The absence of spherical photon orbits as a diagnostic of non-Kerr spacetimes

# The absence of spherical photon orbits as a diagnostic of non-Kerr spacetimes

Kostas Glampedakis Departamento de Física, Universidad de Murcia, Murcia E-30100, Spain Theoretical Astrophysics, University of Tübingen, Auf der Morgenstelle 10, Tübingen, D-72076, Germany    George Pappas Dipartimento di Fisica, “Sapienza” Universitá di Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy
###### Abstract

Photon circular orbits, an extreme case of light deflection, are among the hallmarks of black holes and are known to play a central role in a variety of phenomena related to these extreme objects. The very existence of these orbits when motion is not confined in the equatorial plane is indeed a special property of the separable Kerr metric and may not occur, for instance, in the spacetime of other more speculative ultracompact objects. In this paper we consider a general stationary-axisymmetric spacetime which already possesses an equatorial photon circular orbit and examine under what circumstances spherical or more general, variable-radius, ‘spheroidal’ non-equatorial photon orbits may exist. In addressing this question, we first derive a necessary circularity condition for the existence of such orbits and then go on to show that the presence of spherical orbits is really connected to the separability of the metric (that is, to the existence of a third integral of motion). Specifically, a spacetime that does not admit spherical orbits is necessarily non-separable. In addition, this kind of spacetime may not even admit the more general spheroidal orbits. As concrete examples of this connection between separability and circularity we study null orbits in a variety of known non-Kerr metrics (Johanssen-Psaltis, Johannsen and Hartle-Thorne). The implications of these results for the electromagnetic and gravitational wave signature of non-Kerr objects are briefly discussed.

## I Introduction

The first direct observations of gravitational waves (GWs) by the advanced LIGO/Virgo detector network Abbott:2016blz; Abbott:2016nmj; TheLIGOScientific:2016pea; Abbott:2017vtc; GW170814 saw General Relativity (GR) becoming even more established as the correct theory of gravity. However, these observations, spectacular as they may be, have not yet ruled out alternative to GR theories of gravity nor have they established ‘beyond reasonable doubt’ the Kerr nature of the compact objects involved in the mergers GWverse2018. Indeed, testing the Kerr hypothesis even within GR against the possibility of having some other type of exotic horizonless ultracompact object that could be mistaken for a black hole is by itself a far from easy endeavour, see e.g. Cardoso:2016rao; Cardoso_Pani:2017; Maselli_etal2017; Glampedakis2018. This fascinating prospect provides ample motivation for a more detailed study of how non-Kerr compact objects could manifest themselves under the scrutinous eyes of electromagnetic and gravitational wave observatories.

A not so often emphasized key characteristic of Kerr black holes is the presence of non-equatorial circular photon (or particle) orbits. These are in fact spherical (though not closed) trajectories of constant radius and represent the generalisation of the much more familiar concept of the equatorial circular orbit, the so-called ‘photon ring’. Spherical/circular photon geodesics leave their mark (directly or indirectly) on a variety of phenomena involving black holes.

A black hole illuminated by an external source of light (e.g. a hot accretion flow) casts a shadow that is fringed by a sharp bright ring Bardeen1973; Johannsen2010. The mechanism responsible for the formation of this optical structure is the photon circular orbit which acts as a temporary depository of electromagnetic flux. It is not surprising then that photon circular orbits play a key role in the ongoing effort to capture horizon-scale images of the Sgr supermassive black hole (and of other black holes in our galactic neighborhood) and use them as an observational test of the Kerr metric, see e.g. Johannsen2010; Broderick2014. This program should soon come to fruition with the ongoing observations of the Event Horizon Telescope’s worldwide constellation of radio telescopes.

Much of our intuition about wave dynamics in black hole spacetimes is also based on photon spherical orbits. Quasi-normal mode (QNM) ringdown is intuitively understood in terms of wavepackets temporarily trapped in the vicinity of the photon ring, gradually peeling off towards infinity and the event horizon as they circle the black hole. Indeed, in the eikonal limit of geometric optics the frequency and damping rate of the fundamental QNM are determined, respectively, by the photon orbit’s angular frequency and divergence rate (Lyapunov exponent), see Mashhoon:1985cya; Ferrari:1984zz; Glampedakis:2017 for more details. Similarly, the scattering of plane waves by black holes reveals the presence of a photon ring in the glory pattern of the scattering cross section (see Glampedakis2001 and references therein).

Although the equatorial photon ring is expected to be an ubiquitous orbital feature, present in the spacetime of non-Kerr ultracompact objects such as gravastars and bosons stars, the same may not be true for the off-the-equator spherical orbits. Excluding the idealised case of spherically symmetric systems, the existence of the latter orbits is not guaranteed unless some special conditions are met.

This is precisely the issue addressed in this paper (a more compact discussion of this work can be found in GP_KGprl). Specifically, we ask under what circumstances spherical (of constant radius ) or more general ‘spheroidal’ (of a variable, equatorially-symmetric, radius , where is a meridional coordinate) photon orbits are allowed when one moves away from Kerr to an arbitrary axisymmetric-stationary and equatorial-symmetric spacetime. In this general case we can formulate a necessary ‘circularity condition’ for the existence of the aforementioned orbits. It is then possible to arrive to the remarkable result that a spacetime that has an equatorial photon ring but does not admit spherical orbits is necessarily non-separable (and therefore non-Kerr).

Most non-Kerr spacetime metrics of interest are of course non-separable and therefore spherical photon orbits should not be present in them. Spheroidal orbits, on the other hand, could in principle exist but our results suggest that this may not be the generic situation . Two of the most widely used non-separable metrics in relativistic astrophysics, the deformed Kerr Johannsen-Psaltis metric Johannsen:2011dh and the slow rotation Hartle-Thorne metric Hartle1967; HT68 are found to possess neither spherical nor spheroidal photon orbits. An exception to the rule may be provided by the spheroidal orbits outside black holes with scalar hair discussed in Cunha2017b (where these orbits are dubbed ‘fundamental photon orbits’).

The deeper astrophysical motivation behind this work lies in the aforementioned role played by circular photon orbits in creating a black hole shadow and in the GW ringdown produced in the final stage of black hole mergers. The key element in both phenomena is the ability of trapping photons/wavepackets in orbit around the black hole for a time interval much longer than the system’s dynamical timescale. As our results suggest, this ability could be compromised if the Kerr metric were to be replaced by a non-separable metric that does not support spherical or spheroidal photon orbits. The far-reaching consequence of this conclusion is that non-Kerr ultracompact objects may look very different compared to Kerr black holes with respect to their shadow image and QNM ringdown signal.

The remainder of the paper is organised as follows. Sections II contains the necessary formalism for describing photon geodesics in an axisymmetric-stationary metric. In Section III we focus on circular motion and discuss the distinction between spherical and spheroidal non-equatorial orbits. In Section IV we derive the circularity condition describing these orbits. The direct connection between spherical orbits and the spacetime’s separability is the subject of Section V. In Section VI we search for spherical/spheroidal orbits in three different cases of non-Kerr spacetimes (Johannsen-Psaltis, Johannsen and Hartle-Thorne). A complementary time-domain study of circularity is the subject of Section VII. Finally, in Section VIII we summarise our results and discuss their implications for the observational signature of non-Kerr objects.

## Ii Formalism for general null geodesics

For the general purpose of this paper we consider an arbitrary axisymmetric, stationary and equatorial-symmetric spacetime described by a metric in a spherical-like coordinate system. The resulting line element takes the form

 ds2=gttdt2+grrdr2+2gtφdtdφ+gθθdθ2+gφφdφ2. (1)

Geodesics in this spacetime conserve the energy and the angular momentum component along the symmetry axis (here given per unit mass), , where is the four-velocity along the geodesic. Defining the impact parameter and rescaling the affine parameter we can effectively set and everywhere. These expressions can be inverted to give

 ut=1D(gtφb+gφφ),uφ=−1D(gtφ+gttb),D=g2tφ−gttgφφ. (2)

The location of the horizon (if present) is marked by ; outside the horizon this parameter is positive.

Assuming null geodesics hereafter, the norm leads to

 grru2r+gθθu2θ=1D(gttb2+2gtφb+gφφ)≡Veff(r,θ,b), (3)

where the effective potential shares the same symmetry properties as the metric.

In the case of the Kerr metric, the existence of a Carter constant allows the decoupling of the radial and meridional motion, with (3) becoming a purely radial equation [see Appendix (A) for details]. The ‘miraculous’ property of a third constant is absent in a general axisymmetric-stationary spacetime. Instead one is obliged to work with the second-order geodesic equation, which can be written as:

 ακ≡duκdλ=12gμν,κuμuν. (4)

The -component of this equation is the only one needed here,

 αθ =12[gtt,θ(ut)2+grr,θ(ur)2+gθθ,θ(uθ)2+gφφ,θ(uφ)2+2gtφ,θuφut] =12(grr,θg2rru2r+gθθ,θg2θθu2θ)+12D2[g4tφVeff,θ−g2tφgφφ(gttVeff),θ+gttgφφ{(g2tφ),θ−gttgφφ,θ}Veff +2bgφφ(gtφgtt,θ−gttgtφ,θ)+g2φφgtt,θ−gttgφφgφφ,θ], (5)

For the following discussion of circular orbits we also need to involve the -derivative of (3). This is,

 urgrr(2αr−grr,rg2rru2r−grr,θgrrgθθuruθ)+uθgθθ(2αθ−gθθ,θg2θθu2θ−gθθ,rgrrgθθuruθ)=urgrrVeff,r+uθgθθVeff,θ. (6)

It can be verified that the insertion of [as computed from (4)] into (6) returns a trivial result.

Returning to Eq. (3), one can observe that its quadratic form implies that should act as a zero-velocity separatrix between allowed and forbidden regions for geodesic motion. This follows from

 grr(ur)2+gθθ(uθ)2=0⇔ur=uθ=0⇔Veff=0. (7)

Then () marks the allowed (forbidden) region. Some examples of this are given below in Section VI.2.

## Iii Circular, spherical and spheroidal orbits

It is perhaps instinctive to think of circular motion as that associated with a constant radius . For non-equatorial motion, a more accurate designation for these orbits would be spherical since the trajectory is confined on the surface of a sphere of radius . This is, for example, how Kerr spherical orbits look like in the familiar Boyer-Lindquist coordinates. However, there is a more general way of defining non-equatorial circular orbits, namely, as motion confined on a spheroidal-shaped shell . We shall call this more general type of circular orbit spheroidal. From a mathematical point of view we demand the function to be smooth and expandable in even-order Legendre polynomials,

 r0(θ)=∑ℓβℓP2ℓ(cosθ), (8)

where and are constant coefficients. Note that according to this definition is not required to be single-valued, so that the ‘spheroidal’ shell may actually be torus-shaped (as some of the orbits discussed in Cunha2017b). In this case the orbit could intersect the equatorial plane in two distinct radii instead of one.

Sticking with this general definition of circular motion we find that are ‘locked’ to each other,

 ur=r′0uθ⇒ur=grrgθθr′0uθ, (9)

where a prime stands for a derivative with respect to the argument. In this and the following expressions all functions of are to be evaluated at .

Taking the -derivative of (9),

 αr=grrgθθr′0αθ+u2θg3θθ[gθθgrrr′′0+r′0(gθθgrr,θ−grrgθθ,θ)+(r′0)2(gθθgrr,r−grrgθθ,r)]. (10)

Using (9) in (5),

 αθ =u2θ2g2θθ[grr,θ(r′0)2+gθθ,θ]+12D2[g4tφVeff,θ−g2tφgφφ(gttVeff),θ+gttgφφ{(g2tφ),θ−gttgφφ,θ}Veff +2bgφφ(gtφgtt,θ−gttgtφ,θ)+g2φφgtt,θ−gttgφφgφφ,θ]. (11)

Meanwhile, from Eqs. (3) and (6) we obtain respectively

 [grr(r′0)2+gθθ]u2θ=g2θθVeff, (12) r′0[2αr−r′0u2θg2θθ(grr,θ+grr,rr′0)]+2αθ−u2θg2θθ(gθθ,θ+gθθ,rr′0)=r′0Veff,r+Veff,θ. (13)

It the limit of equatorial motion, and , these two equations reduce to the well-known circular orbit conditions for the potential and its radial derivative,

 Veff(b,r0)=0,2αr=Veff,r ⇒ Veff,r(b,r0)=0. (14)

As discussed in more detail in Glampedakis:2017, these two conditions lead to an equation for the photon ring radius and its associated impact parameter (here the metric is only function of the radius and the upper/lower sign corresponds to prograde/retrograde motion)

 gφφ(g′tt)2+2gtt(g′tφ)2−g′tt(gttg′φφ+2gtφg′tφ)∓2√(g′tφ)2−g′ttg′φφ(gtφg′tt−gttg′tφ)=0, (15) b0=∓1g′tt[√(g′tφ)2−g′ttg′φφ±g′tφ]. (16)

## Iv The circularity condition

The previous equations pertaining general non-equatorial circular motion in an arbitrary axisymmetric-stationary metric can be combined to produce a necessary circularity condition of the functional form .

This constraint originates from Eq. (13) (i.e. essentially the -derivative of ) after using (10)-(12) to eliminate , and , respectively. Once these steps are taken and several terms are combined to form and its derivatives, we arrive at:

 grr(r′0)3(grrVeff),θ+(r′0)2[(gθθgrr,r−2grrgθθ,r)Veff−grrgθθVeff,r] +r′0[(2gθθgrr,θ−grrgθθ,θ)Veff+grrgθθVeff,θ]+gθθ[2grrVeffr′′0−(gθθVeff),r]=0, (17)

where all functions are to be evaluated at . This equation will become our basic tool for searching for spherical/spheroidal orbits in non-Kerr spacetimes (Section VI). It should be noted that (17) is oblivious to the stability of the circular orbit. This extra information is contained in the second derivatives of .

As a sanity check of (17) we consider the Kerr metric in Boyer-Lindquist coordinates with the assumption . The circularity condition reduces to

 EK(r0)≡r20(r0−3M)+a2(M+r0)+ab(r0−M)=0, (18)

which can be identified as one of the two equations that determine non-equatorial Kerr photon orbits (for this leads to the Schwarzschild photon ring ).

This example is indicative of what happens when a spacetime admits orbits: the circularity condition effectively becomes a -independent equation for . As we shall see below in Section VI.4, a similar situation arises in the context of the separable deformed Kerr metric devised by Johannsen Johannsen2013PhRvD.

## V The connection between spherical orbits & separability

The circularity condition (17) provides the means to establish a remarkable result that can be stated as follows: if a stationary-axisymmetric spacetime endowed with an equatorial photon ring is separable (in the sense that it admits a third integral of motion) then it necessarily admits spherical photon orbits (i.e. orbits with ). For the converse to be true an additional condition is required, namely, that the ratio of the metric components takes a factorised form . Then, the existence of spherical orbits in some coordinate system implies a separable spacetime.

A corollary of this proposition is that a spacetime is necessarily non-separable if it possesses an equatorial photon ring but does not admit spherical orbits. The rest of this section is devoted to the derivation of these results; it should be noted that the sphericity-separability connection is not exclusively about photons but it can be extended to the orbits of massive particles (this is discussed in more detail in Appendix C). Nor it is exclusively ‘relativistic’ as it can be shown to hold in the context of Newtonian gravity (see Appendix B).

For a spacetime of the general form (1) with the only non-ignorable coordinates, the Hamilton-Jacobi equation for null geodesics becomes MTW1973,

 (S,r)2grr+(S,θ)2gθθ−Veff=0, (19)

where is Hamilton’s characteristic function. Following the standard separability ansatz LLbook we write . Provided the following conditions hold (here are arbitrary functions of their argument),

 gθθVeff=f1(r)h(θ)+g(θ), (20) gθθgrr=f2(r)h(θ), (21)

we can rearrange (19) as,

 f2(r)(S′r)2−f1(r)=1h(θ)[g(θ)−(S′θ)2]=C. (22)

This demonstrates the separability of the system, with playing the role of the third constant (or ‘Carter constant’). On the same issue of separability, Carter Carter1968CMaPh showed that the Hamilton-Jacobi equation as well as the Schrödinger and scalar wave equations are all separable if the metric of a given spacetime can be put in the ‘canonical’ form (see Frolov2017LRR for a recent review on the subject),

 ds2=ZΔrdr2+ZΔθdθ2+ΔθZ(Prdφ−Qrdt)2+ΔrZ(Qθdt−Pθdφ)2, (23)

where and for (and similarly for the other arbitrary functions). One can easily verify that the canonical metric (23) satisfies the separability conditions (20), (21). As examples of this privileged class of spacetimes we can mention the Kerr and Johannsen metrics (although only the former is a solution of the GR field equations).

We now can make contact with the existence of spherical photon orbits. According to the circularity condition (17) an orbit must satisfy

 (gθθVeff),r|r0=0. (24)

This implies a solution of the general form

 gθθVeff=f1(r)h(θ)+g(θ), (25)

with the additional constraint . Now the expression (25) can be identified as the first separability condition (20), but separability alone cannot enforce . This is the point where we need to invoke the existence of an equatorial photon ring in the spacetime under consideration: for some this requires , with the last equation now becoming a relation , common for both equatorial and non-equatorial motion as it depends neither on nor on [in Kerr, this relation is given by Eq.  (18)].

We have thus established the first half of the proposition, namely, that Hamilton-Jacobi separability entails the existence of spherical photon orbits as long as has roots for some values of . The veracity of the result’s second half rests on Eq. (25), which follows from the circularity condition, in combination with the second separability condition (21) for the form of .

Going beyond our basic result, one can show that spheroidal orbits cannot exist in a separable spacetime. Using in (22), we obtain a pair of decoupled equations,

 f2(grrur)2=f1+C≡Vr(r,b,C),(f2grruθ)2=g−hCh2≡Vθ(θ,b,C). (26)

Assuming first a spheroidal orbit, , these two equations combine to give,

 (r′0)2Vθ=f2(r0)Vr(r0)≡~Vr(r0). (27)

We can then see that both potentials obey , . At the meridional turning points we have which means that . A similar argument can be used in the equatorial plane where due to the assumed symmetry of the orbit, hence leading to . Given that cannot be negative, it can be either zero or increase and subsequently decrease as moves from to the equator. The situation is exactly the same in the lower hemisphere and therefore we should have . Taking the derivative of (27),

 2r′′0Vθ+(r′0)V′θ=~V′r(r0), (28)

we can deduce that . Combined with , it entails that any orbit crossing the equatorial plane can only be an orbit. Therefore, the only possibility is that of spherical orbits.

A subtle point of our discussion on spherical orbits is their inherent coordinate dependence, in the sense that they occur if one uses the appropriate coordinate system. Spherical orbits should not occur if a different coordinate system is employed, but instead one would expect to encounter spheroidal orbits . An example of this situation is provided by the Newtonian Euler potential, see Appendix B.

Moving further up the ladder of generality one would have to consider non-separable axisymmetric-stationary spacetimes and contemplate whether circular orbits of the spheroidal type could be present in them. Assuming the existence of such orbits, one could imagine moving to new ‘adapted’ coordinates so that shells are mapped onto spheres. Now it is well know (see Chapter 3 in wald1984) that any two-dimensional space can be written in a conformally flat form with the new coordinates forming a pair of conjugate harmonic functions (i.e. where the Laplacian refers to the subspace). In such a case the conformal metric trivially satisfies the condition (21). If in addition the conformal coordinates happen to be of the adapted kind (hence admitting spherical orbits), it would follow from the above theorem that the metric is separable, in contradiction with our initial assumption.

Of course this argument would fail in the likely event of an adapted coordinate system not being a member of the conformal family. Similarly, finding an adapted system could be a futile exercise in the case of a multivalued .

In order to see how a set of adapted coordinates could be determined we first note that they should enforce in a surface and therefore,

 r′0u,r+u,θ=0. (29)

The second requirement for the new coordinates is that of orthogonal level surfaces, , which is equivalent to . We can then consider the orthogonality between the tangent vector of the curves and the displacement along the curves. The former vector is found from

 gijTi∇ju=0 ⇒ Tru,r+Tθu,θ=0 ⇒ (Tr,Tθ)=(u,θ,−u,r). (30)

Then from we have,

 grru,θdr−gθθu,rdθ=0 ⇒ r′0grrdr+gθθdθ=0, (31)

where in the last step we used the -equation (29). This expression should stand for and therefore (modulo a constant scale)

 v,θ=gθθ,v,r=r′0grr. (32)

These equations fully determine the function ; together with (29), they are the only constraints that need to be satisfied by the desired adapted coordinate system. Although solving this system of equation is beyond the scope of this work we can notice, for example, that (29) is solved by while Eqs. (32) can be seen to admit solutions of the general form or . In that case the separability condition holds and we again arrive to a contradictory result.

The preceding analysis obviously does not constitute a non-existence proof for spheroidal orbits in non-separable spacetimes 222Indeed, a counter-example may be provided by rotating Proca stars, see Ref.Cunha2017b., but it does suggest that this could be a ‘generic’ situation. Our failure to find spheroidal orbits in the non-separable spacetimes examined below in Section VI lends strong support to this point of view.

## Vi Searching for circular orbits in non-Kerr spacetimes

### vi.1 Strategy

After having established the link between the separability of a given spacetime and the existence of spherical orbits we go on to consider specific examples of both separable and non-separable metrics. This case-by-case analysis comprises the deformed Kerr metrics devised by Johannsen-Psaltis Johannsen:2011dh and Johannsen Johannsen2013PhRvD, and the celebrated Hartle-Thorne metric Hartle1967; HT68 which is the ‘official’ GR solution describing the interior and exterior spacetime of relativistic stars within a slow-rotation expansion scheme. For brevity, hereafter these three metrics will be denoted as ‘JP’, ‘J’ and ‘HT’ respectively. Amongst these metrics only the J is separable (by construction) and admits a Carter-like constant while the other two acquire this property only in their respective Kerr limits.

Our overall strategy is based on a two-pronged approach. In this section we show that the necessary circularity condition for the existence of spheroidal orbits fails in the JP metric, while spheroidal orbits are admitted in the HT metric only when the condition is perturbatively expanded with respect to the spin. In sharp contrast, we find that spherical orbits are allowed in the J metric. This approach, however, leaves some loose ends as it cannot say to what extent orbits are ‘decircularised’. This issue is addressed in Section VII with the help of direct numerical integration of the geodesic equations.

Before embarking on the circularity analysis of the three aforementioned spacetimes it is worth pausing a moment to revisit circular motion in Kerr. This topic is of course well documented and studied in the literature (see e.g. Bardeen:1972fi; Wilkins1972) but for the purpose of completeness a brief discussion can be found in Appendix A.

### vi.2 The Johannsen-Psaltis metric

The JP metric belongs to the broader class of the so-called deformed Kerr metrics, the ‘deformation’ in this instance encoded in the function

 h(r,θ)=ε3M3rΣ2, (33)

where is a constant parameter. In terms of the Kerr metric (listed in Appendix A), the JP metric reads

 gJPtt=(1+h)gKtt,gJPtφ=(1+h)gKtφ,gJPrr=gKrr(1+h)(1+ha2sin2θΔ)−1, gJPθθ=gKθθ,gJPφφ=gKφφ+ha2(1+2MrΣ)sin4θ, (34)

and it is clear that corresponds to the Kerr limit.

The search for spherical/spheroidal orbits is greatly facilitated if we restrict ourselves to a small deformation and work perturbatively with respect to that parameter. We thus consider the ‘post-Kerr’ form of the JP metric.

Assuming spherical orbits, the circularity condition (17) becomes,

 −16(r60+a6cos6θ)(a2−ab+r20)EK(r0)−8ε3M3r30[3a4(4M+3r0) −24a3bM+3a2b2(4M−3r0)+2a2r20(7r0−4M)+8abMr20+r40(5r0−12M)] +cos4θ[8a4ε3M3{a4−a2(b2−6r20)+r3(5r0−8M)}−48a4r20(a2−ab+r20)EK(r0)] +cos2θ[−48a2r40(a2−ab+r20)EK(r0)−32a2ε3M3r0{a4(M+2r0)−2a3bM +a2b2(M−2r0)+2a2r20(r0−3M)+6abMr20−3Mr40}]=0+O(ε23). (35)

where the function was introduced back in Eq. (18). The trigonometric functions can be expressed in Legendre polynomials,

 2112a4ε3M3[a4−a2(b2−6r20)+r30(5r0−8M)]P4(θ) −(a2−ab+r20)EK(r0)[1280a6P6(θ)+1152a4(5a2+11r20)P4(θ) +1760a2(5a4+18a2r20+21r40)P2(θ)+528(5a6+21a4r20+35a2r40+35r60)] +1760ε3a2M3[3a6−3a4b2−2a4r0(7M+5r0)+28a3bMr0+a2r30(60M−13r0) −14a2b2r0(M−2r0)−84abMr30+42Mr50]P2(θ)−616ε3M3[−3a8+a6(3b2+20Mr0+22r20) −40a5bMr0+4a4r0{5b2(M−2r0)+r20(21M+40r0)}−240a3bMr30 +15a2r30{3b2(4M−3r0)+2r20(7r0−6M)}+120abMr50+15r70(5r0−12M)]=0+O(ε23). (36)

An solution exists provided the coefficient of each term vanishes independently. It is straightforward to verify numerically that this is not the case for any . We can thus conclude that spherical photon orbits do not exist in the JP spacetime.

Having failed to find spherical orbits our next objective is to look for the more general spheroidal orbits. In the spirit of our previous post-Kerr approximation we employ an expansion (note that this is equivalent to an expansion in the basis)

 r0(θ)=rK+ε3N∑n=0βncos2nθ+O(ε23), (37)

where are constants and is the Kerr spherical orbit radius i.e. . Upon inserting (37) in the circularity condition, the leading order Kerr terms vanish identically leaving an expression linear in . We again express the trigonometric functions in terms of Legendre polynomials and demand the coefficient of each to vanish separately.

To provide a concrete example, we truncate the expansion (37) at . The resulting circularity condition contains all even-order Legendre polynomials in the range . These provide nine algebraic equations for the four unknowns (the equation originating from has to be discarded because it only makes sense when we push the calculation to a higher ). Symbolically, the resulting system of equations is of the following form:

 P18:f18β3=0,(omitted) (38) P16:f(1)16β2+f(2)16β3=0 (39) P14:f(1)14β1+f(2)14β2+f(3)14β3=0 (40) P12:f(1)12β1+f(2)12β2+f(3)12β3=0 (41) P10:f(1)10β0+f(2)10β1+f(3)10β2+f(4)10β3=0, (42) P8:f(1)8β0+f(2)8β1+f(3)8β2+f(4)8β3+f(5)8=0, (43) P6:f(1)6β0+f(2)6β1+f(3)6β2+f(4)6β3+f(5)6=0, (44) P4:f(1)4β0+f(2)4β1+f(3)4β2+f(4)4β3+f(5)4=0, (45) P2:f(1)2β0+f(2)2β1+f(3)2β2+f(4)2β3+f(5)2=0, (46) P0:f(1)0β0+f(2)0β1+f(3)0β2+f(4)0β3+f(5)0=0, (47)

where the coefficients are polynomials . This is an overdetermined system for and it is a straightforward numerical exercise to show that the equations cannot be satisfied simultaneously. For example, the equation can be solved for and subsequently the equation can be solved to furnish . Inserting these solutions in the equations leads to solutions which are not mutually consistent (in the physically relevant - parameter space) unless . This then causes all the coefficients to vanish. The remaining equations in the above system (for example and ) are not mutually consistent. The situation would essentially be the same if we were to increase the truncation order , for in that case the additional number of equations is balanced by the extra coefficients .

The results of this analysis implies that the non-separable JP metric does not admit spheroidal orbits (unless they are confined in the equatorial plane – in that case Eq. (15) applies and leads to circular orbits, see e.g. Glampedakis:2017).

### vi.3 The Hartle-Thorne metric

We next consider the HT metric Hartle1967; HT68,

 ds2=−eν(1+2h)dt2+eλ(1+2μr−2m)dr2+r2(1+2k){dθ2+sin2θ[dφ−(Ω−ω)dt]2}+O(Ω3), (48)

where denotes the stellar angular velocity. The three metric potentials, , are spherically symmetric functions, the latter representing the usual mass function. The rest of the potentials can be expanded in terms of Legendre polynomials,

 h(r,θ) =h0(r)+h2(r)P2,μ(r,θ)=μ0(r)+μ2(r)P2,k(r,θ)=k2(r)P2,ω(r,θ)=ω1(r)P′1. (49)

In the vacuum exterior of a general relativistic star, the HT metric is most conveniently parametrised in terms of the spin parameter (where is the angular momentum and the mass at ), the quadrupole moment (here expressed in terms of the deviation from the Kerr quadrupole of the same mass and spin parameter), the shift in the mass and, finally, the rescaled radial coordinate :

 m=M,eν=e−λ=1−2x,ω1=Ω−2χMx3,μ0M=χ2(δm−1x3),h0=χ2x−2(1x3−δm), (50) h2 =516χ2δq(1−2x)[3x2log(1−2x)+2x(1−1/x)(1−2/x)2(3x2−6x−2)]+χ2x3(1+1x), (51) k2 =−χ2x3(1+2x)−58χ2δq[3(1+x)−2x−3(1−x22)log(1−2x)], (52) μ2M =−516χ2δqx(1−2x)2[3x2log(1−2x