and for the reference. I should have thought that a general definition of angular momentum requires stronger fall-off conditions than the ones presented in the definition of asymptotic flatness, since otherwise it would be defined together with ADM energy and momentum in most references. But I certainly didn't expect for the problem to be so much trickier; for instance, in the paper you linked it is argued that one cannot choose arbitrary coordinates for then covariance of angular momentum cannot be guaranteed. – Rodrigo Barbosa Jun 26 '11 at 2:10
• On an unrelated topic, I was wondering if you have any thoughts on why ADM energy and momentum are the only possible $1/r$ invariants, or maybe you know a reference that touches on these matters? Thanks again. – Rodrigo Barbosa Jun 26 '11 at 2:13
• Offhand I don't know of any references. Have you checked in ADM's original paper (1961, Phy. Rev.) to see whether they argue it there? For vacuum solutions in the static case, I know there is an argument by Geroch (JMP, 1970), which was extended to the stationary case by Beig (1981, Proc R Soc Lon), which shows clearly that mass and momentum are 1/r, and some angular momentum like quantities are 1/r^2. – Willie Wong Jun 26 '11 at 13:18