Ghost of vector fields in compact stars

Hector O. Silva ,1,* Andrew Coates ,2,† Fethi M. Ramazanoğlu ,2,‡ and Thomas P. Sotiriou 3,§

1Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Mühlenberg 1, 14476 Potsdam, Germany

2Department of Physics, Koç University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey3School of Mathematical Sciences & School of Physics and Astronomy, University of Nottingham,

University Park, Nottingham NG7 2RD, United Kingdom

(Received 18 October 2021; accepted 10 December 2021; published 20 January 2022)

Spontaneous scalarization is a mechanism that allows a scalar field to go undetected in weak gravityenvironments and yet develop a nontrivial configuration in strongly gravitating systems. At the perturbativelevel it manifests as a tachyonic instability around spacetimes that solve Einstein’s equations. The endpointof this instability is a nontrivial scalar field configuration that can significantly modify a compact object’sstructure and can produce observational signatures of the scalar field’s presence. Does such a mechanismexists for vector fields? Here we revisit the model that constitutes the most straightforward generalization ofthe original scalarization model to a vector field and perform a perturbative analysis. We show that a ghostappears as soon as the square of the naive effective mass squared becomes negative anywhere. This resultposes a serious obstacle in generalizing spontaneous scalarization to vector fields.

DOI: 10.1103/PhysRevD.105.024046

I. INTRODUCTION

The first gravitational wave (GW) signal from a compactbinary coalescence detected by the LIGO-Virgo collabo-ration [1] in 2015 opened a new vista into the nonlinear andhighly dynamical regime of gravity. Moreover, and perhapsmore excitingly, GWs now allow us to probe (or constrain)new physics beyond GR and the Standard Model [2–7].This had so far been limited to astronomical probes either inthe weak gravitational field and slow velocity in our SolarSystem or in the strong gravitational field, but smallvelocity and large separation regime of binary pulsars [8,9].In this context, a particularly appealing new physics

scenario is one where new fundamental fields lie “dormant”in weak-gravity environments and yet manage to havesignificant effects in strongly-gravitating bodies and sys-tems. The prototypical theory that achieves this was firstintroduced by Damour and Esposito-Farese [10,11] andinvolves a massless scalar field φ. The theory is describedby the action

SDEF ¼1

16π

Zd4x

ffiffiffiffiffiffi−g

p ðR − 2∇μφ∇μφÞ

þ Sm½Ψm;Ω2DEFðφÞgμν�; ð1Þ

where g is the metric determinant, R is the Ricci scalar, Smis the action of matter fields Ψm, which couple toΩ2

DEFðφÞgμν, with ΩDEF ¼ exp ðβφ2=2Þ; β being a dimen-sionless constant.The scalar field satisfies the field equation

□φ ¼ −4πβΩ4DEFTφ; ð2Þ

where T is the trace of the matter field’s energy-momentumtensor. Equation (2) clearly admits a vanishing scalar fieldas a solution. However this is not the only solution for agiven matter configuration. Linearized scalar field pertur-bations δφ on the background of a neutron star can beshown to obey a wave equation

ð□ − μ2effÞδφ ¼ 0; μ2eff ¼ −4πβΩ4DEFT; ð3Þ

where μ2eff is a position dependent effective mass squared.For a neutron star described by a perfect fluid, T ¼ 3p − ε(where p is the pressure and ε the total energy density).Typically T < 0 and thus these perturbations can becometachyonic when β < 0 [12,13], with only a weak depend-ence on the equation of state [14–16]. Numerical simu-lations show that this linear instability is ultimatelynonlinearly quenched and thus the star becomes

*hector.silva@aei.mpg.de†acoates@ku.edu.tr‡framazanoglu@ku.edu.tr§Thomas.Sotiriou@nottingham.ac.uk

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Open access publication funded by the Max PlanckSociety.

PHYSICAL REVIEW D 105, 024046 (2022)

2470-0010=2022=105(2)=024046(13) 024046-1 Published by the American Physical Society

spontaneously scalarized. Due to Eq. (2), these scalarizedstars coexist with the GR solutions [defined as stars withφ ¼ 0] and, importantly, are energetically favored: thusthey can form dynamically from stellar collapse [14,17–20]or in neutron star binaries [21–26].The Damour–Esposito-Farese scalarization model can-

not lead to black holes scalarization unless the latter isinduced by surrounding matter [27,28]. However, moregeneral models which fashion couplings with the Gauss-Bonnet invariant, have been shown to lead to black holescalarization, controlled by the mass [29,30] or by the spinof the black hole [31–33] and can take place in stellarcollapse [34]. Black hole scalarization can also havepotentially observable effects in binary black hole binaries

]35–37 ] and be induced by other curvature scalars, such asthe Pontryagin invariant [38,39]. The instability leading toscalarization can also be understood from a quantum fieldtheory perspective, see, e.g., Refs. [40–42].A different type of generalization of the Damour–

Esposito-Farese mechanism that has been explored is toextend it to vector fields [43]. Inspired by [10,11], Ref. [43]studied the action

Sv ¼1

16π

Zd4x

ffiffiffiffiffiffi−g

p ðR − FμνFμν − 2μ2vAμAμÞ

þ Sm½Ψm;Ω2vðAμÞgμν�; ð4Þ

where Fαβ ¼ ∇αAβ −∇βAα is the antisymmetric Faradaytensor and Aα is a vector field with bare mass μv.

1 Inanalogy with ΩDEF, the conformal factor is chosen asΩv ¼ exp ðβAμAμ=2Þ, where β is a free parameter of thetheory. The field equation of Aμ is

∇μFμα ¼ ðμ2v − 4πβΩ4vTÞAα: ð5Þ

This equation promotes the bare mass μv of the Proca fieldto what appears to be an effective mass squared μ2eff ¼ zμ2v,where

z ¼ 1 − 4πðβ=μ2vÞΩ4vT; ð6Þ

for linearized vector field perturbations. This effective masssquared can become negative in the presence of densematter as in the theory (1). This property is not specific tothe theory (4), and is shared with other vector-tensortheories with curvature coupling terms [45–48] or disfor-mal couplings [49,50].Based on the similarity between the field equations (2)

and (5) it is natural to expect that in the theory (4), Aμ couldalso become tachyonically unstable around sufficientlycompact neutron stars and a spontaneous vectorizationmechanism exists. Although nonlinear vectorized neutron

star solutions have indeed been shown to exist in [43], theperturbative manifestation of vectorization has not beenexplored yet. This leaves a number of open questionsunanswered. In particular, a massive vector Aμ is known topropagate an additional longitudinal degree of freedom.What is its role in this process? Could vectorization bescalarization in disguise to some extent? More generally,can it be understood intuitively, as is the case for scala-rization, as a tachyonic instability quenched by nonlinear-ities? Answering these questions is important from amodel-building perspective, but also from a phenomeno-logical perspective. They become even more pressing onceone observes that, intuitively speaking, the aforementionedlongitudinal mode gets contributions in its kinetic termfrom the AμAμ terms in the action. That kinetic term willtherefore have a nontrivial structure, which in turn raisesdoubts about whether this mode is well behaved.Motivated by these questions, here we revisit the model

of Ref. [43] from a perturbative perspective and indeeduncover a ghost instability. Therefore vectorization appearsto be fundamentally different from scalarization. It alsostrongly suggests that the time-evolution problem of a starundergoing vectorization is potentially ill-posed, castingserious doubts on the viability of this theory and otherrelated ones. Combined with the work [51] which alsofound ghost (and gradient) instabilities in generalized Procatheories in compact object backgrounds, our work raisesserious questions about the possibility to generalize theoriginal mechanism of Damour and Esposito-Faresebeyond scalars since all proposed vectorization theoriesfeature at least ghost instabilities.The remainder of this paper explains how we arrived at

these conclusions. In Secs. II and III we review the modelintroduced in [43], restore gauge invariance by performingthe Stuckelberg trick and analyse the resulting fieldequations. In Sec. IV we linearize the theory’s action inthe background of a nonrotating, spherically symmetric starand show how ghost instability appears. In Sec. V we liftthe assumption of linearized gauge field perturbations andconsider the complete set of field equations. We show howghosts, which first went unnoticed in [43], arise. In Sec. VIwe summarize our main results.We work with geometrical units c ¼ G ¼ 1 and use the

ð−;þ;þ;þÞmetric signature. Symmetrization of indices isdefined as AðαβÞ ≡ ðAαβ þ AβαÞ=2 and the antisymmetriza-tion by A½αβ� ≡ ðAαβ − AβαÞ=2.

II. A MODEL FOR SPONTANEOUSVECTORIZATION WITH GAUGE SYMMETRY

Action (4) has been constructed in analogy with (1), but acaveat of the resulting tensor-vector theory is absence ofgauge invariance under Aα → Aα þ ∂αλ (λ being a scalarfunction) due to the mass term μ2vAμAμ. To restore gaugeinvariance, and at the same time more easily investigate the

1Ref. [44] is an earlier study of a similar theory that is mainlyconcerned with cosmology.

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different degrees of freedom in the vector field, we canapply the Stueckelberg trick [52]. It consists of introducinga scalar field ψ (the Stueckelberg field) through thesubstitution

Aα → Aα þ μ−1v ∇αψ ; ð7Þ

which results in a scalar-vector-tensor theory,

S ¼ 1

16π

Zd4x

ffiffiffiffiffiffi−g

p ½R − FμνFμν

−2gμνðμvAμ þ∇μψÞðμvAν þ∇νψÞ�þ Sm½Ψm;Ω2ðAα;∇αψÞgμν�; ð8Þ

with conformal factor

lnΩ ¼ β

2μ2vgμνðμvAμ þ∇μψÞðμvAν þ∇νψÞ: ð9Þ

The theory is now gauge invariant under the simultaneoustransformations:

Aα → Aα þ∇αλ; ψ → ψ − μvλ: ð10Þ

We see that ψ can be set to zero by a suitable choice of λand thus the action (4) is a gauge-fixed version ofaction (8).Indeed, for β ¼ 0 the conformal factor Ω becomes unity

and we recover the Stueckelberg theory minimally coupledto gravity (see, e.g., [53]). If we fix a gauge where ψ ¼ 0(we call this the “Proca gauge”), we obtain the non-minimally coupled Einstein-Proca theory of Ref. [43]. Ifwe instead take μv → 0 we obtain the Einstein-Maxwelltheory with the addition of a scalar field. In Proca theory theμv → 0 limit has an apparent discontinuity of the longi-tudinal polarization mode of Aα. In the “Stueckelberged”version of the same theory, the μv → 0 limit is manifestlycontinuous and corresponds to the decoupling between ψand Aα (the latter associated with the usual Maxwelltheory). Note that, when β ≠ 0, maintaining regularity ofΩ requires that β approaches zero at least as fast as μ2v whentaking the limit μv → 0.The action (8) is written in the Einstein frame (thus we

call gαβ the Einstein frame metric). We will refer to gαβ ¼Ω2gαβ as the Jordan frame metric. We will use tildes todenote objects in the Jordan frame, some of which, as T,already appeared in the Introduction.

III. THE FIELD EQUATIONS

The field equations of the theory can be obtained byvarying the action (8) with respect to ψ , Aα and gαβ:

□ψ ¼ −μv∇μAμ þ ð4π=μvÞ∇μðαμAΩ4TÞ; ð11Þ

∇μFμα ¼ μvgμαðμvAμ þ ∂μψÞ − 4πααAΩ4T; ð12Þ

Gαβ ¼ 8πðTeαβ þ Ts

αβ þ TαβÞ; ð13Þ

where,

αμA ≡ ∂ lnΩ∂Aμ

¼ μv∂ lnΩ∂ð∇μψÞ

¼ β

μvðμvAμ þ∇μψÞ; ð14Þ

and we defined the individual energy-momentum contri-butions from “pure electromagnetic” theory Te

μν and fromthe “Stueckelberg contribution” to the action Ts

μν,

Teαβ ¼

1

4π

�FμαFνβgμν −

1

4gαβFμνFμν

�; ð15Þ

Tsαβ ¼

1

4π

�ðμvAα þ∇αψÞðμvAβ þ∇βψÞ

−1

2gαβðμvAμ þ∇μψÞðμvAμ þ∇μψÞ

�: ð16Þ

The Jordan frame energy-momentum tensor of matter fieldsand its trace are defined as

Tαβ ≡ −2ffiffiffiffiffiffi−gp δSm

δgαβ; and T ≡ gμνTμν: ð17Þ

We also have by construction:

∇αFβγ þ∇γFαβ þ∇βFγα ¼ 0: ð18Þ

Going back to Eq. (12) and due to ∇μ∇νFμν ¼ 0, it isconvenient to define a current jα as:

jα ¼ μvgμαðμvAμ þ∇μψÞ − 4πααAΩ4T; ð19Þ

which is conserved

∇μjμ ¼ 0: ð20Þ

In terms of jα we have:

μ2v∇μAμ ¼ −∇μðμv∇μψ − 4παμAΩ4TÞ: ð21Þ

In the absence of matter (T ¼ 0) and in the Proca gauge(ψ ¼ 0), Eq. (21) becomes the Lorenz constraint on Aα ofProca theory. Thus, the field equation for ψ [cf. Eq. (11)]and the Lorenz constraint on Aμ are tightly connected.We can see the first sign of the ghost by introducing a

third metric,

gαβ ¼ z−1gαβ; ð22Þ

in terms of which the scalar field equation becomes

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□ψ ¼ −gμν�μv∇μAν þ

1

2ð∇μ log zÞð∇νψ þ μvAνÞ

�: ð23Þ

This third metric can, in principle, have a signature changein some parts of the spacetime due to the z−1 term. If thishappens, the field will be a ghost in at least some regioncompared to any field which is coupled to a fixed signaturemetric. Another potential problem is the fact that this metricchanges sign by diverging, rather than crossing zero, in asimilar vein discussed in [54,55]. It is unclear whether thereis a rectification for such a problem, or, worse, whether thetheory can evolve from a state where this metric has a fixedsignature to another where the signature changes.It is also instructive to consider the limit μv ¼ 0, with

β → 0 as fast as μ2v. In this limit, the Stueckelberg field ψ isno longer affected by gauge transformations, so Aμ

becomes a gauge field. Then Aμ smoothly decouples fromψ and the matter fields. However, there is still coupling togravity and ψ continues to be coupled to matter. Inparticular, Eq. (23) becomes,

□ψ ¼ −1

2gμνð∇μ log zÞð∇νψÞ; ð24Þ

note that z does not depend on Aα here. So, ψ will become aghost when the gμν metric changes signature and, as it iscoupled to gravity and matter, its ghostly nature is physical.This same procedure is used in the Stueckelberg picture ofProca theory to show that ψ and Aμ decouple and hencethere is no discontinuity as μv ¼ 0 (i.e., no degree offreedom disappears). In this setting, one has β ¼ 0, flatspacetime, and no matter.One may object that ψ can be completely removed by a

gauge choice such as the Proca gauge ψ ¼ 0, and thus theghost can be exorcised. For this reason we will use the restof the paper to assuage any doubts. We will begin byexamining the quadratic Lagrangian for scalar-vectorperturbations around a neutron star GR solution. Doingso wewill find there exists a gauge invariant scalar field thatsuffers the same problems.

IV. PERTURBATIVE ANALYSIS

A. Background spacetime andoverview of the calculation

In this section we explore the test-field limit of our theory,where we study the dynamics of ψ and Aα in a backgroundcorresponding to a stellar solution of Einstein’s field equa-tions, i.e., a solution of the Tolman–Oppenheimer–Volkoff(TOV) equations [56,57] whose line element we write as

ds2 ¼ −eνdt2 þ rr − 2μ

dr2 þ r2ðdθ2 þ sin2 θdϕÞ; ð25Þ

where ν (lapse) and μ (mass function) are functions of theradial coordinate r only.

In Sec. IV B we will linearize the field equations forsmall field perturbation δψ and δAμ at the level of the fieldequations (11)–(12), and show how the ghost arises in thisbackground. Then, in Sec. IV C, we reach the sameconclusion by directly perturbing the Lagrangian, byexpanding it to second order in the fields on the samebackground.

B. Linearized field equations

We are interested in studying the dynamics of δAα andδψ propagating on the background line element (25). Toproceed we decompose δψ and δAα in scalar and vectorharmonics respectively. This is the convenient basis toexpand scalar and vector fields on the unit two-sphere and,thus, in problems with spherical symmetry. We followclosely the presentation by Rosa and Dolan [58], but with aslightly different normalization. More specifically, we writeδψ as

δψ ¼ 1

r

Xlm

σlmðt; rÞYlmðθ;ϕÞ; ð26Þ

where Ylm ¼ Ylmðθ;ϕÞ are the spherical harmonics withl ¼ 0; 1; 2…, and jmj ≤ l. For the vector perturbations, wedecompose δAα as

δAα ¼1

r

X4i¼1

Xlm

ciulmðiÞ ðt; rÞZðiÞlmα ðθ;ϕÞ; ð27Þ

where c1 ¼ c2 ¼ 1, c3 ¼ c4 ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 1Þp

, and ZðiÞlmα

are the vector harmonics given by

Zð1Þlmα ¼ ½1; 0; 0; 0�Ylm; ð28Þ

Zð2Þlmα ¼ ½0; 1; 0; 0�Ylm; ð29Þ

Zð3Þlmα ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lðlþ 1Þp ½0; 0; ∂θ; ∂ϕ�Ylm; ð30Þ

Zð4Þlmα ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lðlþ 1Þp ½0; 0; csc θ∂ϕ;− sin θ∂θ�Ylm: ð31Þ

These functions are orthonormal when integrated on theunit two-sphere, according to the inner product,

ZðZðiÞlm

μ Þ�ημνZði0Þl0m0ν sin θdθdϕ ¼ δii0δll0δmm0 ; ð32Þ

where ηαβ ≡ diag½1; 1; ð1=r2Þ; 1=ðr2 sin2 θÞ�.Under parity inversion x → −x0 (or equivalently, in

spherical coordinates, θ → π − θ and ϕ → ϕþ π), the firstthree harmonics (i ¼ 1, 2, 3) pick a factor of ð−1Þl, whilethe fourth (i ¼ 4) picks a factor of ð−1Þlþ1. We follow theliterature convention and call the former “even parity”

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modes and the latter “odd parity” modes. The scalarperturbation δψ is of even parity.At this point it will be useful to follow a similar

procedure to [51]. We expand the Stueckelberged action(8) around a GR solution to second order in the test fieldapproximation and find,

S2½δA; δψ � ¼1

4π

Zd4x

ffiffiffiffiffiffi−g

p ½2ð∇νδAμÞð∇½μδAν�Þ

− zðμvδAμ þ∇μδψÞðμvδAμ þ∇μδψÞ�; ð33Þ

where,

z ¼ 1 − 4πðβ=μ2vÞT; ð34Þ

which is unity outside the star, where T ¼ 0. Note that wecould have arrived at an action of this form by using theStueckelberg trick in the Proca Lagrangian with a “dressedmass” zμ2v. Therefore, the results of this section apply toany theory whose quadratic Lagrangian can be put in thisform, i.e., where one would naively expect just a screenedProca field prone to develop a tachyonic instability.Substituting the decompositions of δAα and ψ in harmon-ics, results in a Lagrangian, with even and odd-parity sectordecoupled from one another. We look at each sector next.

C. Monopolar even-parity quadratic Lagrangian

We first focus on the monopole perturbations(l ¼ m ¼ 0), which have the lowest instability thresholdand belong to the even-parity sector. Since Y00 ¼ constant,only the i ¼ 1, 2 vector harmonics are defined [58]. Thismeans that we would need to work with three variables σ00,

uð1Þ00 , and uð2Þ00 ,

δAα ¼1

2ffiffiffiπ

p ½u1ðt; rÞ; u2ðt; rÞ; 0; 0�; ð35aÞ

δψ ¼ 1

2ffiffiffiπ

prσðt; rÞ; ð35bÞ

where, to shorten the notation, we use σ ¼ σ00, u1 ¼ uð1Þ00 ,

and u2 ¼ uð2Þ00 hereafter.Inserting Eqs. (35) in the action (33) and integrating over

the angular coordinates leaves us with,

SðeÞ2 ¼Z

dtdre−

ν2

ffiffiffiffiffiffiffiffiffiffiffiffiffir − 2μ

p4πr5=2

�z2μ2v

�r3ðμvru1 þ _σÞ2

r − 2μ

−eνðσ − rðμvru1 þ σ0ÞÞ2�þ r4

2½u01 − _u2�2

�; ð36Þ

where we defined ð·Þ0 ¼ ∂rð·Þ and ð·Þ·

¼ ∂t ð·Þ. It can bereadily verified that under the gauge transformation (10)with λ ¼ l=ð2 ffiffiffi

πp

rÞ that,

σ → σ − μvl; fu1; u2g → fu1; u2g þ f_l=r; ðl=rÞ0g;ð37Þ

and that the action (36) is invariant under this trans-formation. In fact, it can be verified that, the combination,

Φ ¼ _u2 − u01; ð38Þ

is itself gauge invariant (proportional to the l ¼ 0 compo-nent of the electric field). If we introduce the auxiliary fieldϕ such that, on shell, ϕ ¼ r2Φ, we can rewrite Eq. (36) as,

SðeÞ2 ¼Z

dtdre−

ν2

ffiffiffiffiffiffiffiffiffiffiffiffiffir − 2μ

p4πr5=2

�z2μ2v

×

�−eνðσ − rðμvru2 þ σ0ÞÞ2 þ r3ðμvru1 þ _σÞ2

r − 2μ

�

þ 1

2ϕð2r2Φ − ϕÞ

�: ð39Þ

In this formulation, ϕ, u1, and u2 are all nondynamical:their equations of motion can be solved algebraically in theform, e.g.,

u1 ¼ u1½u2; ∂u2;ϕ; ∂ϕ; σ; ∂σ�: ð40Þ

We can then replace this solution directly into the action,“integrating out” whichever field. Integrating out u1 and u2one arrives at an action that is a functional of ϕ alone (allterms involving σ cancel). This transfers all of the dynamicsfrom σ to ϕ. The resulting action has the form,

SðeÞ2 ¼Z

dtdre−

ν2

ffiffiffiffiffiffiffiffiffiffiffiffiffir − 2μ

p4πr5=2

�1

2z

�e−ν _ϕ2 −

�1 −

2μ

r

�ϕ02

−2C×

zr2ϕϕ0 þ

�−zþ z0C1

zr2þ C2

4r3

�ϕ2

��; ð41Þ

where

C× ¼ rðr − 2μÞz0þ z½ðr − 2μÞð4þ rν0Þ − 2rð1 − 2μ0Þ�; ð42Þ

C1 ¼ ν0rðr − 2μÞ þ 2rμ0 − 2μ; ð43Þ

C2 ¼r2ð1 − 2μ0Þ2

r − 2μ− f6rð1 − 2μ0Þð3þ rν0Þ þ 8r2μ00

− ðr − 2μÞ½17þ rν0ð14þ rν0Þ − 4r2ν00�g: ð44Þ

We see immediately that the sign of the kinetic con-tribution changes if z does (and also diverges when zcrosses 0). That is, we have shown that, in this situation,there is a gauge invariant statement of the problemsdiscussed in Sec. II, arising from Eq. (23).

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D. Odd-parity quadratic Lagrangian

Having identified the presence of a ghost in the even-parity sector, it is natural to ask whether such ghosts alsoarise in the odd-parity sector, which contains a singledegree of freedom u4, with multipole l ≥ 1. We find, afterintegration over the angular coordinates,

SðoÞ2 ¼X∞l¼1

Zdtdr

eν2ð1 − 2μ=rÞ−1

2

4πlðlþ 1Þ�e−νð _u4Þ2

−�1 −

2μ

r

�ðu04Þ2 −

�lðlþ 1Þ

r2þ zμ2v

�u24

�; ð45Þ

where we defined u4 ¼ uð4Þl0 and set m ¼ 0 due to thebackground’s spherical symmetry.Hence, we see that u4 is prone to a tachyonic instability

controlled by the same effective mass squared zμ2v alsoresponsible for inducing a ghost instability in the even-parity sector. Indeed, the term between square brackets isthe effective potential for massive vector axial perturbationsfound in [58], Eq. (13), for z ¼ 1. We then conclude that theaxial sector can become tachyonic unstable, but thedominant effect occurs at lower multipole: the ghostinstability in the even-sector.

V. UNVEILING THE GHOSTIN THE PROCA GAUGE

We have identified a ghost instability in the scalar sectorof our theory, however no ghosts were reported in thespontaneous vectorization theory introduced in Ref. [43],or related theories investigated in Refs. [45–48]. In thissection and related Appendices, we will demonstrate thatthese theories contain divergent terms in their field equa-tions irrespective of whether one uses the Stueckelbergtrick to restore gauge symmetry or not.Recall that the Proca gauge (ψ ¼ 0) is equivalent to the

spontaneous vectorization theory of Ref. [43]. Effectively,this gauge undoes the Stueckelberg trick (7) and we onlyneed to consider Eq. (5). Since there is no separate equationfor ψ in this picture and there are no divergent terms in thisfield equation, it is unclear where the ghost lurks. This iselucidated by considering the constraint equation.Since ∇μ∇νFμν ¼ 0 still holds due to the antisymmetry

of Fμν in Eq. (5), we obtain

∇μ½ðμ2v − 4πΩ4βTÞAμ� ¼ 0: ð46Þ

This is the generalized version of the ∇μAμ ¼ 0 constraintfor a minimally coupled Proca field.The puzzling aspect of Eq. (5) is that it does not have any

explicit indication of a ghost, however we now know fromour discussion in Sec. III that the constraint (46) given inthe form of a conserved current in Eq. (20) is also crucial to

understand the time evolution. Indeed, the constraintimposes a time evolution for A0 that will reveal the ghost.2

Let us rewrite the constraint in terms of z [cf. Eq. (6)],

∇μðzAμÞ ¼ 0: ð47Þ

We can convert the covariant derivatives to partial deriv-atives to obtain

∂0ðffiffiffiffiffiffi−g

pzA0Þ ¼ −∂ið

ffiffiffiffiffiffi−g

pzAiÞ; ð48Þ

where i runs over the spatial coordinates. We see that thistime-evolution equation has divergent terms due to thebehavior of z, even if all fields other than A0 are regular.Outside any matter distribution z ¼ 1 and we require z < 0in some part of spacetime if we want an astrophysicalobject to vectorize. Since z is continuous, it has to vanish atsome point. There is no symmetry to ensure that

ffiffiffiffiffiffi−gpzA0

vanishes where z vanishes since z and its derivatives do notvanish at the same spacetime points in general. This means,A0 will generically diverge even if

ffiffiffiffiffiffi−gpzA0 stays regular.

Alternatively, we can move the z term outside the derivativeon the left-hand side, which means that now the coefficientof the leading time derivative of A0 vanishes at certainpoints. This means that the divergent terms we observed inthe ghost instabilities of ψ manifest themselves not directlyin the field equation (5), but in the constraint equation (46),or equivalently, in Eq. (48).The dynamics of A0 implied by Eq. (48) is first order in

time, thus not strictly of the same nature of the waveequation obeyed by ψ. Nonetheless, the change of sign inthe time derivative leads to an analogous pathology. Thiscan be understood by recasting the field equation (5) into anexplicitly hyperbolic form.Let us start by rearranging the constraint (47) as

∇μðzAμÞ ¼ 0 ⇒ ∇μAμ ¼ −Aμ∇μ ln jzj ð49Þ

Next, we manipulate Eq. (5) as follows

zμ2vAα ¼ ∇μFμα;

¼ ∇μ∇μAα −∇μ∇αAμ;

¼ □Aα −∇α∇μAμ − RμνμαAν;

¼ □Aα þ∇αðAμ∇μ ln jzjÞ − RαμAμ; ð50Þ

2Note that A0 is not a dynamical degree of freedom in thestandard Hamiltonian sense [59]. The zeroth-component of theequation of motion (5), is not a time-evolution equation; itimposes an elliptic constraint on A0 in terms of the othercomponents of the vector and matter fields. However, one canindirectly calculate how A0 evolves in time through the evolutionof these other degrees of freedom, which can be obtained by theconstraint.

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where we related the commutator of two covariant deriv-atives to the Riemann tensor in the third line, and used theconstraint equation (49) in the fourth line. We finally obtain

□Aα þ ð∇μ ln jzjÞ∇αAμ ¼ MαμAμ; ð51Þ

where we defined the mass-squared tensor

Mαβ ¼ zμ2vgαβ þ Rαβ −∇α∇β ln jzj: ð52Þ

We should be cautious about the fact that z contains Aα

terms [inside the conformal factor; cf. Eq. (6)], which,strictly speaking, means that∇α∇β ln jzj also belongs to theprincipal part of the differential equation. However, forperturbative values of Aα, such as in a fixed backgroundcalculation of Sec. IV, this dependence can be ignored toleading order and Mαβ becomes a proper mass-squaretensor. Hence, Eq. (51) can be viewed as a generalizedmassive wave equation.Equation (51) has a divergent mass term due to various

factors of z−1 on its right-hand side. We have the option ofmoving these factors to the left-hand side, which means theprincipal part becomes z□Aμ. This is a field equation proneto a ghost instability since z changes sign as we discussedbefore in Eq. (23). One can also analyze the equation ofmotion for each vector harmonic, which likewise leads todivergent effective mass terms.The behavior of z is slightly modified for a vector field

with no intrinsic mass, μv ¼ 0. In this case Eq. (6), andcorrespondingly Eq. (47), are modified as

z ¼ −4πβΩ4vT ¼ −4πβΩ4

vð3p − εÞ; ð53Þ

where we assume the neutron star matter to behave as aperfect fluid with Jordan frame total energy density ε andpressure p as before. We see that z vanishes outside the starand is generally negative within it; thus it never crosseszero. However, there are still divergences.The first case of the divergence in the field equations for

μv ¼ 0 occurs at the surface of the neutron star. Therelevant part of the TOV equations for a sphericallysymmetric star is [56,57],

dpdr

¼ −εμ

r2

�1þ p

ε

��1þ 4πpr3

μ

��1 −

2μ

r

�−1: ð54Þ

In the outer layers of the star one has p ≪ ε and 4πpr3 ≪ μ[60,61], which allows us to approximate Eq. (54) as

dpdϱ

¼ −ρg; ð55Þ

where we approximated the total energy-density as equal tothe rest mass density (ε ≈ ρ), introduced the proper radiallength ϱ [related to the coordinate radius r as

dϱ=dr ¼ ð1 − 2μ=rÞ−1=2], and defined the “local gravita-tional acceleration” g ¼ ðμ=r2Þð1 − 2μ=rÞ−1=2 [60].Focusing on the outer envelope of the star [62], we can

approximate the spacetime as being Schwarzschild, i.e.,μ ≈M and ν ¼ lnð1 − 2M=rsÞ in Eq. (25), where M is themass and rs the radius of the star. We can further introducea local proper depth ʓ ¼ ðR − rÞð1 − 2M=rsÞ−1=2, in termsof which we can recast Eq. (55) as,

dpdʓ

¼ gsρ; ð56Þ

i.e., the equation of a plane-parallel atmosphere with arelativistic-corrected surface gravity gs ¼ gðrsÞ ¼ ðM=rsÞð1 − 2M=rsÞ−1=2. In the outermost stellar layers, the maincontribution to the pressure is due to a nonrelativisticdegenerate electron gas, for which Eq. (56) can be solvedexactly (see Ref. [60], Sec. 6.9), yielding the scaling ρ ∝ з3=2

and, within the same assumptions, T ∝ −4πβʓ3=2. This, inturn, means that ∇μ ln jzj and the effective mass diverge onthe surface, completing our argument. The same reasoningcan in principle be applied to other systems which have aninterface of vacuum and matter, suggesting that any suchinterfaces would lead to a divergence in the vector fieldequations in general. These divergences at the surface of thestar are not exclusive to the vector-tensor model consideredhere, but are known to also arise, albeitwith a different origin,in Palatini fðRÞ [63–65] and in Eddington-inspired Born-Infeld [66] theories. See also [67,68].The second case of divergence in the field equations for

μv ¼ 0 is related to massive neutron stars. Although T isnegative in general, it can switch sign and become positivein the core of such stars for some equations of state (seee.g., [69–72]). This means that z vanishes somewhereinside the star [cf. Eq. (53)], where our previous resultsfor the μv ≠ 0 case directly apply.Overall, the above discussion provides a heuristic tool to

identify ghosts in spontaneous vectorization theories. If thespacetime dependent μ2eff vanishes in nonvacuum regions in atheory with field equation ∇μFμα ¼ μ2effA

α, this genericallyleads to divergent terms in the explicitly hyperbolic fieldequations. In other words, despite the appearances and thenaming we used, μeff is not the effective mass of all physicaldegrees of freedom. A careful analysis reveals that the trueeffective mass diverges as in Eq. (51), which was overlookedin the original spontaneous vectorization theory of Ref. [43]and other similar theories. We work this out explicitly inAppendixA (for theHellings-Nordtvedt vector-tensor theory[73,74] studied in [45]) and in Appendix B (for the vector-Gauss-Bonnet theory of [46,48]).

VI. CONCLUSIONS

We revisited the tensor-vector gravity model proposed inRef. [43] and explored the vectorization process using

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perturbation theory. This was done by working with agauge invariant, Stueckelberg version of the theory andcomplemented with an analysis of the Lorenz constraint inthe Proca gauge. In analogy with scalarization, one wouldexpect to see the vector field develops a tachyonicinstability, which is then quenched nonlinearly, and thisprocess gives rise to the vectorized configurations found inprevious work. Instead, we have uncovered a ghostinstability. This results demonstrates quite clearly thatthe strong resemblance of this model of vectorization tothe Damour–Esposito-Farese model of scalarization is infact rather misleading and a phase transition process that isphysically similar to scalarization does not take place.A potential way out may exist if one can tame the ghost

instability nonlinearly, similar to the quenching of thetachyonic instability in scalarization. Indeed, “ghost-basedspontaneous tensorization” has been investigated [75]. Inthe vectorization model studied here, ghosts appear inad-vertently, and there is no explicit derivative coupling beforethe introduction of the Stueckelberg mechanism. Yet, if anonlinear quenching mechanism exists, it could, in prin-ciple, suppress the ghost. Note that the z term in Eq. (47)that controls the instability approaches its GR value ofz ¼ 1 when AμAμ → ∞ (for β < 0). Hence, a solution withlarge vector field values can lead to a case where z > 1

everywhere. This possibility was recently investigated foraction (4) in Ref. [76], and all computed static andspherically symmetric vectorized neutron stars were shownto still carry ghost or gradient instabilities. Hence, there isno sign of a quenching of the instabilities so far.The main issue however with the ghost instabilities we

investigated is that it is not known whether their timeevolution can be done. Even if a vector field growing tolarge values might quench the ghost, it is not clear if thevery time evolution of the vector field that leads to growthcan be formulated as a well-posed initial value problem dueto the divergent terms such as those in Eq. (51). Theresolution of this issue requires a mathematical analysis ofthe partial differential equations we have, which is beyondthe scope of this work. We remark that these are notproblems in the Proca limit of our model and in the absenceof matter, in which numerical relativity simulations havebeen performed, e.g., in Refs. [77–79].Spontaneous vectorization theories with restored gauge

symmetry were also conceived using the Higgs mechanismrather than the Stueckelberg mechanism [80], inspired bythe gravitational Higgs mechanism [81–83]. However, thistheory [80] also has divergent terms in its field equationsakin to Eq. (50), hence, it is susceptible to the same ill-posedness problems we discussed here.We worked on the specific theory of Eq. (5), but other

spontaneous vectorization models in the literature havesimilar field equations where ∇μFμα directly appears as theprincipal part [44–49]. Hence, a constraint can be obtainedthe same way as we did, which leads to divergent terms

using the arguments in Sec. Vor related ones, as we show inAppendices A and B.Lastly, we stress that our results are relevant for most

known extensions of spontaneous scalarization to otherfields, not just the vectors, and our study can be consideredas a first step to obtain a no-go theorem for extendingspontaneous scalarization to other fields. For vector fields,Garcia-Saenz et al. [51] has identified the presence of ghostand gradient instabilities in the background of compactobjects in a broad class of generalized Proca theories[84,85]. Similar concerns were also raised in the contextof cosmology in Ref. [86]. Going beyond vector fields, allknown formulations of nonminimally coupled spin-2 fieldsthat could spontaneously grow are known to lead to ghostinstabilities as well [75]. Likewise, p-form fields also havethe same constraint structure we discussed in Sec. V, hencethey suffer from similar divergent terms [87]. Spontaneousgrowth of spinor fields as it was introduced in Ref. [88] alsocontains divergent terms.The only potential exception to our long list of prob-

lematic theories is a second form of spontaneous spinori-zation theory proposed in Ref. [89], whose equations ofmotion are not known to feature divergences. It remains tobe seen if other well-posed theories exist. If this is the case,understanding what distinguishes these theories at a fun-damental level from the problematic ones may lead to aproper no-go theorem for arbitrary generalizations ofspontaneous scalarization.

ACKNOWLEDGMENTS

We thank Leonardo Gualtieri, Kirill Krasnov, HelviWitek and Jun Zhang for discussions. T. P. S. acknowledgespartial support from the STFC Consolidated Grants No. ST/T000732/1 and No. ST/V005596/1. F. M. R. and A. C.were supported by Grant No. 117F295 of the Scientific andTechnological Research Council of Turkey (TÜBİTAK).A. C. acknowledges financial support from the EuropeanCommission and TÜBİTAK under the CO-FUNDED BrainCirculation Scheme 2, Project No. 120C081. F. M. R. isalso supported by a Young Scientist (BAGEP) Award ofBilim Akademisi of Turkey. H. O. S. thanks the hospitalityof the University of Nottingham where this work started.The authors also acknowledge networking support by theGWverse COST Action CA16104, “Black holes, gravita-tional waves and fundamental physics”. Some of ourcalculations were performed with the Mathematica pack-ages XPERT [90] and XCOBA, parts of the XACT/XTENSORsuite [91,92].

APPENDIX A: THE HELLINGS-NORDTVEDTTHEORY

In this Appendix we apply the approach of Sec. V toexamine the field equations in the Hellings-Nordtvedt

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[73,74] vector-tensor theory studied in Ref. [45] as avectorization model.In this theory, the vector field obeys the field equation,

∇μFμα −1

2ωRAα −

1

2ηRα

μAμ ¼ 0; ðA1Þ

where ω and η are dimensionless coupling constants.Let us first obtain a generalized Lorenz constraint

satisfied by Aα by taking a covariant derivative ofEq. (A1) and using ∇ν∇μFνμ ¼ 0,

∇μðωRAμ þ ηRμνAνÞ ¼ 0: ðA2Þ

We can expand this equation and replace the Ricci tensorwith the Einstein tensor and the Ricci scalar. The resultingconstraint equation is,

∇μAμ þ Aμ∇μ ln jωRj þ2η

ηþ 2ω

1

RGμ

ν∇μAν ¼ 0: ðA3Þ

We can now return to Eq. (A1), write Fαβ in terms of Aα,follow the same steps that lead to Eq. (50), and find:

□Aα −∇α∇μAμ −�1

2ωRgαμ þ

�1þ η

2

�Rαμ

�Aμ ¼ 0:

ðA4Þ

At last, using Eq. (A3) we obtain,

□Aα þ∇μ lnðjωRjÞ∇αAμ þ∇α

�2η

ηþ 2ω

1

RGμ

ν∇μAν

�

−MαμAμ ¼ 0; ðA5Þ

where

Mαβ ¼1

2ωRgαβ þ

�1þ η

2

�Rαβ −∇α∇β ln jωRj; ðA6Þ

which should be compared against Eq. (52). Note that inEq. (A5) the last term in the first line is also second order,hence, it contributes to the principal part of the differentialequation in addition to the wave operator. Hence, thisequations is not in an explicitly hyperbolic form, and wecannot immediately identifyMαβ as a squared-mass tensorwhose eigenvalues are related to the effective masses of theindividual degrees of freedom. However, such identifica-tion is possible in the special case η ¼ 0 in which theproblematic term vanishes and then:

Mðη¼0Þαβ ¼ ðω=2ÞRgαβ þ Rαβ −∇α∇β ln jωRj: ðA7Þ

We see, by comparing with Eqs. (A7) and (52), that ωRplays the role of z. We then conclude that a ghost arises forthe same reasons discussed in Sec. V.For the general case η ≠ 0 it is more convenient to

analyse the constraint (A2) which we write as,

∇αðΞαβAβÞ ¼ 0; ðA8Þ

where

Ξαβ ¼ ηGα

β þ ðωþ η=2ÞRδαβ: ðA9Þ

Let us focus on the perturbative regime where thebackground metric is fixed and the Einstein equationshold, i.e., Gαβ ¼ 8πTαβ [45]. For a static, sphericallysymmetric perfect fluid star with energy density ε andpressure p,

Ξαβ ¼ 4πη½ðε − pÞδαβ − 2ðεþ pÞδα0δ0β�

þ 8πωðε − 3pÞδαβ; ðA10Þ

which is diagonal. The constraint can then be written as

∂0ðffiffiffiffiffiffi−g

pΞ0

0A0Þ ¼ −Xk

∂kðffiffiffiffiffiffi−g

pΞk

kAkÞ; ðA11Þ

where we wrote the summation over the spatial coordinatesk explicitly to avoid confusion. This means the diagonalelements have the role of a generalized z in the masslesscase in Eq. (53). We see that ∂0A0 has a contribution in theform of

∂0A0 ¼ −∂rðΞr

rÞΞ0

0

Ar þ… ðA12Þ

The behavior of this term is given by the dependence of theenergy density and the pressure on the radial coordinate atthe surface of the star. We normally encounter power lawdependence in stars due to the TOV equations as wementioned in relation to Eq. (53). Hence, ∂0A0 divergesfor generic configurations of Aμ.We conclude by noticing that the constraint equations of

disformally coupled vector-tensor theories of Refs. [49,50]have a similar structure to Eq. (A8), which would lead tosimilar results in terms of divergences.

APPENDIX B: VECTOR-GAUSS-BONNETTHEORY

In this Appendix we apply the approach of Sec. V toexamine the field equations in the vector-Gauss-Bonnettheory introduced in Ref. [46], and further studied inRef. [48]. The motivation behind these theories is to

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generalize the spontaneous scalarization of black holes[29,30] to vector fields.In this theory, the vector field obeys the field equation,

∇μFμα ¼ vAα − fGAα; ðB1Þ

with

v ¼ 1

2

dVðAμAμÞdðAμAμÞ ; f ¼ 1

2

dFðAμAμÞdðAμAμÞ ; ðB2Þ

where V is the vector field’s self-interaction potential, Fprescribes the coupling between the vector field and G, theGauss-Bonnet invariant. For small field perturbations, thepotential V and coupling function F considered in Ref. [48]reduce to:

v ¼ μ2v; f ¼ β=2; ðB3Þ

where μ2v is the bare mass of Aα and β a coupling constant.As with Eq. (5), one can identify an “effective masssquared” μ2eff ¼ zμ2v, but where now z ¼ 1 − ðβ=μ2vÞG=2.We can now proceed in the same manner as in Sec. V to

obtain

□Aα þ ð∇μ ln jzjÞ∇αAμ −MαμAμ ¼ 0; ðB4Þ

where

Mαβ ¼ μ2vzgαβ −∇β∇α ln jzj; ðB5Þ

[compare against Eqs. (51)–(52)] where the absence of theRicci tensor is due to the assumption of the GR background

being Ricci flat [48]. Therefore, this theory suffers from aghost instability as the one considered in Ref. [43].For a Schwarzschild black hole, G is positive every-

where. Thus, for v ¼ 0 the above argument cannot berepeated verbatim. However, G, and hence the effectivemass squared, changes sign in some regions outside theevent horizon of black holes with dimensionless spin ≳0.5[93]. Therefore, these commonly encountered astrophysicalsystems lead to divergent field equations in such theories.Neutron stars also feature divergences for the case of

v ¼ 0. On a fixed general relativistic background, theGauss-Bonnet invariant of a static spherically symmetricperfect fluid star of energy density ε and pressure p is givenby [30]

GðrÞ ¼ 48μ2

r6− 128π

�2πpþ μ

r3

�ε; ðB6Þ

which is positive definite outside the star. On the otherhand, near the center of the star r ¼ rc, the TOV equationsimply that the mass function is approximately

μc ¼ μðrcÞ ≈4

3πεcr3c; ðB7Þ

where εc ¼ εðrcÞ is the central energy density. We then findthat in the star’s center,

Gc ≈ −256π2ðpc þ εc=3Þεc; ðB8Þ

where pc ¼ pðrcÞ is the central pressure. The right-handside of Eq. (B8) is negative meaning that G, and thus μ2eff ,change sign within the star, numerically found to happennear the surface [30].

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