# Gravitational lensing beyond geometric optics: I. Formalism and observables

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## Abstract

The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of high-frequency fields are shown to scale with different powers of their frequencies. Additionally, finite-frequency corrections are identified for a number of conservation laws and observables. Among these observables are a field’s energy and momentum densities, as well as several candidates for its corrected “propagation directions”.

## Keywords

Wave propagation Gravitational lensing Gravitational waves Geometric optics## 1 Introduction

Nearly all astronomical observations involve, fundamentally, measurements of electromagnetic or (more recently) gravitational radiation. However, these waves carry with them an imprint of the spacetime through which they travel. The spacetime geometry provides a kind of “transfer function” that relates the intrinsic properties of a source to its radiated fields. Such relations must be understood if an object’s properties are to be accurately inferred from distant measurements of its fields. If a source has already been characterized, its radiation might instead be used to probe the intervening geometry, and thus the matter which contributes to it—matter which might not be bright enough to observe directly. For these reasons and others, gravitational lensing has become a standard tool with which to extract information from astronomical observations.

Much of the theory of gravitational lensing which is used in practice may be viewed as an elaboration on the particle-like laws of geometric optics: Light travels along null geodesics, intensity variations are determined by the changing cross-sectional areas of ray bundles, and polarization vectors are parallel transported. These simple statements beget a remarkable variety of applications [1, 2, 3, 4]. However, the laws of geometric optics are an approximation. Electromagnetic fields are more properly described as solutions to Maxwell’s equations, and gravitational waves as solutions to Einstein’s equation. While the full complexities of these equations may often be ignored, there are exceptions. For example, it is well-known in ordinary optics [5, 6, 7] that the geometric approximation breaks down completely at caustics—a result which has also had astrophysical implications [1, 8, 9]. In other contexts, wave-optical corrections may be small but still detectable, in which case they might supply information which is different—and therefore complementary to—that which can be learnt from geometric optics alone.

Wave-optical effects may be viewed as frequency-dependent corrections to the frequency-independent laws of geometrical optics. Apparent source locations, intensities, phases, and polarization states might all depend on the frequencies at which a source is observed. Any such quantity measured at a sufficiently-high characteristic frequency \(\omega \) may be viewed as a geometric-optics result plus relative corrections which scale like, e.g., \(\omega ^{-1}\). Somewhat more precisely, these corrections scale like \((\omega \ell )^{-1}\), where \(\ell \) is a relevant lengthscale. Several lengthscales may be present simultaneously and different ones can be relevant for different observables. In simple cases, \(\ell \) might represent a notion of distance between a source and its observer: That geometric optics breaks down at caustics may be understood in this context by noting that the “source-centered area distance” \(r_\mathrm {a}\) goes to zero at caustics and \(\ell \sim r_\mathrm {a}\) for some contributions to some observables. More generally, \(\ell \) can be a nontrivial composite of different lengthscales. For example, some corrections associated with fields of mass \(\mu \ne 0\) which are radiated by a source at affine distance *r* can involve the lengthscale \(\ell \sim (\mu ^2 r)^{-1}\); fractional corrections to geometric optics grow with distance for massive fields. More generally (and even for massless fields) a relevant \(\ell \) might be a highly nontrivial nonlocal combination of different lengthscales—including those associated with the spacetime geometry and with details of the particular field under consideration. A systematic development of the underlying theory is thus required in order to understand precisely when such effects might be interesting. This paper begins on the path to such a development.

More directly, the purpose here is to provide general expressions which allow the propagation of high-frequency scalar, electromagnetic, and gravitational waves to be characterized in general spacetimes. While the basic equations governing geometric optics and its corrections have been discussed elsewhere [10, 11, 12] from a general spacetime perspective, very few of their higher-order consequences appear to have been explored. Some discussions which do go beyond geometric optics have appeared in various contexts, although most of these have employed a different “pseudo-Euclidean” approach which is restricted to weakly-curved spacetimes [1, 13, 14, 15, 16].

The discussion here is intended to be largely self-contained, and therefore begins by reviewing the equations which govern geometric optics and its corrections. Mathematically, these equations transform the partial differential equations which control the underlying fields into a hierarchy of algebraic constraints and ordinary differential equations along null geodesics. These are used to derive wave-optical corrections to field strengths, curvature perturbations, stress–energy tensors, and conservation laws—in arbitrary spacetimes and for arbitrary polarization states. Several types of “propagation direction” are identified and discussed. For some such definitions, multiple directions can arise simultaneously; these experience relative corrections which scale like \(\omega ^{-1/2}\) instead of, e.g., \(\omega ^{-1}\), implying that they are particularly sensitive to wave-optical effects. Frequency dependencies of the different tensorial components of electromagnetic and gravitational waves are determined as well, resulting in what are essentially local peeling results. Throughout, we emphasize connections between the various types of fields considered here. When, for example, can aspects of an electromagnetic problem be reduced to those of an effective scalar problem?

*Notation*—Sign and index conventions follow those of Wald [17]. Units are used in which \(G=c=1\) and the number of spacetime dimensions is fixed at four. In several cases, a complex field is considered despite that it is only its real component which is considered to be physical. These fields are distinguishing by using an upper-case symbol to denote the real quantity and a lower-case one for its complex counterpart; \(F_{ab} = {\text {Re}}f_{ab}\), for example.

## 2 Scalar fields

*R*denotes the Ricci scalar associated with this background, \(\Box \equiv \nabla ^a \nabla _a\), and the field mass \(\mu \) and the curvature coupling \(\xi \) are constants. Approximate solutions may be found by restricting the geometry, the initial data for \(\Psi \), or the spacetime region of interest. Here, we place no significant restrictions on the geometry, nor do we require that the field be evaluated in any special location. Instead, we restrict the initial data in the sense of imposing a high-frequency ansatz. The associated approximation is systematic in the sense that geometric optics is recovered as the first term in an easily-derived perturbative expansion. While there are systems in which the laws of geometric optics arise without any significant frequency restrictions [18, 19, 20], these are largely special cases wherein no relevant lengthscale exists which might be used to decide whether a particular frequency is large or small.

Physically, the connection between high frequencies and geometric optics may be understood by noting that discontinuities in the field—perhaps jumps representing bits of information transmitted from a source to a waiting receiver—may be expected to obey geometrical laws. The essential structure of these discontinuities is however determined by shorter wavelengths than any scales which might be associated either with the background geometry or the curvature of a wavefront; geometric optics should thus be recovered at high frequencies. To motivate that high-frequency assumptions are not only sufficient but also “not too strong,” recall that Huygens’ principle is valid essentially just for massless fields propagating in very particular spacetimes [21, 22]: Except in special cases, finite-frequency data is known to propagate in timelike as well as null directions—a process which cannot be described by the geometric-optics expectation that information travels only along null geodesics. The geometric picture must therefore fail unless interference can be counted upon to suppress propagation in timelike directions. This type of suppression is exactly what occurs at high frequencies.

*a priori*well-defined without reference to an observer, and no observer naturally presents itself (except in special spacetimes). We proceed instead by applying a WKB ansatz, where the frequency is simply identified with an expansion parameter \(\omega \). More precisely, consider a 1-parameter family of real solutions \(\Psi (x;\omega )\) to the Klein–Gordon equation which can be expanded asymptotically as real components of a complex series with the form

^{1}

*m*may be arranged to satisfy the Klein–Gordon equation up to terms of order \(\omega ^{-m}\) as \(\omega \rightarrow \infty \). This result is obtained by substituting the ansatz for \(\psi \) into the field equation (1) and equating equal powers of \(\omega \), a method which appears first to have been introduced in an optical context by Sommerfeld and Runge [23]. We now apply it for Klein–Gordon fields in general spacetimes.

### 2.1 Geometric optics

*L*may be viewed as an

*ordinary*differential operator along each null ray tangent to \(k^a\), implying that (4) may be treated as an ordinary differential equation—or transport equation—for \(\mathcal {A}_0\) along the rays. Amplitudes evaluated on distinct rays thus propagate independently of one another; cross-ray interaction does not exist at this order. The field mass \(\mu \) and the curvature coupling \(\xi \) are also irrelevant at this order.

### 2.2 Corrections to geometric optics

*L*arises for all

*n*implies that it is impossible for the aforementioned failure of Huygens’ principle to ever be taken into account by the WKB ansatz. So-called tail effects, which involve the propagation of fields in timelike directions, thus fail to be taken into account not only by geometric optics, but also by all of its corrections in integer powers of \(\omega ^{-1}\). This is mathematically consistent in the sense that the expansion is intended only to be an asymptotic approximation; it cannot be used to describe effects which are, e.g., exponentially suppressed as \(\omega \rightarrow \infty \). Tails are examples of such effects. In this context, they are intrinsically non-perturbative.

*r*(

*x*) for which

*m*(ignoring homogeneous solutions which can always be added to the \(\mathcal {A}_n\) if no initial conditions are imposed).

In summary, asymptotic approximations for real high-frequency solutions \(\Psi = {\text {Re}}\psi \) of the Klein–Gordon equation may be generated by combining the ansatz (2) for a complex \(\psi \) with the eikonal equation (3) and the transport equations (4) and (14) [or (15)]. These results convert the partial differential equation which governs \(\Psi \) into a collection of ordinary differential equations for the \(\mathcal {A}_n\). Similar equations have been been obtained before for ordinary optics in flat spacetime and in the presence of nontrivial materials [6, 23, 25, 26], and also for electromagnetic and gravitational waves propagating in vacuum in generic background spacetimes [7, 10, 11, 12, 27].

### 2.3 Observables

^{2}parallel to \(k^a\) at leading nontrivial order. Noting that \(k^a\) is null and the trajectory of a massive test particle must be timelike, the leading-order force inevitably changes a particle’s rest mass while also accelerating it along (or against) the direction of propagation. Forces transverse to \(k^a\) may appear at higher orders. For example, if \(\mathcal {A}_0 \ne 0\),

^{3}[31]

### 2.4 Conservation laws

## 3 Electromagnetic fields

A WKB ansatz may be used to understand electromagnetic fields just as it can for Klein–Gordon fields. There are at least two interesting ways to proceed: One of these works directly with the field strength \(F_{ab}\) [10, 11, 32] while the other fixes a gauge and expands a vector potential \(A_a\) [12, 33]. The latter approach is adopted here due to its similarity with the Klein–Gordon case.

### 3.1 Geometric optics and its corrections

*L*is given by (5) and we have set \(\mathcal {A}^{-1}_a \equiv 0\) for simplicity. Unlike in the scalar case, the electromagnetic amplitudes are algebraically constrained; not all solutions to the transport equations are physically admissible. Nevertheless, if the constraints (43) are satisfied on an initial hypersurface, (42) guarantees that they remain satisfied along all rays emanating from that hypersurface.

Solving (42) and (43) for all amplitudes up to some order *m*, the result may be substituted back into (41) and the series truncated at that order. This results in an approximation for \(A_a = {\text {Re}}a_a\) which solves both equations in (40) up to terms of order \(\omega ^{-m}\). However, this does not necessarily imply that the full Maxwell equation \(\nabla ^b F_{ab} = 0\) is satisfied up to terms of this same order; see “Appendix A”. Despite this, it is straightforward to determine which terms are needed in order to consistently compute different observables up to whichever order is desired.

### 3.2 Field strengths

### 3.3 Polarization

^{4}

If \(\mathcal {F}^0_{ab}\ne 0\), it is somewhat imprecise to identify \(e_a\) as *the* polarization state of the electromagnetic wave, as any modification \(e_a \mapsto e_a + \chi k_a\) results in the same leading-order field. It is less ambiguous to say instead that the null 2-form \(k_{[a} e_{b]}\) encodes a wave’s leading-order polarization state. The space of physical polarization states associated with a nonzero \(\mathcal {F}^0_{ab}\) at a point may be identified with the space of 2-forms \(k_{[a} e_{b]}\) for which \(k \cdot e = 0\) and \(e \cdot \bar{e} = 1\), modulo overall phases (which can always be absorbed into redefinitions of \(\mathcal {A}_0\)). This space is two-dimensional, so leading-order fields may be characterized by two independent polarization states. Linear polarization may be defined at a point to correspond to cases in which \(|e \cdot e|^2 = 1\), which implies that \(e_{[a} \bar{e}_b k_{c]} = 0\). Circular polarization may instead be characterized by \(e \cdot e = 0\). If a field is linearly or circularly polarized at a point, (51) implies that it retains that characteristic along the entire ray which passes through that point.

### 3.4 Newman–Penrose scalars and peeling

### 3.5 Directions associated with the field

One of the most basic characteristics of the geometric-optics field is its propagation direction \(k^a\), and it is natural to ask how this might be corrected at finite frequencies. In the scalar context, the factorization (10) of the leading-order field suggested the corrected direction \(\hat{k}^a\), as given by (12), and the physical interpretation of this guess was confirmed^{5} by the factorization (28) of the field’s stress–energy tensor, and especially by the momentum density (30). Unfortunately, the same simple arguments fail in the electromagnetic context. The problem is essentially that an electromagnetic field has several scalar components, and each of these may suggest a different effective phase. Worse, it is shown below that the electromagnetic stress–energy tensor does not remain in geometric-optics form beyond leading order: While the direction of the subleading 4-momentum density is indeed corrected relative to \(k^a\), that correction can be observer-dependent for an electromagnetic field. It thus appears that although geometric optics remains “essentially valid” even at subleading order for stress–energy tensors associated with Klein–Gordon fields, the same cannot be said for electromagnetic fields.

Despite this, a considerable literature has grown up around ascribing helicity-dependent corrections to propagation directions associated with circularly-polarized fields in curved spacetimes [38, 39, 40, 41, 42, 43, 44]. In some of these cases [38, 39], different components of the electromagnetic field are evaluated with respect to a certain frame and then factorized to motivate corrections to the eikonal equation. It is not made clear how these results are directly interpretable as propagation directions, and in any case they depend upon the chosen frame. Other approaches note that there are cases in which the Mathisson–Papapetrou equations govern the linear and angular momenta of a “photon,” and that its trajectory may be deduced by combining these equations with an appropriate centroid (or spin supplementary) condition. While the momenta \(P^a\) and \(S^{ab}\) of suitable classical wavepackets are indeed governed by the Mathisson–Papapetrou equations, imposing a supplementary condition such as \(P_a S^{ab} = 0\) (as in, e.g., [40]) may be shown to fail even for plane-fronted waves in flat spacetime; that condition constrains only one component of the centroid, not three.^{6} Different spin-supplementary conditions are motivated in [41, 42] and shown to imply that spinning massless particles move on null geodesics.

The many different approaches and conclusions in these papers and others appear to be symptoms of the fact that it is not necessarily meaningful to define *a* direction of propagation beyond leading order. While momentum densities and beam centroids do shift at finite frequencies, it can be misleading to ascribe these and other phenomena to a single “corrected propagation direction;” different directions might arise for different phenomena. The point of view adopted here is that the single propagation direction associated with geometric optics splits into two at finite frequencies. Both directions must be taken into account in order to describe observables beyond geometric optics^{7}.

*direction*is defined to be a congruence tangent to any nonzero null vector field \(k'^a\) which satisfies

*vector*. Multiplying one principal null vector by any nonzero scalar results in another principal null vector but the same principal null direction. Besides their direct interpretation as eigenvectors of \(F_{ab}\), principal null vectors are also eigenvectors of a field’s (full, non-averaged) stress–energy tensor. As mentioned above, the geometric-optics field strength admits exactly one principal null direction, namely that determined by the ray congruence tangent to \(k^a\). At higher orders, this single direction generically splits into two.

*z*, whence

*z*, with solutions

*z*determines, via (74), a real principal null direction associated with \(F_{ab}\).

It is clear from (78) that the single leading-order principal null direction splits into two whenever \(\sigma \ne 0\). This dependence on the shear is reminiscent of—although different from—Robinson’s theorem [35, 50], which non-perturbatively relates shear-free null geodesic congruences to null electromagnetic fields (i.e., fields which admit only one principal null direction). This theorem implies in particular that if \(\sigma \ne 0\), there does not exist an exact Maxwell field whose principal null congruence is tangent to \(k^a\). One might therefore suspect that the 1-parameter family of fields associated with the high-frequency approximation cannot all be null if the leading-order approximation for their principal null vectors has nonzero shear. However, it does not appear to imply a particular order at which nonzero shear forces the principal null directions to split.

If \(\sigma = 0\), the Newman–Penrose scalars satisfy the peeling result (71) and the first correction to the principal null directions may be seen from (76) to scale like \(\omega ^{-1}\), not \(\omega ^{-1/2}\). Computing this correction explicitly would require evaluating \(\Phi _0\) to one higher order than in (65), which we do not do. Nevertheless, the principal null directions may be seen to again split into two, except in special cases where \(\Phi _1^2 = \Phi _0 \Phi _2\). Indeed, this latter condition is sufficient (at all orders) to imply that there exists only a single principal null direction.

Although we have explicitly computed principal null directions only through \(\mathcal {O}(\omega ^{-1/2})\) and only for circularly-polarized fields, closely-related directions are determined below, through \(\mathcal {O}(\omega ^{-1})\) and for general polarization states; cf. (86). These are the eigenvectors of the field’s averaged stress–energy tensor. The distinction between these directions and the principal null directions may be seen by noting that if a real null vector field is an eigenvector of \(f_{ab}\), it is also an eigenvector of \(F_{ab}\), \(T_{ab}[F_{cd}]\), and \(\langle T_{ab} \rangle \). However, while real eigenvectors of \(F_{ab}\) are also eigenvectors of \(T_{ab} [ F_{cd}]\), they are not necessarily eigenvectors of \(f_{ab}\) or \(\langle T_{ab} \rangle \). Despite this difference in general, the eigenvectors of \(\langle T_{ab} \rangle \) calculated below do agree with the principal null vectors for circularly-polarized fields through \(\mathcal {O}(\omega ^{-1/2})\).

### 3.6 Stress–energy tensors and other quadratic observables

*two*null vectors—its eigenvectors. Inspired by the principal null vectors (78), one might expect that these eigenvectors differ from one another by a \(\sigma \)-dependent term which scales like \(\omega ^{-1/2}\). This is indeed the case: Eq. (84) may be rewritten as

^{8}The additional complication of the generic electromagnetic problem may therefore be dropped in these cases.

*two*measurable frequencies by analogy with (31), namely

### 3.7 Conservation laws

Exact Maxwell fields are known to also admit a large number of conservation laws which are not of the types discussed here [51, 52, 53]. While there is no obstacle to also expanding these at high frequencies, their physical interpretations are less clear.

## 4 Gravitational waves

### 4.1 Geometric optics and its corrections

*L*is the transport operator (5). These results hold for all \(n \ge 0\). High-frequency metric perturbations \(H_{ab} = {\text {Re}}h_{ab}\) may be constructed by solving (101) and (102) for the amplitudes \(\mathcal {A}^n_{ab}\) and then substituting the results into (100).

### 4.2 Curvature perturbations

*c*and

*f*in this expression while employing (107) allows \(\mathcal {R}^n_{abcd} k^d\) to be written in terms of lower-order curvature coefficients. Separately, the \(\mathcal {R}^n_{abcd}\) may also be shown to satisfy a number of transport equations. These have a rather complicated form in general, but reduce to \(L \mathcal {R}^0_{abcd} = 0\) in the \(n=0\) case.

### 4.3 Polarization

^{9}the \(\chi _a\) in (115), implying that it is only the \(e_\pm \) coefficients which contribute to \(\mathcal {R}_{abcd}^0\). These describe the two polarization states of the gravitational wave in the circularly-polarized basis \(\{ m_a m_b, \bar{m}_a \bar{m}_b \}\). If \(\mathcal {R}^0_{abcd} \ne 0\), there is no loss of generality in normalizing such that

### 4.4 Newman–Penrose scalars

### 4.5 Principal null directions

*z*which parametrizes that rotation,

*z*recovers the principal null congruences. It is a quartic equation, so there are four such congruences in general.

*z*oscillate with the same frequency as \(h_{ab}\). Moreover, \(\delta \Psi _4\) oscillates through zero at this frequency for linearly-polarized waves. These complications may be avoided by restricting considerations to circularly-polarized waves. Choosing \(m_a\) such that \(e_+ = 1\) and \(e_- = 0\) in (54), it follows from (119), (122), and (123) that

### 4.6 Other observables

In the electromagnetic context, propagation directions were associated in Sect. 3.6 not only with the principal null directions of \(F_{ab}\), but also with its averaged stress–energy tensor. It is much less clear that a similar calculation would be physically interesting for gravitational waves. While Isaacson’s stress–energy tensor [27, 57, 58] may be interpreted as explaining the averaged gravitational backreaction due to a high-frequency gravitational wave, existing and proposed methods of gravitational-wave detection do not directly probe this; the perturbed curvature is instead the most natural observable. Moreover, it is not clear that Isaacson’s stress–energy is meaningful in a regime where expansions are performed beyond geometric optics while nonlinearities in Einstein’s equation are ignored; it is derived assuming specific relations between a wave’s amplitude, its frequency, and an external lengthscale—relations which are not necessarily appropriate to the finite-frequency discussions considered here.

^{10}The prototypical example is the Bel–Robinson tensor, which is a rank-4, divergence-free tensor field which is quadratic in the Weyl tensor. Its definition does not depend on any type of approximation or averaging procedure. If a high-frequency expansion is nevertheless applied to the perturbed Bel–Robinson tensor, the leading-order result may be shown to be

## 5 Relating different types of high-frequency fields

*L*is the operator (5),

*B*is a multi-index appropriate to the field under consideration, and \(\mathcal {D} = \Box + \ldots \) is the hyperbolic operator associated with the appropriate field equation; Eqs. (4), (14), (42) and (102) are all in this form. In the electromagnetic and gravitational cases, the amplitudes must also satisfy the algebraic constraints (43) and (101). At this level, it may appear that there is very little difference between the various types of fields considered here. We now discuss to what extent differences do exist, and also when similarities may be exploited to simplify calculations. When, for example, does solving an effective scalar (or electromagnetic) problem suffice to understand a problem which is physically electromagnetic (or gravitational)?

### 5.1 Leading-order amplitudes

The clearest cases in which such simplifications arise are those which depend only locally^{11} on the \(n=0\) amplitudes. These leading-order amplitudes locally determine all of geometric optics, but also much more than this: All approximations for the Newman–Penrose scalars given in Sects. 3.4 and 4.4 are written solely in terms of the \(n=0\) amplitudes and the geometric-optics propagation direction \(k^a\), even though it is only \(\Phi _2\) and \(\delta \Psi _4\) which characterize geometric-optics fields. Similarly, the variously-defined corrections (12), (78), (86), (130), and (137) to \(k^a\) are locally written using only leading-order quantities. An understanding for how \(\mathcal {A}_0\), \(\mathcal {A}^0_a\), and \(\mathcal {A}_{ab}^0\) relate to one another provides new insights into these quantities and others.

The first such relation is associated with the fact that solving a scalar problem automatically solves aspects of electromagnetic and gravitational problems, in the sense that if a leading-order scalar amplitude \(\mathcal {A}_0\) is known, (50), (52), (112), and (116) imply that its square \(|\mathcal {A}_0|^2 = \mathcal {A}_0 \bar{\mathcal {A}}_0\) also determines the squares \(g^{ab} \mathcal {A}^0_a \bar{A}_b^0\) and \(g^{ab} g^{cd} \mathcal {A}^0_{ab} \bar{A}_{cd}^0\) of leading-order electromagnetic and gravitational amplitudes. These quantities appear in the averaged stress–energy and superenergy tensors at leading order. In fact, the entirety of the averaged electromagnetic stress–energy tensor may be determined at this order by solving a scalar problem; see (85). This is also true for the leading-order average (131) of the Bel–Robinson tensor in the gravitational case, at least if^{12} \(e^{a}{}_{a} = 0\). In geometric optics, observables such as the averaged energy and momentum densities and the propagation direction may thus be understood purely by solving scalar problems; the additional complexities of the electromagnetic and gravitational amplitudes do not affect these quantities at leading order.

*g*” superscript has been inserted to distinguish the gravitational components. The term involving \(\chi ^a\) is irrelevant at leading order, so there is a sense in which the gravitational polarization components are simply products of the underlying electromagnetic components: If at least one of the electromagnetic waves used here is circularly polarized, so is the resulting gravitational wave. If both waves are circularly polarized but with opposite helicities, the associated gravitational wave vanishes at leading order—as already expected from (145).

The correspondence (144) might be interpreted as a kind of classical double copy. Two electromagnetic solutions are “copied into,” or “squared” to produce a single gravitational solution. This language is borrowed from quantum field theory, where it is known that under certain conditions, gravitational scattering amplitudes look like gauge-theory amplitudes “squared” [63]. Such results have inspired significant discussion of classical analogs in which solutions to gauge-theory equations generate solutions to gravitational equations (often coupled to non-gravitational fields); see [64, 65] and references therein. Much of the discussion on the classical gravitational side has been confined to metrics of Kerr–Schild type, i.e. those in which a background is deformed by adding to it a term with the form \(V k_a k_b\), where \(k_a\) is null. The double copy given by (144) includes at least the *pp*-waves in this class; see “Appendix B.2”. However, the correspondence given here between electromagnetic and gravitational solutions in fact holds in general in geometric optics. In special cases where geometric optics is exact—as for *pp*-waves—it extends to exact solutions. However, given that the non-classical double copy results are associated with scattering, it is perhaps reasonable to expect a classical analog to be generic mainly in the geometric-optics regime which is so central to scattering calculations.

### 5.2 Higher-order amplitudes

*can*be derived between different theories—involves scalar theories with different masses \(\mu \) or curvature couplings \(\xi \). These possibilities behave identically at the level of the leading-order amplitudes, but not more generally. Concentrating on the \(n=1\) case, suppose that one leading-order amplitude \(\mathcal {A}_0\) is known and that this is used to determine corrected amplitudes associated with two different types of scalar field. Letting \((\mu ,\xi )\) and \((\mu ',\xi ')\) be the parameters which characterize those fields, (14) implies their corrected amplitudes must satisfy

*r*which satisfies (16). If \(\mathcal {A}_1\) and \(\mathcal {A}'_1\) are assumed to coincide on a hypersurface \(r = r_0\), where \(k \cdot \nabla r_0 =0\),

*r*. Amplitudes associated with different curvature couplings instead grow with the integral of the Ricci scalar. While these terms may be large for radiation emitted by distant sources, their primary effect is to shift the phase of \(\psi '\) relative to that of \(\psi \): Differing masses result in the phase shift \(\frac{1}{2} ( \mu '^2 - \mu ^2 ) (r - r_0)/\omega \), and if the vacuum Einstein equation (97) holds, differing curvature couplings produce the additional phase difference \(2 \Lambda (\xi '- \xi ) (r-r_0)/\omega \). These shifts do not, however, affect intensities as given by (21) and (32). More generally, it follows from (28) that the entirety of the averaged stress–energy tensor is independent of \(\mu \) and \(\xi \) to the orders computed here: \(\langle T'_{ab} \rangle = \langle T_{ab} \rangle + \mathcal {O}(\omega ^0)\). Masses and curvature couplings do affect at least the trace (27) of this tensor at the following order, although not via any differences between \(\mathcal {A}_1\) and \(\mathcal {A}'_1\).

## 6 Discussion

We have derived a number of general features of high-frequency scalar, electromagnetic, and gravitational waves propagating on curved background spacetimes, focusing on observables, physical intuition, and also relations between these different types of fields. However, no specific applications were considered. The purpose has been instead to set the stage for further exploration.

While it would be straightforward to use the results presented here to compute corrections to geometric optics in various scenarios, subsequent papers in this series will take a more foundational approach. Two basic questions will be addressed before considering the details associated with any specific systems: First, how do changes in the background metric affect observables? General invariance properties of the underlying equations will be shown to provide powerful tools with which to address this question. Second, we ask how the measured properties of a radiated field can be related to intrinsic properties of its source. Alternatively, how should initial data be specified for the various transport equations? Although the space of possibilities is large in general, gravitational lensing is typically concerned with compact sources. In this context, the initial data problem simplifies considerably. We shall discuss how this occurs and how the relevant data can be related to a source’s intrinsic properties.

## Footnotes

- 1.
Various other WKB-like ansätze may be considered. For example, the amplitude might be replaced with a single \(\omega \)-independent function while the exponent is instead expanded in powers of \(\omega ^{-1}\). This naturally leads to the consideration of exponents which are not necessarily purely imaginary, thus allowing evanescent waves and other exponentially-suppressed phenomena to potentially be understood. Such an ansatz nevertheless comes with considerable complications, and is not considered here. These complications are especially severe when considering electromagnetic or gravitational fields, in which case one must resort to “phases” which take the form of higher-rank tensor fields.

- 2.
The force acting on a scalar test particle with charge

*q*which is immersed in a (real) field \(\Psi \) is known to be \(q \nabla _a \Psi \), where the notion of force used here refers to the rate of change of a particle’s momentum. Note that force is sometimes defined instead in terms of a particle’s rest mass multiplied by its 4-acceleration, which can be different. Regardless, the expression here may be motivated using the actions discussed in, e.g., [28, 29]. It may also be derived by appending a source term to the wave equation (1) and then applying stress–energy conservation using (23); see [30] for a full discussion of the minimally-coupled case, also including extended-body and self-interaction effects. - 3.
The factor of \(-2\) in the Lagrangian quoted here is a matter of convention. Changing it would modify the factor of \(1/4\pi \) in (23).

- 4.
If \(e_{[a} k_{b]}=0\), it follows from (45) that \(\mathcal {F}_{ab}^0 = 0\). This does not imply, however, that such cases are necessarily unphysical. Nonzero field strengths may be generated at higher orders by zeroth-order amplitudes with this property; see “Appendix B.1”. We nevertheless assume \(e_{[a} k_{b]} \ne 0\) unless otherwise noted.

- 5.
Other criteria may nevertheless be used to be obtain other generalizations of \(k^a\). For example, (18) suggests a different (though rapidly varying) direction based on the forces which act to test charges.

- 6.
This follows from applying the standard definitions for \(P^a\) and \(S^{ab}\) (see, e.g., [27]) to a stress–energy tensor proportional to \(k^a k^b\), where \(k^a\) is null and constant. Separately, it may be seen directly that the equations of motion in [40] are ill-defined in flat spacetime. This is explained there by saying that massless spinning particles are “delocalized” in that case. However, narrow beams in flat spacetime clearly

*are*localizable; the connection with classical wavepackets is therefore unclear. - 7.
Geometric intuition must still be treated with caution. Even with two directions at hand, most results cannot be described as an incoherent sum of two geometric-optics expressions with different propagation directions. While the directions we consider are well-defined, it is debatable whether or not it is useful to refer to them as

*propagation*directions. - 8.
Some differences remain in the sense that the \(|\mathcal {A}_0 + \omega ^{-1} \mathcal {A}_1|^2\) which appear in \(\langle A^2 \rangle \) and \(\langle \Psi ^2 \rangle \) can behave somewhat differently for scalar versus vector amplitudes. This is discussed in Sect. 5.2 below. Furthermore, if an electromagnetic field is linearly polarized but \(\chi \ne 0\), the only change to these statements is that the component of the propagation direction proportional to \(k_a\) might change: \(\hat{k}^+_a = \hat{k}^-_a = \hat{k}_a + \omega ^{-1} (\ldots ) k_a\).

- 9.
While \(\chi _a\) cannot affect the leading-order curvature—which implies also that the trace of \(e_{ab}\) cannot affect it—these statements do not necessarily apply at higher orders. See the example in “Appendix B.2”.

- 10.
Superenergy tensors may also be associated with non-gravitational fields; see, e.g., [59] and references therein.

- 11.
It follows from (142) that up to homogeneous solutions, all higher-order amplitudes may be viewed as functionals of the \(n=0\) amplitudes. However, these functionals are nonlocal in general; they involve integrals along null geodesics. Nevertheless, there are many cases in which the dependence relevant to a particular observable at a particular order reduces to a local function of the leading-order amplitude and a finite number of its derivatives.

- 12.
This caveat is not essential. It may be removed by modifying the normalization condition (116).

## Notes

### Acknowledgements

I thank Yi-Zen Chu, Sam Dolan, and Justin Vines for valuable discussions.

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