Gravitational Instantons, old and new

The Schwarzschild metric is given by

The apparent singularity at $r=2m$ corresponds to an event horizon, and can be removed by a coordinate transformation. The singularity at $r=0$ is essential as the squared norm of the Riemann tensor blows up as $r^{-6}$. An attempt to get rid of this singularity by removing the origin $r=0$ from the space–time leads to a geodesically incomplete metric.

The Euclidean Schwarzschild metric [31] is obtained by setting $t=i\tau$, and restricting the range of $r$ to $2m<r<\infty$. Set $\rho=4m\sqrt{1-2m/r}$. Near $\rho=0$ the metric takes the form

This metric is flat and regular as long as the imaginary time $\tau$ is periodic with the period $8\pi m$. This period is inverse proportional to the Hawking temperature of the black hole radiation (Fig 1). Although this was not how the Hawking temperature was first discovered, the instanton methods gave rise to a derivation simpler than the original calculation based on the Bogoliubov transformation [23, 24]. In a similar manner the non–extreme Kerr black hole can be turned into the Euclidean Kerr instanton with the period of the imaginary time proportional to the inverse of the surface gravity. In the case of the extreme Kerr solution the surface gravity vanishes and the extreme Kerr instanton does not exist.

Before introducing the next example let us define the left–invariant one–forms $(\sigma_{1},\sigma_{2},\sigma_{3})$ on $S^{3}=SU(2)$ by

where $0\leq\theta\leq\pi,0\leq\phi\leq 2\pi,0\leq\psi\leq 4\pi.$ They satisfy

In terms of these one–forms the flat metric on $\mathbb{R}^{4}$ is given by

The Taub–NUT instanton [31] is

Introducing a coordinate $\rho$ by $r=m+\frac{\rho^{2}}{2m}$ shows that, near $r=m$, the metric (2.2) approaches the flat metric (2.1) and so $r=m$ is only a coordinate singularity. The Riemann curvature of the metric (2.2) is anti-self–dual (ASD); it satisfies

where $\varepsilon_{abcd}=\varepsilon_{[abcd]}$ is a chosen volume–form on $M$. The ASD condition in particular implies the vanishing of the Ricci–tensor. This follows from taking the trace of (2.3). It also shows that the metric (2.2) has no Lorentzian analogue, as the Riemann tensor of a metric in signature $(3,1)$ is ASD iff the metric is flat. For large $r$ the metric $g_{TN}$ is the $S^{1}$ bundle over $S^{2}$ with Chern number equal to $1$ - this is the Hopf fibration with the total space $S^{3}$.

The ASD Taub–NUT example (2.2) motivates the following definition

The asymptotic form of an ALF metric is

where the integer $n$ is the Chern number of the $S^{1}$ bundle. If the $S^{1}$–bundle is trivial, so that $n=0$, the ALF metric is called asymptotically flat (AF). Euclidean Schwarzschild and Euclidean Kerr metrics are AF. According to the Lorentzian black hole uniqueness theorems of Hawking, Carter, D. Robinson, and others [51], the Kerr family of solutions exhausts all AF solutions to the Einstein equations with $\Lambda=0$. These theorems gave rise to the Riemannian ‘black hole uniqueness’ conjecture stating that the Euclidean Schwarzschild and Kerr are the only AF gravitational instantons [38]. This conjecture is now known to be false. We shall return to it in §4.

The ASD Taub–NUT instanton, and other ALF metrics can be uplifted to the so–called Kaluza–Klein monopoles in $4+1$–dimension [28, 48] with the product metric

The Kaluza–Klein reduction of $ds^{2}$ along the Killing vector $\partial/\partial\psi$ gives a monopole–type solution to the Einstein–Maxwell-dilaton theory in $(3+1)$ dimensions.

The Eguchi–Hanson (EH) instanton [19, 20] is given by

with $r>a$. Setting $\rho^{2}=r^{2}\left[1-(a/r)^{4}\right]$ we find that, near $r=a$, the metric is given by

This metric is regular as long as the ranges of the angles are

Thus, although for $r\rightarrow\infty$, the Eguchi–Hanson metric approaches (2.1), given the allowed range of $\psi$ this metric is not asymptotically Euclidean, but corresponds to a quotient $\mathbb{R}^{4}/\mathbb{Z}_{2}$. The Eguchi–Hanson example motivates the following

The anti–self–dual ALE metrics are the best understood class of gravitational instantons. This is due to the following

The Eguchi–Hanson metric corresponds to the case $A_{2}$, where $\Gamma=\mathbb{Z}_{2}$. It is not known [25, 42] whether there exist non self–dual or anti–self–dual ALE Ricci–flat metrics.

We shall start with a definition

The almost complex structure gives rise to a decomposition

of the complexified tangent bundle into eigen-spaces of $I$ with eigenvalues $\pm i$. One says that $I$ is a complex structure iff these eigenspaces are integrable in the sense of the Frobenius theorem, i. e.

A theorem of Newlander and Nirenberg justifies the terminology: $I$ is a complex structure iff there exists a holomorphic atlas so that $M$ is a two–dimensional complex manifold. For example, if $M=\mathbb{R}^{4}$ and

then (3.1) holds and the complex atlas on $M=\mathbb{C}^{2}$ consists of complex coordinates $z_{1}=x_{1}+ix_{3},z_{2}=x_{2}+ix_{4}$ and $T^{1,0}M=\mbox{span}\{\partial/\partial z_{1},\partial/\partial z_{2}\}$.

We shall now assume that $(M,g)$ is a Riemannian four–manifold with almost–complex structure $I$. We say that the metric $g$ is

• •

  Hermitian if $g(X,Y)=g(IX,IY)$.

• •

  Kähler if $I$ is a complex structure, and $d\Omega=0$, where $\Omega(X,Y)=g(X,IY)$.

• •

  hyper–Kähler if it is Kähler w.r.t. three complex structures $I_{1},I_{2},I_{3}$ such that

  $$I_{1}I_{2}=I_{3},\quad I_{2}I_{3}=I_{1},\quad I_{3}I_{1}=I_{2}.$$

For example, if $M=\mathbb{R}^{4}$ then the metric $g=|dz_{1}|^{2}+|dz_{2}|^{2}$ is hyper–Kähler with

The importance of hyper–Kähler metrics in the study of gravitational instantons comes from the fact that locally, and with the choice of orientation which makes the Kähler forms self–dual (SD), the Riemann tensor of $(M,g)$ anti–self–dual (ASD) iff $(M,g)$ hyper–Kähler. Therefore the ASD gravitational instantons are complete hyper–Kähler metrics. Compact hyper–Kähler metrics are far more rare. There is the four–dimensional torus with a flat metric, and the elusive $K3$ surface whose existence follows from Yau’s proof [57] of the Calabi conjecture. Finding the explicit closed form of a metric on a $K3$ surface is one of the biggest open problems in the field.

Let $(V,A)$ be respectively a function, and a one–form on $\mathbb{R}^{3}$. The metric

is hyper–Kähler (and therefore ASD and Ricci flat) with the Kähler forms given by

iff the Abelian Monopole Equation

holds (here $\star_{k}$ is the Hodge operator on $\mathbb{R}^{k}$ taken w.r.t the flat metric and a chosen volume form). This equation follows from the closure condition $d\Omega_{i}=0$, and implies that the function $V$ is harmonic on $\mathbb{R}^{3}$. The general Gibbons Hawking ansatz (3.2) is characterised by the hyper–Kähler condition together with the existence of a Killing vector $K$ which Lie–derives all Kähler forms. The Cartesian coordinates $(x_{1},x_{2},x_{3})$ in (3.2) arise as the moment maps, i. e. $K{\begin{picture}(0.833,0.8)\put(0.15,0.08){\line(1,0){0.35}} \put(0.5,0.08){\line(0,1){0.5}} \end{picture}}\Omega_{i}=dx_{i}$.

The multi–centre metrics correspond to a choice

where $V_{0}$ is a constant, and ${\bf x}_{1},\dots,{\bf x}_{N}$ are position vectors of $N$ points in $\mathbb{R}^{3}$. The special cases of (3.4) are

• •

  $V_{0}=0,N=1$ give the flat metric.

• •

  $V_{0}=0,N=2$ give the Eguchi–Hanson metric (2.4) albeit in a different coordinate system. $V_{0}=0$ and $N>2$ correspond to the general $A_{N}$ ALE instantons.

• •

  $V_{0}\neq 0,N=1$ give the Taub–NUT metric (2.2). $V_{0}\neq 0,N>1$ correspond to the $A_{N}$ ALF instantons.

Let $f$ be a quartic polynomial with four real roots. Set

The family of metrics

is Ricci–flat for any choice of the parameters $(a_{0},\dots,a_{4},\nu,k)$. Two out of five parameters $(a_{0},\dots,a_{4})$ can be fixed by scalings, so (4.1) is a five–parameter family. The Riemann curvature is regular if the range of $(x,y)$ is restricted to the rectangle on Figure 2, where $r_{1}<r_{2}<r_{3}<r_{4}$ are the roots $f$.

To avoid the conical singularities, and ensure the asymptotic flatness one makes a choice

This leads to a two parameter family of AF instantons on $M=\mathbb{CP}^{2}\setminus S^{1}$.

The Chen–Teo metrics (4.1) admit two commuting Killing vectors $K_{i}=\partial/\partial\phi^{i}$ where $\phi^{i}=(\phi,\tau)$. Any metric with two commuting Killing vectors can locally be put in the form

where $J=J(r,z)$ is a $2$ by $2$ symmetric matrix, and the $(r,z)$ coordinates are defined by

The space of orbits of the $T^{2}$ action is the upper half–plane $\mathbb{H}=\{(r,z),r>0\}$ with the boundary $\partial\mathbb{H}$ where rank$(J(0,z))<2$. Generically this rank is equal to $1$. It vanishes at the turning points $z_{1},z_{2},\dots,z_{N}$ where both Killing vectors vanish. These turning points divide the $z$–axis into $(N+1)$ rods [29]

In the Lorentzian case these rods correspond to horizons or axes of rotation, and in the Riemannian case they are axes. The rod data associated to (4.3) consists of a collection of $(N+1)$ rods, together with the lengths $(z_{k}-z_{k-1}),k=2,\dots,N$ of the finite rods, and the constant rod vectors $V_{2},\dots,V_{N}$ such that $V_{k}$ vanishes on the rod $I_{k}$. Each of these vectors can be expanded as $V_{k}=V_{k}^{1}K_{1}+V_{k}^{2}K_{2}$, and then the admissibility condition [30] is

While the rod structure does not uniquely determines the metric of the instanton, it specifies the topology of the underlying four–manifold [43]. The number of turning points is equal to the Euler signature. In the Chen–Teo case there exist thee turning points, so that $\chi(M)=3$ for the Chen–Teo instanton. Closing up the semi–infinite rods gives the triangular rod structure of $\mathbb{CP}^{2}$ with three turning points as the triangle vertices, and three rods as sides. Joining the rods adds $S^{1}\times\mathbb{R}^{3}$ to $M$, and so $M=\mathbb{CP}^{1}\setminus{S^{1}\times\mathbb{R}^{3}}\cong\mathbb{CP}^{1}\setminus{S^{1}}$. The signature of the Chen–Teo family is $1$.

The Ricci–flat condition on (4.3) reduces to the Yang equation

Once a solution to this equation has been found, the conformal factor $\Omega$ can be found by a single integration.

The Yang equation (4.4) also arises as a reduction of anti–self–dual Yang-Mills equations [55, 53]. To see it, consider the complexified Minkowski space $M_{\mathbb{C}}=\mathbb{C}^{4}$, with coordinates $(W,Z,\widetilde{W},\widetilde{Z})$ such that the metric and the volume form are

Let $\Phi\in\Lambda^{1}(M_{\mathbb{C}})\otimes\mathfrak{sl}(2)$, and $F=d\Phi+\Phi\wedge\Phi$. The anti–self–dual Yang–Mills (ASDYM) equations are $F=-\star_{4}F$ (now $\star_{4}$ is taken w.r.t. the flat metric on $\mathbb{C}^{4}$), or

The first two equations imply the existence of a gauge choice such that

The final equation in (4.5) holds iff

Setting

and performing a symmetry reduction $J=J(r,z)$ reduces (4.7) to (4.4).

The twistor correspondence for ASDYM is based on an observation that ASDYM condition is equivalent to the flatness of a connection $\Phi$ on $\alpha$–planes in $M_{\mathbb{C}}$

The twistor space $PT\equiv\mathbb{CP}^{3}\setminus\mathbb{CP}^{1}$ is the space of all such planes. It can be covered by two open sets, with affine coordinates $(\mu,\nu,\lambda)$ in an open set where $\lambda\neq\infty$. Points in $M_{\mathbb{C}}$ correspond to rational curves (twistor lines) in $PT$, and points in $PT$ correspond to $\alpha$–planes in $M_{\mathbb{C}}$. The conformal structure on ${M_{\mathbb{C}}}$ is encoded in the algebraic geometry of curves in $PT$: $p_{1},p_{2}$ are null separated iff $L_{1},L_{2}$ intersect.

The connection between twistor theory and ASDYM is provided by the following

To read off the solution (4.7) from this Theorem cover $PT$ with two open sets: $U$, where $\lambda\neq\infty$ and $\widetilde{U}$ where $\lambda\neq 0$. The bundle $E$ is then characterised by its patching matrix: $P=P(\mu,\nu,\lambda)$. The triviality on twistor lines implies that there exists a splitting $P=P_{U}{{P}_{\widetilde{U}}}^{-1}$, where $P_{U}$ and ${P}_{\widetilde{U}}$ are holomorphic and invertible matrices on $U$ and $\widetilde{U}$ respectively. The incidence relation (4.8) implies that $P$ is constant along the vector fields $\{\partial_{\widetilde{Z}}-\lambda\partial_{W},\partial_{\widetilde{W}}-\lambda\partial_{Z}\}$. Applying this to the splitting relation, and using the Liouville theorem implies the existence of $\Phi\in\Lambda^{1}(M_{\mathbb{C}})\otimes\mathfrak{sl}(2)$ such that

where $H=P_{U}(\lambda=0),\widetilde{H}=P_{\widetilde{U}}(\lambda=\infty)$. This is gauge equivalent to (4.6) with

Let us go back to the toric Ricci–flat metrics. For any of the Killing vectors $K$ we can find its twist potential: a function $\psi$ such that

Another solution to the Yang equation (4.4) then arises from a Bäcklund transformation

Pick a rod on which $K$ is not identically zero. The following has been established in [21, 56, 41]: The patching matrix for the bundle $E$ from Theorem 4.1 is an analytic continuation of $P(z)\equiv J^{\prime}(0,z)$:

The splitting procedure leads, via (4.9), to $J^{\prime}(r,z)$ from which $J(r,z)$ can be recovered.

This patching matrix can be found for the Chen–Teo family [18]. It is given by

where $C_{1},C_{2},C$ monic cubics, $Q$ quadratic, with coefficients depending on the Chen–Teo parameters. Examining the outer rod and the asymptotics near $z=\infty$ gives

where $m$, $n$ are mass and nut parameters. For Chen–Teo instanton with (4.2) we find

in agreement with [37]. In general, the patching matrix $P$ of the form (4.10) where $C,C_{1},C_{2}$ are monic polynomials of degree $N$ and $Q$ is a polynomial of degree $N-1$ subject to $\mbox{det}(P)=-1$ lead to Ricci–flat ALF metrics with $N+1$ rods and $N$ turning points. The ALE metrics with $N+1$ rods can also be constructed, but from a different ansatz [50, 15].

The ALE and ALF classes of gravitational instantons have been defined in (2.2) and (2.3) in terms of the asymptotic quotients of $\mathbb{R}^{4}$ and asymptotic $S^{2}$ fibrations respectively. There is an alternative and unifying definition in terms of the volume growth of a ball of large radius $R$. It is of orders $R^{4}$ and $R^{3}$ for respectively ALE and ALF. This classification gives rise to more families of instantons: ALG and ALG* the volume growth $R^{2}$, ALH with the volume growth $R$, and ALH* with the volume growth $R^{4/3}$ [5, 32, 6]. Unlike the ALE and ALF, these new families do not contain any examples which are known analytically in closed form. It is however the case that all classes are asymptotically described by the Gibbons–Hawking form (3.2) with the harmonic function given by

Therefore the metrics are locally asymptotic to $\mathbb{R}^{k}\times T^{4-k}$ with $k=4$ for ALE, $k=3$ for ALF, $k=2$ for ALG and $k=1$ for ALH. Let us focus on the ALH* case, and perform an affine transformation of $x_{3}$, such that $V=x_{3}$ in the Gibbons–Hawking ansatz (3.2). The coordinate $x_{3}$ is on the base $\mathbb{R}$ of the fibration $M\rightarrow\mathbb{R}$. The fibres are Nil 3–manifolds fibering over $T^{2}$ with periodic coordinates $(x_{1},x_{2})$ with the fibre coordinate $\tau$. The one–form $A$ in the ansatz (3.2) is such that $dA=dx_{1}\wedge dx_{2}$ is the volume form on $T^{2}$. Setting $x_{3}=r^{2/3}$ and rescalling $(x_{1},x_{2},\tau)$ by constants yields

The volume form is $\mbox{vol}=r^{1/3}dr\wedge dx_{1}\wedge dx_{2}\wedge d\tau$, so that the volume growth is indeed $\int_{M}\mbox{vol}\sim R^{4/3}$ if the range of $r$ is bounded by $R$.

The gravitational instantons exist in the Einstein–Maxwell theory. Unlike the pure Einstein case, there exist many asymptotically flat solutions in the multi–centred class. These solutions arise as analytic continuations of the Israel–Wilson and Majumdar–Papapetrou black holes (see [54, 58, 12]), and are given by

where $V$ and $\widetilde{V}$ are harmonic functions on $\mathbb{R}^{3}$, and the one–form $A$ satisfies

The Maxwell field is given by

If $\tilde{V}=1$ then (5.2) reduces to the monopole equation (3.3) and the metrics (5.1) are Ricci flat, and coincide with the Gibbons–Hawking ansatz (3.2). If

with $V_{0},\widetilde{V}_{0},a_{m},\tilde{a}_{m},{\bf x}_{m},{\bf\tilde{x}}_{m}$ constant and $N,\widetilde{N}$ integers. In particular if $V_{0}=\widetilde{V}_{0}\neq 0,N=\widetilde{N}$ and $\sum a_{m}=\sum\tilde{a}_{m}$ then the metrics (5.1) are AF. The Riemannian Majumdar–Papapetrou metrics have $V=\widetilde{V}$ and purely magnetic Maxwell field $F=-2\star_{3}dV$. See [12] for other choices which lead to AE, ALE and ALF solutions.

There also exist Einstein–Maxwell instantons with no Lorentzian counterpart, and anti–self–dual Weyl curvature [39, 40]. An example is the Burns metric

It is the unique scalar–flat Kähler metric on the total space of the line bundle $\mathcal{O}(-1)\rightarrow\mathbb{CP}^{1}$. It is also an AE Einstein–Maxwell gravitational instanton, with the self–dual part of the Maxwell field strength given by the Kähler form, and its anti–self–dual part given by the Ricci form. It is one of few gravitational instantons where the isometric embedding class is known: It has been shown in [16] that (5.3) can be isometrically embedded in $\mathbb{R}^{7}$, but not in $\mathbb{R}^{6}$.

The twistor non–linear graviton approach of Penrose [46] parametrises holomorphic anti–self–dual Ricci flat metrics in terms of complex three-folds with 4–parameter family of rational curves and some additional structures. The Riemannian version of this correspondence have been given by Atiyah, Hitchin and Singer [2], where the twistor space is the six–dimensional manifold arising as an $S^{2}$–bundle over a Riemannian manifold $(M,g)$. Each fiber of the $S^{2}$–fibration parametrises the almost–complex structures in $M$. The twistor space is itself an almost–complex manifold, and its almost–complex structure is integrable iff (with respect to a chosen orientation on $M$) the Weyl tensor of $g$ is ASD.

This formulation is well suited to the study of gravitational instantons. In particular the ALE class can be fully characterised twistorially [33, 35, 36, 34]. In this case there exists a holomorphic fibration $PT\rightarrow{\mathcal{O}}(k)$ for some integer $k$. If $k=2$, then the associated instanton admits a tri–holomorphic Killing vector and belongs to the $A_{N}$ Gibbons–Hawking class (3.2), [49]. If $k>2$ then in general $(M,g)$ does not admit a Killing vector, but it admits tri–holomorphic Killing spinor which leads to a hidden symmetry of the associated heavenly equations [13, 14].

Euclidean quantum gravity which gave rise to the initial interest in gravitational instantons in the late 1970 does not any more aspire to the status of a fundamental theory of quantum gravity. According to Gary Gibbons’s interesting account [27], it never did. And yet it is the only theory of quantum gravity with experimental predictions, including the black hole thermodynamics. In this theory the gravitational instantons dominate the Euclidean path integral. So if a quantum gravity theory exists, and if it reduces to Einstein’s general relativity in the classical limit, then Euclidean quantum gravity is here to stay, and will occupy a place similar to that the WKB approximation has in the quasi-classical limit relating the quantum mechanics to Newtonian physics. This short, and subjective review has focused on recent, and not so recent, mathematical development. It remains to be seen what role will the gravitational instantons play in physics in the years to come.