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**On the boundary conditions for gravitational and gauge fields at spatial infinity.**
*(English)*
Zbl 0553.53050

Asymptotic behavior of mass and space-time geometry, Proc. Conf., Corvallis/Oreg. 1983, Lect. Notes Phys. 202, 95-109 (1984).

[For the entire collection see Zbl 0539.00014.]

The paper reviews three results concerning the asymptotic behavior of fields at spatial infinity in general relativity and gauge theories. The first has to do so with the supertranslation ambiguities in the definition of angular momentum of isolated gravitating systems. It is pointed out that, under the usual \(3+1\) boundary conditions à la Arnowitt, Deser, Misner, angular momentum is not well-defined; in a generic space-time, the procedure can be made to yield any value of angular-momentum one wants! On the other hand, using a manifestly 4- dimensional framework, the author and Hansen have shown how one can strengthen the boundary conditions to remove these ambiguities. We summarize results obtained from a systematic \(3+1\) reduction of this framework.

The second result concerns the ”supertranslation ambiguities” in the definition of electric charge in non-Abelian gauge theories which were first discussed in detail by Tafel and Trautman. We point out an additional problem: even if one were to fix a gauge by hand and define the electric charge, under the usual boundary conditions, the electric charge fails to be conserved under boosts (although it is conserved under time-translations.) Following the Ashtekar-Hansen procedure from general relativity, we show that one can strengthen the boundary conditions in gauge theories such that the charge is defined unambiguously and conserved under translations and boosts. However, the stronger conditions do not permit magnetic monopoles and the problems discussed above are open in presence of monopole configurations.

The last result concerns general relativity again. In addition to the supertranslations, there exist the so-called ”logarithmic ambiguities” in the choice of the background flat metric. These do not affect the values of energy, momentum and angular momentum. Nonetheless, their presence is aesthetically unappealing. A procedure is sketched to remove these ambiguities by mildly strengthening the boundary condition on the asymptotic Weyl curvature. Details will appear in the Bergmann Festschrift issue of Foundations of Physics (1985).

The paper reviews three results concerning the asymptotic behavior of fields at spatial infinity in general relativity and gauge theories. The first has to do so with the supertranslation ambiguities in the definition of angular momentum of isolated gravitating systems. It is pointed out that, under the usual \(3+1\) boundary conditions à la Arnowitt, Deser, Misner, angular momentum is not well-defined; in a generic space-time, the procedure can be made to yield any value of angular-momentum one wants! On the other hand, using a manifestly 4- dimensional framework, the author and Hansen have shown how one can strengthen the boundary conditions to remove these ambiguities. We summarize results obtained from a systematic \(3+1\) reduction of this framework.

The second result concerns the ”supertranslation ambiguities” in the definition of electric charge in non-Abelian gauge theories which were first discussed in detail by Tafel and Trautman. We point out an additional problem: even if one were to fix a gauge by hand and define the electric charge, under the usual boundary conditions, the electric charge fails to be conserved under boosts (although it is conserved under time-translations.) Following the Ashtekar-Hansen procedure from general relativity, we show that one can strengthen the boundary conditions in gauge theories such that the charge is defined unambiguously and conserved under translations and boosts. However, the stronger conditions do not permit magnetic monopoles and the problems discussed above are open in presence of monopole configurations.

The last result concerns general relativity again. In addition to the supertranslations, there exist the so-called ”logarithmic ambiguities” in the choice of the background flat metric. These do not affect the values of energy, momentum and angular momentum. Nonetheless, their presence is aesthetically unappealing. A procedure is sketched to remove these ambiguities by mildly strengthening the boundary condition on the asymptotic Weyl curvature. Details will appear in the Bergmann Festschrift issue of Foundations of Physics (1985).

### MSC:

53C80 | Applications of global differential geometry to the sciences |

81T08 | Constructive quantum field theory |

83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |