Gravitational instanton

Properties

A four-dimensional Kähler–Einstein manifold has a self-dual Riemann tensor.

Equivalently, a self-dual gravitational instanton is a four-dimensional complete hyperkähler manifold.

Gravitational instantons are analogous to self-dual Yang–Mills instantons.

Taxonomy

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.

Examples

It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S³ (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by

σ 1 = sin ⁡ ψ d θ − cos ⁡ ψ sin ⁡ θ d ϕ σ 2 = cos ⁡ ψ d θ + sin ⁡ ψ sin ⁡ θ d ϕ σ 3 = d ψ + cos ⁡ θ d ϕ .

Taub–NUT metric

d s 2 = 1 4 r + n r − n d r 2 + r − n r + n n 2 σ 3 2 + 1 4 ( r 2 − n 2 ) ( σ 1 2 + σ 2 2 )

Eguchi–Hanson metric

The Eguchi–Hanson space is important in many other contexts of geometry and theoretical physics. Its metric is given by

d s 2 = ( 1 − a r 4 ) − 1 d r 2 + r 2 4 ( 1 − a r 4 ) σ 3 2 + r 2 4 ( σ 1 2 + σ 2 2 ) .

where r ≥ a 1 / 4 . This metric is smooth everywhere if it has no conical singularity at r → a 1 / 4 , θ = 0 , π . For a = 0 this happens if ψ has a period of 4 π , which gives a flat metric on R⁴; However for a ≠ 0 this happens if ψ has a period of 2 π .

Asymptotically (i.e., in the limit r → ∞ ) the metric looks like

d s 2 = d r 2 + r 2 4 σ 3 2 + r 2 4 ( σ 1 2 + σ 2 2 )

which naively seems as the flat metric on R⁴. However, for a ≠ 0 , ψ has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R⁴ with the identification ψ ∼ ψ + 2 π , which is a Z₂ subgroup of SO(4), the rotation group of R⁴. Therefore the metric is said to be asymptotically R⁴/Z₂.

There is a transformation to another coordinate system, in which the metric looks like

d s 2 = 1 V ( x ) ( d ψ + ω ⋅ d x ) 2 + V ( x ) d x ⋅ d x ,

where ∇ V = ± ∇ × ω , V = ∑ i = 1 2 1 | x − x i | .

(For a = 0, V = 1 | x | , and the new coordinates are defined as follows: one first defines ρ = r 2 / 4 and then parametrizes ρ , θ and ϕ by the R³ coordinates x , i.e. x = ( ρ sin ⁡ θ cos ⁡ ϕ , ρ sin ⁡ θ sin ⁡ ϕ , ρ cos ⁡ θ ) ).

In the new coordinates, ψ has the usual periodicity ψ   ∼   ψ + 4 π .

One may replace V by

V = ∑ i = 1 n 1 | x − x i | .

For some n points x i , i = 1, 2..., n. This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid conical singularities). The asymptotic limit ( r → ∞ ) is equivalent to taking all x i to zero, and by changing coordinates back to r, θ and ϕ , and redefining r → r / n , we get the asymptotic metric

d s 2 = d r 2 + r 2 4 ( d ψ n + cos ⁡ θ d ϕ ) 2 + r 2 4 [ ( σ 1 L ) 2 + ( σ 2 L ) 2 ] .

This is R⁴/Z_n = C²/Z_n, because it is R⁴ with the angular coordinate ψ replaced by ψ / n , which has the wrong periodicity ( 4 π / n instead of 4 π ). In other words, it is R⁴ identified under ψ   ∼   ψ + 4 π k / n , or, equivalnetly, C² identified under z_i ~ e 2 π i k / n z_i for i = 1, 2.

To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C²/Z_n. According to Yau's theorem this is the only geometry satisfying these properties. Therefore this is also the geometry of a C²/Z_n orbifold in string theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation) [1].

Gibbons–Hawking multi-centre metrics

d s 2 = 1 V ( x ) ( d τ + ω ⋅ d x ) 2 + V ( x ) d x ⋅ d x ,

where

∇ V = ± ∇ × ω , V = ε + 2 M ∑ i = 1 k 1 | x − x i | .

ϵ = 1 corresponds to multi-Taub–NUT, ϵ = 0 and k = 1 is flat space, and ϵ = 0 and k = 2 is the Eguchi–Hanson solution (in different coordinates).