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- In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean. There are many methods for constructing gravitational instantons, including the Gibbons–Hawking Ansatz, twistor theory, and the hyperkähler quotient construction. (en)
- 重力インスタントン（じゅうりょく - ）とは、以下の3つの性質を持つ4次元リーマン多様体のことである。 1.
* リッチ平坦 2.
* （self-dual）なリーマン曲率テンソルをもつ 3.
* 無限遠で局所的に平坦（asymptotically locally flat）である （しかし実は、2. ならば 1. が言える。） あるいは、もっと広い意味で、3. を満たしリッチ曲率が計量に比例している（いわゆる宇宙定数がある）ものを言う。 ヤン・ミルズ理論のインスタントンとの類似から、そう呼ばれる。ALE（Asymptotically Locally Euclidean）空間とも呼ばれる。 (ja)
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- 重力インスタントン（じゅうりょく - ）とは、以下の3つの性質を持つ4次元リーマン多様体のことである。 1.
* リッチ平坦 2.
* （self-dual）なリーマン曲率テンソルをもつ 3.
* 無限遠で局所的に平坦（asymptotically locally flat）である （しかし実は、2. ならば 1. が言える。） あるいは、もっと広い意味で、3. を満たしリッチ曲率が計量に比例している（いわゆる宇宙定数がある）ものを言う。 ヤン・ミルズ理論のインスタントンとの類似から、そう呼ばれる。ALE（Asymptotically Locally Euclidean）空間とも呼ばれる。 (ja)
- In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of th (en)
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- Gravitational instanton (en)
- 重力インスタントン (ja)
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