We investigate parallel spinors on the Eguchi–Hanson metrics and find the space of complex parallel spinors is complex two-dimensional. For the metrics of Eguchi–Hanson type with the zero scalar curvature, we separate variables for the harmonic spinors and obtain the solutions explicitly.
Zhuohua Cai, Xiao Zhang. The Dirac equation on metrics of Eguchi–Hanson type[J]. Communications in Theoretical Physics, 2023, 75(5): 055002. DOI: 10.1088/1572-9494/acc7f3
1. Introduction
Eguchi–Hanson metrics were constructed in [1, 2], which are complete four-dimensional Ricci flat, anti-self-dual ALE Riemannian metrics on R≥0 × P3. They are one type of gravitational instanton, and play important roles in the Euclidean approach of gravitational quantization [3]. Since then, gravitational instantons including the Eguchi–Hanson metrics attract much attention both in geometry and in general relativity.
In order to investigate the physical behaviour of spin-$\tfrac{1}{2}$ particles in general relativity, Chandrasekhar solved the Dirac equation in Kerr spacetimes by separating variables [4]. We refer to [5, 6] and references therein for further developments of the Dirac equation in various spacetimes. In [7], Sucu and Ünal solved the Dirac equation on Eguchi–Hanson and Bianchi VII0 gravitational instanton metrics. In particular, the solutions are obtained as the product of two hypergeometric functions for the Eguchi–Hanson gravitational instanton metric.
As it actually relates to manifold's scalar curvature, it is interesting to study the Dirac equation on much more general metrics of Eguchi–Hanson type with the zero scalar curvature. This type of metric was constructed by LeBrun using the method of algebraic geometry [8] and by the second author solving an ordinary differential equation [9] on R≥0 × S3/Zd (d > 2), and they provide the counter-examples of Hawking and Pope's generalized positive action conjecture.
In this note we first introduce the Eguchi–Hanson metrics, the metrics of Eguchi–Hanson type with the zero scalar curvature as well as the spin connections on them. Then we investigate the parallel spinors and the harmonic spinors. As the existence of a nontrivial parallel spinor on a Riemannian spin manifold implies that the manifold is Ricci-flat, parallel spinors only exist on the Eguchi–Hanson metrics, and we find the space of complex parallel spinors is complex two-dimensional. For the metrics of Eguchi–Hanson type with the zero scalar curvature, we separate variables for the harmonic spinors and obtain the solutions explicitly.
We refer to [10] for an introductory knowledge of spin geometry. It is clear that the underline manifolds of both metric (2.5) and metric (2.6) are spin and their complex spinor bundles are complex four-dimensional equipped with spin connection
Spinor Φ is parallel if ∇XΦ = 0 for any tangent vector X. Any Riemannian metric carried a parallel spinor must be Ricci-flat, see, e.g. [10]. Thus it is interesting to investigate the parallel spinors for the Eguchi–Hanson metric.
Parallel spinors for the Eguchi–Hanson metric (2.5) are complex constant spinors which form a complex two-dimensional space.
So ${{\rm{\Phi }}}_{1}$ and ${{\rm{\Phi }}}_{2}$ are arbitrary complex constants and the proof of the theorem is complete.
4. Harmonic spinors
Harmonic spinors are the solutions of the Dirac equation
$\begin{eqnarray*}{ \mathcal D }=\displaystyle \sum _{k=1}^{4}{e}_{k}\cdot {{\rm{\nabla }}}_{{e}_{k}}=0.\end{eqnarray*}$
We shall find the explicit solutions of harmonic spinors on metric (2.6) of Eguchi–Hanson type with zero scalar curvature (d > 2). In the case d = 2, harmonic spinors on Eguchi–Hanson metric (2.5) were solved by Sucu and Ünal in [7], and the solutions can be obtained in terms of hypergeometric functions.
Let positive integer $d\gt 2$. There exists harmonic spinors
Solving these ordinary differential equations, we obtain (4.1) and (4.2). This gives the proof of theorem.
Finally we study the singular sets of harmonic spinors constructed in theorem 4.1. The singularity occurs when at least one of the power indices of r − r0 is negative in (4.1) or at least one of the power indices of $\sin \displaystyle \frac{\theta }{2}$ or $\cos \displaystyle \frac{\theta }{2}$ is negative in (4.2). The relations between polar coordinates {r, θ, ψ, φ} and Cartesian coordinates {x0, x1, x2, x3} of the metric (2.6) are [9]
The harmonic spinor Φ, constructed in theorem 4.1, as well as its norm $| {\rm{\Phi }}| $ are singular at $r={r}_{0}$. Moreover, it holds that
(1) If $-\tfrac{d}{2}\lt {a}_{{mn}}^{-}\lt 0$, $-\tfrac{d}{2}\lt {b}_{{mn}}^{-}\lt 0$, v1, v2, v3, v4 and $| {\rm{\Phi }}| $ are singular either at ${x}_{1}{x}_{2}$-plane, or at ${x}_{3}{x}_{0}$-plane;
(2) If $-\tfrac{d}{2}\lt {a}_{{mn}}^{-}\lt 0$, ${b}_{{mn}}^{-}\geqslant 0$, either v1, v2, v3, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$-plane, or v1, v3, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane;
(3) If $-\tfrac{d}{2}\lt {a}_{{mn}}^{-}\lt 0$, ${b}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, either v1, v2, v3, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$- plane, or v2, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane;
(4) If ${a}_{{mn}}^{-}\geqslant 0$, $-\tfrac{d}{2}\lt {b}_{{mn}}^{-}\lt 0$, either v2, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$-plane, or v1, v2, v3, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane;
(5) If ${a}_{{mn}}^{-}\geqslant 0$, ${b}_{{mn}}^{-}\geqslant 0$, v2, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$-plane, or v1, v3, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane,
(6) If ${a}_{{mn}}^{-}\geqslant 0$, ${b}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, v2, v4, $| {\rm{\Phi }}| $ are singular either at ${x}_{1}{x}_{2}$- plane or at ${x}_{3}{x}_{0}$-plane;
(7) If ${a}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, $-\tfrac{d}{2}\lt {b}_{{mn}}^{-}\lt 0$, either v1, v3, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$-plane, or v1, v2, v3, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane;
(8) If ${a}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, ${b}_{{mn}}^{-}\geqslant 0$, v1, v3, $| {\rm{\Phi }}| $ are singular either at ${x}_{1}{x}_{2}$-plane, or at ${x}_{3}{x}_{0}$-plane;
(9) If ${a}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, ${b}_{{mn}}^{-}\leqslant -\tfrac{d}{2}$, either v1, v3, $| {\rm{\Phi }}| $ are singular at ${x}_{1}{x}_{2}$-plane, or v2, v4, $| {\rm{\Phi }}| $ are singular at ${x}_{3}{x}_{0}$-plane.
If any ui or vi goes to $\infty $ at some subset of the space, the harmonic spinors and their norm become singular there. The power orders of $r-{r}_{0}$ in ui are
Therefore the rest parts of the theorem follow from the above formulas, (4.7) and the fact that $\sin \tfrac{\theta }{2}$, $\cos \tfrac{\theta }{2}$ cannot be both zero.
The authors would like to thank the referees for valuable suggestions. This work is supported by Chinese NSF grants 11731001, the special foundations for Guangxi Ba Gui Scholars and Junwu Scholars of Guangxi University.
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