Ultracompact vector stars

Gianmassimo Tasinato

Physics Department, Swansea University, SA28PP, United Kingdom

###### Abstract

We analytically investigate a new family of horizonless compact objects in
vector-tensor theories of gravity, called ultracompact vector stars. They are
sourced by a vector condensate, induced by a non-minimal coupling with
gravity. They can be as compact as black holes, thanks to their internal
anisotropic stress. In the spherically symmetric case their interior resembles
an isothermal sphere, with a singularity that can be resolved by tuning the
available integration constants. The star interior smoothly matches to an
exterior Schwarzschild geometry, with no need of extra energy momentum tensor
at the star surface. We analyse the behaviour of geodesics within the star
interior, where stable circular orbits are allowed, as well as trajectories
crossing in both ways the star surface. We analytically study stationary
deformations of the vector field and of the geometry, which break spherical
symmetry, and whose features depend on the vector-tensor theory we consider.
We introduce and determine the vector magnetic susceptibility as a probe of
the star properties, and we analyze how the rate of rotation of the star is
affected by the vector charges.

##  1 Introduction

The new experimental opportunities offered by gravitational wave science
motivate the theoretical analysis of compact objects in theories beyond
General Relativity (GR). Encouraging us to take a broader view on
gravitational interactions, their study can address, by means of concrete
examples, open issues in our understanding of gravity in strong-field regimes.
For instance: establish the validity of no-hair [1] and cosmic censorship [2]
hypothesis. Clarify the extremal properties of self-gravitating compact
objects [3], as a function of their content. Determine the most convenient
methods for testing GR against alternative theories of gravity [4]. See e.g.
[5] for a review on these and related subjects. Examples of exotic compact
objects relevant for our discussion are: boson stars [6, 7, 8], first
introduced as regular solutions of Einstein-Klein-Gordon equations (see e.g.
[9, 10] for reviews), and then shown to exist also in the vector-tensor case
[11, 12, 13]. Gravastar, supported by negative pressure fluids [14, 15, 16,
17] (see e.g. [18] for a general discussion). The singular isothermal sphere
[19, 20, 21, 22, 23], a self-similar configurations supported by a perfect
fluid in its interior which contains a naked singularity [24], and which has
been investigated also in the context of exceptions to the cosmic censorship
hypothesis (see e.g. [25] for a review).

In this work we present and investigate new analytic solutions describing a
family of compact objects in vector-tensor theories of gravity, which we dub
ultracompact vector stars. They correspond to a vector condensate, induced by
a non-minimal coupling with gravity [26, 27, 28, 29, 30] that break the vector
Abelian symmetry. The Lagrangian for the system has no additional parameters
with respect to an Einstein-Maxwell system. The vector field can be
interpreted as a dark photon with relevant applications for dark energy (see
e.g. [31, 32]), but it can also be motivated by recent theoretical advances in
characterising scenarios of ultralight vector dark matter [33, 34, 35, 36, 37]
(see e.g. [38] for a recent review). We refer also to [39, 40] for recent
reviews on the physics of dark photons, and their phenomenological
consequences.

The self-gravitating objects can be as compact as black holes, thanks to their
internal anisotropic stress: we discuss their properties in section 2. In
fact, a free parameter controls their compactness, spanning from Minkowski
space to the black hole solutions first studied in [41] (see [42, 43, 44, 45,
46, 47, 48, 49, 50] for further developments). In the spherically symmetric
case, their external geometry corresponds to the Schwarzschild solution as in
GR, even if they are characterised by an electric-type charge. The solution
has no horizon, since its Schwarzschild radius is located inside the star. The
interior part of the solution resembles a singular isothermal sphere, and the
singularity at the star centre can be resolved by tuning some of the available
integration constants. The interior configuration is smoothly matched to the
exterior geometry of the star, with no need of extra energy momentum tensor
localised at the star surface.

By studying the behaviour of geodesics and of vector and metric perturbations,
we investigate how to distinguish vector star solutions from self-gravitating
configurations in GR. In fact, the simplicity of our solutions allow us to
analytically study in detail their properties. The study of geodesics in
section 3 reveals new features with respect to Schwarzschild black holes in
GR, thanks to the possibility to cross the star surface in both directions. We
find new stable circular orbits for time-like geodesics whose location depend
on the star compactness. We also study geodesic trajectories that enter,
bounce, and leave from the star interior, with the time spent to complete the
process depending on the star properties. For null-like geodesics, we relate
as [51] the presence of unstable circular orbits (light-rings) with parameters
controlling the compactness of the object (see also [18] for a comprehensive
review on how to distinguish exotic compact objects from black hole
configurations).

In section 4 we analyse stationary, parity-odd fluctuations of the system. The
inclusion of magnetic vector perturbations allows us to switch on a magnetic-
type charge, which backreacts on the geometry at the linearised level, and
breaks the spherical symmetry forcing the star to slowly rotate. The perturbed
geometry depends on the features of the vector profile, making the exterior
configuration distinguishable from their GR counterparts. The rotation rate
depends both on the values of magnetic and electric charge, and the interior
region of the object is dragged by the external rotation. We study the
response of the vector field profile to an external magnetic field, and the
distinctive features of the induced magnetic susceptibility.

We conclude in section 5, discussing possible future directions for studying
the properties of vector star configurations.

##  2 Spherically symmetric vector stars

###  2.1 The set-up and the field equations

We consider a vector-tensor theory described by the following Lagrangian
density (we set $M_{\rm Pl}\,=\,c\,=\,\hbar\,=\,1$)

${\cal
L}\,=\,\frac{R}{2}-\frac{1}{4}\,F_{\mu\nu}^{2}+\frac{\beta}{4}\,V_{\mu}\,V_{\nu}\,G^{\mu\nu}\,,$
(2.1)

with $V_{\mu}$ the vector field and
$F_{\mu\nu}\,=\,\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ the associated
field strength. The last term, proportional to the dimensionless parameter
$\beta$, controls a ghost-free [26, 27, 28] non-minimal coupling of vector
fields with the Einstein tensor $G_{\mu\nu}$. This coupling with gravity
breaks the Abelian gauge symmetry $V_{\mu}\to V_{\mu}-\partial_{\mu}\chi$,
with $\chi$ an arbitrary scalar field. Notice that we do not include a mass
term for the vector: the gauge symmetry breaking is only due to coupling with
gravity. It would be interesting to find symmetry arguments protecting the
structure of the theory governed by Lagrangian (2.1), for example along the
lines of [52]. While Lagrangians as (2.1) have been studied at length in the
context of dark energy, it would be interesting to explore possible
applications for dark matter, as for the ultralight vector dark matter
scenarios of [33, 34, 35, 36, 37].

From now on, we make the choice

$\beta=1\,,$ (2.2)

since it is the simplest option for finding novel configurations with
interesting properties. This choice implies that we do not have additional
free parameters with respect to the standard Einstein-Maxwell system. In this
section we focus on spherically symmetric solutions associated with the metric
element

$\displaystyle ds^{2}\,=\,g_{\mu\nu}\,dx^{\mu}dx^{\nu}$ $\displaystyle=$
$\displaystyle-A(r)\,dt^{2}+\frac{dr^{2}}{B(r)}\,+r^{2}\,d\theta^{2}+r^{2}\,\sin^{2}{\theta}\,d\phi^{2}\,,$
(2.3)

and an electric-type vector field Ansatz

$\displaystyle V_{\mu}dx^{\mu}$ $\displaystyle=$
$\displaystyle\alpha_{0}(r)\,dt+\Pi(r)\,dr\,,$ (2.4)

associated with a dark electric charge for the object. In section 4 we extend
our discussion to configurations with magnetic-type charges, and solutions
breaking spherical symmetry. Notice that the metric can be influenced by the
vector radial vector profile $\Pi(r)$, since the theory we consider breaks the
Abelian gauge symmetry and the radial vector component is not removable by a
gauge transformation. The field equations associated with our Ansätze of eqs
(2.3) and (2.4) are (all quantities depend on the radial direction $r$ only)

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Pi\,\left(A^{\prime}-\frac{A}{rB}\left(1-B\right)\right)\,,$
(2.5) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\hskip-8.5359pt\frac{\alpha_{0}^{\prime
2}+\partial_{r}\alpha_{0}^{2}}{4A}+\frac{\alpha_{0}^{2}(B-1)}{8r^{2}\,A\,B}+\left(\frac{4+3B\Pi^{2}}{8\,r\,A}-\frac{\alpha_{0}^{2}}{8\,r\,A^{2}}\right)\,A^{\prime}+\frac{B\left(4+(3B-1)\,\Pi^{2}\right)-4}{8\,r^{2}\,B}\,,$
(2.6) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{r}\,\left[\frac{r^{2}\,B^{1/2}\,\alpha_{0}^{\prime}}{A^{1/2}}-\frac{r\,\alpha_{0}}{A^{1/2}\,B^{1/2}}\right]+B^{-1/2}\,\partial_{r}\left(\frac{r\,\alpha_{0}}{A^{1/2}}\right)+\frac{\alpha_{0}\,(B-1)}{2\,A^{1/2}\,B^{1/2}}\,,$
(2.7)

and

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{r}\left[B^{3/2}\,\Pi^{2}+B^{1/2}\left(1-\frac{\alpha_{0}^{2}}{4\,A}\right)\right]+B^{1/2}\,\partial_{r}\left(\frac{\alpha_{0}^{2}}{4A}\right)+\frac{r\,B^{1/2}\alpha_{0}^{\prime
2}}{A}$ (2.8)
$\displaystyle+\frac{B^{1/2}\,\left(B-1\right)\,\Pi^{2}}{2r}+\frac{\alpha_{0}^{2}\,(B-1)}{2r\,B^{1/2}\,A}+\frac{2B-2}{r\,B^{1/2}}\,.$

The condition (2.5) identifies two branches of solutions. If we were
considering $\beta$ a free parameter, the branch with $\Pi(r)\,=\,0$ would be
continuously connected with the Reissner-Nordström configuration, when sending
$\beta$ to zero [41]. We concentrate here on the second branch with
$\Pi(r)\,\neq\,0$, that exists because of the non-minimal coupling with
gravity in the Lagrangian (2.1). Eqs (2.5) and (2.6) are algebraic equations
and dictate the conditions

$\displaystyle B(r)$ $\displaystyle=$
$\displaystyle\frac{A(r)}{A(r)+r\,A^{\prime}(r)}\,,$ (2.9)
$\displaystyle\Pi^{2}(r)$ $\displaystyle=$
$\displaystyle\frac{r\,\alpha_{0}^{2}(r)}{A(r)}\,\left(\frac{A^{\prime}(r)}{A(r)}-2\frac{\alpha_{0}^{\prime}(r)}{\alpha_{0}(r)}-r\,\frac{\alpha_{0}^{\prime
2}(r)}{\alpha_{0}^{2}(r)}\right)\,.$ (2.10)

The Ricci scalar associated with the metric (2.3), and with the condition
(2.9), is

${\text{Ricci
scalar}}\,=\,\frac{\left(2A(r)-r\,A^{\prime}(r)\right)\left(2A^{\prime}(r)+rA^{\prime\prime}(r)\right)}{2r\,\left(A(r)+r\,A^{\prime}(r)\right)}\,,$
(2.11)

and usually diverges at the origin $r=0$, unless $A(r)$ acquires a specific
profile (more on this later). Once we implement the conditions (2.9) and
(2.10), it is straightforward to check that a solution of eqs (2.5)–(2.8) is
111A version of these solutions was presented in [43]. See also the discussion
in [44].

$\displaystyle\alpha_{0}(r)$ $\displaystyle=$
$\displaystyle\frac{2\,Q\,R}{r}+2\sigma+\frac{2\,\left(1-\sigma\right)}{1+\gamma}\,\left(\frac{r}{R}\right)^{\gamma}\,,$
(2.12) $\displaystyle A(r)$ $\displaystyle=$
$\displaystyle\sigma^{2}-\frac{2M}{r}+\frac{2\,\sigma\,\left(1-\sigma\right)}{(1+\gamma)}\,\left(\frac{r}{R}\right)^{\gamma}+\frac{\left(1-\sigma\right)^{2}}{(1+2\gamma)}\,\left(\frac{r}{R}\right)^{2\gamma}\,,$
(2.13) $\displaystyle B(r)$ $\displaystyle=$
$\displaystyle\frac{A(r)}{\left(\sigma+\left(1-\sigma\right)(r/R)^{\gamma}\right)^{2}}\,,$
(2.14)

where $Q$, $\gamma$ and $\sigma$ are dimensionless constant parameters, while
$M$, $R$ dimensionful parameters. The solution for $\Pi(r)$ is cumbersome and
we do not write it explicitly: it is directly obtained plugging eqs (2.12) and
(2.13) in eq (2.10). Choosing $\gamma=0$, $\sigma=1$ one finds the black hole
solutions of [41]: the geometry corresponds to a stealth Schwarzschild space-
time, with a non-vanishing profile for $\alpha_{0}(r)$ and $\Pi(r)$ and a
horizon at $r\,=\,2M$. Turning on $\gamma$ and $\sigma-1$ allows us to
determine new families of horizonless configurations: these parameters have a
transparent physical interpretation, that we are going to discuss in what
comes next.

###  2.2 The simplest ultracompact vector star configuration

We proceed asking: Can we use the solutions (2.12)–(2.14) as building blocks
to construct horizonless objects, with compactness ${\cal C}$ in the interval
$0\leq{\cal C}<1/2$, and with an asymptotically flat exterior geometry?

We are going to answer affirmatively to the question, by discussing a simple
analytical configuration with the desired properties. The horizonless object
can be as compact as a Schwarzschild black hole, violating the Buchdahl bound
[53] thanks to the anisotropic stress characterising its interior. For this
reason we call the solution ultracompact vector star. (See e.g. [54] and
references therein for a recent analysis of consequences of compact objects
with pronounced anisotropic stress tensor.) The star interior geometry is
sourced by a vector condensate, induced by the non-minimal couplings with
gravity in Lagrangian (2.1). As we will learn, the simplicity of the
configuration will allow us to analytically investigate many of its
properties.

#### The geometry of the vector star

We assume that the geometry is separated in two regions, smoothly matching at
a radial position $R$, which we interpret as the radius of the compact object.
In fact, we aim at finding a configuration with a continuous transition from
an interior to an exterior region at $r\,=\,R$, with no need of additional
localised energy-momentum tensor at the star surface $r=R$.

For characterizing the interior region $r\leq R$ we take the configuration
(2.12), (2.13), and set $M\,=\sigma\,=\,0$ focussing on $\gamma\geq 0$. The
line element $ds^{2}\,=\,g_{\mu\nu}dx^{\mu}dx^{\nu}$, with
$g_{\mu\nu}\,=\,{\rm diag}(-A(r),\,1/B(r),\,r^{2},\,r^{2}\sin^{2}{\theta})$,
results

$\displaystyle ds_{\rm(int)}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{(1+2\gamma)}\,\left(\frac{r}{R}\right)^{2\gamma}\,dt^{2}+(1+2\gamma)\,{dr^{2}}\,+r^{2}\,d\theta^{2}+r^{2}\,\sin^{2}{\theta}\,d\phi^{2}\,.$
(2.15)

The coefficients of the vector field configuration,
$V_{\mu}dx^{\mu}\,=\,\alpha_{\rm(int)}(r)\,dt+\Pi_{\rm(int)}(r)\,dr$, are

$\displaystyle\alpha_{\rm(int)}(r)$ $\displaystyle=$
$\displaystyle\frac{2Q_{I}\,R}{r}+\frac{2}{1+\gamma}\,\left(\frac{r}{R}\right)^{\gamma}\,,$
(2.16) $\displaystyle\Pi^{2}_{\rm(int)}(r)$ $\displaystyle=$ $\displaystyle
4\,\left(1+2\gamma\right)^{2}\,\left[\left(\frac{Q_{I}\,R^{1+\gamma}}{r^{1+\gamma}}+\frac{1}{1+\gamma}\right)^{2}-\frac{1}{1+2\gamma}\right]\,,$
(2.17)

where $Q_{I}$ the vector charge in the interior geometry. The solution
(2.15)-(2.17) is valid for any radius $r\leq R$: for ensuring that the profile
for $\Pi_{\rm(int)}(r)$ is real in this interval, we impose the following
inequality on the charge $Q_{I}$:

$Q_{I}\,\geq\,\left(\frac{1}{\sqrt{1+2\gamma}}-\frac{1}{1+\gamma}\right)\,.$
(2.18)

The function in the right hand side of eq (2.18) is positive and has a maximum
for $\gamma\simeq 5.22$, with value $0.13$: a $Q_{I}\,>\,0.14$ then ensures
that (2.18) is satisfied for any value of $\gamma$. The geometry described by
eq (2.15) is quite simple, and corresponds to a self-similar space-time
controlled by the constant parameter $\gamma$. In fact, the parameters have
been chosen in such a way to relate our geometry to the one of the singular
isothermal sphere [19, 20, 21, 22]. Under a scale transformation
$r\to\lambda\,r$, $t\to\lambda^{1-\gamma}\,t$, the metric scales as
$ds^{2}\,\to\,\lambda^{2}\,ds^{2}$. Examples of self-similar geometries have
been found in the literature [24, 23], sourced by a perfect fluid equation of
state.

In our case the interior geometry is sourced by vector field condensate,
induced by the non-minimal couplings of the vector with gravity: the
corresponding energy-momentum tensor associated with the vector field
222Notice that we could also include further sources of internal stress
tensor, for example an additional perfect fluid: the solutions can be extended
to include this case as well. can be computed straightforwardly using our
Ansatz for the interior star configuration. It is anisotropic and can be
expressed as

$\displaystyle T_{\mu}^{\,\nu}\,=\,{\rm
diag}\left(-\rho,\,p_{r},\,p_{t},\,p_{t}\right)\,,$ (2.19)

with

$\displaystyle\rho_{r}$ $\displaystyle=$
$\displaystyle\frac{A(r)\left(2A^{\prime}(r)+rA^{\prime\prime}(r)\right)}{r\,\left(A(r)+rA^{\prime}(r)\right)^{2}}\,=\,\frac{2\gamma}{(1+2\gamma)\,r^{2}}\,,$
(2.20) $\displaystyle p_{r}$ $\displaystyle=$ $\displaystyle 0\,,$ (2.21)
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{r\,A^{\prime}(r)}{4\,A(r)}\,\rho_{r}\,=\,\frac{\gamma^{2}}{(1+2\gamma)\,r^{2}}\,=\,\frac{\gamma}{2}\,\rho_{r}\,.$
(2.22)

While the first of the equalities in the previous equations applies to any
solution of the system (2.5)–(2.8) (selecting the branch with non-vanishing
profile $\Pi(r)$), the remaining ones are specialized to the internal
configuration (2.15). Notice that the tangential pressure is larger than the
radial one, $p_{t}\geq p_{r}$, hence we have the opportunity to overcome
Buchdahl theorem [53, 18].

The interior geometry (2.15) is singular, since it has a singularity at the
origin $r=0$ as it can be readily checked computing the Ricci scalar. (Self-
similar configurations are being discussed as possible candidates for the
exceptions to the cosmic censorship hypothesis, see e.g. [25] for a review.)
In our system, the singularity can be resolved turning on the parameter
$\sigma$ in eqs (2.12) and (2.13): we will study this option in section 2.3.
We find this feature interesting, since the singularity can be resolved
without changing the initial action.

For characterizing the exterior region $r\geq R$ we instead set $\gamma=0$,
$\sigma=1$ in eqs (2.12) and (2.13). The resulting line element corresponds to
a Schwarzschild geometry

$\displaystyle ds_{\rm(ext)}^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)\,dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}\,+r^{2}\,d\theta^{2}+r^{2}\,\sin^{2}{\theta}\,d\phi^{2}\,.$
(2.23)

The vector field profile
$V_{\mu}dx^{\mu}\,=\,\alpha_{\rm(ext)}(r)\,dt+\Pi_{\rm(ext)}(r)\,dr$ is

$\displaystyle\alpha_{\rm(ext)}$ $\displaystyle=$ $\displaystyle
2+\frac{2\,Q_{E}\,R}{r}\,,$ (2.24) $\displaystyle\Pi^{2}_{\rm(ext)}$
$\displaystyle=$
$\displaystyle\frac{4\,Q_{E}^{2}\,R^{2}+8M\,r+8Q_{E}\,R\,r}{(r-2M)^{2}}\,.$
(2.25)

The quantity $Q_{E}$ corresponds to an electric-type charge for the star
configuration, as measured by an external observer. In order to smoothly match
the interior and exterior geometries at radius $R$, we select the exterior
mass $M$ and the exterior charge $Q_{E}$ as

$\displaystyle Q_{E}$ $\displaystyle=$
$\displaystyle{Q}_{I}-\frac{\gamma}{1+\gamma}\,,$ (2.26) $\displaystyle M$
$\displaystyle=$ $\displaystyle\frac{\gamma}{1+2\gamma}\,R\,,$ (2.27)

where $Q_{I}$ is the vector charge in the interior, as appearing in eq (2.16).
Interestingly, these conditions automatically ensure that the profiles for
$A$, $B$, $\alpha_{0}$, $\Pi$ are continuous at $r=R$, as well as 333 Notice
that it is not necessary to impose that the first derivatives of $B$ and $\Pi$
are continuous at the surface $r=R$, since these components on the radial
direction along which we match the geometries. This can be checked by
computing the Israel junction conditions. Alternatively, we can integrate
equations (2.5)-(2.8) along the radial direction, over a small interval
$R-\epsilon$, $R+\epsilon$ for an infinitesimal $\epsilon$. One can check that
the consistency of the equations requires the continuity of all functions at
$r=R$, but only of the first derivatives of $A$, $\alpha_{0}$ at that
position. the first derivatives of $A$, $\alpha_{0}$, with no need of extra
energy momentum tensor at the matching surface position. This differs from
gravastar configurations, which make use of a crust of energy momentum tensor
for connecting the exterior with the interior geometry [14]. Using this
property, in what follows we define the star surface as the position $r=R$
where the star interior and exterior geometries smoothly match.

Figure 1: The time-time metric component $A(r)$ (see our metric Ansatz in eq
(2.3)) both in the interior and exterior of the star. We make the following
choices (from left-to-right and top-to-bottom) of the exponent $\gamma$:
$\gamma=10.2,\,1.2,\,0.2,\,0.02$. Red dashed line: the Schwarzschild radius
$r=2M$. Black dashed line: the position of the star surface $r=R$.

We turn off possible additional contributions associated with parameters
$\sigma-1$, $\gamma$ in eq (2.13) in the exterior region of the geometry,
since we find that only the Schwarzschild profile in eq (2.23) allows us to
consistently satisfy the matching conditions at the sphere radius $R$. We
interpret this fact as a manifestation of a no-hair condition for this set-up,
so that in the spherically symmetric case the exterior geometry is only
characterised by the mass $M$, while it is independent from the charge
$Q_{E}$. We will learn in section 4 that the story is more complex (and
interesting) when breaking spherical symmetry, since then the geometry depends
on magnetic and electric charges.

#### The vector star compactness

We find the inequality

$r_{\rm Schw}\equiv 2M\,=\,\frac{2\,\gamma\,R}{1+2\,\gamma}\,\leq\,R\,,$
(2.28)

showing that the Schwarzschild radius $r_{\rm Schw}$ is inside the star radius
$R$. We conclude that our configuration has no horizons. Hence the non-
negative parameter $\gamma$ has a transparent physical interpretation. It
controls the compactness

${\cal C}\,\equiv\,M/R$ (2.29)

of the object we are examining. We find

$0\,\leq\,{\cal C}\,=\,\frac{\gamma}{1+2\gamma}\,<\,\frac{1}{2}\,.$ (2.30)

Hence this quantity spans from $0$ for $\gamma=0$ (flat space) to $1/2$ for
$\gamma=\infty$, which is the same compactness of a Schwarzschild black hole.
As anticipated above, the vector star can violate the Buchdahl bound ${\cal
C}\leq 4/9$, thanks to a sizeable anisotropic stress.

We represent in Fig 1 the time-time metric component $A(r)$, together with the
position of the would-be Schwarzschild radius, for different choices of
$\gamma$. As $\gamma$ becomes smaller and smaller, the geometry approaches
flat space. In the opposite case, the larger the $\gamma$ is, the nearer the
Schwarzschild radius approaches from inside the surface of the star. In the
limit $\gamma\to\infty$ the interior geometry is not well defined, at least in
Boyer-Lindquist coordinates, since the metric profile for $A$ becomes flatter
and flatter (see Fig 1, upper left panel). However, as will discuss in what
comes next, this limit can be meaningful in specific contexts and
applications.

###  2.3 Resolving the singularity in the star interior

The singularity at the star origin $r\,=\,0$ can be resolved by switching on
the parameter $\sigma$ within the interval $0\leq\sigma\leq 1$ in eqs (2.13),
at least for large enough values of $\gamma$. In order to obtain a
configuration with a smooth matching at the star surface, the interior
geometry and vector field solutions are given by (the cumbersome expression
for $\Pi(r)$ can be determined plugging the next formulas in eq (2.10)):

$\displaystyle\alpha_{0}(r)$ $\displaystyle=$
$\displaystyle\frac{2\,Q_{I}\,R}{r}+2\sigma+\frac{2\,\left(1-\sigma\right)}{1+\gamma}\,\left(\frac{r}{R}\right)^{\gamma}\,,$
(2.31) $\displaystyle A(r)$ $\displaystyle=$
$\displaystyle\sigma^{2}+\frac{2\,\sigma\,\left(1-\sigma\right)}{(1+\gamma)}\,\left(\frac{r}{R}\right)^{\gamma}+\frac{\left(1-\sigma\right)^{2}}{(1+2\gamma)}\,\left(\frac{r}{R}\right)^{2\gamma}\,,$
(2.32) $\displaystyle B(r)$ $\displaystyle=$
$\displaystyle\frac{A(r)}{\left[\sigma+\left(1-\sigma\right)(r/R)^{\gamma}\right]^{2}}\,.$
(2.33)

The exterior geometry is again given by the Schwarzschild expressions (2.23),
but this time the mass and the outside vector charge are given by

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{(1-\sigma)\,\gamma\,(1+\gamma+\sigma\,\gamma)}{(1+\gamma)(1+2\gamma)}\,R\,,$
(2.34) $\displaystyle Q_{\rm E}$ $\displaystyle=$
$\displaystyle{Q}_{I}-\frac{\gamma(1-\sigma)}{1+\gamma}\,,$ (2.35)

in order to continuously match the interior with the exterior. This
configuration is typically less compact than the one of section 2.2, since now
the compactness parameter ${\cal C}$ spans between ${\cal C}=0$ for
$\gamma=0$, to ${\cal C}\,=\,(1-\sigma^{2})/2$ for $\gamma\to\infty$ (recall
that $0\leq\sigma\leq 1$).

The curvature invariants are now no more necessarily singular for $r\to 0$. In
fact, computing for example the Ricci scalar by means of eq (2.11) we find

$\displaystyle\hskip-17.07182pt{\text{Ricci
scalar}}\,=\,\frac{2\,r^{\gamma-2}\,\gamma\,(1-\sigma)}{\left[\sigma+(1-\sigma)\,r^{\gamma}\right]^{3}}\,\left[\sigma^{2}+\frac{\sigma\,(2-\gamma)\,(1-\sigma)}{1+\gamma}\,r^{\gamma}+\frac{(1-\gamma)\,(1-\sigma)^{2}}{1+2\gamma}\,r^{2\gamma}\right]\,.$
(2.36)

Hence, if $\gamma\geq 2$, the Ricci scalar can be non-singular at the origin,
when turning on any positive value for $\sigma$. The other curvature
invariants built in terms of the Riemann and Ricci tensors show a similar
behaviour.

Figure 2: The time-time metric profile $A(r)$ of eq (2.32) evaluated nearby
the origin, choosing $\gamma=2$ and $\sigma=0.2$ (left panel) and $\sigma=0.8$
(right panel).

We represent in Fig 2 the profile for $A(r)$, which makes manifest that this
function is non-vanishing at the origin and can be continued for negative $r$.
One might explore these configurations as possible wormhole solutions [55,
56], and study their properties and their stability. Notice that the parameter
$\sigma$ here is free in the interval $0\,\leq\,\sigma\,\leq\,1$: it would be
interesting to determine physical motivations for its size and how to relate
it with observable properties, as for some wormhole configurations [57]. We
postpone these questions to future work.

##  3 Geodesics

We learned in the previous section that – when considering spherically
symmetric configurations – the exterior geometry of our vector star solutions
corresponds to a Schwarzschild space-time. In this and the next section we
discuss possible methods for distinguishing our system from GR configurations,
and for probing the interior properties of the star.

We start studying time-like and null-like geodesics, as a probe of the
internal geometry of the star. The behaviour of geodesics around our
configuration can be richer 444The topic in a similar context has been
recently studied in the work [23]. than what occurs in a Schwarzschild
geometry within GR, also thanks to the possibility of having geodesics
crossing the star surface in both directions, probing its interior.

In fact, the analytic study of geodesics is important for understanding the
global and causal properties of the geometry under consideration. For time-
like geodesics, we show that there can be additional stable circular orbits
with respect to Schwarzschild black holes. Geodesics can cross the star
surface in both directions, probing its compactness. For null-like geodesics,
we study the connections between the compactness of the configuration, and the
existence of unstable circular orbits, the light rings associated with a
photosphere [51]. We assume that the probe particle experiencing the geometry
is uncharged under the vector field $V_{\mu}$.

We explore the case $\sigma=0$ and focus on the configurations of section 2.2.
Using standard textbook methods it is straightforward to obtain the relevant
geodesic equations. We study the trajectory of a probe particle at radial
position $r(\tau)$, with $\tau$ indicating the particle proper time. We
introduce the dimensionless combination

$y(\tau)\,\equiv\,\frac{r(\tau)}{R}\,,$ (3.1)

with $R$ indicating the radius of the star.

For the case of time-like geodesics, when $y>1$ the particle is outside the
star, and experiences a Schwarzschild space-time characterised by a mass $M$
related with $\gamma$ by eq (2.27). The particle geodesics is controlled by
the equation

$\frac{1}{2}\left(\frac{dy}{d\tau}\right)^{2}+U_{\rm ext}(y)\,=\,\frac{\cal
E}{R^{2}}\,\,\,,\hskip 28.45274pt{\text{for $y\geq 1$.}}$ (3.2)

The geodesic potential is

$\displaystyle U_{\rm ext}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\,\left[-\frac{M}{R\,y}+\frac{\ell^{2}}{2\,y^{2}}-\frac{M\,\ell^{2}}{R\,y^{3}}\right]\,,$
(3.3) $\displaystyle=$
$\displaystyle\frac{1}{(1+2\gamma)\,R^{2}}\,\left[-\frac{\gamma}{y}+\frac{\ell^{2}\,(1+2\gamma)}{2\,y^{2}}-\frac{\gamma\,\ell^{2}}{y^{3}}\right]\,.$

In eqs (3.2)-(3.3) the quantity ${\cal E}$ is a parameter associated with the
conserved energy of the probe particle, and $\ell\,=\,L/R$ is a convenient
dimensionless combination, with $L$ indicating the conserved angular momentum
of the particle.

The particle trajectory in the interior of the star, $y\leq 1$, is instead
described by the equation

$\frac{1}{2}\left(\frac{dy}{d\tau}\right)^{2}+U_{\rm int}(y)\,=\,\frac{\cal
E}{R^{2}}\,\,,\hskip 28.45274pt{\text{for $y\leq 1$.}}$ (3.4)

The interior potential is

$\displaystyle U_{\rm int}(y)$ $\displaystyle=$
$\displaystyle\frac{\left(1+\gamma\right)^{2}}{2\,R^{2}\,(1+2\gamma)}\times$
(3.5)
$\displaystyle\hskip-8.5359pt\,\left[y^{\frac{2\gamma}{(1+\gamma)}}+\left(\frac{\ell^{2}}{\left(1+\gamma\right)}+\frac{\gamma^{2}}{1-\gamma^{2}}\right)\,{y^{-\frac{2(1-\gamma)}{1+\gamma}}}-\frac{1+\gamma\left(3+\ell^{2}-\gamma(2+\ell^{2})\right)}{\left(1-\gamma^{2}\right)\,\left(1+\gamma\right)}\right]\,.$

Figure 3: Particle potential for time-like geodesics. Left panel:
$\gamma=\ell=1/8$. Right panel $\gamma=2$, $\ell=1.4$. Black dashed line: star
surface. Red dashed line: Schwarzschild radius. On the right we include in
magenta dashed line the innermost stable circular orbit position calculated
within GR, and located at $y\,=\,6\,M\,R$. Both panels makes manifest that
there are extrema of the geodesic potential located within the star radius.

The integration constants characterising the interior potential (3.5) have
been chosen so to ensure that the potential $U_{\rm int}(y)$, together with
its first derivative 555For the special case $\gamma=1$ it is not possible to
ensure continuity of the first derivative of the potential at the star
surface. , is continuously connected with the potential $U_{\rm ext}(y)$ at
the star surface $y=1$. When $\gamma<1$ the interior potential is unbounded
and diverges at plus infinity at the origin. This implies that a time-like
geodesics entering the star can not reach the singularity, and bounces back to
the exterior (more on this later). We represent in Fig 3 the entire profile of
the geodesic time-like potential, for two representative values of $\gamma$.

We can now study more quantitatively the main features of time-like geodesics,
using our expressions (3.3) and (3.5) for the potentials. In the exterior part
of the geometry $y\geq 1$ (or $r\geq R$), the potential has extrema if the
following condition is realised

$\ell\,>\,\sqrt{12}\,\frac{\gamma}{1+2\gamma}\,.$ (3.6)

When this inequality is satisfied, the extrema are located at the positions

$y_{0}\,=\,\frac{\ell^{2}\,(1+2\gamma)}{2\,\gamma}\left[1\,\pm\,\sqrt{1-\frac{12\gamma^{2}}{\ell^{2}(1+2\gamma)^{2}}}\right]\,.$
(3.7)

The plus choice corresponds a maximum of the potential (always located in the
star exterior), the minus choice is a minimum of the potential. The innermost
stable circular orbit (ISCO) in the star exterior is at

$y_{\rm isco}\,=\,\frac{6\gamma}{1+2\,\gamma}\,,$ (3.8)

and is outside the star if we choose $\gamma>1/4$.

Considering now the interior geometry, we find that the potential (3.5) admits
an extremal point at

$y_{0}^{\frac{2}{1+\gamma}}\,=\,\frac{\gamma^{2}+(1-\gamma)\,\ell^{2}}{\gamma(1+\gamma)}\,.$
(3.9)

The position $y_{0}$ lies within the star interior if the following condition
on $\ell$ is satisfied

$\displaystyle\ell^{2}$ $\displaystyle<$
$\displaystyle\frac{\gamma}{1-\gamma}\hskip 28.45274pt{\text{for
$\gamma<1$}}\,,$ (3.10) $\displaystyle\ell^{2}$ $\displaystyle<$
$\displaystyle\frac{\gamma^{2}}{\gamma-1}\hskip 28.45274pt{\text{for
$\gamma>1$}}\,.$ (3.11)

The value $\gamma=1$ is special case, since the extremum is always at $y=1/2$
(regardless of the value of $\ell$). When $\gamma>1/4$, by tuning $\ell$, we
can have local stable minima for the time-like geodesic potential both in the
star interior and the star exterior, as shown in Figure 3 (right panel).

It is interesting to study time-like geodesics which cross the star surface in
both directions: we can use them for probing the star interior. In this case,
the analysis of the particle trajectories can help in determining the
properties of the internal geometry. The simplest option are radial plunge
orbits, that start at rest from plus infinity with zero angular momentum, and
enter inside the star radius. Let us then choose ${\cal E}=\ell=0$ in the
geodesic evolution equations. If we further choose $\gamma<1$, the interior
potential (see eq (3.5)) has an infinite barrier at the origin, which prevents
the trajectory from falling into the singularity. A massive particle arriving
from infinitely far away enters into the star region with finite velocity at
the star radius, and changes the direction of its speed at the zero of the
internal potential $U_{\rm int}$. Then it bounces, crossing the star surface
with the same velocity (in opposite direction) as it enters, and then travels
back to infinite distance from the star. In such a situation, the propert time
$\tau_{\rm int}$ spent within the star in the ‘bouncing process’ depends only
on $\gamma$, and on the star radius. An observer far away from the star, who
measures the time for radial plunge orbits to reach the star and bounce back,
can then infer the star compactness.

The proper time $\tau_{\rm int}$ spent within the star interior is given by
the integral

$\displaystyle\frac{\tau_{\rm int}(\gamma,R)}{R}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{R}\,\int_{y_{\star}}^{1}\,\frac{dy}{\sqrt{-U_{\rm
int}(y)}}\,,$ (3.12) $\displaystyle=$
$\displaystyle\frac{\sqrt{4+8\gamma}}{1+\gamma}\,\int_{y_{\star}}^{1}\,\frac{dy}{y^{\frac{\gamma-1}{\gamma+1}}}\,\left[\frac{\left(1+3\gamma-2\gamma^{2}\right)\,y^{\frac{2(1-\gamma)}{(1+\gamma)}}}{(1-\gamma)(1+\gamma)^{2}}-y^{\frac{2}{(1+\gamma)}}-\frac{\gamma^{2}}{1-\gamma^{2}}\right]^{-1/2}\,,$
(3.13)

where $y_{\star}$ is the zero of the potential in the interior – i.e. the zero
of the integrand function within square parenthesis in eq (3.13). In general,
this expression needs to be integrated numerically, although for some special
values of $\gamma$ an analytical integration is possible.

For example, for $\gamma=1/2$ we get

$\displaystyle T_{\rm int}(1/2,R)$ $\displaystyle=$
$\displaystyle-\frac{4\,\sqrt{2}}{3}\,\left(1+\frac{4}{3}\,\arccos\left[-\frac{1}{\sqrt{37}}\right]\right)\,R\,,$
(3.14) $\displaystyle\simeq$ $\displaystyle 2.48\,R\,.$ (3.15)

Given its dependence on the value of $\gamma$, the time spent by a particle
travelling along the geodesics in the interior geometry of the star
configuration can be used as a tool for studying the star properties, as its
compactness that depends on $\gamma$ and $R$ through relation (2.30).

The equation for null-like geodesics results

$\frac{1}{2}y^{\prime
2}+\frac{1}{R^{2}}\,\frac{\ell^{2}}{2\,y^{2}}\left(1-\frac{2\gamma}{(1+2\gamma)\,y}\right)\,=\,\frac{{\cal
E}^{2}}{R^{2}}\,,$ (3.16)

in the exterior. (The various quantities have the same meanings as in the
previous time-like case.) Imposing continuity of the function $y(\tau)$ and
its first derivatives at the star surface, we find the following equation for
null-like geodesics in the interior

$\frac{1}{2}y^{\prime
2}+\frac{1}{R^{2}}\,\frac{\ell^{2}}{2\,(1+2\gamma)}\left(-\gamma+(1+\gamma)\,y^{-\frac{2(1-\gamma)}{(1+\gamma)}}\right)\,=\,\frac{{\cal
E}^{2}}{R^{2}}\,.$ (3.17)

These results indicate that the interior geometry has no extremal points for
null-like geodesics. The exterior potential admits one maximum at

$y_{\rm ext}\,=\,\frac{3\gamma}{1+2\gamma}$ (3.18)

If $\gamma>1$, $y_{\rm ext}$ is located in the exterior region of the star.
Hence, if this condition is satisfied, the null-like geodesic potential have a
maximum which can be associated with a light-ring and the photosphere: the
situation is then very similar to GR. For $\gamma<1$ there is no local maximum
instead, and there instead is an infinite barrier for light rays: the
situation is then similar to what we discussed in the case of time-like
geodesics. The value $\gamma=1$ is special because the potential is constant.
The condition for having a photosphere, $\gamma>1$, corresponds to the
condition of considering ultracompact objects, ${\cal C}>1/3$, in agreement
with the general analysis of [51].

##  4 Beyond spherical symmetry

We have seen that the behaviour of geodesics can be richer with respect to
what occurs in a Schwarzschild geometry, since geodesics can probe the star
interior. In this section we go beyond spherical symmetry, considering the
dynamics of fluctuations of the vector field and of the metric tensor, both
outside and inside the star. We will learn that their properties are sensitive
to vector charges (both electric and magnetic-type ones) as well as the star
compactness. They can then offer probes for distinguishing a vector star from
a black hole, as well as to investigate applications of no-hair arguments in
this context.

We focus on parity-odd, spin-1 stationary fluctuations: they are easier to
deal with analytically, and can lead to distinctive effects in systems (like
ours) with a pronounced anisotropic stress in the internal stress-tensor of
the star. Our aims are as follows:

  1. 1.

First, in section 4.1 we analytically investigate the response of the vector-
field profile $V_{\mu}$ to magnetic-type, spin-1 perturbations, which add
magnetic-type charges to the vector star solution. We determine the structure
of the dipolar magnetic field which can be sourced by the star configuration,
as well as the vector magnetic susceptibility to the application of an
external field. The characteristic response of the vector can be used as a
probe of the star properties, playing a role analog to neutron star Love
numbers [58, 59, 60] for distinguishing exotic compact objects from black
holes [61].

  2. 2.

Then, in section 4.2 we focus on metric fluctuations, and study the behaviour
of the corresponding parity-odd spin-1 deformations. They break spherical
symmetry, and can be sourced by magnetic-type deformations of the vector
profile studied in section 4.1. Their properties depend on the electric and
magnetic charges, showing that – when breaking spherical symmetry – the
exterior geometry becomes sensitive to the properties of the vector
configuration. We are also able to characterize the properties of fluctuations
in the interior of the star, and their dependence on the vector charge.

###  4.1 Magnetic charge, and the vector magnetic susceptibility

We start studying the dynamics of parity-odd, magnetic fluctuations of the
vector field profile $V_{\mu}$, that contribute to the total combination

$V_{\mu}\,dx^{\mu}+\delta V_{\mu}\,dx^{\mu}\,,$ (4.1)

defining the total vector field background$+$fluctuations. We focus on this
section on the vector fluctuations $\delta V_{\mu}$ only: we study their
backreaction on the metric in section 4.2. The first part of eq (4.1),
$V_{\mu}\,dx^{\mu}$, contains the background electric-type components, as in
eq (2.4). Its profile in the internal and external regions of space-time are
given in eqs (2.16) and (2.24) (we focus on the case of singular internal
geometry of section 2.2). On top, we have the second contribution to eq (4.1).
Parity-odd, spin-1 stationary fluctuations are parameterised by

$\delta
V_{\mu}\,dx^{\mu}\,\equiv\,\,\frac{a_{\varphi}(r)}{\sin\theta}\,\left(\partial_{\varphi}Y_{\ell
m}(\theta,\varphi)\right)\,d\theta-\,{a_{\varphi}(r)}\,{\sin\theta}\,\left(\partial_{\theta}Y_{\ell
m}(\theta,\varphi)\right)\,d\varphi\,,$ (4.2)

in terms of a function $a_{\varphi}(r)$. In this formula the $Y_{\ell m}$ are
the scalar spherical harmonics, and their gradients are associated with the
spin-1 spherical harmonics (see e.g. [62, 63]) we are interested in. They are
characterised by a multipole index $\ell$ ($\ell\geq 1$) and an azymuthal
index $m$: in our study of fluctuations around spherically symmetric
configurations, there is no dependence on the index $m$ hence we understand it
from now on. We assume that the quantities $a_{\varphi}(r)$ are small. We
obtain the linearised equations in the exterior ($r\geq R$) and interior
$(r\leq R)$ of the star as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
a_{\varphi}^{\prime\prime}+\frac{2Q_{E}\,R}{r(Q_{E}\,R+r)}\,a_{\varphi}^{\prime}-\frac{2Q_{E}R+\ell(\ell+1)\,r}{r(Q_{E}\,R+r)^{2}}\,a_{\varphi}\,=\,0\hskip
28.45274pt,\,\,$ (4.3) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
a_{\varphi}^{\prime\prime}+\frac{Q_{I}(2+\gamma-\gamma^{2})-3\gamma(r/R)^{1+\gamma}}{r\left(Q_{I}(1+\gamma)+(r/R)^{1+\gamma}\right)}\,a_{\varphi}^{\prime}+$
$\displaystyle+$
$\displaystyle\hskip-2.84544pt\frac{2\,Q_{I}^{2}\,\gamma\,\left(1+\gamma\right)+Q_{I}(r/R)^{1+\gamma}\left(1+\gamma\right)\left(1+2\gamma\right)-2(r/R)^{2+2\gamma}\left(\ell+\gamma(\ell-1)\right)\left(1+\ell+(\ell+2)\gamma\right)}{2\,r^{2}\left(Q_{I}(1+\gamma)+(r/R)^{1+\gamma}\right)^{2}}\,a_{\varphi}$

The internal electric charge $Q_{I}$ is related with the external one $Q_{E}$
by the relation (2.26). Interestingly, in both the exterior and interior
regions the equations governing the vector fluctuations $a_{\varphi}$ are
decoupled from the parity-odd metric perturbations 666For the case $\ell\geq
2$ we need to substitute a constraint condition into the equations – more on
this in section 4.2.. However, as we will learn, the vector modes
$a_{\varphi}$ backreact on the metric at the linearised level, breaking
spherical symmetry. Notice that also the exterior equation (4.3) depends on
the star electric charge, indicating that the behaviour of the external
magnetic deformation is controlled by $Q_{E}$.

The two equations (4.3) and (4.1) admit two distinct exact solutions, each
depending on two integration constants. Fixing the integration constants in
the interior so that the solutions continuously match – together with their
derivative – at the star surface $r\,=\,R$, we get

$\displaystyle a_{\varphi}(r)$ $\displaystyle=$
$\displaystyle\frac{P_{\varphi}\,R}{r}\,\left(Q_{E}+r/R\right)^{1-\ell}+\frac{S_{\varphi}\,R}{r}\,\left(Q_{E}+r/R\right)^{2+\ell}\,,$
(4.5) $\displaystyle a_{\varphi}(r)$ $\displaystyle=$
$\displaystyle\frac{P_{\varphi}\,R}{r}\,\left(\frac{Q_{E}(1+\gamma)+\gamma+(r/R)^{1+\gamma}}{1+\gamma}\right)^{1-\ell}$
(4.6)
$\displaystyle+\frac{S_{\varphi}\,R}{r}\,\left(\frac{Q_{E}(1+\gamma)+\gamma+(r/R)^{1+\gamma}}{1+\gamma}\right)^{2+\ell}\,,$

for two constants $P_{\varphi}$, $S_{\varphi}$, respectively in the exterior –
eq (4.5) – and in the interior – eq (4.6) – of the star configuration.

We now study two applications of the previous general solutions. In the first
case, we interpret the solutions (4.5)-(4.6) as controlling a magnetic dipolar
potential of the star configuration, with a magnetic field strength decaying
at infinity. Hence, we select $\ell=1$ and $S_{\varphi}\,=\,0$: the vector
potential component $\delta V_{\varphi}$ becomes

$\displaystyle\delta V_{\varphi}$ $\displaystyle=$ $\displaystyle-
a_{\varphi}(r)\,\sin\theta\,\left(\partial_{\theta}Y_{10}(\theta)\right)\,,$
(4.7) $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{4\pi}}\,P_{\varphi}\,\frac{R}{r}\,\sin^{2}\theta\,,$

for any $r>0$. This leads to a dipolar, parity-odd magnetic field deformation
$B_{\hat{r}}$ along the radial direction, which can be computed through the
vector field strength using standard formulas

$B_{\hat{r}}\,=\,\frac{\partial_{\theta}\,V_{\varphi}}{r^{2}\,\sin\theta}\,=\,P_{\varphi}\,\sqrt{\frac{3}{\pi}}\,\frac{R}{r^{3}}\,\cos\theta\,.$
(4.8)

The dipolar magnetic field of the star is then controlled by the magnetic
dipolar charge $P_{\varphi}$.

In the second case, we study the response of the system to an external
constant dipolar magnetic field, whose radial component behaves at large
distances ($r\to\infty$) as

$B_{\hat{r}}^{\rm(ext)}(r\to\infty)\,=\,C_{0}\,\cos\theta\hskip
14.22636pt\Rightarrow\hskip
14.22636pta_{\varphi}^{\rm(ext)}(r\to\infty)\,=\,C_{0}\,\sqrt{\frac{\pi}{3}}\,r^{2}\,,$
(4.9)

for a small constant $C_{0}$. In passing from the left to the right of the
arrow in eq (4.9), we made use of the definitions in eqs (4.7)-(4.8).

In studying the star vector response to the external field (4.9) – a
phenomenon that we call vector magnetic susceptibility – we impose for
definiteness that the magnetic-type fluctuations $a_{\varphi}$ do not blow-up
at the origin $r=0$, so to fix the last of the integration constants in eq
(4.6). The regularity condition requires that $P_{\varphi}$ and $S_{\varphi}$
in eq (4.6) are related by the condition

$\frac{S_{\varphi}}{P_{\varphi}}\,=\,-\frac{\left(1+\gamma\right)^{1+2\ell}}{\left(Q_{E}(1+\gamma)+\gamma\right)^{1+2\ell}}\,.$
(4.10)

Requiring to match the asymptotic value of the magnetic field as in eq (4.9),
we fix the last of the integration constants, and find the following solution
for $a_{\varphi}$:

$\displaystyle a_{\varphi}(r)$ $\displaystyle=$ $\displaystyle
C_{0}\,\sqrt{\frac{\pi}{3}}\,\frac{R^{3}}{r}\,\left(Q_{E}+r/R\right)^{3}\left[1-\left(\frac{R}{r}\right)^{3}\frac{\left(Q_{E}+\gamma/(1+\gamma)\right)^{3}}{\left(Q_{E}\,R/r+1\right)^{3}}\,\right]\,.$
(4.11)

While the part outside the squared parenthesis matches the boundary condition
at spatial infinity, the part inside the parenthesis controls to the dipolar
susceptibility of the external magnetic field. Interestingly, such vector
magnetic susceptibility depends both on the electric charge $Q_{E}$ and the
parameter $\gamma$ controlling the star compactness. This quantity is
analogous, although not identical, to the gravitational magnetic
susceptibility as studied in various works in the context of studies of tidal
deformability for black holes – see e.g. [64, 65, 66, 63].

Figure 4: Plots of the vector magnetic susceptibility of eq (4.12), as
function of $\gamma$ (left panel) and $Q_{E}$ (right panel). Left:
$Q_{E}=1/10$ (black); $Q_{E}=1$ (red dashed); $Q_{E}=10$ (blue dotted). Right:
$\gamma=1/10$ (black); $\gamma=1$ (red dashed); $\gamma=10$ (blue dotted).

Defining the vector magnetic susceptibility ${\cal S}_{m}$ as the opposite of
the second term in the squared parenthesis of eq (4.11), evaluated at the star
surface $r=R$, we get

${\cal
S}_{m}\equiv\frac{\left(Q_{E}+\gamma/(1+\gamma)\right)^{3}}{\left(Q_{E}\,+1\right)^{3}}\,.$
(4.12)

Interestingly, this quantity approaches the value ${\cal S}_{m}=1$ for very
large $\gamma$, showing that in the limit of ultracompact objects approaching
the compactness of a black hole, the susceptibility behaves as the one of a
conducting sphere.

We plot in Fig 4 examples of the dependence of ${\cal S}_{m}$ on $Q_{E}$ and
$\gamma$, for some representative cases. It would be interesting to find
measurable observables sensitive to the value of ${\cal S}_{m}$, which can
represent distinctive probes of the properties of the star – its charges and
compactness.

###  4.2 Parity-odd stationary deformations of the metric, and star rotation

After discussing the perturbations in the vector field, we now focus on
parity-odd, stationary fluctuations of the metric, along the lines of the
classical analysis of [67]. To slightly reduce the length of our formulas, in
this section we set the star radius $R=1$.

We decompose the metric in a background part and perturbations:

$g_{\mu\nu}dx^{\mu}dx^{\nu}+h_{\mu\nu}dx^{\mu}dx^{\nu}\,.$ (4.13)

In this equation, $g_{\mu\nu}dx^{\mu}dx^{\nu}$ corresponds to the spherically
symmetric background solution with Ansatz (2.3), with exterior and interior
metric components studied in section 2. The quantity
$h_{\mu\nu}dx^{\mu}dx^{\nu}$ controls the parity-odd, spin-1 stationary metric
fluctuations in Regge-Wheeler gauge:

$\small{h_{\mu\nu}=\begin{pmatrix}0&0&-h_{0}(r)\sin\theta^{-1}\,\partial_{\varphi}&h_{0}(r)\sin\theta\,\partial_{\theta}\\\
0&0&-h_{1}(r)\sin\theta^{-1}\,\partial_{\varphi}&h_{1}(r)\sin\theta\,\partial_{\theta}\\\
-h_{0}(r)\sin\theta^{-1}\,\partial_{\varphi}&-h_{1}(r)\sin\theta^{-1}\,\partial_{\varphi}&0&0\\\
h_{0}(r)\sin\theta\,\partial_{\theta}&h_{1}(r)\sin\theta\,\partial_{\theta}&0&0\\\
\end{pmatrix}}Y(\theta,\varphi).$ (4.14)

The metric fluctuations break the spherical symmetry of the system. The
angular dependence of the solution is controlled by derivatives of the scalar
spherical harmonics $Y_{\ell m}(\theta,\varphi)$ ($\ell\geq 1$) which define
spin-1 spherical harmonics as in the previous section. It is straightforward
to determine the expression for the linearized equations controlling the
radial profiles of the metric components $h_{0}(r)$ and $h_{1}(r)$, and how
they are sourced by the magnetic-type vector fluctuations studied in section
4.1. For the case $\ell=1$, the function $h_{1}(r)$ is a gauge mode and can be
set to zero; for $\ell\geq 2$, the equation of $h_{1}$ is algebraic, and can
be solved as a function of $h_{0}$ and $a_{\varphi}$, both in the exterior and
the interior of the star. For example, in the star exterior we find

$h_{1}(r)\,=\,\frac{r}{r-2M}\,\frac{\sqrt{Q_{E}^{2}+2Mr+Q_{E}r}}{Q_{E}+r}\,h_{0}(r)-\frac{r\,\sqrt{Q_{E}^{2}+2\,M\,r+Q_{E}\,r}}{2(Q_{E}+r)^{2}}\,a_{\varphi}(r)\,.$
(4.15)

Plugging this expression into the equation for $a_{\varphi}$ we find equation
(4.3), that – as explained in section 4.1 – is decoupled from metric
fluctuations. The quantity $h_{0}(r)$ is controlled by the following equation
in the star exterior, valid for any $\ell\geq 1$ (we understand the dependence
on $r$):

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
h_{0}^{\prime\prime}-\frac{4Q_{E}}{r\,(Q_{E}+r)}\,h_{0}^{\prime}+\frac{\left(6Q_{E}^{2}+4\,Q_{E}\,r-\ell(\ell+1)r^{2}\right)}{r^{2}\,\left(Q_{E}+r\right)^{2}}\,h_{0}+\frac{4M-2r}{Q_{E}+r}\,a_{\varphi}^{\prime\prime}-\frac{4M}{r}\,\frac{a_{\varphi}^{\prime}}{Q_{E}+r}$
$\displaystyle+$
$\displaystyle\frac{a_{\varphi}}{r\,(Q_{E}+r)^{3}}\left[Q_{E}^{2}\left(1+\frac{3}{2}\ell(\ell+1)\right)+2Mr+\left(1+\frac{3}{2}\ell(\ell+1)\right)Q_{E}r-\ell(\ell+1)(M-2r)\,r\right]\,.$

Notice that the metric component $h_{0}(r)$ is sourced by the magnetic-type
vector perturbation $a_{\varphi}(r)$. The equation for $h_{0}$ in the star
interior can also be derived quite easily, but is longer and we do not write
it explicitly.

Let us study explicit examples of solutions. Focus on the dipolar case
$\ell=1$, $m=0$ (and $h_{1}=0$), and on perturbations that decay at infinity.
As first noticed in [67], the metric then describes a slowly rotating
configuration, with $h_{0}(r)$ controlling the stationary rotation of the
geometry through an off-diagonal metric component

$g_{t\varphi}\,dt\,d\varphi\,=\,-2\,\left(\sqrt{\frac{3}{4\pi}}\,\frac{h_{0}(r)}{r^{2}}\right)\,r^{2}\,\sin^{2}\theta\,dt\,d\varphi\,.$
(4.17)

It is straightforward to solve eq (LABEL:eqh0stext) for $h_{0}(r)$ in the star
exterior, taking as source the magnetic-type solution we found in eq (4.5)
with $\ell=1$ and $S_{\varphi}=0$. We find for $r\geq 1$

$-\sqrt{\frac{3}{4\pi}}\,\frac{h_{0}(r)}{r^{2}}\,=\,\frac{a\,M}{(Q_{E}+r)^{3}}+\sqrt{\frac{3}{\pi}}\,\frac{P_{\varphi}\,M}{Q_{E}\,r^{3}}\,,$
(4.18)

with $a$ an integration constant associated with the rotation of the geometry
(the Kerr parameter), and $P_{\varphi}$ the intrinsic magnetic dipolar charge
as given in eq (4.7) above. This metric coefficient should be compared with
the one found in the slowly-rotating limit of the Kerr solution in GR:

$-\sqrt{\frac{3}{4\pi}}\,\frac{h^{\rm
GR}_{0}(r)}{r^{2}}\,=\,\frac{a\,M}{r^{3}}\,.$ (4.19)

Comparing (4.18) with (4.19), we find that the magnetic charges $P_{\varphi}$
and $Q_{E}$ influence the geometry as vector hairs, and contribute in breaking
the spherical symmetry of the configuration to an axial one. Interestingly,
$Q_{E}$ appears at the denominators: we interpret this fact as a consequence
of the specific non-minimal couplings of the vector to gravity. The quantity
$Q_{E}$ contributes to modulate the $1/r^{3}$ decay at large distances of the
angular momentum contribution proportional to $a$. Instead,
$P_{\varphi}/Q_{E}$ adds a new contribution to $h_{0}$, absent in the vacuum
GR case, being induced by the magnetic dipolar charge of the star
configuration.

The peculiar dependence on the radial coordinate $r$ in eq (4.18) can lead to
interesting effects, distinctive of the vector-tensor system under
consideration. Let us suppose that $a$ and $P_{\varphi}$ have opposite sign
(say $a$ is positive) and call

$\nu^{3}\,\equiv\,-\sqrt{{3}/{\pi}}\,{|P_{\varphi}|\,M}/{(Q_{E}\,a)}$ (4.20)

We assume $0\,\leq\,\nu\,<\,1$. We can rewrite eq (4.18) as

$-\sqrt{\frac{3}{4\pi}}\,\frac{h_{0}(r)}{r^{2}}\,=\,\frac{a\,M}{r^{3}}\left(\frac{r^{3}}{(Q_{E}+r)^{3}}-{\nu^{3}}\right)\,.$
(4.21)

While for large values of $r$ the right hand side of (4.21) is positive, as we
approach the star the sign of $h_{0}(r)$ changes. This occurs at

$r_{\star}\,=\,\frac{\nu\,Q_{E}}{1-\nu}$ (4.22)

which is larger than the star radius $R=1$ by choosing a large enough $Q_{E}$.
This fact implies that the direction of rotation of the space-time changes as
we approach the star, due to to the opposite contributions to rotation of the
Kerr parameter $a$ and the magnetic charge $P_{\varphi}$. It would be
interesting to further explore the consequences of this phenomenon, and
whether can be used to find distinctive probes of this system.

The discussion of the interior part of the geometry is particularly simple in
the case $P_{\varphi}\,=\,0$: let us focus for simplicity on this case for the
rest of the section, although analytic expressions for $P_{\varphi}\,\neq\,0$
can also be found. This limit implies that the exterior dipolar configuration
is described by eq (4.18) with no magnetic-charge contribution. The interior
solution $r\leq 1$ of the field equations for $h_{0}$ that continuously match
with the exterior for $r\geq 1$ is

$\displaystyle-\sqrt{\frac{3}{4\pi}}\,\frac{h_{0}(r)}{r^{2}}$ $\displaystyle=$
$\displaystyle\frac{a\,M\,(1+\gamma)^{3}}{\left[Q_{E}(1+\gamma)+\gamma(1-\sigma)+\,r^{1+\gamma}(1-\sigma)+\,r\,\sigma\,(1+\gamma)\right]^{3}}\,,$
(4.23)

where we selected as interior background configuration the solution described
in section 2.3. The interior part of the star is dragged by the rotation of
the configuration: the dragging effects are also modulated by the electric
charge $Q_{E}$ and depend on the star compactness through the parameter
$\gamma$. While in this example we focussed on the case $\ell=1$, the cases
$\ell\geq 2$ can also be straightforwardly analysed, at least in absence of
intrinsic magnetic charge, and the previous formulas generalise by changing
the exponents from $3$ to $\ell+2$.

In summary, the study of properties of parity-odd stationary metric
fluctuations reveal a very rich structure that goes beyond what found in
solutions of GR in empty space, even if the exterior star geometry is given by
Schwarzschild space-time. Both the exterior and the interior metric elements
are indeed sensitive to the vector properties. A detailed analysis of time-
dependent perturbations, as well as parity-even modes, is left for future
studies.

##  5 Outlook

In this work we presented and investigated new analytic solutions describing a
family of horizonless compact objects in vector-tensor theories of gravity,
dubbed ultracompact vector stars. They are sourced by a vector condensate,
induced by a non-minimal coupling with gravity. They can be as compact as
black holes, thanks to their internal anisotropic stress. In the spherically
symmetric case, their external geometry corresponds to the Schwarzschild
solution as in GR, even if they can be characterised by an electric-type
charge. The interior part of the solution resembles a singular isothermal
sphere, and the singularity at the star centre can be resolved by tuning some
of the available integration constants. The interior configuration is smoothly
matched to the exterior geometry of the star, with no need of extra stress-
tensor on the star surface.

We investigated features of our systems that allow one to distinguish vector
star objects from GR black hole solutions. We analytically studied the
behaviour of geodesics trajectories within the star interior – where new
stable circular orbits are allowed – as well as geodesics crossing in both
ways the star surface. We analytically investigated parity-odd stationary
fluctuations that break spherical symmetry and can assign a magnetic charge to
our vector star configurations.

It would be interesting to further develop our analysis along the following
directions:

  * •

Study time-dependent fluctuations – both odd and even parity – and investigate
the stability the horizonless vector star objects. This topic is particularly
interesting since black hole configurations in vector-tensor theories of
gravity have been shown to have instabilities [68, 69] which might be cured
(or not) in more general configurations like ours.

  * •

Study mechanisms of formation of vector stars from a process of gravitational
collapse, including further sources of energy momentum tensor in the form of
standard matter. Gravitationally bound solitons made of dark-matter vector
bosons – in a Proca theory – have been recently numerically investigated in
[70, 71, 72, 73], and would be interesting to pursue similar studies in our
context. In fact, non-minimal couplings of the vector with gravity are allowed
(and expected) once the vector Abelian symmetry is broken.

  * •

Determine solutions with a large rotation parameter in the exterior geometry –
the analog of Kerr – and large magnetic charge. Investigate the properties and
phenomenology of the resulting magnetically-charged configurations, for
example taking inspiration from [74].

  * •

Determine additional field-theory and cosmological motivations for the vector-
tensor theories we consider and their possible extensions for dark matter and
dark energy.

We leave these points to future investigations.

## Acknowledgements

It is a pleasure to thank Mustafa Amin and Ivonne Zavala for discussions. The
work is funded by the STFC grant ST/T000813/1. For the purpose of open access,
the author has applied a Creative Commons Attribution licence to any Author
Accepted Manuscript version arising.

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