Non-Relativistic Twistor Theory and Newton–Cartan Geometry

We develop a non–relativistic twistor theory, in which Newton–Cartan structures of Newtonian gravity correspond to complex three–manifolds with a four–parameter family of rational curves with normal bundle O⊕O(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O} \oplus \mathcal {O}(2)}$$\end{document}. We show that the Newton–Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton–Cartan connections can nevertheless be reconstructed from Merkulov’s generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non–trivial on twistor lines. The resulting geometries agree with non–relativistic limits of anti-self-dual gravitational instantons.


Introduction
Over the last 6 years there has been large interest in the AdS/CFT correspondences providing the gravity duals of non-relativistic gauge theories relevant in solid-state physics [15]. A non-relativistic field theory should have a dual made out of a nonrelativistic theory of gravity. The mathematical structure of such a theory-the Newton-Cartan (NC) space-time-is rather baroque. It consists of a torsion-free non-metric connection ∇, a parallel degenerate metric h and a parallel one-form θ in the kernel of this metric [2,8,32].
In this paper we shall construct a non-relativistic twistor theory for Newton-Cartan space-times. There are several equivalent definitions of twistors for flat Minkowski space, but they lead to nonequivalent pictures once a non-relativistic limit is taken. One could define twistors as spinors for the conformal group SU (2,2) and then take a Wigner contraction corresponding to a limit where the speed of light becomes infinite [12]. This approach has to be abandoned in the search for an analogue of a curved twistor correspondence [25] as a general space time does not admit any conformal isometries. Thus the linear/group theoretic structures of the twistor space become irrelevant.
We shall instead concentrate on the holomorphic geometry of the twistor space and its one-dimensional complex submanifolds. In the non-relativistic limit the normal bundle of the twistor curves corresponding to space-time points jumps from O(1) ⊕ O (1) to O ⊕ O (2), where O(n) is the holomorphic line bundle over CP 1 with the Chern class n. This results in the conformal structure becoming degenerate. We shall show that the Newtonian space-times are unstable under the general Kodaira deformation of the twistor complex structure.
The Newton-Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non-trivial on twistor lines. The resulting geometries agree with non-relativistic limits of anti-self-dual gravitational instantons.
The paper is organised as follows: In the next Section we shall review the Newton-Cartan geometry, and show that in the presence of the time-independent Coriolis force, the general Newton-Cartan connection satisfying the Einstein equations is determined by a couple of harmonic functions on R 3 . In Sect. 3 we shall explain how the NC structures arise as limits from general relativity. We shall give examples of NC limit of the Lorentzian Taub-Nut metric, anti-self-dual (ASD) gravitational waves, and Gibbons-Hawking gravitational instantons. The one-parameter family of anti-self-dual Ricci flat metrics depending on a harmonic function V and a one-form A on R 3 such that ∇V = ∇ ∧ A admits a Newtonian limit (h = lim →0 (g −1 ), θ = dτ, ∇ = lim →0 ∇ g ) with the unparametrised geodesic equations of ∇ given by d 2 x/dτ 2 = −(1/2)∇V . In Sect. 4 we shall introduce the spinor calculus adopted to a 3+1 splitting, leading to Newtonian spinors. In Sect. 5 we shall review the relativistic twistor theory, and proceed to construct its non-relativistic limit. This will be done by realising the relativistic twistor space as an affine line bundle over the total space of the bundle O(2) → CP 1 , taking a limit of the resulting patching matrix (Sect. 5.2.1) and then (Sect. 5.2.2) by directly analysing the incidence relation for a complexified Minkowski space M C with a speed of light c = 0. We shall prove the following Theorem 1.1. Let P T c be a one-parameter family of rank-two holomorphic vector bundles over CP 1 determined by a patching matrix (1) If c is finite, then the holomorphic sections of P T c → CP 1 have normal bundle O(1) ⊕ O (1). The four-dimensional moduli space M C of these sections carries a flat conformal structure such that two points in M C are null separated iff the corresponding sections intersect at one point in P T c . (2) If c = ∞ then the holomorphic sections of P T c → CP 1 have normal bundle O ⊕ O (2). The four-dimensional moduli space M C of these sections admits a fibration M C → C defined by a closed one-form θ ∈ 1 (M C ), and a contravariant metric h of rank 3, which is non-degenerate on ker(θ ).
(3) For any c = 0 (finite or not) there exists an involution σ : PT c → PT c which restricts to an antipodal map on each section, and such that the σ -invariant sections form a real four-manifold M with structures as in (1) and (2).
This Theorem will be established in Sect. 5.2.1. We shall explicitly construct the sections of PT c that will allow us to identify c with the speed of light. If c = ∞ then F ∞ = diag(1, λ −2 ) and PT ∞ = O ⊕ O (2), but for finite c there exist holomorphic splitting matrices H and H such that F = H diag(λ −1 , λ −1 ) H −1 and PT c = O(1) ⊕ O (1). For all non-zero c there exists a holomorphic fibration PT c → O (2). This fibration is trivial if c = ∞, which gives rise to a global twistor function on PT ∞ and consequently to a closed one-form θ on M C . The conformal freedom in θ and h are both fixed by a holomorphic canonical one-form on the base of the fibration PT ∞ → CP 1 .
In Sect. 6 we shall discuss the Kodaira deformation theory of the complex structure underlying the non-relativistic twistor space. We shall exhibit a class of deformations that do not preserve the type of the normal bundle of twistor curves, and lead to Gibbons-Hawking metrics. This is a consequence of the non-vanishing of the obstruction group is the normal bundle to non-relativistic twistor lines. We shall also discuss deformations for which the twistor curves do not change their holomorphic type.
In Sect. 7 we shall use Merkulov's relative deformation theory to give a construction of Newtonian connections on M with no Coriolis term. We shall establish Theorem 1.2. There is a 1-1 correspondence between line bundles over P T ∞ which are trivial on σ -invariant sections of P T ∞ → CP 1 , and Newtonian connections on M.
It may appear that all Theorem 1.2 does is reinterpret the Penrose transform between σ -invariant sections of H 1 (PT ∞ , O) and solutions to the Laplace equation on R 3 . In fact, we are saying much more than that. The Newtonian connections of Theorem 1.2 will arise as a family of morphisms 2) where N F is the normal bundle to the correspondence space in the product manifold M C × PT ∞ and where L p = CP 1 is a rational curve in PT ∞ corresponding to a point p in the complexified Newtonian space-time M C . The contribution from the H 0 (CP 1 , O(2)) = C 3 factor in (1.2) corresponds (by the Serre duality) to a zero-rest-mass field on M C arising from a gravitational potential V in the Gibbons-Hawking family (1.1) with = c −2 . In Sect. 8, rank-two vector bundles that restrict to O ⊕ O(2) on twistor curves will lead to connections with non-zero Coriolis term.
In Appendix 1 we shall exhibit and interpret a contour integral formula for solutions to the 2+1 Schrodinger equation 2m∂ u ψ = i(∂ x 2 + ∂ y 2 )ψ. Finally in Appendix 2 we shall show how the spin connection in the Nonlinear Graviton construction arises as a Ward transform of a rank-two holomorphic vector bundle over the relativistic deformed twistor space.

Newton-Cartan Gravity
The trajectories of test particles in Newtonian physics with a chosen universal time t are integral curves x = x(t) of the system of ODEs where V : R 3 → R is the Newtonian potential. One normally interprets these equations as giving rise to curved paths in a fixed three-dimensional space. The geometric perspective of Cartan [2] differs in that the particle trajectories are instead regarded as geodesics of some connection in a four-dimensional space time M. To introduce the necessary structures consider a four-manifold M equipped with a triple (h, θ, ∇), where h is a degenerate contravariant metric of signature (0, +, +, +), θ is a one-form which belongs to a kernel of h (when viewed a map from T * M to T M), and ∇ is a torsion-free affine connection on M with covariant derivative ∇ which keeps h and θ parallel. In local coordinates The last two conditions do not specify ∇ uniquely. The most general connection satisfying (2.2) is parametrised by an otherwise irrelevant choice of time-like vector U such that θ a U a = 1 and a choice of two-form F. The most general connection such that where ∂ a = ∂/∂ x a , and h ab is a degenerate metric uniquely determined by the conditions where = a dx a is an arbitrary one-form, and Thus, if ∇ is a Newtonian connection (so that d F = 0), then one can locally set F = 0 and instead work with a non-trivial vector field U . This approach is adapted in [29]. The absence of torsion of ∇, and the consistency conditions (2.2) imply the existence of a function t : M → R such that θ = dt. The three-dimensional distribution spanning the kernel of θ is integrable in the sense of the Frobenius theorem, and the manifold M admits a fibration over a real line R called the time axis, with a coordinate t M → M/ker (θ ) = R.
Thus the space-time M in the Newton-Cartan theory is a fibre bundle over a universal time axis equipped with a Riemannian metric h on the fibers.
The Einstein field equations are where R ab is the Ricci tensor of ∇, the function ρ : M → R is the mass density and G is the gravitational constant. These equations in particular imply that the metric h on threedimensional spatial fibres is flat. This allows an introduction of the Galilean coordinates x a = (t, x) such that (in general indices a, b, . . . run from 0 to 3 and i, j, . . . run from 1 to 3), Let us spell out some physical consequences of the Newton-Cartan formalism. In general, the free falling particles follow the geodesics of ∇ which however do not need to be affinely parametrised. The statement that two events are simultaneous in Newtonian physics is invariant and coordinate independent. This is a consequence of the existence of the one-form θ and the resulting fibration M → R. On the other hand one can not say that two events occur at the same point in 3-space but at different times. To be able to compare two non-simultaneous events one needs to make an additional choice of Galilean coordinates such that the spatial part of the Newton-Cartan connection vanishes. The existence of such coordinates is a consequence of ∇h = 0, and the flatness of h which itself follows from the field Eq. (2.5). Moreover the distance between two events is only invariantly defined if these events are simultaneous.

Example 1: Newtonian connection.
Take h to be a flat metric on the spatial fibres, i. e. h i j = δ i j , θ = dt and consider i 00 = δ i j ∂ j V with all other components of ∇ equal to zero. In this case F = dt ∧ dV, and the geodesics of ∇ parametrised by t are the integral curves of the Newton equations of motion (2.1). The coordinate transformation where R is an element of SO (3) which is allowed to depend on t, gives rise to the 'Coriolis forces' given by the connection componentsˆ i 0 j =Ṙ k j R ik . A combination of an inertial forceä and a centrifugal forceRx also appear as components of the connection asˆ The form (2.1) of the geodesic equations is preserved only ifṘ = 0,ä = 0.

Example 2: Newton-Cartan connection with the Coriolis force.
The Newtonian connection of the previous subsection can be modified to a Galilean connection which includes the general space-time dependent Coriolis force B = B(x, t). In this case the closed two-form F is and the corresponding non-zero connection components are where θ = dt, h ab is a degenerate metric of signature (0, 1, 1, 1) such that h ab θ b = 0 which in our case is chosen to be diag(0, 1, 1, 1). The field equations together with the Trautman condition imply B = ∇W , for some function W , and At this stage we can not assume that E = −∇V . This can be either put in by hand, or derived from an asymptotic condition that B → B(t) at spatial infinity. This condition, together with the uniqueness theorem for the Laplace equation implies that B can be set to zero. This is because if which can be set to zero by a choice of an orthogonal matrix R(t). Alternatively, the closure of F implies that E arises from a potential if B does not depend on t. If E = −∇V , then so that the connection is determined by two harmonic functions. In Sect. 8 we shall show how two independent harmonic functions arise from holomorphic vector bundles over a Newtonian twistor space PT ∞ which restrict to a non-trivial vector bundle O ⊕ O(2) on each twistor line.

Newton-Cartan Geometry as a Limit of General Relativity
In this section we shall show how Newton-Cartan geometry arises as a degenerate limit of General Relativity [3,19]. Let (M, g( )) be a family of pseudo-Riemannian manifolds parametrised by > 0, and such that where h ab is a contravariant tensor on M of signature (0, +, +, +). The following result has been established in [19].
Theorem 3.1. Let θ = θ a dx a be an -independent one-form such that h ab θ a = 0, normalised by k ab θ a θ b = −1.
If the one-form θ is closed, then the Levi-Civita connection of g( ) has a well defined limit ∇ as goes to zero. It is a Newton-Cartan connection (2.3) with d F = 0.
The Newton-Cartan structure resulting from Theorem 3.1 is (h, ∇, θ). The Ricci tensor of g( ) also has a well defined limit, and choosing a perfect fluid energy-momentum tensor for the relativistic theory leads (in the limit) to the field Eq. (2.5). See [3] for details of this derivation.
Another way to proceed [11] is to define a one-parameter family of structures {h ab , t ab , ∇} such that (t, h) are parallel with respect to ∇ and t ab h bc = − δ c a . For > 0 this gives pseudo-Riemannian geometry with g ab = −t ab / , and for = 0 the Newton-Cartan theory with t ab = ∇ a t∇ b t, and θ a = ∇ a t.

Examples.
We shall now give three examples of Newtonian limits corresponding to particular solutions of Einstein equations. The first example is a minor modification from [11] (where some factors appear to be wrong), and corresponds to a non-zero Coriolis force and a Newton-Cartan structure of the type (2.7). The remaining two examples give limits of anti-self-dual solutions and thus correspond to analytic continuation of the Newton-Cartan theory. The resulting geodesic equations can be analytically continued to Newton's equations of motion despite the fact that ASD gravitational instantons in Riemannian signature, or ASD pp-waves in neutral signature do not admit Lorentzian analytic continuations. Our motivation for considering the limits of ASD solutions will become clear in Sect. 7.2 when we construct non-relativistic twistor spaces from a limiting procedure.
3.1.1. Newtonian limit of Taub-NUT Consider a one-parameter family [11] of Lorentzian Taub-NUT metrics, with the mass parameter m, and the NUT charge a Employing the Cartesian coordinates (x, y, z) r = x 2 + y 2 + z 2 , θ = arccos z and the Coriolis force Both forces arise from scalar potentials V and W = −a/r , but only W is a harmonic function. The two potentials satisfy which implies the Ricci-flat condition (2.8).

Newtonian limit of gravitational instantons
Let V and A be respectively a harmonic function, and a one-form on R 3 which satisfy the Abelian monopole equation where * is the Hodge endomorphism of the flat metric on R 3 . The one-parameter family of Gibbons Hawking metrics [13] has anti-self-dual Weyl tensor and is Ricci-flat. Conversely, any Riemannian ASD Ricci-flat (and so hyper-Kähler) four-manifold admitting a Killing vector preserving the hyper-Kahler structure is of the form (3.1). The Newtonian limit of (3.1) is where all other components of a bc vanish. For example V = 1/r corresponds to the ASD Taub-NUT metric. It is an asymptotically locally flat (ALF) gravitational instanton. The metric (3.1) is everywhere regular, and asymptotically approaches the non-trivial S 1 fibration over S 2 with the monopole number 1. In the Newtonian limit the fibration becomes trivial, so that the connection is asymptotically flat, but the genuine singularity arises at r = 0.

Newtonian limit of anti-self-dual plane waves
The most general ASD Ricci-flat metric admitting a parallel Killing vector is of the form [27] If the coordinates (w, z, x, y) on M, and the arbitrary function γ = γ (w, y) are taken to be real, then the metric has signature (2, 2). Set and redefine γ . This, for a given γ , gives a one-parameter family of metrics with a Newtonian limit where now γ = γ (t, y). This structure can, for certain γ s, be analytically continued to a Newtonian theory, 1 where the signature of h is (0, 1, 1, 1). 1 Consider a different way of taking the limit, and set z = t/ √ + u, y = t/ √ − u so that

Newtonian Spinors
Let (M, g) be a (pseudo) Riemannian four-manifold. Locally there exist complex ranktwo vector bundles S and S over M equipped with covariantly constant symplectic structures ε and ε parallel w.r.t the Levi-Civita connection and such that is a canonical bundle isomorphism, and for p 1 , p 2 ∈ (S) and q 1 , q 2 ∈ (S ). We use the conventions of Penrose and Rindler [26], where the spinor indices are capital letters, unprimed for sections of S and primed for sections of S . For example p A denotes a section of S, and q A a section of (S ) * . The symplectic structures ε AB and ε A B (such that ε 01 = ε 0 1 = 1) are used to lower and raise the spinor indices according to The properties of the complex conjugation on spinors depends on the signature of g.
• If g is Lorentzian, then the spinor complex conjugation is a map S → S given by Linear SL(2, C) transformations on S induce Lorentz transformations on vectors, and SO(3, 1) = SL(2, C)/Z 2 . • If g is Riemannian, then the complex conjugation preserves the type of spinors, i.e.
it maps sections of S to sections of S and sections of S to sections of S . It is given by In the Riemannian case the structure group is not simple, and SO(4, R) = SU (2) × SU (2) /Z 2 , where the spin groups SU (2) and SU (2) act linearly on sections of S and S respectively. Now assume that g has Lorentzian signature, and chose a time-like unit vector T to perform the 3+1 split of g. This vector gives a map from S to S given by [28] where we have chosen ∂t .
This appears to be different than (3.4), but isn't, as the coordinate is the normalised basis of S. Under this map the isomorphism (4.1) becomes where 2 (S ) is a rank-one vector bundle, and Sym 2 (S ) is a rank-three vector bundle isomorphic to a bundle of self-dual two-forms on M.
The isomorphism (4.5) gives the orthogonal decomposition: The symmetric matrix V (A B ) represents a three-vector in the space t =const and the multiple of ε A B is the component in normal direction. Consider the flat metric In our discussion of the Newtonian limit of general relativity in Sect. 3 we needed to single out a time-like one-form θ a = ∇ a t in the kernel of the three-metric h. We shall use the dual vector-field to perform the 3+1 decomposition (4.6), and construct the isomorphism (4.3). Therefore the two-component spinor formalism for Newton-Cartan structures involves only one type of spinors (which we choose to be sections of S ). It is natural to ask whether the Newton-Cartan connection on T M is induced from some spin connection on S → M. The answer to this question appears to be 'no' which is related to the fact that the Galilean boosts do not have a double cover. A Newton-Cartan connection takes its values in the Lie algebra of the Galilean group. However we find that the homogeneous Galilean transformations can not be realised spinorially as It is nevertheless possible to construct a spin connection in the Newtonian limit, although to achieve a non-zero result the speed of light has to be regarded as a part of the constant metric on the fibres of T * M in the Cartan formalism. To see it consider the post-Newtonian limit of GR given by Employing the spin frame (ce V /c 2 dt, dx i ) yields a spin connection which vanishes in the limit c → ∞. Let us instead consider so that (4.11) is given by g = η ab e a e b . Then the Cartan equations de a + ω a b ∧ e b = 0 together with a limit c → ∞ yield the non-vanishing connection given by

Review of relativistic twistor theory.
In this section we shall recall basic facts about twistor theory, following [4,26].
The theory was put forward by Roger Penrose [23] in the late 1960s. The primary motivation was to unify general relativity and quantum theory in a non-local theory based on complex numbers, but a good starting point for our discussion is Penrose's contour integral formula for wave equation in Minkowski space [10,24] φ(x, y, z, t) where ⊂ CP 1 is a closed contour and the function f is holomorphic on an intersection of two open sets in CP 1 away from poles inside . Differentiating inside the integral shows that The formula (5.1) gives real solutions to the wave equation in Minkowski space from holomorphic functions of three variables, and the next step is to recognise these variables as coordinates in some complex three-dimensional manifold. The five dimensional space P N of light rays in the Minkowski space is a hypersurface in a three-dimensional complex manifold PT = CP 3 −CP 1 called the projective twistor space. Let (ω 0 , ω 1 , π 0 , π 1 ) ∼ γ (ω 0 , ω 1 , π 0 , π 1 ), γ ∈ C * with (π 0 , π 1 ) = (0, 0) be homogeneous coordinates of a twistor. The Minkowski space and the twistor space are linked by the incidence relation Fig. 1. Twistor relativistic incidence relation. Two points on an α-plane in M C correspond to two twistor lines passing through a point in PT where x μ = (t, x, y, z) are coordinates of a point in Minkowski space. It then follows that if two points in Minkowski space are incident with the same twistor, then they are separated by a null geodesic. The spinor form of the incidence formula above is where the displacement vector x A A is given by (4.8).
Let Z = (ω A , π A ) represent homogeneous coordinates of a twistor. The nonprojective twistor space C 4 − C 2 is equipped with a Hermitian inner product The signature of is (+ + −−) and endomorphisms of C 4 preserving as well as the orientation of C 4 form a group SU (2, 2) which is locally isomorphic to the conformal group of the Minkowski space. The projective twistor space is divided into three regions depending on the sign of (Z ,Z ). The hypersurface is preserved by the conformal transformations of the Minkowski space. Fixing the coordinates (x, y, z, t) of a space-time point p in (5.3) gives a projective line L p = CP 1 in PT . If the coordinates of p are real, then L p is contained in the hypersurface P N . Conversely, fixing a point in P N gives a light-ray in the Minkowski space.
Points in PT /P N can be interpreted in terms of the complexified Minkowski space M C = C 4 where they correspond to null two-dimensional planes (α-planes) with self-dual tangent bi-vector. This, again, is a direct consequence of (5.3) where now the coordinates x A A are complex (Fig. 1).
There is a CP 1 worth of α-planes through each point of M C . For any π A ∈ CP 1 a corresponding α-plane is spanned by a two-dimensional twistor distribution Finally we can give a twistor interpretation of the contour integral formula (5.1). Consider a function f = f (ω A /π 1 , π 0 /π 1 ) which is holomorphic on an intersection of two open sets covering PT (one of these sets is defined by π 1 = 0 and the other by π 0 = 0). Restrict f to a rational curve L p ⊂ P N and integrate f along a contour in L p . This yields (5.1) with λ = π 0 /π 1 . The function f , when regarded as defined on the non-projective twistor space, must be homogeneous of degree −2.

Non-relativistic twistor theory.
Our construction of a non-relativistic twistor space for the flat Newtonian space time will proceed in two steps. We shall first represent the relativistic twistor space introduced in the last section as an affine line bundle over O (2), and show that in the limit c → ∞ the normal bundle of sections of PT → CP 1 jumps (2). This will be done is Sect. 5.2.1. In Sect. 5.2.2 we shall relate the jumping lines approach to a non-relativistic limit of the 3+1 split of the twistor incidence relation.
and set λ = π 0 /π 1 on U ∩ U . The Birkhoff-Grothendieck theorem states that any rank-k holomorphic vector bundle over CP 1 is isomorphic to a direct sum of line bundles for some integers m 1 , m 2 , . . . , m k . Moreover the transition matrix F : C * → G L(k, C) of this bundle can be written as where H : U → G L(k, C) and H : U → G L(k, C) are holomorphic. Let PT c −→ CP 1 be a one-parameter family of rank-two vector bundles determined by a patching matrix where c is a constant. If c = ∞ then F ∞ = diag(1, λ −2 ) is the patching matrix for PT ∞ = O ⊕ O (2) with H and H in (5.9) both equal to the identity matrix. From now on we shall identify the constant c with the speed of light, and refer to PT ∞ as the Newtonian twistor space. If c = ∞ then if the speed of light is non-zero and finite. This is the relativistic twistor space, with the holomorphic sections of PT c → CP 1 parametrised by points in the complexified Minkowski space M C .
Let x A A given by (4.8) be a displacement vector of a point in M C from the origin, and let T A A be a unit vector (4.4) with respect to the flat metric (4.7). Set and define inhomogeneous coordinates (Q, T ) and ( Q, T ) in pre-images of U and U in PT c by On the pre-image of U ∩ U in PT c we have Note that on PT ∞ there exists a global twistor function of weight 0, given by T = T . where Q and T are given by (5.12), and the variation δ is w. r.t the moduli (x, y, z, t) and not λ. If c is finite, then the conformal structure on M C is determined by det(dx A A ) = 0, and the conformal factor is fixed by a two-form d Q ∧ dT on the fibres of PT c . If c = ∞, then the conformal structure on Minkowski space degenerates in the following way: The equation δT = 0 implies that t = const, and in the Newtonian limit the null separation implies simultaneity of events. We also deduce the existence of a canonical closed oneform dT on PT ∞ which yields θ ≡ dt on M C . This is the clock of Newton-Cartan theory which defines a fibration of M C over the time axis

Restricting the inhomogeneous coordinates to a twistor line ω
The section of N (L p ) will have a single zero if the two roots of the remaining equation δ Q = 0 coincide. Evaluating the discriminant of the quadratic form leads to a degenerate conformal structure on M C , which is determined by h = dx 2 + dy 2 + dz 2 on threedimensional fibres of (5.13). The conformal factor is fixed by the canonical O(2)-valued one-form ε A B π A dπ B on the fibres of PT ∞ → CP 1 by h ab = ε A (C ε D )B . The relation between h ab and h ab is provided by a unique choice of a vector field U such that U θ = 1, and h ab U a = 0. This gives a degenerate contravariant metric on T * M C such that h ab θ a = 0. The discussion of the intrinsic definition of the connection is postponed to Sect. 7. Now we shall move on to discuss the reality conditions and establish point (3) in Theorem 1.1. The Euclidean real slice of the complexified Minkowski space M C is characterised by an involution σ : PT c → PT c given by so that σ 2 = Id on the projective twistor space. This involution has no fixed points, because on the non-projective twistor space it squares to minus identity, and so for any ]. These are the real twistor curves, which correspond to points in the Euclidean slice of M C . These σ -invariant sections are characterised by so that t = −t, and t = iτ is purely imaginary. Thus the real Newtonian twistor lines in PT ∞ are where (x, y, z, τ ) are real coordinates on M.
It is worth remarking that in the limit when c → ∞ the real (i.e. σ -invariant) twistor curves lie on the hypersurface (5.6). To see it consider the inner product c := c −1 , where is given by (5.5). Then Thus the Newtonian twistor space is divided into two regions Re(T ) > 0 and Re(T ) < 0 separated by the five-dimensional space P N ∞ of null twistors Re(T ) = 0. The difference between the Lorentzian reality conditions (twistor curves contained in P N c ) and Riemannian reality conditions (twistor curves invariant under an anti-holomorphic involution) disappears in the Newtonian limit c → ∞.
To sum up, we have exhibited a realisation of the relativistic twistor space O(1)⊕O(1) as an affine line bundle over the total space of the line bundle O(2) → CP 1 . In the limit c = ∞ the holomorphic structure of PT c changes discontinuously: The total space of PT c , and the normal bundle of the twistor curves both jump to O ⊕ O (2). Note that this jumping is a purely holomorphic phenomenon: the bundles PT c are all the same for any c finite or not from the topological perspective. Our findings agree with the analysis of [31] (as well as with the recent work [17]), where the jump from O(1) ⊕ O(1) to O ⊕ O(2) was interpreted as a singularity of the conformal structure.

Non-relativistic incidence relation
Combining the (3 + 1) splitting (4.9) with the incidence relation (5.4) yields where x A B is given by (4.9), and ω A = T A A ω A . This equation blows up in the limit c → ∞ but the twistor function ω = ω A T A B π B remains finite. If a pair (x A B 0 , t 0 ) solves the twistor Eq. (5.16), then so does for any constant spinor ρ A ∈ (S ). The image of the two dimensional twistor distribution (5.7) tangent to α-planes under (4.6) is In the Newtonian limit this gives an integrable 2-plane distribution (the mini-twistor distribution) on the constant time three-dimensional fibres of M → C The non-relativistic twistor space PT ∞ is the three-dimensional space of leaves of D in the correspondence space M C × CP 1 . To specify a leaf pick a point on the time axis, and choose a two plane (a point in a mini-twistor space O(2)) which is null with respect to the holomorphic metric dx 2 + dy 2 + dz 2 on the fibres of (5.13). Thus, in the Newtonian limit the α-planes become space-like, and they lie on the fibres of (5.13) (Fig. 2).
as the last map is, in the spinor notation, given by clearly projecting onto O(2) ⊕ O, and the decomposition of C 4 into C 3 ⊕ C is given by (4.6). This is a complexified picture. If the reality conditions (4.2) and (5.15) are imposed, then the intersection of a complex α-plane with a real τ = const slice is a real straight line-a geodesic of the spatial metric-with a tangent vector π (A π B ) (Fig. 3).
The twistor curves invariant under the involution (5.14) correspond to points in the flat Newtonian space-time M = R 4 . The points in PT ∞ correspond to oriented straight lines on the fibres of M → R. Thus two real curves L 1 and L 2 in PT ∞ either do not intersect, or intersect at two points Z and σ (Z ) which correspond to a straight line (with two possible orientations) joining p 1 and p 2 in M.

Kodaira Deformations
A way to introduce the curvature on M C in relativistic twistor theory is to deform the complex structure of PT c . The Kodaira theorems [18] guarantee that, under the additional assumption of the deformation being 'small', the deformed twistor space will still admit a four-dimensional moduli space M C of rational curves with normal bundle O(1) ⊕ O(1). The Nonlinear Graviton theorem of Penrose [25] then implies that M C admits a holomorphic conformal structure with anti-self-dual (ASD) Weyl tensor, and moreover that all ASD conformal structures correspond to some deformed twistor spaces. If the deformation preserves the fibration of PT c over CP 1 , and the symplectic two-form ε AB dω A ∧ dω B on the fibres of this fibration, then the resulting conformal structure contains a Ricci-flat metric.
In this section we shall show that the Kodaira deformation theory, when applied to a Newtonian twistor space PT ∞ , in general leads to the jumping lines phenomenon: the deformed twistor curves have normal bundle O(1) ⊕ O(1), and thus M C carries a nondegenerate metric, rather than a Newton-Cartan structure. We can phrase it by saying that the Newtonian space-times are unstable under the general Kodaira deformations, but we should note that this instability is a purely holomorphic feature of the underlying twistor space, and not a dynamical process. Therefore where t does not depend on π by the Liouville theorem and thus can be used as a coordinate on the moduli space of curves in L. The deformed CP 1 s (in a patch containing λ = 0) are Here ι A is a spinor (a choice of which arises because of the non-uniqueness of the splitting). The contours and are homologous to the equator of the twistor curve L p = CP 1 in an intersection of two open sets covering L p , and such that − surrounds the point ρ A = π A in L p (Fig. 4).  (6.4) and the one form A A B = (A B ) together with a function V satisfy the monopole equation is not a twistor distribution, as it does not Lie-derive ω. The deformed twistor distribution (also called a Lax pair) is where e A B are four vector fields on M C . This is a jumping line phenomenon: the deformation changes the normal bundle to O(1) ⊕ O (1). The vector fields in the Lax pair (6.6) give rise, by e A B e A B to a conformally rescaled Gibbons-Hawking metric. Imposing the reality conditions t = iτ , where τ ∈ R and choosing a conformal factor V gives the ASD, Ricci-flat metric Therefore the deformation (6.1) does not preserve the holomorphic type of twistor curves. where α, δ ∈ H 1 (CP 1 , O), O(2)). Thus as the second entry in the sum does not vanish.

A twisted photon construction for
Newtonian space-times. The argument above shows that general Kodaira deformations of PT ∞ do not preserve the type of the normal bundle. We have seen that a deformation which does not preserve the one-form dT certainly leads to jumping lines. We can look at the restricted class of deformations which preserve dT , and thus do not change the normal bundle. This requires the coefficient β in the matrix (6.7) to vanish. One way to achieve it is to take the twistor space to be a product C×Z, where Z is a two-dimensional complex manifold with a three-parameter family of curves with normal bundle O (2). Thus Z is a mini-twistor space for a general Einstein-Weyl structure in three dimensions [16], and the corresponding M is a product of an Einstein-Weyl three-manifold with a line. This is not the structure we are seeking, as the metric (or in fact a conformal structure) on the three-dimensional spatial slices of M will in general be curved, whereas we require it to be flat. In the remainder of this section we shall explore another possibility, and deform the relation between a projective and non-projective twistor space in a way which preserves the one-form dT . This is analogous to the construction of [5] which in turn was motivated by Ward's twisted photon construction [33].
The canonical bundle of PT ∞ restricted to a twistor line is O(−4). The argument works for PT c with any non-zero c, as restricting the canonical bundle κ → PT c to a line L p with a normal bundle N (L p ) gives as Let us assume that the line bundle L = κ * ⊗O(−4) is non-trivial on PT ∞ , but trivial when restricted to twistor lines, and deform the twistor space in a way which preserves the fibration over CP 1 . Thereforeπ where f =h − h is the cohomology class defining L which gives rise to the abelian monopole (6.3). This cohomology class coincides with that from Sect. 6.1 and its splitting is given by (6.2). The deformed twistor distribution Lie-derives e −h π A . To this end we note that the ϒ term can be removed from the Lax pair by a Mobius transformation of π iff is a gradient: If A B = ∇ A B U , then setting In this case the deformed twistor distribution in Frobenius integrable for any function U .

Newtonian Connections from Merkulov's Relative Deformation Theory
The original Non-Linear Graviton construction of Penrose [25] yields Ricci-flat ASD metrics from the holomorphic geometry of associated twistor spaces. In a far reaching generalisation of this construction Merkulov [20,21] developed a twistor correspondence between complete analytic families of complex submanifolds on a given complex manifold, and a class of torsion-free affine connections (which he called the -connections) on the moduli space of these submanifolds. This has subsequently led to the celebrated solution of the holonomy problem [22]. In this section we shall adapt Merkulov's construction to the Newtonian twistor correspondence, and show that the twistor space PT ∞ introduced earlier in this paper leads to a family of Newton-Cartan connections depending on five arbitrary functions. Two of these functions can be set to zero by fixing the overall conformal factors in the degenerate Newtonian metric h, and the associated clock one-form θ in the kernel of h. The remaining three functions correspond to the gravitational force. By analysing the non-relativistic limit of the Gibbons-Hawking twistor space we shall show that the gravitational force arises from a harmonic gravitational potential which is given by a choice of a holomorphic line-bundle over PT ∞ trivial on twistor lines.
Let J p k be an ideal of holomorphic functions on M C which vanish to order k at p ∈ M C . Therefore T p M C = (J p /J p 2 ) * . The second-order tangent bundle T [2] M C is defined as a union over all points in M C of second order tangent spaces T [2] p M C ≡ (J p /J p 3 ) * . Thus an element of (T [2] p M C ) * consists of the first two non-zero terms of a Taylor expansion of a function vanishing at p, and a section of T [2] M C is a second order linear differential operator V [2] = V a (x)∂ a + V ab (x)∂ a ∂ b . We also have an exact sequence , 0), and (V a , V ab ) → V ab . A torsion-free affine connection ∇ on T M C is then equivalent to a linear map where γ (V [2] ) = (V a + a bc V bc )∂ a for some functions a bc which are to be identified with the Christoffel symbols of ∇. In [20,21] Merkulov proposed a twistor construction of a class of maps (7.1). In some cases (e.g. the Nonlinear Graviton construction) Merkulov's construction leads to a unique connection. We will show that in the case of the non-relativistic twistor theory a unique Newtonian connection in Merkulov's class is provided by a natural choice of a holomorphic line bundle over PT ∞ .
The general construction, adapted to three-dimensional complex manifolds PT c which fiber holomorphically over CP 1 , and admit a moduli space M C of holomorphic curves can be summarised as follows. Let U and U be a covering of PT c by two open sets which is a deformation of the covering (5.11). Thus (Q, T, λ) are holomorphic coordinates in U and ( Q, T ,λ) are holomorphic coordinates on U. The patching relation on the overlap We shall make use of the Kodaira isomorphism, which relates vector fields on M C to global sections of the normal bundle N F to the correspondence space where L p = β • α −1 ( p) = CP 1 is the rational curve in PT c corresponding to a point in M C . The patching for this normal bundle is given by where the subscript F means the pull-back to the correspondence space. To exhibit the Kodaira isomorphism take a vector field given by a differential operator V = V a ∂ a on F, and apply it to the twistor functionsw A (x a , λ) to obtain a global section of N F given by The map (7.1) arises by taking a section of T [2] M and similarly applying the associated differential operator to the twistor functions. We find The term involving F A BC is an obstruction to (7.4) representing a global section of N F ; only if F A BC splits as for some 0-cochain {σ,σ } of N F ⊗ 2 N * F will we arrive at In solving the splitting problem (7.5) we have constructed a global section of N F out of a section of T [2] M, and via the Kodaira isomorphism we have therefore constructed a vector field out of a section of T [2] M. This is the map (7.1), and we can read off the connection components from One can check that c ab transforms correctly as an affine connection under a change of coordinates.
The splitting problem (7.5) is solvable for any patching in the isomorphism class N F iff The procedure is however non-unique (and hence leads to a family of connections) Otherwise an element of this cohomology group can be added to {σ,σ }. In the relativistic case N F | L p = O(1) ⊕ O(1) which implies that (7.7) holds, H 0 CP 1 , N F ⊗ 2 N * F = 0, and therefore the splitting is unique.
Example. Merkulov's connection for ASD plane waves To illustrate how this procedure works in practice we will consider the simple twistor space (originally due to Sparling) described by the infinitesimal deformation of the flat relativistic twistor space PT c given by (A4), where f = (ω 0 ) 4 /(4π 0 π 1 ). Integrating the deformation equations and introducing the inhomogeneous coordinates on the deformed twistor space by The holomorphic splitting of these relations, together with the Liouville theorem imply the existence of coordinates (w, z, x, y) on M C such that the twistor curves pulled back from U to F are The ASD Ricci-flat metric resulting from the Nonlinear Graviton construction is of the form (3.3) for γ = −3 y 2 . We shall find the connection on M C directly. We find and all other F A BC = 0. The splitting problem (7.5) can be solved uniquely to givẽ σ 1 00 = −6 w, σ 1 00 = 6 y, and all other {σ A BC } = 0, leading to a connection whose only non-vanishing components are x wy = x yw = − z ww = −6 y.
We have recovered the Levi-Civita connection of (3.3).

Newtonian connections from P T ∞ .
We shall now construct the general class of connections on the moduli space M C of twistor lines in PT ∞ = O ⊕ O (2). Using (5.10) and (7.2) we find The condition F A BC = 0 implies that the connections will be constructed only out of global sections of N F ⊗ 2 N * F . In the relativistic case this would imply c ab = 0, but Therefore we expect c ab to depend on five arbitrary functions. The global sections of N F ⊗ 2 N * F which constitute the Newtonian 0-cochain {σ,σ } ∞ are of the form (with all other components of {σ A BC } ∞ set to zero) for five arbitrary functions (E A B , χ, ) on M C . Using (7.10) and (7.6), and imposing the reality conditions (5.15) yields the non-vanishing connection components where we used the isomorphism (4.5) to replace E A B ∈ (S S ) by a spatial vector E i ∈ (T M C ). The vector E i yields the gravitational attraction, whilst and χ are consistent with ∇h = 0 and ∇θ = 0, where the degenerate metric h, and clock one-form θ are defined up to some conformal factors. When we fix these conformal factors as in Sect. 5.2.1, so that h = diag(0, 1, 1, 1, ) and θ = dτ , then = χ = 0. In particular, we note that this construction cannot generate a Coriolis force. What remains, then, is a connection whose non-zero i τ τ components are determined by a global section {σ Q T T } ∞ of O (2). Using the Serre duality between the elements of H 1 (CP 1 , O(−4)) and H 0 (CP 1 , O(2)) we can fix this connection by where g (−4) is an element of H 1 (PT ∞ , O(−4)) restricted to a twistor line (5.12). The resulting E A B satisfies the zero-rest-mass field equation ∂ A C E A B = 0. In what follows we shall demonstrate that the expression (7.12) arises naturally from a limiting procedure applied to the unique splitting of the 0-cochain corresponding to the Gibbons-Hawking metric.

Proof of Theorem 1.2. In Sect. 3.1.2 we have demonstrated that any Newtonian
connection with the only non-zero components given by i τ τ = δ i j ∇ j V arises as a Newtonian limit of the Gibbons-Hawking metric (3.1). The twistor space of (3.1) is an affine line bundle over the total space of O(2) described by a cohomology class f ∈ H 1 (CP 1 , O). This is in fact what we have recovered in our analysis in Sect. 6.1 leading to the jumping phenomenon and the Kodaira instability of the Newtonian twistor space. Let us assume that the cohomology class is represented by a function f = f (Q, λ, T ) in a patch (5.10) in PT c , and define with f = 0 giving the patching relation for the undeformed relativistic twistor space PT c (compare 5.11). We then have For this twistor space we have N F = O(1) ⊕ O(1) (for finite, non-zero c) and so one can find the unique solution {σ,σ } c to the splitting problem (7.5). The solution is given by where is a contour enclosing ρ A = 0, and ω = (  (2) and giving rise to a non-zero i τ τ via Merkulov's procedure above. This provides a way of fixing the Newtonian 0-cochain σ A BC : we identify it with the c → ∞ limit of the Gibbons-Hawking 0-cochain The only non-vanishing components are given by σ Q T T andσ Q T T in (7.10), where E A B is a zero-rest-mass field on M C . Comparing this limit with (7.12) and replacing ∂ A B by ρ A ρ B ∂/∂ω inside the integral yields where f is an element of H 1 (PT ∞ , O) restricted to a twistor line ω = x A B ρ A ρ B and V is the harmonic function which (up to a constant multiple) gives the Newtonian potential in agreement with (3.2). We conclude that the limiting procedure yields a canonical element of H 1 (PT ∞ , O(−4)) in (7.12). It is given by g (−4) = −(1/4)∂ 2 f /∂ω 2 . To complete the proof we shall show that f ∈ H 1 (PT ∞ , O) gives rise to a line bundle ν : E → PT ∞ which is trivial on twistor curves. Let U = {(Q, T, π A ), π 1 = 0} and U = {(Q, T, π A ), π 0 = 0} be a covering of PT ∞ and let Example Consider a one-parameter family of Gibbons-Hawking metrics with linear potential [7,14] where = c −2 . This family is ASD and Ricci flat for all > 0. In the limiting case → 0 the metric blows up, the inverse metric degenerates to δ i j , but the Levi-Civita connection has the finite limit with the only non-zero components given by z τ τ = 1/2. The cohomology class defining the line bundle from Theorem 1.2 gives rise to the monopole (6.4) with  .7)). Thus we can identify the connection with a couple of gauge fields on M, and construct it from a generalisation of Ward transform [33] adapted to Newtonian settings. The gauge fields needed to reconstruct the gravitational and Coriolis parts of the connection consist of a couple of abelian monopoles. The one-forms in the monopoles will give rise to an electric and magnetic fields on M, and these fields will in turn give rise to a connection. This is a legitimate approach, as electromagnetic forces are inertial in the Newton-Cartan theory: the equations of motion of a charged particle under the influence of the electric field E and magnetic field 2B

Newton-Cartan Connections from Holomorphic Vector Bundles
can be reinterpreted as a geodesic motion corresponding to a connection where now E is the gravitational acceleration, and B is the Coriolis force-compare (2.7). The field B is a gradient of a harmonic function W , and (assuming that B is timeindependent) E = −∇V is also a gradient, but V is not necessarily harmonic. However W 2 + V is harmonic. Let M C be a moduli space of holomorphic sections of the Newtonian twistor space be the five dimensional correspondence space. Here the twistor line L p = CP 1 is a rational curve in PT ∞ with normal bundle O ⊕ O(2) corresponding to p ∈ M C . The correspondence space can be identified with the projective spin bundle P(S ). Let μ : S → F denote the corresponding fibration. Thus a pre-image of any twistor line in F can be pulled back to S , where μ * (L p ) = C 2 − {0}.
Consider a vector bundle E → PT ∞ such that where n > m ≥ −1. Therefore the restriction of E to a twistor line is a non-trivial vector bundle, and the pull back to a pre-image of L p in F of the patching matrix of E can not be split in a way which leads to the Ward twistor correspondence [33]. However pulling back E to the total space of S , and restricting it to a pre-image C 2 − {0} of L p in S is a trivial bundle. The corresponding pulled-back G L(2, C) patching matrix F = F(Q, π A , T ), where Q is given by (5.12), satisfies 2 are holomorphic in U andŨ respectively, where now U andŨ are pre-images of the standard covering of L p in C 2 − {0}. Note that H andH are not homogeneous on C 2 − {0} and so do not descend down to F. Let ∇ A := π B ∂/∂ x (A B ) be the a rank-two twistor (α-plane) distribution on S , such that the three-dimensional space of leaves of the rank-three distribution {∇ 0 , ∇ 1 , π B /∂π B } in S is PT ∞ . Therefore ∇ A F = 0, and the matrix components 2 The splitting matrices H and H can be derived from those in (5.9) using (π 0 /π 1 ) −m 0 0 (π 0 /π 1 ) −n = (π 0 ) −m 0 0 (π 0 ) −n (π 1 ) −m 0 0 (π 1 ) −n −1

.
of H −1 ∇ A H are homogeneous functions on the fibres of S , with the coefficients given by functions on M. Let us rewrite this condition as Here (v −m , r −n ) and (w −m , s −n ) denote a pair of local sections of E, where (v −m , w −m ) are homogeneous of degree −m and (r −n , s −n ) are homogeneous of degree −n when regarded as functions on the fibres of S → M. The polynomials (φ A , ψ A , κ A , δ A ) will give rise to potentials for higher spin fields on M, i.e. sections of various powers of S → M satisfying some field equations [30]. To construct these fields, and find the corresponding field equations rewrite (8.2) as Therefore, as ∇ A is homogeneous of degree one, we conclude that φ A and ψ A are homogeneous of degree one, κ A is homogeneous of degree (n − m + 1), and finally δ A = 0. Equivalently The SL(2, C) irreducible components give rise to two spin-2 fields, two functions, one spin-(n − m) field and one spin (n − m + 2) field.
To find the field equations satisfied by these potentials, we shall impose the integrability conditions on (8.3) arising from ∇ A ∇ A = 0. Contracting each equation in (8.3) with ∇ A and using (8.3) to eliminate the derivatives of H gives Decomposing this last set of equations into irreducible parts yields the final system (which is the non-relativistic version of the Sparling equations [30]) If c = ∞, then E is trivial when restricted to twistor lines, and its Ward correspondence gives a solution to anti-self-dual Yang-Mills field on the complexified Minkowski space. In the Newtonian case (1) corresponds to n = 1, m = −1 in (8.2). The resulting potentials are two abelian monopoles, one spin-4 field and one spin-2 field subject to Eq. (8.4). One can now identify the field strengths of these potentials with various components of the Newton-Cartan connection. Note that it is consistent to set γ A B C D = 0. where L p = CP 1 is a twistor curve corresponding to p ∈ M C . Assume that the ASD metric on M C is vacuum, and so the twistor space fibres over CP 1 . Let f ∈ H 1 (CP 1 , O(2)). To construction the connection on S consider the inifinitesimal deformationω and let U ∈ (E), so that U = U A A π A ∂/∂ω A . The relatioñ To construct the splitting F = H H −1 consider H = 1 + G, H = 1 + G so that, to the first order in , The freedom in splitting the RHS is measured by elements of H 0 (CP 1 , O). Choosing a constant spinor ι A yields ι.π (ι.ρ)(π.ρ) ∂ 2 f ∂ω C ∂ω B ρ.dρ.

The usual Liouville argument gives
for some matrix A A which does not depend on π A . Equivalently Therefore the linearised Ward transform gives a connection This is consistent, in the linearised 2nd Plebanski gauge, with the expression for the ASD Weyl spinor as [6] C ABC (in Plebanski's gauge the quadratic term in the connection on S contracts to zero, and the connection on S vanishes).