Thermodynamics of quantum photon spheres

M. C. Baldiotti*

Departamento de Física, Universidade Estadual de Londrina, 86051-990 Londrina, Paraná, Brazil

Walace S. Elias†

Instituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970 São Paulo, SP, Brazil

C. Molina‡

Escola de Artes, Ciências e Humanidades, Universidade de São Paulo,Av. Arlindo Bettio 1000, 03828-000 São Paulo, SP, Brazil

Thiago S. Pereira§

Departamento de Física, Universidade Estadual de Londrina, 86051-990 Londrina, Paraná, Brazil(Received 7 October 2014; published 19 November 2014)

Photon spheres, surfaces where massless particles are confined in closed orbits, are expected to becommon astrophysical structures surrounding ultracompact objects. In this paper a semiclassical treatmentof a photon sphere is proposed. We consider the quantum Maxwell field and derive its energy spectra.A thermodynamic approach for the quantum photon sphere is developed and explored. Within thistreatment, an expression for the spectral energy density of the emitted radiation is presented. Our resultssuggest that photon spheres, when thermalized with their environment, have nonusual thermodynamicproperties, which could lead to distinct observational signatures.

DOI: 10.1103/PhysRevD.90.104025 PACS numbers: 04.70.Dy, 05.30.Jp

I. INTRODUCTION

General relativity predicts the existence of regions wherelight is confined in closed orbits. These structures, theso-called photon spheres or light spheres, are expected to becommon astrophysical objects surrounding ultracompactbodies [1–5]. Although black holes are natural candidatesto create light lines and light surfaces, other bodies couldsupport such objects. Initially considered as a particularfeature of the Schwarzschild spacetime, the photon sphereconcept was generalized and found in a broad class of staticand spherically symmetric geometries [6,7]. Given anapproximate spherical symmetry, staticity, and reasonableenergy conditions, photon spheres should be present, evenconsidering extensions of Einstein’s relativity [6,7].More recently, it is observed a renewed interest in the

physics of photon spheres. For instance, the problem of thecharacterization of the photon spheres in several geometrieswas treated, for example, in [6–8]. The connection betweenphoton sphere parameters and quantities associated to theperturbative dynamics around black holes has been recentlyexplored, for example, in [5,9,10]. From the observationalpoint of view, light sphere and light ring phenomenology isalso an issue. For instance, light ring astrophysical sig-natures are explored in [11,12].

The physical framework we consider is commented inthe following. We assume that a spherically symmetric andstatic distribution of matter generates a photon sphere, withthe photons propagating in vacuum or in optically trans-parent media. We also assume some exchange of photonsof the photon sphere with the surrounding environment, insuch a way that the photon sphere is in thermal equilibriumwith the environment. The photons are considered asmetastable entities, approximately free bosonic particleswith a finite average lifetime in the photon sphere.We propose a semiclassical treatment for quantum

photons in light spheres, with the quantized electromag-netic field in a classical photon sphere background. Byconsidering Maxwell’s electrodynamics in usual sphericalcoordinates and in a suitable gauge, energy spectra for thequantum physical modes are derived. Our approach sug-gests that light spheres populated by photons in thermalequilibrium with their environment have distinct thermo-dynamic properties, which could lead to observablesignatures.The structure of this paper is presented in the following.

In Sec. II we review the notation and comment on some keycharacteristics of the photon spheres, emphasizing theclassical energy spectrum of this system. A quantumtreatment for the electromagnetic field in the photon sphereis introduced in Sec. III. Quantum energy spectra areobtained for the two physical polarizations of the field,and the connections with the classical limit are discussed.In Sec. IV the photon sphere is presented as a thermal

*baldiotti@uel.br†walace.elias@usp.br‡cmolina@usp.br§tspereira@uel.br

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bosonic system, and its thermodynamics is characterized.Some final remarks are made in Sec. V. We use signatureð−;þ;þ;þÞ and natural units with G ¼ ℏ ¼ c ¼ kB ¼ 1throughout this paper.

II. PHOTON SPHERES IN SPHERICALLYSYMMETRIC SPACETIMES

In the present work, we are interested in static andspherically symmetric spacetimes. These geometries areequipped with four Killing vector fieldsKt,Kx,Ky, andKzsatisfying

½Kx;Ky� ¼ Kz; ½Ky;Kz� ¼ Kx; ½Kz;Kx� ¼ Ky;

ð1Þ

and

½Kt;Kx� ¼ ½Kt;Ky� ¼ ½Kt;Kz� ¼ 0: ð2Þ

We are assuming the existence of a “static region” in thespacetime, where Kx, Ky, Kz are spacelike and Kt istimelike. For these spacetimes, a coordinate systemðt; r; θ;ϕÞ may be defined in the static region, such thatKt ¼ ∂=∂t, r is the “areal radius” and ðθ;ϕÞ are the usualangular coordinates that cover S2 surfaces, invariant underthe action of the diffeomorphisms associated toKx,Ky, andKz. In terms of this coordinate system, the line element iswritten as

ds2 ¼ gttðrÞdt2 þ grrðrÞdr2 þ r2ðdθ2 þ sin2θdϕ2Þ; ð3Þ

with −gttðrÞ > 0 and grrðrÞ > 0 in the static region.We will also assume that the spacetime is asymptotically

flat and that the compact object modeled by this geometryhas no electric charge. In this case, the metric functionsbehave as −gttðrÞ → 1 and g−1rr ðrÞ → 1 in the limit r → ∞.Considering the spacetime described by Eq. (3), the photonsphere is a two dimensional surface generated by nullgeodesics that describe circular orbits. The photon sphereradius is denoted in the present work as R. If the geometrymodels a black hole, the photon sphere will be locatedoutside the event horizon, always in the static region. Thespacetimes selected by the conditions imposed include notonly the Schwarzschild geometry but also many other casesof interest.Given affine parametrized null geodesics xμðλÞ, four

constants of motion can be constructed: E, Lx, Ly, and Lz,associated to the Killing fields Kt, Kx, Ky, and Kz,respectively. If the geometry is asymptotically flat, theseconstants can be interpreted as energy and angular momen-tum components associated to the geodesic, as seen by astatic observer far from the compact object. For geodesicsin the photon sphere, E, Lx, Ly, and Lz obey the classicalconstraint [13]:

E2 ¼ −gttðRÞL2

R2; ð4Þ

with L2 ¼ L2x þ L2

y þ L2z . The constant L2 can be inter-

preted as the (classical) squared total angular momentum ofthe photon in the geodesic.For any given geodesic in the photon sphere, it is always

possible to choose a coordinate system such that this nullorbit is located in the equatorial plane (θ ¼ π=2). In thiscase, the equation of motion which describes the geodesicis [13]

�drðλÞdλ

�2

¼ VeffðrðλÞÞ; ð5Þ

where the effective potential Veff is given by

VeffðrÞ ¼1

grrðrÞ�þ E2

−gttðrÞ−L2

r2

�: ð6Þ

The photon sphere radius R is such that VeffðRÞ ¼ 0and dVeffðRÞ=dr ¼ 0. A basic fact about photon spheresis that the classical trajectories that form this structureare usually, but not necessarily, unstable. Spacetimes wherethe photon sphere orbits are stable are presented, forexample, in [14,15]. For unstable photon orbits, it ispossible to estimate an average lifetime τ for the nullcircular geodesics [9],

τ ¼�−gttðRÞR2

2L2

d2VeffðrÞdr2

����r¼R

�−1=2: ð7Þ

However, it should be pointed out that the averagelifetime τ must be used cautiously. It is possible to showthat, in spacetimes of interest, the time averaged Lyapunovexponent vanishes and the orbit behaves as if there were noinstability, even though it clearly has an unstable region inphase space (this is a general feature of a larger class ofdynamical systems involving the so-called homoclinicorbits [16,17]). As a result, photons in the light spheremay have long lifetimes, even when they are characterizedby unstable effective potentials.At this point, although we are not restricted to this

specific scenario, it is illustrative to particularize ourdiscussion considering the photon sphere prototype,present in the Schwarzschild spacetime. For this geometry,−gttðrÞ ¼ g−1rr ðrÞ ¼ 1 − 2M=r. In this case R ¼ 3M, andtherefore a photon sphere can be supported by a sphericalcompact body of mass M, with a radius smaller than 3M.The Schwarzschild black hole is one possibility for such anobject. An estimate for the lifetime of a photon in thisphoton sphere is τ ¼ 3

ffiffiffi3

pM. The relevant point about this

calculated value for τ is the fact that it is finite andproportional to the compact body mass. But again, itshould be used cautiously. Null geodesics in Schwarzschild

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spacetime are one example where the averaged Lyapunovexponent vanishes [16,17]. More concretely, null geodesicsspiral around r ¼ R if the classical constraint in Eq. (4) isapproximately obeyed, as can be seen by considering thesolutions presented in Eqs. (231) and (238) in Chap. 3 of[13], for instance. In this case, the null-mass particles mayhave quasicircular trajectories, circling many times aroundthe photon sphere, arbitrarily close the surface r ¼ R [13].The preceding example shows that photons can be expectedto be in the light spheres for quite long times in relevantastrophysical scenarios, such as large black holes invacuum.

III. ENERGY SPECTRA OF PHOTON SPHERES

The picture of quantum photons around a nonchargedcompact object will be made more precise consideringastrophysical situations where the photons in the lightsphere have a long lifetime, and therefore are essentiallyconfined to the surface r ¼ R. The photon sphere is in theoutside region of the black hole (if one is present) and theradiation emitted by the photon sphere is detected by adistant observer. As a consequence, issues associated to thefield behavior at the event horizon [18] are not a problemhere. We will effectively quantize the electromagnetic fieldin the three dimensional manifold S2 ×R.The typical strategy in the quantization of the electro-

magnetic field, based on plane wave expansions, is notconvenient to us due to the geometry of the photon sphere.For our purposes, a spherical representation of the field isbetter suited. However, given the gauge arbitrariness, it isnot obvious that in this representation the electromagneticfield can be decomposed in two independent modes(polarizations). This issue was already considered in[13,19] in classical contexts involving curved spacetimes.The quantum field theory treatment was developed, forexample, in [20,21,21–23]. In this paper we will adapt thetreatment developed in [20,22], considering the electro-magnetic quanta in the photon sphere.We proceed to the quantum treatment with the intro-

duction of the Maxwell field, minimally coupled to thegeometry. From the classical electromagnetic tensor Fμν,the potential Aμ is defined as

Fμν ¼ ∇μAν −∇νAμ: ð8Þ

Following [22], we adopt a modified Feynman gauge, withthe Lagrangian density written as

LF ¼ ffiffiffiffiffiffi−g

p �−1

4FμνFμν −

1

2G2

�; ð9Þ

where g ¼ det½gμν� and

G ¼ ∇μAμ þ KμAμ; ð10Þ

Kμ ¼ ð0; g0ttðrÞ=½gttðrÞgrrðrÞ�; 0; 0Þ: ð11Þ

The gauge condition then reads G ¼ 0.For our purposes, only the physical modes (according to

[22]) are relevant. An important result is that, consideringthe modified Feynman gauge defined by Eqs. (9)–(11), theelectromagnetic potential has a scalar and a vector physicalmodes [24]. These “polarizations” will be observables ofthe theory. In the photon sphere, the scalar mode potentialAscμ has only one non-null component,

Ascμ ¼ ð0; Asc

r ; 0; 0Þ: ð12Þ

On the other hand, the vector mode potential Avecμ can be

written as

Avecμ ¼ ð0; 0; Avec

θ ; Avecϕ Þ: ð13Þ

In the following, we will proceed to the quantization of theelectromagnetic field in r ¼ R.

A. Scalar and vector physical modes

For the quantization of the potentials Ascr and Avec

i ,corresponding respectively to the scalar and vector sectorsin the photon sphere, we construct the one-particle Hilbertspaces [25] Hsc

1 and Hvec1 associated to the (scalar and

vector mode) photons. We take as Hsc1 the set of functions

Hsc1 ∶S2 → C that have, as a dense subset, the collection of

functions f that can be expanded as

f ¼Xlm

flmðtÞYlmðθ;ϕÞ; ð14Þ

where Ylm are the spherical harmonics.In a similar way, we define the one-particle Hilbert space

Hvec1 , associated to the (vector mode) photons, as the set of

functions Hvec1 ∶vecðS2Þ → C that have, as a dense subset,

the collection of vectors ~fi that can be expanded in vectorspherical harmonics,

~fi ¼Xlm

~flmðtÞYðlmÞi ðθ;ϕÞ; ð15Þ

with vec ðS2Þ denoting the S2 vector bundle and YðlmÞi the

vector spherical harmonics [20,22].With Hp

1 defined, for p ¼ sc for the scalar modeand p ¼ vec for the vector mode, the one-particleHamiltonian can be constructed, based on the Killingvector field Kt, as

Hp ¼ i∂∂t ; p ∈ fsc; vecg: ð16Þ

That is, our notion of energy is being defined bystatic observers which follow integral curves of Kt. The

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one-particle angular momentum operators Lpx , L

py , and Lp

z

are given by

Lpx ¼ i

�− sinϕ

∂∂θ − cot θ cosϕ

∂∂ϕ

�; ð17Þ

Lpy ¼ i

�cosϕ

∂∂θ − cot θ sinϕ

∂∂ϕ

�; ð18Þ

Lpz ¼ i

∂∂ϕ ; ð19Þ

where p ∈ fsc; vecg in Eqs. (17)–(19).From the one-particle sector, we construct the Fock

space associated with the photons in the photon sphere withthe usual procedure. Taking into account Eq. (16), weobserve that

Ylmðθ;ϕÞe−iϵt; Y�lmðθ;ϕÞeþiϵt; ð20Þ

are positive and negative energy modes which span a densesubset of Hsc

1 . We define creation and annihilation oper-ators, ðascϵlmÞ† and ascϵlm, such that the quantum version ofthe electromagnetic potential in the photon sphere, theoperator Asc

r , can be expanded as

Ascr ¼

Xj

½ascðjÞYlmðθ;ϕÞe−iϵt þ ðascðjÞÞ†Y�lmðθ;ϕÞeþiϵt�;

ð21Þ

with ðjÞ≡ ðϵlmÞ. In addition, based on the definition ofHvec

1 , we note that

YðlmÞi ðθ;ϕÞe−iϵt; ðYlm

i Þ�ðθ;ϕÞeþiϵt; ð22Þ

are positive and negative energy modes which span a densesubset ofHvec

1 . Creation and annihilation operators ðavecϵlmÞ†and avecϵlm are introduced such that Avec

i , with i ∈ fθ;ϕg, canbe expanded as

Aveci ¼

Xj

½avecðjÞ YðlmÞi ðθ;ϕÞe−iϵt

þ ðavecðjÞ Þ†ðYðlmÞi Þ�ðθ;ϕÞeþiϵt�; ð23Þ

with ðjÞ≡ ðϵlmÞ, consonant to the notation in Eq. (21).According to the usual conventions, we denote by Hp,

Lpx , Lp

y , and Lpz the Hamiltonian and (orbital) angular

momentum operators acting in the Fock space. Distinctionfrom their one-particle counterparts is made by context.With the Casimir operator ðLpÞ2 given by

ðLpÞ2 ¼ ðLpx Þ2 þ ðLp

y Þ2 þ ðLpz Þ2; ð24Þ

expressions for Hp and ðLpÞ2 in terms of the creation andannihilation operators apðiÞ and ðapðiÞÞ† are

Hp ¼Xi

ϵðapðiÞÞ†apðiÞ; ð25Þ

ðLscÞ2 ¼Xi

lðlþ 1ÞðascðiÞÞ†ascðiÞ; ð26Þ

ðLvecÞ2 ¼Xi

½lðlþ 1Þ − 1�ðavecðiÞ Þ†avecðiÞ : ð27Þ

Quanta defined by ðapðiÞÞ† and apðiÞ have well-defined energyϵ and squared angular momentum, the later quantity withmagnitude lðlþ 1Þ for the scalar and ½lðlþ 1Þ − 1� forthe vector modes.We set as the quantum constraint for the scalar and vector

sectors of the multiparticle photon sphere

½Hp; ½Hp; Apk �� ¼

−gttðRÞR2

½ðLpÞ2; Apk �; ð28Þ

with p ∈ fsc; vecg and k ∈ fr; θ;ϕg. The validity ofproposed relations in Eq. (28) will be justified by theirclassical limit. As will be seen in the end of this section,from Eq. (28) classical field equations will be obtained.From these equations, in the geometrical optics limit, theclassical constraint in Eq. (4) can be recovered. It followsfrom the quantum constraints in Eqs. (28), consideringthe results in Eqs. (25)–(27), that the scalar and vectormode photon energies ϵ assume values in discrete setsfϵpl ;l ¼ 1; 2;…g labeled by l, where

ϵscl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−gttðRÞ

pR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 1Þ

p; ð29Þ

ϵvecl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−gttðRÞ

pR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 1Þ − 1

p: ð30Þ

These are the photon sphere energy spectra of the electro-magnetic scalar and vector sectors. The relations (29)and (30), respectively for the scalar and vector modequanta, are the quantum version of the classical constraintin Eq. (4).

B. Geometrical optics limit

As a consistency check, let us interpret the obtainedresults in the geometrical optics limit. The results in thissection are important for the proper justification of theproposed quantum constraints in Eq. (28).In the classical limit, the quantum constraints imply that

the classical scalar and vector mode potentials Ascr , Avec

θ ,and Avec

ϕ satisfy Klein-Gordon-type equations (at thephoton sphere),

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∂2Φ∂t2 ¼ −gttðRÞ

R2~∇2Φ; ð31Þ

where ~∇2 is the Laplace operator on S2,

~∇2 ¼ 1

sin2θ∂2

∂ϕ2þ 1

sin θ∂∂θ

�sin θ

∂∂θ

�: ð32Þ

Taking the eikonal ansatz,

Φ ¼ Φ0eiS; ð33Þwith the appropriate eikonal conditions [26], we obtain

∇μS∇μS ¼ 0: ð34ÞIt is straightforward to show that not only is ∇μS a nullvector, as indicated in the eikonal equation (34), but alsothat its integral curves are null geodesics.Moreover, in the eikonal limit we have l ≫ 1, and

therefore the scalar and vector electromagnetic spectracoincide in this limit,

ðϵlÞ2 ¼ −gttðRÞl2

R2; ð35Þ

as seen from Eqs. (29) and (30). Relation (35) shows that inthe eikonal limit, a (scalar or vector) photon in a light ray,with orbital angular momentum l and energy ϵl, obeys theclassical constraint in Eq. (4), as it should by consistencywith the geometrical optics limit.

IV. THERMODYNAMICS OF PHOTON SPHERES

In the treatment for the photons introduced in Secs. IIand III, the quantum electromagnetic field was assumed tobe free, allowing for no direct coupling among the photonsin the photon sphere. Still, in many astrophysical situationsof interest, some interchange of photons from the photonsphere with the surrounding environment is expected. Infact, perturbations in the photon sphere would take awayphotons from this region, which is consistent with a finiteaverage lifetime for photons in the light sphere. At the sametime, the photon sphere continuously captures externalphotons, as long as they satisfy the constraint in Eq. (4).The astrophysical scenario assumed is a densely populatedphoton sphere, in dynamical thermal equilibrium with itsenvironment.We consider then a light sphere populated by photons

with a well-defined energy, subjected to the Bose-Einsteinstatistics, in thermal equilibrium with its surroundings.Moreover, the number of photons is not conserved.Therefore, the total macroscopic energy U of the photonsphere at temperature T is

U ¼X

p∈fsc;vecg

X∞l¼1

Xþl

m¼−l

ϵpl

expðϵplT Þ − 1

: ð36Þ

Because of spatial isotropy, the sums in m can be readilycalculated,

U ¼X∞l¼1

ð2lþ 1Þ�

ϵscl

expðϵsclT Þ − 1þ ϵvecl

expðϵveclT Þ − 1

�; ð37Þ

where, for convenience, we have separated the contribu-tions of the two modes, presented in Eqs. (29) and (30).The relevant macroscopic thermodynamic quantities are

defined only if a proper thermodynamic limit can beestablished [27]. In this limit, it is considered the behaviorof the photon sphere as its area A and the number ofphotons N tend to infinity. It is also required that thedensity ratio N=A is bounded. In order to characterizethe thermodynamics of the photon sphere, we considerthe energy differences Δϵsc ≡ ϵsclþ1 − ϵscl and Δϵvec ≡ϵveclþ1 − ϵvecl of the scalar and vector modes, respectively.In the thermodynamic limit they coincide,

Δϵ ¼ Δϵsc ¼ Δϵvec ¼ −gttðRÞð2lþ 1Þ2ϵR2

; ð38Þ

where it was used in Eq. (38) the fact that R is large andΔl=R is small in the thermodynamic limit. Therefore, wecan rewrite Eq. (37) as a Riemann sum, in the form

U ¼Xϵ

4R2

−gttðRÞϵ2

expðϵTÞ − 1Δϵ; ð39Þ

with ϵ ∈ fϵscl g ∪ fϵvecl g, according to Eqs. (29) and (30). Interms of the photon sphere area A ¼ 4πR2, Eq. (39) iswritten as

UA

¼Xϵ

1

−gttðRÞπϵ2

expðϵTÞ − 1Δϵ: ð40Þ

We now take the thermodynamic limit [27] of Eq. (40),in which R is large and U=A is bounded. From Eqs. (29)and (30), we see that the maximum value of Δϵ for a givenl, Δϵmax ¼ maxfΔϵscg ∪ fΔϵvecg, can be made arbitrarilysmall as R is larger. Moreover, the partial sums in Eq. (40)are well defined as Δϵmax is taken arbitrarily smaller.Therefore, the Riemann sum in Eq. (40) tends to theRiemann integral in the thermodynamic limit (with a fixedvalue for ϵ). Finally, in the limit of large ϵ, the properRiemann integral tends to an integral in the form

UA

¼Z

∞

0

ρðϵÞdϵ; ð41Þ

with ρ given by

ρðϵÞ ¼ 1

−gttðRÞπϵ2

expðϵTÞ − 1: ð42Þ

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The spectral energy density ρ for the radiation emitted bythe photon sphere is qualitatively different from the usualPlanck distribution that might be expected, and couldprovide observational signatures of the photon sphere.This is one of the main results in this paper.As a side remark, we point out that the spectral

distribution in Eq. (42), although having the same formof the analogous result in the two-torus [28], was obtainedhere considering a thermodynamic treatment on a two-sphere, a topological and geometrical distinct object.Moreover, in the two-torus setup, the question of theproper decomposition of the electromagnetic field in thespherical representation is absent. In the two-sphere, this isa nontrivial issue.From the result in Eq. (42), we observe that the radiation

emitted by a photon sphere should have a distinct profile,when compared to the emission of a usual star. Forinstance, the emitted total energy of the photon sphere isgiven by

U ¼ σT3; ð43Þwhere σ is a constant. This is the modified Stefan-Boltzmann law for the quantum photon sphere.

V. FINAL REMARKS

In the present work we considered spherically sym-metric and asymptotically flat geometries, modelingultracompact bodies capable of maintaining light inclosed orbits. We derived quantum and thermodynamicproperties of photon spheres in thermal equilibrium withthe environment. The electromagnetic field in the photonsphere was considered in a second quantization scheme,and its energy spectra derived. The associated thermo-dynamics suggests an observational signature that couldbe used to distinguish photon spheres from other astro-physical objects.The results obtained here are very general within the

specified premises. For instance, the Einstein fieldequations are not used, and in this sense only kinematicaspects of gravity are assumed in the present work. Arelevant point is that gravity manifests itself only throughthe redshift factor −gttðRÞ, according to Eqs. (29), (30),and (42).It should be pointed out that the quantum constraint in

Eq. (28) is not unique. Other (nonequivalent) quantiza-tions, and consequently other dynamics, are possible.Nonetheless the proposal in Eq. (28) is “robust” in the

sense that it correctly gives expected limits, consideringboth the one-particle sector and the eikonal regime. Infact, the nonunique character of the quantization pro-cedure is not a peculiarity of our work, but a generalissue in the quantization process with Fock spaces.Usually the problem of the vacuum choice, when treatingquantum theories in Minkowski spacetime, can beresolved by the imposition of invariance by the actionof the Poincaré group (and the mass-shell condition). Inour case, this procedure can be applied since we havean asymptotic flat limit. Therefore, one argument show-ing that our constraint is satisfactory is the waveequation (31), which corresponds to the field equationfor the electromagnetic field in the Feynman gauge.This equation is covariant (for each potential compo-nent), and tends to a Klein-Gordon equation with theMinkowski metric considering large values of R. Thissuggests that the vacuum of the quantum field theoryobtained with Eq. (28) is well defined. Any otherquantization, which respects covariance and the mass-shell condition (with a redshift correction), will beunitarily equivalent.One assumption made in this work is the consideration

that the compact objects maintaining the photon spherehave no electric charge. This condition was implicitlyused when we postulated that the electromagnetic pertur-bations do not couple with the gravitational ones [23].Still, the treatment of photon spheres around chargedcompact objects should be possible if the electromag-netic-gravitational compound modes in [13,29,30] areconsidered.Further generalizations could be made with the consid-

eration of asymptotically de Sitter or anti–de Sitter space-times. These could be interesting in cosmological setups orAdS/CFT applications. In fact, the assumption of asymp-totically flatness is not needed for any of the classicalresults presented, and could possibly be relaxed if a properquantization scheme is used. In this case, the energydefinition and the proper choice of observers wouldbecome a relevant issue. Work along those lines is currentlyunder way.

ACKNOWLEDGMENTS

This work was partially supported by Conselho Nacionalde Desenvolvimento Científico e Tecnológico (CNPq) andFundação de Amparo à Pesquisa do Estado de São Paulo(FAPESP), Brazil.

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