A coupling prescription for post-Newtonian corrections in Quantum Mechanics

The interplay between quantum theory and general relativity remains one of the main challenges of modern physics. A renewed interest in the low-energy limit is driven by the prospect of new experiments that could probe this interface. Here we develop a covariant framework for expressing post-Newtonian corrections to Schr\"odinger's equation on arbitrary gravitational backgrounds based on a $1/c^2$ expansion of Lorentzian geometry, where $c$ is the speed of light. Our framework provides a generic coupling prescription of quantum systems to gravity that is valid in the intermediate regime between Newtonian gravity and General Relativity, and that retains the focus on geometry. At each order in $1/c^2$ this produces a nonrelativistic geometry to which quantum systems at that order couple. By considering the gauge symmetries of both the nonrelativistic geometries and the $1/c^2$ expansion of the complex Klein--Gordon field, we devise a prescription that allows us to derive the Schr\"odinger equation and its post-Newtonian corrections on a gravitational background order-by-order in $1/c^2$. We also demonstrate that these results can be obtained from a $1/c^2$ expansion of the complex Klein--Gordon Lagrangian. We illustrate our methods by performing the $1/c^2$ expansion of the Kerr metric up to $\mathcal{O}(c^{-2})$, which leads to a special case of the Hartle--Thorne metric. The associated Schr\"odinger equation captures novel and potentially measurable effects.


Introduction
General Relativity (GR) is a well-established theory that has been thoroughly tested in many experiments [1], but all tests beyond the Newtonian limit so far have been limited to the classical domain.Usually GR is required to describe physics at very large scales, such as in astrophysical observations or cosmology, but for tests that interface quantum mechanics, laboratory experiments are becoming increasingly relevant [2][3][4][5][6][7].Several experimental routes have recently been proposed to test how general relativity affects the quantum dynamics and imprints signatures in genuine quantum observables at low energies and in the weak limit beyond Newtonian gravity [8][9][10][11][12][13][14].However, such tests in which GR interfaces quantum mechanics, and for which both theories are required, have not yet been realised as the relevant scales are still difficult to reach.An exception is the Newtonian limit: one class of experiments involves matter-wave superpositions in the gravitational field that experience a quantum phase shift due to the presence of the Newtonian gravitational potential [2,4,[15][16][17].Another class of such experiments are bound states in the Newtonian potential of Earth that results in a potential well and discrete energy levels for the bouncing neutrons [18,19].For such experiments Newtonian gravity is entirely sufficient and is typically incorporated by the addition of the Newtonian potential term in the Schrödinger equation.
With the rapid advent of ever more precise measurements of gravitational effects in quantum mechanical systems, developing a systematic framework that combines the laws of quantum mechanics with General Relativity beyond the Newtonian limit is of major interest.We stress that this is not a theory of quantum gravity, but rather a way to capture the gravitational effects of the background spacetime on the quantum systems.Among the myriad of applications of such a framework, let us highlight the effects of gravitational waves on quantum systems [20], post-Newtonian phase shifts [8], entanglement generated by time dilation in composite quantum systems [9], single-photon phase-shifts due to the Shapiro delay [10], decoherence of quantum superpositions due to time dilation [11], as well as quantum formulations of the Einstein equivalence principle [21].While these effects can be described without a systematic framework, some of them have only recently been predicted due to a new-found focus on low energy systems, such as composite quantum systems in the presence of gravity beyond the Newtonian limit [9,11,12,21,22].This has sparked renewed interest in how lowenergy systems interface gravity [23][24][25][26][27] and how this may be probed.Such results highlight the interest in a systematic exploration of this limit, as new and overlooked effects can arise when complex quantum systems start interfacing this regime in laboratory experiments.
The purpose of this paper is to lay down the foundations of a covariant framework that utilises recent advances in nonrelativistic geometry to construct a quantum mechanical theory that takes into account gravitational effects that arise from fixed GR backgrounds.The ultimate goal is to devise a coupling prescription that gives rise to the Schrödinger equation for the centre of mass degrees of freedom of a quantum system coupled to a fixed post-Newtonian background geometry at any given order in 1/c.For both Newtonian gravity and GR, such minimal coupling prescriptions are well-known: in the case of GR, minimal coupling instructs us to replace the Minkowski metric with the background metric η → g, and to replace derivatives with covariant derivatives ∂ → ∇.For Newtonian gravity the minimal coupling to Newton-Cartan geometry follows from coupling the wave function to the metric data and mass gauge field of Newton-Cartan geometry in a manner that respects all the local symmetries of Newton-Cartan geometry (see, e.g., equation (117)).It is therefore natural to ask: what is the analogue of minimal coupling for quantum mechanics in the intermediate regime between Newton and Einstein?While a full answer to this question is likely to involve the 1/c expansion of the Poincaré algebra and its representations, this paper considers the 1 1/c2 expansion of Lorentzian geometry and complex Klein-Gordon theory to construct a theory of quantum mechanics on post-Newtonian backgrounds order-by-order in 1/c 2 .
The time evolution of the quantum mechanical wave function Ψ on 3 is described by the Schrödinger equation where we set ħ h = 1.The Hamiltonian operator H encodes the kinetic energy and the potential energy, and the simplest Hamiltonian that describes a particle of mass m in a gravitational field generated by another body of mass M located at the origin is where G is Newton's constant and ∆ is the Laplacian.Quantum systems described by this Hamiltonian exhibit gravitationally induced phase-shifts that have been measured with neutrons [15,28] and atoms [2,4,16,17,29].Such experiments confirm that the Newtonian interaction can be included in the usual quantum formalism as above, in the same way as any other potential.But one can also obtain the above Hamiltonian starting from a fully relativistic picture: Kiefer and Singh showed in [30] how the Klein-Gordon equation on curved space-time leads to the above Hamiltonian in the weak-field and nonrelativistic limit.Building on these results, Lämmerzahl studied how post-Newtonian corrections and electromagnetic interactions yield a modified Hamiltonian to first order in c −2 in [31].Such corrections can, for example, yield modified phase-shifts [8] which are yet to be observed.More recently, composite quantum systems have also become of interest, where the internal dynamics is affected by gravity through time-dilation and offers new prospects for experimental studies with quantum delocalised systems.The relevant coupling can be derived by simply using the mass-energy equivalence (or sometimes called the mass-defect) in the above mentioned results [9,11,23], as also confirmed from first-principles derivations [24,25].
The basic lesson of General Relativity is that gravity is geometry: gravitational effects arise due to the curvature of the underlying spacetime.This remains fundamentally true for nonrelativistic gravity, where characteristic velocities are small compared to the speed of light.This geometric perspective is not emphasised in the approaches outlined above, but maintaining a geometric view helps highlight how fundamental GR concepts manifest themselves at the relevant scale and illuminates how unique aspects of GR affect the quantum theory.This, in turn, leads to a deeper understanding of how GR and Quantum Mechanics interface conceptually.
What does change in the nonrelativistic regime, however, is the notion of geometry.In the case of General relativity, the underlying geometry is Lorentzian (or pseudo-Riemannian) geometry.Nonrelativistic gravity, on the other hand, is described by non-Lorentzian geometry of Newton-Cartan type.Originally developed by Cartan more than a hundred years ago to provide a geometric framework for Newton's law of gravity [32,33], Newton-Cartan geometry has since been generalised by considering the formal expansion of Lorentzian geometry in inverse powers of the speed of light c [34][35][36][37][38][39] (see also [40][41][42][43]).These more general geometries exist at any order in c and share the same underlying Galilean geometric structure (τ µ , h µν ) consisting of a one-form τ µ and a symmetric tensor h µν with signature (0, 1, 1, 1) whose kernel is spanned by τ µ , i.e., h µν τ ν = 0, where Greek letters represent spacetime indices, µ, ν, • • • = (t, 1, 2, 3).This Galilean structure is what replaces the more familiar metric g µν and its inverse g µν in Lorentzian geometry.To set up the nonrelativistic expansion, we split the metric and its inverse according to which is reminiscent of the "3 + 1 split" of General Relativity [44].The components T µ , T µ , Π µν and Π µν are then expanded in inverse powers of c, for example, where we recognise the Galilean structure (τ µ , h µν ) appearing at leading order (LO): The LO geometry is Galilean [38].
Here, as in the rest of this work, we expand in even inverse powers of c, i.e., we perform a 1/c 2 expansion, for simplicity.As we consider higher order corrections in 1/c 2 , more and more subleading fields such as m µ are included in the geometric description, and their transformation properties are governed by the corresponding 1/c 2 expansion of the local Lorentz transformations and diffeomorphisms (which can be formulated in terms of a 1/c 2 expansion of the Poincaré algebra supplemented with appropriate curvature constraints).These higher order "gauge" fields encode gravitational effects; for example, the time component of m µ is Newton's gravitational potential that features in (2).
The Schrödinger equation ( 1) is nonrelativistic in the sense discussed above: it is only valid when the particle moves slowly compared to the speed of light and the energies are lower than required for particle production.It is well known that it is possible to turn the Klein-Gordon equation on a Lorentzian background into an equation with the same structure as the Schrödinger equation in position space L 2 ( 3 ) by making a WKB-like ansatz for the Klein-Gordon field and expanding in inverse powers of c [13,27,30,31,45].
Our framework builds on these works and complements them by showing that nonrelativistic geometry provides an organising principle behind these expansions, which were previously either highly specific [31] or generic [13].It is interesting to note that the "wave functions" generated by this procedure are not wave functions in the sense of Born, since their inner product is not the standard L 2 ( 3 ) norm.This is because the would-be wave functions inherit the inner product defined on (the positive-frequency part of) the Klein-Gordon solution space, and a field redefinition is required for this to reduce to the L 2 ( 3 ) inner product.
For simplicity, and due to its physical relevance, we take the Galilean structure to be flat, which in Cartesian coordinates amounts to τ µ = δ t µ and h µν = δ µ i δ ν j δ i j with i = (1, 2, 3) a spatial index.Now, both the metric and the wave function are assumed to be analytic in 1/c 2 and hence have well-defined 1/c 2 expansions.Including terms that are one order higher in 1/c 2 means including three extra fields: one from the wave function and one from T µ and Π µν , respectively (cf., Eq. ( 4)).While this preponderance of fields obscures the underlying structure, their transformation properties are all inherited from the relativistic theory and follow from a 1/c 2 expansion of the relativistic gauge symmetries.These gauge symmetries allow us to iteratively write down the Schrödinger equation coupled to a curved post-Newtonian background at any order in 1/c 2 by making sure that all terms in the equation transform correctly under these gauge symmetries.This requires us to derive expressions for covariant derivatives at each order in 1/c 2 , which take on increasingly complicated forms as we go to higher and higher orders in 1/c 2 .This allows us, at least in principle, to write down the Schrödinger equation coupled to an arbitrary post-Newtonian background at any order in 1/c 2 .
Rather than deriving the form of the Schrödinger equation by starting from the flat space result (1) and requiring that it transforms correctly under gauge transformations introduced by coupling to post-Newtonian gravity order-by-order in 1/c 2 , we may also start directly from the Klein-Gordon Lagrangian and expand it in powers of 1/c 2 .At low orders in 1/c 2 , this was also considered in [38].We show that this produces the same Schrödinger equation as our algebraic/gauge-theoretic prescription.
To illustrate our techniques in a concrete setting, we work out the nonrelativistic expansion of the Kerr metric in Boyer-Lindquist form, where in the process of the 1/c 2 expansion we perform a coordinate transformation from oblate spherical coordinates to ordinary spherical coordinates.This leads to the Lense-Thirring metric with an additional term proportional to J 2 where J is the angular momentum.This metric is also the Hartle-Thorne approximation of the Kerr solution.This defines a nonrelativistic geometry to which we may couple the Schrödinger equation using the formalism that we develop.This gives rise to a quantum Hamiltonian that takes into account the gravitational effects from both the mass and the rotation.If we set the rotation equal to zero, we get the 1/c 2 expansion of the Schwarzschild metric in Schwarzschild coordinates.These coordinates are related to isotropic coordinates via a c-dependent coordinate transformation, and we connect our expansion to the 1/c 2 expansion of the Schwarzschild metric in isotropic coordinates, which are the coordinates used in [31].
The paper is structured as follows.In Section 2, we review and further develop the formalism of 1/c 2 expansions.We show how the 1/c 2 expansion leads to a universal Galilean structure at LO, and how the subleading fields that appear in the expansions of (4) encode the information of the Lorentzian spacetime to the given order in 1/c 2 .We then discuss the gauge symmetries of these fields in Section 2.2, where we also consider flat Galilean structures.
In Section 3, we discuss how a WKB-like ansatz for the Klein-Gordon field leads to a Schrödinger-like equation, which in the limit c → ∞ becomes the free Schrödinger equation.Using our results for the gauge structure of nonrelativistic geometry, we then develop a framework in Section 3.1 that allows us to derive the Schrödinger equation on a gravitational background order by order in 1/c 2 .In Section 3.4, we show show how to pass from the inner product of the Klein-Gordon fields to the L 2 ( 3 ) inner product by performing a backgrounddependent field redefinition.We then expand the Klein-Gordon Lagrangian in Section 3.5 and demonstrate that this leads to the same equations of motion as those we obtained using bottom-up methods in Section 3.1.
We then turn our attention to an explicit example in Section 4. We begin by performing the 1/c 2 expansion of the Kerr metric in Boyer-Lindquist coordinates in Section 4.1, leading to a generalised version of the Lense-Thirring metric which takes the form of a nonrelativistic geometry.Having identified the geometric structures, we then apply the formalism we developed in the first part of the paper to write down the Schrödinger equation on this background in Section 4.2.
We conclude with a discussion and outlook in Section 5. We have included Appendix A, which explicitly recovers previous results in the literature using the formalism we develop here.In this appendix, we furthermore discuss subtleties that arise when performing coordinate transformations that explicitly depend on c.We illustrate this by considering the Schrödinger equation on a Schwarzschild background expressed in either Schwarzschild or isotropic coordinates, which are related by a c-dependent rescaling of the radial direction.

Nonrelativistic expansion of spacetime geometry
It is well known that the nonrelativistic limit of a relativistic theory may be obtained by expanding in inverse powers of the speed of light c.Rather than coupling to familiar Lorentzian spacetimes, i.e., pseudo-Riemannian geometries of signature (−1, 1, 1, 1) in four spacetime dimensions, these expanded theories couple to spacetimes that arise by expanding the Lorentzian spacetimes in 1/c.
In this section, we expand Lorentzian geometry in powers of 1/c 2 .Such systematic expansions in inverse powers of the speed of light were considered in [34] (see also [41,42]), and, more recently, an appropriately truncated expansion of the expanded geometry was used to write down an action for nonrelativistic gravity [35,37,38] (see also the review [39]).These geometries do not possess a Lorentzian metric, but rather come equipped with a Galilean structure consisting of a nowhere vanishing corank one "spatial metric" and a nowhere vanishing "clock" one-form.These geometries generalise Newton-Cartan geometry, which was originally conceived by Cartan [32,33] (see, e.g., [46,Ch. 12] and [47] for a pedagogical introduction) to provide a geometric framework in which to formulate Newton's equations of motion in a covariant way, just as Lorentzian geometry provides the geometric framework underlying Einstein's equation.To distinguish this original Newton-Cartan geometry from the one employed in the formulation of off-shell nonrelativistic gravity [35,37,38], the latter geometry was dubbed "type II torsional Newton-Cartan geometry".

1/c 2 expansion of Lorentzian geometry
Consider a (d + 1)-dimensional2 manifold M equipped with a Lorentzian metric g µν (µ, ν = 0, 1, . . ., d).We split the metric into timelike and spacelike components as follows with a similar relation holding for the inverse metric g µν The objects T µ and Π µν and their inverses satisfy the relations We emphasise that this still describes a Lorentzian structure: The above is just a reparameterisation of the metric g µν and its inverse.To turn this into a "nonrelativistic" (NR) geometry, we formally Taylor expand the fields T and Π in powers of 1/c 2 . 3Note that concrete applications of this scheme requires the existence of a suitable characteristic velocity v ch ≪ c such that the formal 1/c 2 expansion turns into an expansion in the dimensionless parameter ε = v 2 ch /c 2 .Hence, the geometric fields T µ and Π µν are expanded as Here at each order new fields are introduced, which will be discussed further below.The field τ µ is known as the clock 1-form and measures the proper time T along any curve γ in the resulting nonrelativistic geometry When τ∧ dτ = 0, in which case τ gives rise to a foliation in terms of hypersurfaces of absolute simultaneity, the field h µν measures spatial distances on these hypersurfaces when pulled back to the leaves of the foliation.The condition τ ∧ dτ = 0 is implied by the Einstein equations for suitable matter [38].The expansions of T µ and Π µν mean that the metric expands according to [38] where For the inverse structures, we have similar expansions The relations (7) imply that the leading order (LO) fields satisfy Figure 1: A cartoon of Newton-Cartan geometry.When dτ = 0, there exists an absolute time t.Then Σ 1 and Σ 2 are equal-time hypersurfaces which are equipped with a Riemannian metric, namely h µν restricted to the spatial surface.The inverse timelike vielbein v µ is an observer-dependent (since it transforms under Galilean boosts) vector that points away from equal-time hypersurfaces.
Together, the fields (τ µ , h µν ) define a Galilean structure.As we will see in Section 2.2, these fields are inert under local tangent space transformations.Subleading fields, such as m µ and Φ µν can be considered as "gauge fields" that are defined on a nonrelativistic spacetime.These subleading fields are part of the 1/c 2 corrected geometry and are dynamical fields in a theory of nonrelativistic gravity [38].The causal structure of a Galilean geometry is entirely determined by the properties of the clock form [49][50][51][52]; we will be interested in the case when τ is (locally) exact in which case there exists a notion of absolute time: That is, the proper time T in (9) between any two points in the nonrelativistic spacetime is the same regardless of the curve γ that connects them (see Figure 1).An exact clock form is required to obtain the Newtonian limit of GR [38].In fact, we will see in Section 3.5 that, at least in the absence of an external electromagnetic field, the clock form is determined by the WKB phase that defines the relation between the Klein-Gordon field and the nonrelativistic wave function.This observation was also made in [38], where it was shown that various (bosonic) matter field theories, including electromagnetism, have actions that expand in such a way that no torsion is generated.The relations (7) furthermore imply that the subleading fields that appear in T µ and Π µν are entirely determined by the subleading fields that appear in T µ and Π µν .Explicitly, we have where we will not need the explicit forms of X ρ , Xµ , X ρσ , Xρσ and where we defined In deriving these expressions, we have used the fact that the LO relations (13) imply that any symmetric contravariant 2-tensor X µν may be decomposed as In Section 3.5, we will consider the 1/c 2 expansion of actions defined on Lorentzian backgrounds, which involve the integration measure −g d d+1 x, where g = det(g µν ) is the determinant of the metric.For the square root of the determinant of the metric, we write where E expands in powers of 1/c 2 as where e = det(τ µ τ ν + h µν ) defines the integration measure ed d+1 x of the Galilean structure.

Gauge structure & flat LO geometry
The subleading fields that appear in the expansion of T µ and Π µν up to (and including) nextto-leading order (NLO) are τ µ , h µν , m µ , Φ µν .These data define a type II torsional Newton-Cartan geometry [35,38].In order to determine the metric at order c −2 we need to include the NNLO field B µ from the expansion of T µ .In this section, we work out the transformation properties of these fields, which will allow us to uniquely fix the structure order-by-order of the Schrödinger equation coupled to a nonrelativistic geometry up to a given order in 1/c 2 .
In particular, the local tangent space symmetries of (d + 1)-dimensional Lorentzian geometry form the Lorentz group SO(d, 1).By expanding the corresponding Lie algebra so(d, 1) in powers of 1/c 2 , one obtains, after suitable quotienting, the local tangent space algebra of the nonrelativistic geometry at that order (see [35,38,53] for more details).
To elucidate the local tangent space structure, it is useful to decompose Π µν that appears in the decomposition (5) in terms of spatial vielbeine E a µ as where a, b = 1, . . ., d are spatial tangent space indices.The vielbeine have a 1/c 2 expansion of the form which means that which implies The metric transforms under diffeomorphisms infinitesimally generated by a vector Ξ µ as δ Ξ g µν = L Ξ g µν , where L denotes the Lie derivative.The vector Ξ µ has a 1/c 2 expansion of the form [38] We are demanding that the diffeomorphisms preserve the 1/c 2 expansion properties of the metric.
The LO diffeomorphism ξ µ will behave as diffeomorphisms in the nonrelativistic geometry, while the subleading diffeomorphisms will not: instead, they admit an interpretation as extra gauge symmetries in the theory.In addition to diffeomorphisms, the vielbeine T µ and E a The local rotations and boosts have 1/c 2 expansions of the form where λ a b is a local rotation in the nonrelativistic geometry, while λ a is a Galilean boost.Again, we are assuming that the local Lorentz transformations preserve the 1/c 2 expansion of the vielbeine.The subleading boosts η a and rotations σ a b act as gauge symmetries on the NLO fields.Combining all these transformations, we get At this stage, it is useful to introduce an inverse vielbein e where ηa = η a + λ b π b ν e ν a and where we used which follows from (21) and the definition λ µ = λ a e a µ .We can use this to eliminate π a µ from the gauge transformations of Φ µν and B µ in favour of Φ µν since it allows us to write We will assume throughout that the clock one-form τ is closed, i.e. (dτ Since we will not allow for non-contractible closed timelike loops, this is equivalent to saying that τ is exact, i.e., that time is absolute.Using that dτ = 0 the transformations in ( 26) can be written as where we defined This is the form of the gauge transformations that we will work with in what follows.
We will furthermore assume that the LO geometry described by τ µ and h µν is a flat.This means that we can go to a Cartesian coordinate system in which where we split the spacetime index according to µ = (t, i), where now i, j, k, The residual gauge transformations of this gauge choice are all the transformations for which δτ µ = 0 and δh µν = 0 where the transformation is given in (31).This means that ξ t = cst, λ t = 0, λ i = −∂ t ξ i and ∂ i ξ j + ∂ j ξ i = 0.The latter equation can be solved by hitting it with ∂ k , leading to ∂ k ∂ i ξ j + ∂ j ξ i = 0, where ξ i = ξ i .For the flat spatial geometry, where indices are raised and lowered by a Kronecker delta, we do not distinguish between raised and lowered indices.Next, we write down all three cyclic permutations of this equation by permuting the indices i, j, k.Adding two of these and subtracting the third leads to This solves ∂ i ξ j + ∂ j ξ i = 0 provided λ i j = −λ j i .The residual gauge transformations are thus time-dependent translations a i (t) and time-dependent rotations λ i j (t).The vectors ξ = ξ t ∂ t + ξ i ∂ i with (ξ t , ξ i ) as above are Killing vectors in the sense that they obey L ξ τ µ = 0 = L ξ h µν = 0.These Killing vectors form the Coriolis algebra [54].
Omitting the time-dependent rotations, it follows from (31) that the subleading fields for a flat LO geometry transform as where a i only depends on t and where Λ, ηi , ζ i and χ are arbitrary.We note that we can always set Φ i t = 0 by fixing the ηi gauge transformation which describes a subleading local boost.The residual gauge transformations have an ηi that can be solved by setting δΦ i t = 0.
This however makes the transformation of B i a bit more complicated, and hence we will refrain from doing this.These transformations will play a key rôle in the next section.The next ingredient we need is the notion of a complex scalar field, i.e., the wave function, that is defined on the flat spacetime as described above.This requires that we understand the 1/c 2 corrections to the wave function as well as the transformation properties under diffeomorphisms and subleading diffeomorphisms.This allows us to define covariant derivatives acting on the wave function from which we can build equations of motion that couple the wave function to the 1/c 2 expanded geometry.This will be the subject of the next section.

Gravitational corrections in quantum mechanics
It is well known that a complex scalar φ KG that obeys the Klein-Gordon equation in Minkowski space gives rise to a wave function Ψ that satisfies the Schrödinger equation upon making the decomposition [13,25,27,30,31,45] where m is the mass of the complex scalar, which also becomes the mass of the Schrödinger field in the nonrelativistic quantum mechanics picture.The Galilean absolute time t that appears in the exponential factor defines the clock form via In Minkowski space, the Klein-Gordon equation for a free scalar field reads where η µν = (−c −2 , δ i j ) is the (inverse) Minkowski metric.Using the decomposition (36), the equation above becomes If we expand the field Ψ = ψ (0) + c −2 ψ (2) + O(c −4 ) we obtain the LO and NLO Schrödinger equations In the rest of this section we will design a coupling prescription that allows us to couple these equations to a NC geometry plus its 1/c 2 correction.

LO Schrödinger equation
In order to describe modifications to Schrödinger's equation (40) due to relativistic effects and gravity, we must include 1/c 2 corrections in its formulation.In this section, we develop a framework that allows us to obtain Schrödinger's equation using the geometric framework developed in Section 2. We make the simplifying assumption that the Galilean structure is flat, cf., (33).As discussed above, to derive the 1/c 2 corrections, we assume that The Klein-Gordon field φ KG transforms under infinitesimal diffeomorphisms generated by Ξ µ (cf., ( 23)) as a scalar, i.e., where Ξ µ expands as in (23).As we saw in Section 2.2, the residual temporal LO diffeomorphisms ξ t that preserve the LO Cartesian structure (33) (see also (37) above) are just constant shifts.In what follows, we take ξ t = 0 since this particular transformation will not be useful when fixing the form of the Schrödinger equation.This means that the decomposition of the Klein-Gordon field in (36), and its expansion in (42), combined with the transformation (43) and the expansion ( 23), lead to where Λ and χ are defined in (32) and where we used (34).We also used that the global time t is inert (since we took ξ t = 0), and where we omitted the time-dependent rotations λ i j (t) since they will not be needed in what follows.The transformations of ψ (0) and ψ (2) are of course such that transforms like a scalar field under general c-dependent coordinate transformations.These transformations can also be understood as follows.Under a global time translation t ′ = t + t 0 , x ′i = x i we have (cf.( 42)) If we assume that t 0 is small, then to first order in t 0 we obtain If we gauge this symmetry by replacing t 0 with −Ξ t and expand the latter in 1/c 2 as follows we obtain (44) (with a i = 0 = ζ i ) where we also expanded Ψ in c −2 .In the expansion of Ξ t we omitted the LO term ξ t since this was also not considered in (44) as this is just a constant since we are not changing coordinates at LO.What this shows is that the appearance of the Λ and χ terms in ( 44) is due to time reparametrisations in GR.Similarly, the presence of ζ i (and a i ) is dictated by spatial coordinate transformations in GR.The field ψ (0) transforms as a complex scalar field with respect to the LO diffeomorphisms and it has a (linear) local U( 1) transformation acting on it whose parameter Λ comes from NLO time reparametrisations (see [55] for related observations).The U(1)-like transformations with parameters {Λ, χ} will play an important rôle in the construction of suitable gauge covariant derivatives, which allow for a natural formulation of Schrödinger's equation coupled to the 1/c 2 expanded geometries described in Section 2.
The Schrödinger equation for ψ (0) can be formulated as Oψ (0) = 0 where O is an operator that does not depend on ψ (0) .The object Oψ (0) (without setting it to zero) should transform like ψ (0) .We will denote Oψ (0) by "LO Eq.", i.e., the leading order equation, and since this should transform like ψ (0) we demand that δ(LO Eq.) = a i ∂ i (LO Eq.) − imΛ(LO Eq.) . ( The LO Schrödinger equation should contain (40).In other words, we must construct an object X that enters the LO equation as in such a way that (49) holds.
In order to construct this X , it is useful to introduce a gauge covariant derivative D µ that acts on the LO wave function ψ (0) as The combination D µ ψ (0) transforms as where we used equations (35a), (35b) and (44).By the δ a transformation we mean all the a i -dependent terms in the transformations of (35a), (35b) and (44).We note that the a idependent terms have two origins: one is from Lie derivatives with respect to residual LO diffeomorphisms acting on the gauge field m µ and the other is from the residual Galilean boosts with parameter λ t = 0, λ i = −∂ t a i .When we say the derivative D µ is covariant we mean here with respect to the Λ gauge transformation.We also need the double spatial covariant derivative which transforms as The usefulness of the covariant derivative (51) stems from the property that it is constructed precisely such that if we replace all ordinary derivatives in (40) with covariant derivatives, we automatically make sure that the LO equation transforms covariantly under Λ transformations.Thus, the equation where k is a real constant that will be fixed shortly, transforms correctly under Λtransformations.We must also check that the LO equation transforms correctly under timedependent translations a i (that preserve the frame choice h i t = 0 which is affected by a compensating local Galilean boost transformation with λ i = −∂ t a i ), and using (52b) and (54b), we find that the LO equation ( 55) transforms as δ a (LO Eq.) = a j ∂ j (LO Eq.) , provided we take k = 1, in accordance with (49).This means that X in ( 50) is given by The LO equation ( 55) is defined up to the addition of any terms that by themselves transform as in (49).The minimal choice is to set these terms to zero.

NLO Schrödinger equation
The NLO equation is an equation for the NLO wave function ψ (2) of the form Oψ (2) + Õψ (0) where O and Õ are operators independent of ψ (0) and ψ (2) .By (44) we would like this to transform as δ(NLO Eq.) = a i ∂ i (NLO Eq.) − imΛ(NLO Eq.) + Λ∂ t (LO Eq.) − imχ(LO Eq.) The first line corresponds to homogeneous terms and the second line to inhomogeneous terms.Adopting the same approach as for the LO equation, we can guarantee the correct transformation properties under the gauge transformations {Λ, χ} if we express the NLO equation in terms of a covariant derivative that transforms in the same way as ψ (2) with respect to the {Λ, χ} transformations.By combining the transformations (35a)-(35f) with ( 44), we find that transform correctly, i.e., where δ gauge = δ Λ + δ χ denotes the combined gauge transformation.The reason we take is because we need the B i to ensure that we have the right transformations under Λ and χ, but B i shifts under the transformation with parameter ηi .However, so does Φ i t , and therefore the difference is invariant under ηi .Since no other fields transform under ηi this is the only way to ensure that the expressions built from these covariant derivatives will be inert under this transformation.Furthermore, since Φ i t is inert under both Λ and χ we do not spoil these transformation properties of the covariant derivative.The ηi transformations admit an interpretation as subleading local Lorentz boosts; more generally, the LO local Lorentz boosts are Galilean transformations with parameter λ i and the subleading corrections are described by ηi .We need all equations of motion to be invariant with respect to these LO and subleading boosts.The double spatial covariant derivative is which transforms as This means that we can tentatively write the NLO equation as where k is a real constant and where Y represents any additional terms that ensure that (58) will hold.In order to produce the correct inhomogeneous terms in the second line of (58) that involve Λ and χ, we need to set k = 1.However, we will find it instructive to delay setting k equal to unity.The Y term should be inert under the χ transformation and transform under the Λ transformation as δ Λ Y = −imΛY .Furthermore, the Y term must be such that the whole equation transforms correctly under the ζ i and a i transformations as well.How do we find such an expression for Y in (63)?If we look at (35a)-(35f), we see that the B t , B i and Φ i t gauge fields also transform under the ζ i gauge transformation.These are subleading diffeomorphisms.With respect to these transformations equation ( 63) transforms as If we choose k = 1 then the first term on the right hand side is equal to the LO equation (which is the last term in ( 58)) and furthermore we can get rid of the second term on the right hand side.There is thus a cancellation between terms coming from the D t ψ (2) term and terms coming from the D i D i ψ (2) term that involve ∂ t ζ i .This cancellation is important because there seems to be no terms that can be added to Y that would be able to cancel a term proportional to ∂ t ζ i D i ψ (0) .We wanted to highlight this, but from now on we will set k = 1.The remaining terms can be cancelled by choosing Y to be where Ỹ is inert under the ζ i and χ gauge transformations and transforms as follows under the Λ gauge transformations Since Φ i j is inert under Λ and χ gauge transformations we have It then follows that for this choice of Y and with k = 1 the combination (63) transforms like in (58) for all gauge transformations.It is left to check that this combination also transforms correctly under the a i transformation and to fix Ỹ .Before we fix Ỹ we mention that we could have added to equation (61) the term −χ k i j D k ψ (2) where χ k i j is given by Such a term is reminiscent of a Levi-Civita connection but for the NLO diffeomorphisms generated by ζ i .The second term in parentheses in the expression for Y is in fact just χ i i j .Since the Ỹ term is inert under χ and ζ i and transforms covariantly under Λ it can only be built out of covariant derivatives of ψ (0) .Demanding that the NLO equation transforms correctly under the a i transformations we find where we defined as the field strength of m µ .This field strength also arises as the commutator of two covariant derivatives (51) acting on the LO wave function The term Ŷ in equation ( 69) is any term that is inert under χ and ζ i transformations and that transforms as under the Λ and a i transformations.The Ŷ is not needed to make the NLO equation transform correctly and so the minimal choice is to set Ŷ = 0, which we will do in what follows.The final NLO equation is thus The first two terms and the last term in this equation follow directly from ( 41) by replacing ordinary derivatives by covariant ones.The remaining terms then follow from covariance with respect to the ζ i and residual ξ i transformations.

From Cartesian to spherical coordinates
We have chosen to work in Cartesian coordinates to keep things simple.On the other extreme one could work in an arbitrary coordinate system and study how the LO and NLO Schrödinger equations transforms under LO diffeomorphisms.This will be done in Section 3.5, but only at the level of the Lagrangian.It is often convenient to work with different LO coordinate systems, in particular spherical coordinates.The latter would arise naturally when looking at 1/c 2 expansions of the Schwarzschild geometry in Schwarzschild coordinates.In this section we discuss how we can transform the previous Cartesian results for the LO and NLO Schrödinger equations to an arbitrary new set of spatial coordinates.At the end of the section, we provide an explicit example by introducing spherical coordinates which we will use in Section 4.
Our derivation of the Schrödinger equation above involved Cartesian coordinates for the flat LO geometry.In this section, we change coordinates from spatial Cartesian coordinates (x, y, z) to an arbitrary set of spatial coordinates; note that the absolute time t is unaffected by this change of coordinates.Denoting the Cartesian coordinates by x 1 = x, x 2 = y and x 3 = z and the new coordinates by x ′i = (x ′1 , x ′2 , x ′3 ), the relation between the components of the flat space metric in Cartesian coordinates h i j (x) = δ i j and the components of the metric in the primed coordinates Unlike in Cartesian coordinates, the indices i, j, . . . in the primed coordinates are raised and lowered with h ′i j (the inverse of h ′ i j ) and h ′ i j .We write the square root of the determinant of the metric in the primed coordinates as Changing coordinates, the Laplacian becomes where ψ (0) (t, x) = ψ ′ (0) (t, x ′ ), and where ∇ ′ i is the Levi-Civita connection in the primed coordinates with ∂ ′ i = ∂ ∂ x ′i .We also have that where m ′ i obeys m i d x i = m ′ i d x ′i and where m ′i = h ′i j m ′ j .This means that the LO equation with flat LO geometry in the primed coordinates becomes LO Eq. = iD ′ t ψ ′ (0) + where and where m ′ t (t, x ′ ) = m t (t, x).Following the same line of reasoning, we can write the NLO equation in primed coordinates as follows where and where 78) and (81) are valid in any coordinate system that we choose to represent a flat 3-dimensional Euclidean space.In the remainder of this section, we choose the primed coordinate to be spherical coordinates (r, θ , φ) and give explicit formulae that will be useful in Section 4. In that case, the relation between the coordinate systems is We again emphasise that the absolute time t is unaffected by this change of coordinates.The components of the metric in spherical coordinates are h ′ i j (x ′ ) = diag(1, r 2 , r 2 sin 2 θ ), which means that the measure becomes h ′ = r 2 sin θ .

The inner product & field redefinitions
The wave function Ψ defined in (36) comes with a non-standard inner product up to a given order in 1/c 2 , and we must perform a field redefinition to bring it to the standard L 2 form which allows for the usual probabilistic interpretation of the norm of the wave function.More precisely, the inner product 〈Ψ|Ψ〉 descends from the Klein-Gordon inner product, as we will now show.Assuming that the Lorentzian spacetime (M , g), in which the Klein-Gordon theory is defined, is globally hyperbolic and stationary, the Klein-Gordon inner product of two different positive frequency solutions ϕ KG and ψ KG is given by (see, e.g., [56]) where Σ is a Cauchy hypersurface defined by t = cst with outward pointing timelike unit normal vector n µ = −g t t −1/2 g µν ∂ ν t, while γ µν is the induced metric on the hypersurface Σ whose determinant satisfies γn µ = −g g µν ∂ ν t.As in Eq. ( 36), we make the following decomposition of the Klein-Gordon fields4 which means that the inner product becomes Using the relations we obtain where D µ ψ (0) is defined in equation (51).For a flat LO geometry in Cartesian coordinates (33) the inner product (87) becomes We note that this inner product (87) is Galilean boost invariant.It is straightforward to see that the inner product is invariant under the χ transformations.To see that the inner product is invariant under the ζ i transformations we observe that the terms at order c −2 transform into a total derivative.We assume that the boundary terms arising from applying Stokes' theorem vanish.To see the invariance under the Λ transformation we have to integrate by parts the transformation of the m i terms (which couple to the spatial part of the U(1) Noether current) and use the LO equation of motion.We would like the inner product to take the standard L 2 ( d ) form This can be achieved if we define where X and X i are arbitrary real objects (that drop out of the inner product when integrating by parts) and where the dots denote terms proportional to the LO equation of motion.There are no obvious choices for the X and X i terms that would make the redefinition simpler.With X i = X = 0, the redefinition of the NLO wave function takes the form where we defined the operator We find that ψ(2) transforms as follows under the gauge transformations {Λ, χ, We will next define a gauge covariant derivative Dµ that acts on ψ(2) so that Dµ ψ(2) has the same transformation properties under Λ and χ as ψ (2) .A convenient choice is to define the covariant derivative in the same way as we defined ψ(2) in (91), i.e., 5Dµ ψ(2 where it is useful to note that D µ , D ν on any number of covariant derivatives acting on ψ (0) is equal to imM µν times that same set of covariant derivatives acting on ψ (0) .It can be explicitly verified that We also need an expression for the double contracted spatial covariant derivative, and we can use the same trick for this, namely we define Explicitly, this is given by This quantity transforms as We can thus recast the NLO equation (73) in terms of ψ(2) entering the standard inner product (89) as where we have used that the terms on which the Ô operator acts precisely combine to give the LO equation of motion.Thus, when imposing the LO equation of motion, the NLO equation written in terms of the redefined fields (91) assumes the same functional form as the NLO equation written in terms of the original fields (73).As a last remark, we note that the NLO equation involving the wave function with the standard inner product of nonrelativistic Quantum Mechanics (101) in spherical coordinates takes the form NLO Eq. = i D′ t ψ′ (2) + where and where In writing the above, we used the notation introduced in Section 3.3 where a prime indicates that we are using spherical coordinates.

The 1/c 2 expansion of the Klein-Gordon Lagrangian
In this section, we expand the Lagrangian for a complex scalar field in powers of 1/c 2 .This leads to an off-shell formulation of the theory we developed above.Furthermore, we will no longer restrict to a flat LO geometry, and we will see that the theory can only couple on-shell to LO geometries that admit a notion of absolute time, i.e., ∂ [µ τ ν] = 0 (this was also observed in [38]).For other examples of theories obtained by 1/c 2 expansions as well as more details about the general framework of 1/c 2 expansions, we refer to [35,36,38,39,53,57].The Klein-Gordon Lagrangian for a complex scalar field is Just like in (36), we expand the Klein-Gordon field according to and we will see that θ (0) is related to absolute time.The wave function Ψ admits an expansion of the form Using equations ( 6) and ( 17) for the inverse metric and the metric determinant, we can write the Lagrangian as Using furthermore the expansions ( 12) and ( 18), the Klein-Gordon Lagrangian expands as where the Lagrangians at orders c 4 and c 2 are given by The Lagrangian at order c 4 gives the equation when varying6 ψ (0) .Upon imposing this equation, the Lagrangian L O(c 4 ) vanishes identically.This same equation is imposed by ψ (2) in L O(c 2 ) , and combined with the equation from ψ (0) , we find and the Lagrangian L O(c 2 ) again vanishes identically when imposing these equations.Together, Eqs. ( 113) and (114) imply that This equation tells us that this theory can only be defined on backgrounds with a notion of absolute time, which in an appropriate gauge is given by t = −θ (0) /m. 7oing forward, we will impose this condition at the level of the Lagrangian.Had we not done so, they would have been reproduced as equations of motion for subleading components of Ψ.This means that the LO Lagrangian, which appears at order c 0 , can be written as which is the Schrödinger model of [58] (see also [59]).We can rewrite this in terms of covariant derivatives as follows The equation of motion obtained by varying with respect to ψ ⋆ (0) is where with the "extrinsic curvature" 8 given by K µν = − 1 2 $ v h µν , where $ denotes the Lie derivative.This extrinsic curvature is symmetric and spatial, i.e., When the LO geometry is flat, the LO equation of motion (118) reduces to (55) if we choose Cartesian coordinates.Using equations ( 12), ( 14), and ( 18) for the expansions of the inverse metric and the metric determinant, we can obtain the NLO Lagrangian from (109) in which we set ∂ µ θ (0) = −mτ µ .The result can be written as where In writing these expressions, we used the covariant derivative with a spacetime index acting on ψ (2) as which in flat space reproduces (59), and where we remind the reader that Φ t t = 0 as follows from equation (22).We emphasise that we cannot just use LNLO to compute the NLO equation of motion since that would miss terms arising from integrating by parts the second term in (121) when varying ψ (0) and ψ ⋆ (0) .When the LO geometry is flat, the NLO Lagrangian takes the form and the variation with respect to ψ ⋆ (0) produces the NLO equation plus a contribution proportional to the LO equation of motion9 where NLO Eq. is given in (73).The final step in our derivation of the NLO Schrödinger equation involves identifying the wave function with the standard inner product of nonrelativistic Quantum Mechanics as explained in Section 3.4.Generalising the result (91) to curved space is straightforward and produces the relation This step can be implemented at the level of the NLO Lagrangian, which we discuss further in Appendix A.

Quantum mechanics in a Kerr background
In this section, we study the Schrödinger equation on a nonrelativistic approximation of the Kerr background.By expanding the Kerr metric in powers of 1/c 2 , we obtain a "generalised" Lense-Thirring metric that is valid beyond the regime of slow rotations.By identifying the appropriate nonrelativistic geometry, we can apply the formalism developed in Section 3 to write down the LO and NLO Schrödinger equation on this background.

From Kerr to Lense-Thirring
The Kerr metric in Boyer-Lindquist coordinates can be thought of as a deformation of Minkowski spacetime written in terms of oblate spherical coordinates where a is a fixed length.The flat metric in oblate spherical coordinates is where We can write the Kerr metric as where where J and M are the angular momentum and mass, respectively, of the Kerr background.We assume J and M to be independent of c (see [48] for alternative choices).This implies that the metric can be expanded in c −2 , i.e., without using odd powers of c −1 .The definition of the oblate spherical coordinates (128) implies the following relation 10 Hence, surfaces in 3 of constant R form oblate spheroids.The solution to this equation has a 1/c 2 expansion of the form 10 It also implies that which shows that surfaces of constant Θ are hyperboloids of revolution.For a = 0 this becomes the equation of a cone.
where r 2 = x 2 + y 2 +z 2 is the radial coordinate of a spherical coordinate system, and where we used that Θ = θ + O(c 2 ).The relation between the Cartesian coordinate z and the spherical oblate angular coordinate Θ in (128c) combined with the expansion of R in (135) tells us that where θ = cos −1 (z/r) is the polar angle coordinate of a spherical coordinate system.Making factors of c explicit in (131) and expanding to order c −2 , using our results above, we get where is the second order Legendre polynomial.We remark that this metric is the 1/c expansion of the Hartle-Thorne metric [60] specified to the case of the Kerr black hole.It would be interesting to consider 1/c expansions of the general Hartle-Thorne metric which is an approximate solution outside a rotating object.In addition to mass and angular momentum, the Hartle-Thorne metric contains a quadrupole moment as a free parameter.
If we assume that the black hole rotates slowly, J ≪ 1, so that we can ignore the J 2 term, the above reduces to the Lense-Thirring metric.Setting J = 0 lands us on the 1/c 2 expansion of the Schwarzschild metric, which we consider in Section A.2.The geometric data of the nonrelativistic geometry is given by Thus, the LO geometry is flat in spherical coordinates (cf., Section 3.3).All the rotational aspects (terms proportional to J) are captured by the B µ gauge field.This means that the LO Schrödinger equation does not notice the rotation.

The LO and NLO Schrödinger equation on a Kerr background
On a Kerr background, where the geometric structure up to NLO is given by (139), the LO equation of motion (78) becomes The NLO equation (102) becomes where we dropped the prime we used earlier for fields in spherical coordinates.Using the LO equation of motion (140), we can write the double covariant time derivative as we find that where f is any radial function is Hermitian (ignoring issues with boundary terms or fall off conditions for the fields φ, ψ).The Laplacian in spherical coordinates is where ∆ S 2 is the Laplacian on the unit 2-sphere.
Using the above we can see that the radial part of the last three terms in (147) conspire, as indeed they must since the KG theory is Hermitian, to form the Hermitian combination (152) If we denote by ⃗ L the Hermitian angular momentum operator with components L x , L y , L z then we have Finally, if we define, as usual, the momentum p i = −i∂ i in Cartesian coordinates then we get where we wrote r −1 ∂ 2 r + 1 r ∂ r = r −3 r∂ r (r∂ r ) as −r −3 x i p i x j p j .This is our final result for the Hamiltonian of a spinless particle in a Kerr background up to order c −2 .The J L z coupling in the Hamiltonian has also appeared in, e.g., Refs.[61][62][63] which consider the Lense-Thirring effect in quantum mechanics.The result above includes further novel effects of order J 2 which may potentially be measurable.

Discussion & outlook
We conclude with a discussion of our results and an overview of future directions.We have presented a general framework based on symmetries for deriving the Schrödinger equation on a given gravitational background that can in principle be applied at any order in 1/c 2 and for a wide range of metrics.This relied on recent advances in the description of nonrelativistic geometry using covariant 1/c 2 expansions, which allowed us to use symmetry to uniquely fix the form of the equations (up to non-minimal terms).Complementary to this "bottom-up" perspective, we showed that it is also possible to get these equations by 1/c 2 expanding the Klein-Gordon Lagrangian.We then used this formalism to write down the Schrödinger equation on a Kerr background up to O(c −2 ), which led to a generalised Lense-Thirring geometry and to a novel Hamiltonian on this geometry.
With the ultimate goal of deriving a general minimal coupling prescription that allows us to write down the Schrödinger equation on a post-Newtonian geometry at arbitrary order in 1/c, this work paves the way for many interesting avenues of research.A general minimal coupling prescription should, in particular, make it immediately clear what the covariant derivatives at any given order are, and so likely requires us to understand better the representation theory of the 1/c 2 expansion of the Poincaré algebra.An immediate generalisation of the methods we develop would be to include odd powers; i.e., to consider a 1/c expansion rather than a 1/c 2 expansion.This will lead to a different geometric structure compared to Section 2, and would allow for the expansion of a much more general class of metrics, including metrics in Kerr-Schild form and metrics that include retardation effects such as pp waves.Moreover, the inclusion of electromagnetism and spin would allow us to apply our formalism to a much broader class of physical systems.
In this work, we have only discussed single particles.It would be very interesting to extend the formalism to describe composite systems.It was recently shown that when going beyond particles that are fully described by a single parameter m, new effects can arise.For example, systems that keep track of time, and are in quantum superpositions that are delocalised over a region in the gravitational field, will experience time-dilation induced entanglement between the internal and external degrees of freedom [9,22].This can result in new effects that can be probed in experiments, such as decoherence of superpositions or dephasing of clocks [9,11,26].Importantly, such effects only arise within the quantum framework when post-Newtonian corrections are included, and thus their observation amounts to a test of GR in an entirely new domain.A general geometric formalism that includes such effects will be able to highlight what aspects of the theory are probed and how to design novel tests that go beyond the current paradigms.The methods presented in this paper are ideally suited to isolate fundamental principles that can become accessible in such experiments, and to pave the way for novel experimental designs to probe the elusive interplay between quantum systems and general relativity.

µ
transform under local Lorentz transformations (Λ a b , Λ a ), where Λ a b is a local rotation and Λ a is a local boost, as[38]