A discussion of polyhomogeneity (asymptotic expansions in terms of $1/r$ and$\ln r$) for zero-rest-mass fields and gravity and its relation with theNewman-Penrose (NP) constants is given. It is shown that for spin-$s$zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic termsin the asymptotic expansion appear naturally if the field does not obey the``Peeling theorem''. The terms that give rise to the slower fall-off admit anatural interpretation in terms of advanced field. The connection between suchfields and the NP constants is also discussed. The case when the backgroundspacetime is curved and polyhomogeneous (in general) is considered. The freefields have to be polyhomogeneous, but the logarithmic terms due to theconnection appear at higher powers of $1/r$. In the case of gravity, it isshown that it is possible to define a new auxiliary field, regular at nullinfinity, and containing some relevant information on the asymptotic behaviourof the spacetime. This auxiliary zero-rest-mass field ``evaluated at futureinfinity ($i^+$)'' yields the logarithmic NP constants.
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机译：讨论了零静质量场和重力的多均质性（以$ 1 / r $和$ \ ln r $表示的渐近展开）及其与Newman-Penrose（NP）常数的关系。结果表明，对于在Minkowski时空上传播的自旋$ s $零静质量场，如果该场不服从``Peeling定理''，则渐近扩展中的对数项自然会出现。导致下降变慢的术语承认对高级领域的自然解释。还讨论了此类字段与NP常数之间的关系。考虑背景时空是弯曲且多态的（通常）的情况。自由场必须是多齐性的，但是由于连接而导致的对数项出现在更高的$ 1 / r $次方。在重力的情况下，显示出可以定义一个新的辅助场，该辅助场规则为零无穷大，并且包含有关时空渐近行为的一些相关信息。该辅助零剩余质量字段``在futureinfinity（$ i ^ + $）上求值''产生对数NP常数。
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