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| {{Expert-subject|Physics|date=November 2008}}
| | Hi there. Let me start by introducing the author, her title is Myrtle Cleary. North Dakota is where me and my husband live. Hiring is his occupation. Doing ceramics is what my family members and I enjoy.<br><br>my blog post - [http://Www.Hotporn123.com/user/RPocock Www.Hotporn123.com] |
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| In [[mathematical physics]] and [[differential geometry]], a '''gravitational instanton''' is a four-dimensional [[complete metric|complete]] [[Riemannian manifold]] satisfying the [[vacuum]] [[Einstein equation]]s. They are so named because they are analogues in [[quantum gravity|quantum theories of gravity]] of [[instanton]]s in [[Yang–Mills theory]]. In accordance with this analogy with [[instanton#Instantons in Yang–Mills theory|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 manifold|hyperkähler 4-manifolds]], and in this sense, they are special examples of [[Einstein manifold]]s. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum [[Einstein equation]]s with ''positive-definite'', as opposed to [[Pseudo-Riemannian manifold|Lorentzian]], metric.
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| 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.
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| There are many methods for constructing gravitational instantons, including the [[Gibbons–Hawking Ansatz]], [[twistor theory]], and the [[hyperkähler quotient]] construction.
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| == Properties ==
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| * A four-dimensional [[Kähler manifold|Kähler]]–[[Einstein manifold]] has a self-dual [[Riemann tensor]].
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| * Equivalently, a self-dual gravitational instanton is a four-dimensional complete [[hyperkähler manifold]].
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| * Gravitational instantons are analogous to [[Instanton|self-dual Yang–Mills instantons]].
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| == Taxonomy ==
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| By specifying the 'boundary conditions', i.e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as '''asymptotically locally Euclidean spaces''' (ALE spaces), '''asymptotically locally flat spaces''' (ALF spaces). There also exist ALG spaces whose name is chosen by induction.
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| == Examples ==
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| It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the [[three-sphere]] '''S'''<sup>3</sup>
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| (viewed as the group Sp(1) or SU(2)). These can be defined in terms of [[Euler angles]] by
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| :<math>
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| \sigma_1 = \sin \psi \, d \theta - \cos \psi \sin \theta \, d \phi
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| </math>
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| :<math>
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| \sigma_2 = \cos \psi \, d \theta + \sin \psi \sin \theta \, d \phi
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| </math>
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| :<math>
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| \sigma_3 = d \psi + \cos \theta \, d \phi.
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| </math>
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| === Taub–NUT metric ===
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| {{main|Taub–NUT metric}}
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| :<math>
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| ds^2 = \frac{1}{4} \frac{r+n}{r-n} dr^2 + \frac{r-n}{r+n} n^2 {\sigma_3}^2 + \frac{1}{4}(r^2 - n^2)({\sigma_1}^2 + {\sigma_2}^2)
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| </math>
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| === Eguchi–Hanson metric ===
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| The [[Eguchi–Hanson space]] is important in many other contexts of geometry and theoretical physics. Its metric is given by
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| : <math>
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| ds^2 = \left( 1 - \frac{a}{r^4} \right) ^{-1} dr^2 + \frac{r^2}{4} \left( 1 - \frac{a}{r^4} \right) {\sigma_3}^2 + \frac{r^2}{4} (\sigma_1^2 + \sigma_2^2).
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| </math>
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| where <math>r \ge a^{1/4}</math>.
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| This metric is smooth everywhere if it has no [[Gravitational singularity#Conical singularities|conical singularity]] at <math>r \rightarrow a^{1/4}</math>, <math>\theta = 0, \pi</math>. For <math>a = 0</math> this happens if <math>\psi</math> has a period of <math>4\pi</math>, which gives a flat metric on '''R'''<sup>4</sup>; However for <math>a \ne 0</math> this happens if <math>\psi</math> has a period of <math>2\pi</math>.
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| Asymptotically (i.e., in the limit <math>r \rightarrow \infty</math>) the metric looks like
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| :<math> ds^2 = dr^2 + \frac{r^2}{4} \sigma_3^2 + \frac{r^2}{4} (\sigma_1^2 + \sigma_2^2) </math>
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| which naively seems as the flat metric on '''R'''<sup>4</sup>. However, for <math>a \ne 0</math>, <math>\psi</math> has only half the usual periodicity, as we have seen. Thus the metric is asymptotically '''R'''<sup>4</sup> with the identification <math>\psi\, {\sim}\, \psi + 2\pi</math>, which is a [[Cyclic group|Z<sub>2</sub>]] [[subgroup]] of [[SO(4)]], the rotation group of '''R'''<sup>4</sup>. Therefore the metric is said to be asymptotically
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| '''R'''<sup>4</sup>/'''Z'''<sub>2</sub>.
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| There is a transformation to another [[coordinate system]], in which the metric looks like
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| :<math> ds^2 = \frac{1}{V(\mathbf{x})} ( d \psi + \boldsymbol{\omega} \cdot d \mathbf{x})^2 + V(\mathbf{x}) d \mathbf{x} \cdot d \mathbf{x},</math>
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| where | |
| <math> \nabla V = \pm \nabla \times \boldsymbol{\omega}, \quad V = \sum_{i=1}^2 \frac{1}{|\mathbf{x}-\mathbf{x}_i| }.
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| </math>
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| :(For a = 0, <math>V = \frac{1}{|\mathbf{x}|}</math>, and the new coordinates are defined as follows: one first defines <math>\rho=r^2/4</math> and then parametrizes <math>\rho</math>, <math>\theta</math> and <math>\phi</math> by the '''R'''<sup>3</sup> coordinates <math>\mathbf{x}</math>, i.e. <math>\mathbf{x}=(\rho \sin \theta \cos \phi, \rho \sin \theta \sin \phi,\rho \cos\theta) </math>).
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| In the new coordinates, <math>\psi</math> has the usual periodicity <math>\psi\ {\sim}\ \psi + 4\pi.</math>
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| One may replace V by
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| :<math>\quad V = \sum_{i=1}^n \frac{1}{|\mathbf{x} - \mathbf{x}_i|}.</math>
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| For some ''n'' points <math>\mathbf{x}_i</math>, ''i'' = 1, 2..., ''n''.
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| This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid [[Gravitational singularity#Conical singularities|conical singularities]]). The asymptotic limit (<math>r\rightarrow \infty</math>) is equivalent to taking all <math>\mathbf{x}_i</math> to zero, and by changing coordinates back to r, <math>\theta</math> and <math>\phi</math>, and redefining <math>r\rightarrow r/\sqrt{n}</math>, we get the asymptotic metric
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| :<math> ds^2 = dr^2 + \frac{r^2}{4} \left({d\psi\over n} + \cos \theta \, d\phi\right)^2 + \frac{r^2}{4} [(\sigma_1^L)^2 + (\sigma_2^L)^2]. </math>
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| This is '''R'''<sup>4</sup>/'''Z'''<sub>''n''</sub> = '''C'''<sup>2</sup>/'''Z'''<sub>n</sub>, because it is '''R'''<sup>4</sup> with the angular coordinate <math>\psi</math> replaced by <math>\psi/n</math>, which has the wrong periodicity (<math>4\pi/n</math> instead of <math>4\pi</math>). In other words, it is '''R'''<sup>4</sup> identified under <math>\psi\ {\sim}\ \psi + 4\pi k/n</math>, or, equivalnetly, '''C'''<sup>2</sup> identified under ''z''<sub>''i''</sub> ~ <math>e^{2\pi i k/n}</math> ''z''<sub>''i''</sub> for ''i'' = 1, 2.
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| To conclude, the multi-center Eguchi–Hanson geometry is a [[Kähler manifold|Kähler]] Ricci flat geometry which is asymptotically '''C'''<sup>2</sup>/'''Z'''<sub>n</sub>. According to [[Calabi–Yau manifold|Yau's theorem]] this is the only geometry satisfying these properties. Therefore this is also the geometry of a '''C'''<sup>2</sup>/'''Z'''<sub>n</sub> [[orbifold]] in [[string theory]] after its [[Gravitational singularity#Conical singularities|conical singularity]] has been smoothed away by its "blow up" (i.e., deformation) [http://arxiv.org/abs/hep-th/9603167].
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| === Gibbons–Hawking multi-centre metrics ===
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| <math>
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| ds^2 = \frac{1}{V(\mathbf{x})} ( d \tau + \boldsymbol{\omega} \cdot d \mathbf{x})^2 + V(\mathbf{x}) d \mathbf{x} \cdot d \mathbf{x},
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| </math>
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| where
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| <math>
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| \nabla V = \pm \nabla \times \boldsymbol{\omega}, \quad V = \varepsilon + 2M \sum_{i=1}^{k} \frac{1}{|\mathbf{x} - \mathbf{x}_i | }.
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| </math>
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| <math>\epsilon = 1</math> corresponds to multi-Taub–NUT, <math>\epsilon = 0</math> and <math>k = 1</math> is flat space, and <math>\epsilon = 0</math> and <math>k = 2</math> is the Eguchi–Hanson solution (in different coordinates).
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| == References ==
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| * Gibbons, G. W.; [[Stephen Hawking|Hawking, S. W.]], ''Gravitational Multi-instantons''. Phys. Lett. B 78 (1978), no. 4, 430–432; see also ''Classification of gravitational instanton symmetries''. Comm. Math. Phys. 66 (1979), no. 3, 291–310.
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| * Eguchi, Tohru; Hanson, Andrew J., ''Asymptotically flat selfdual solutions to Euclidean gravity''. Phys. Lett. B 74 (1978), no. 3, 249–251; see also ''Self-dual solutions to Euclidean Gravity''. Ann. Physics 120 (1979), no. 1, 82–106 and ''Gravitational instantons''. Gen. Relativity Gravitation 11 (1979), no. 5, 315–320.
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| * [[Peter B. Kronheimer|Kronheimer, P. B.]], ''The construction of ALE spaces as hyper-Kähler quotients''. J. Differential Geom. 29 (1989), no. 3, 665–683.
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| [[Category:Riemannian manifolds]]
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| [[Category:Quantum gravity]]
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| [[Category:Mathematical physics]]
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| [[Category:4-manifolds]]
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Hi there. Let me start by introducing the author, her title is Myrtle Cleary. North Dakota is where me and my husband live. Hiring is his occupation. Doing ceramics is what my family members and I enjoy.
my blog post - Www.Hotporn123.com