Static spacetimes with a Finsler angular sector

TL;DR

This paper explores static spacetimes with Finslerian angular sectors, deriving model-independent photon orbit relations, exact Finslerian effects, and revisiting black hole solutions to highlight Finsler features.

Contribution

It introduces a kinematical framework for Finsler angular sectors in static spacetimes and critically analyzes existing black hole solutions for Finsler properties.

Findings

Derived model-independent relations for photon orbits.

Obtained exact Finslerian conserved quantities and effects.

Revealed overlooked Finsler features in existing black hole solutions.

Abstract

We consider static spacetimes in spherical coordinates whose angular sector is described by a Finsler metric rather than the standard round metric on $S^2$. Our first contribution is kinematical: maintaining arbitrary lapse and radial factors $e^{\nu(r)}$, $e^{\vartheta(r)}$, and relying solely on Killing symmetries and the null constraint, we derive model--independent relations for circular photon orbits and the effective dynamics. By specializing the angular sector to Randers sphere of constant positive flag curvature, we obtain exact expressions for the conserved angular charge, the critical impact parameter and we quantify a Finslerian Sagnac--type effect. Our second contribution is dynamical: we examine the field equations used in the literature to determine $(e^{\nu},e^{\vartheta})$. We revisit the family of hairy black holes in \cite{Nekouee2025}, demonstrating that the analysis therein neglects crucial non-reversible Finsler features. Furthermore, we show that the solutions presented as new reproduce previously known results in \cite{Ovalle2021}.

Equations

$$g_{\mu \nu }(x,y)=\frac{1}{2}\frac{{\partial }^{2}L}{\partial y^{\mu }\partial y^{\nu }}(x,y),\mu ,\nu \in \{0,\dots ,3\},\text{for all }(x,y)\in \mathcal{A},$$

$$L(t,r,\theta ,\varphi ,y^t,y^r,y^{\theta },y^{\varphi })=e^{\nu (r)}(y^t{)}^2-e^{\vartheta (r)}(y^r{)}^2-r^2{\overline{F}}^2(\theta ,\varphi ,y^{\theta },y^{\varphi }),$$

$$\mathcal{A}:=\{(x,y)\in TM\setminus \{0\}:(y^{\theta },y^{\varphi })\ne (0,0)\}.$$

$$\alpha (\theta ,\varphi ;y^{\theta },y^{\varphi })$$

$$\beta (\theta ,\varphi ;y^{\theta },y^{\varphi })$$

$$h_K=\frac{1}{K}(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),W=\epsilon {\partial }_{\varphi },0\le \epsilon <\sqrt{K}.$$

$${\lambda }_K(\theta ):=1-\frac{{\epsilon }^2}{K}{\sin }^2\theta .$$

$$F_K(\theta ,\varphi ;y^{\theta },y^{\varphi })=\frac{1}{\sqrt{K}{\lambda }_K(\theta )}\sqrt{{\lambda }_K(\theta )(y^{\theta }{)}^2+{\sin }^2\theta (y^{\varphi }{)}^2}-\frac{\epsilon {\sin }^2\theta }{K{\lambda }_K(\theta )}{y}^{\varphi }.$$

$${\mathrm{Isom}}_0(\overline{F})=\{\phi \in \mathrm{Isom}(h_K):{\phi }_{\ast }W=W\},\mathfrak{kill}(\overline{F})=\{X\in \mathfrak{so}(3):[X,W]=0\}.$$

$$L(\gamma ,\dot{\gamma })=\kappa$$

$$E(x,y):={\partial }_y(\frac{1}{2}L)(x,y)[{\partial }_t]=e^{\nu }{y}^t.$$

$$\dot{r}=0,\frac{1}{2}{e}^{\nu (r)}{\nu }^{\prime }(r){\dot{t}}^2=r{\overline{F}}^2(\theta ,\varphi ,\dot{\theta },\dot{\varphi }).$$

$$r_C{\nu }^{\prime }(r_C)=2.$$

$$E^2{e}^{-\nu (r_C)}=r_C^2{\overline{F}}^2(\theta (s),\varphi (s),\dot{\theta }(s),\dot{\varphi }(s)).$$

$$\dot{t}=E{e}^{-\nu (r)}.$$

$$E^2{e}^{-\nu (r_C)}=r_C^2{\overline{F}}^2.$$

$$e^{\nu (r)}=1-\frac{2M+\alpha \ell /K}{Kr}+\alpha e^{-Kr/M},$$

$$\frac{3(2M+\alpha \ell /K)}{K{r}_C}-2=\alpha e^{-K{r}_C/M}\left(2+\frac{K{r}_C}{M}\right).$$

$${\overline{\mathcal{J}}}_X(x,y):=\frac{\partial \overline{L}}{\partial y^A}(x,y)X^A=\overline{F}(x,y)\frac{\partial \overline{F}}{\partial y^A}(x,y)X^A,$$

$$\frac{\partial \overline{F}}{\partial y^A}=\frac{1}{\alpha }{a}_{AB}{y}^B+b_A,$$

$${\overline{\mathcal{J}}}_X(\gamma ,\dot{\gamma })=\overline{F}(\sigma ,\dot{\sigma })(\frac{a(\dot{\sigma },X)}{\alpha (\sigma ,\dot{\sigma })}+\beta (X))$$

$$\overline{\mathcal{J}}={\overline{F}}_K(\sigma ,\dot{\sigma })(\frac{a_{\varphi \varphi }\dot{\varphi }}{\alpha (\sigma ,\dot{\sigma })}+b_{\varphi }).$$

$$\alpha (\sigma ,\dot{\sigma })=\frac{1}{\sqrt{K}{\lambda }_K}\sqrt{{\lambda }_K{\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2},a_{\varphi \varphi }=\frac{{\sin }^2\theta }{K{\lambda }_K^2},b_{\varphi }=-\frac{\epsilon {\sin }^2\theta }{K{\lambda }_K},$$

$$e^{\nu (r_C)}{\dot{t}}^2=r_C^2{\overline{F}}_K^2(\theta ,\varphi ;\dot{\theta },\dot{\varphi }).$$

$$E=r_C\sqrt{e^{\nu (r_C)}}.$$

$$\mathcal{J}=\frac{\partial (\frac{1}{2}L)}{\partial y^{\varphi }}(\gamma ,\dot{\gamma })=-r_C^2{\overline{F}}_K(\sigma ,\dot{\sigma })\frac{\partial {\overline{F}}_K}{\partial y^{\varphi }}(\sigma ,\dot{\sigma })=-r_C^2\overline{\mathcal{J}}.$$

$$b_C=\frac{r_C{\sin }^2\theta }{K{\lambda }_K\sqrt{e^{\nu (r_C)}}}|\frac{\dot{\varphi }}{{\lambda }_K\alpha (\sigma ,\dot{\sigma })}-\epsilon |.$$

$$b_C^{\pm }=\frac{r_C}{K{\lambda }_K^{\ast }\sqrt{e^{\nu (r_C)}}}|\sqrt{K}\mathrm{sgn}(\dot{\varphi })-\epsilon |=\frac{r_C}{K{\lambda }_K^{\ast }\sqrt{e^{\nu (r_C)}}}(\sqrt{K}\mp \epsilon ),$$

$$\mathcal{J}=r^2(1-ϵ{)}^{-2}\dot{\varphi }.$$

$$\mathcal{J}=-r^2(\dot{\varphi }\mathcal{H})\mathcal{H}=-r^2\dot{\varphi }{\mathcal{H}}^2,$$

$$\mathcal{H}:=\frac{\text{sgn}(\dot{\varphi })\sqrt{K}-ϵ}{K{\lambda }_K^{\ast }}.$$

Full text

Our second contribution is dynamical: we examine the field equations used in the literature to determine $(e^{\nu},e^{\vartheta})$. We revisit the family of hairy black holes in [13], demonstrating that the analysis therein neglects crucial non-reversible Finsler features. Furthermore, we show that the solutions presented as new reproduce previously known results in [14].

1. Introduction

The quest to understand gravity beyond Einstein’s general relativity motivates geometric extensions of spacetime. Among them, Lorentz–Finsler geometry relaxes the quadratic restriction on the metric by replacing it with a positively homogeneous, of degree $2$ Lagrangian, while retaining key structures such as causal cones and Noether symmetries. Because the metric depends on both position and direction, anisotropy becomes a built-in feature. This provides a natural framework to model departures from local isotropy or Lorentz invariance within a well-defined causal theory [1, 2, 3, 4, 5].

An interesting class of Lorentz–Finsler spacetimes arises when the Finslerian structure is confined to specific sectors of the geometry. In this work, we consider static spacetimes in spherical coordinates where the sphere $S^{2}$ (also called here angular sector) carries a positive, strongly convex, Finsler metric [6] while the temporal and radial sectors remain Lorentzian. This hybrid structure provides a tractable framework for exploring Finslerian effects while maintaining a structure familiar from general relativity [7]. The study of black hole solutions in this class initiated with the work of Li and Chang [8]. These solutions naturally raise questions about how the direction–dependent metric structure affects fundamental black hole properties such as photon spheres, shadows, and thermodynamic characteristics.

The choice of the Finsler metric on the angular sector is not arbitrary, but the mathematical freedom available in the Finslerian setting is remarkably different from the Riemannian case. In Riemannian geometry, the Killing–Hopf theorem severely constrains the possible metrics of constant curvature on a two-sphere: up to diffeomorphisms and an overall scale factor, there is only the round metric. In the Finslerian setting, Bryant and Shen [9, 10] demonstrated that there exist infinitely many non-isometric Finsler metrics on $S^{2}$ with the same constant positive flag curvature. This remarkable non-uniqueness opens up a rich landscape of possibilities absent in the Riemannian case and suggests that Finslerian angular sectors could encode additional physical degrees of freedom not present in general relativity. The choice of metrics with constant flag curvature is strongly motivated when one considers a Finslerian generalization of the Einstein’s field equations proposed by Rutz [11] (originally, just for vacuum solutions) and based on the Finslerian Ricci scalar. In fact, the constant flag curvature condition gives that the Finslerian Ricci scalar of the angular metric is constant as well, simplifying the field equation.

Among metrics with constant flag curvature, Randers metrics –which can be understood as Riemannian metrics perturbed by a one-form– are particularly appealing due to their connection with Zermelo’s navigation problems and their relatively simple geodesic structure [12]. Within this family one finds both metrics with all geodesics closed and metrics admitting just two closed geodesics (the so-called Katok examples).

Our work has two main objectives. First, we provide a mathematically rigorous treatment of null geodesics in static Lorentz–Finsler spacetimes with a Finsler angular sector (focusing in particular on Randers ones), carefully considering how the anisotropic structure affects the conserved quantities. We show that while the condition for circular photon orbits (the photon sphere) remains formally identical to the Lorentzian case, the critical impact parameter acquires direction–dependent corrections that encode the Finslerian nature of the angular geometry. A Sagnac–type effect, where co-rotating and counter-rotating photons on the photon sphere have different periods as measured by the observer $\partial_{t}$ is also described.

Second, we discuss some critical aspects of [13]; as shown in Remark 3.6, the equatorial restriction is not “without loss of generality” even for Randers metrics. Moreover, some results in [13] overlook orientation effects due to non-reversibility. More important, we show that the claimed new solutions in [13] are essentially identical to the hairy black hole solutions previously obtained by Ovalle et al. [14] using gravitational decoupling in the Lorentzian setting.

The paper is organized as follows. In Section 2, we establish the Lorentz–Finsler framework and introduce the specific form of static Finsler spacetimes with angular Randers structure having constant flag curvature. Section 3 examines circular geodesics and derives the generalized photon sphere condition; moreover we show how the Randers structure induces a Finslerian Sagnac effect. In Section 4, we show that it is possible to discriminate between a stationary Lorentzian and a static Lorentz–Finsler metric under shared optical data, i.e. when the two metrics provide the same Sagnac–type delay, by evaluating the proper time of a worldline in the two geometries. Section 5 discusses the Finslerian field equations and their simplification for constant flag curvature angular metrics. In particular, Subsection 5.2 provides a critical analysis of [13] establishing proper attribution of results contained there. Finally, Section 6 presents our conclusions.

2. Lorentz–Finsler Randers framework

In a Finsler spacetime, the fundamental geometric object is not a metric tensor but rather a function $L:TM\to\mathbb{R}$ defined on the tangent bundle $TM$ of the four dimensional manifold $M$. This function, which we call the Lorentz–Finsler structure or metric, must satisfy certain conditions to provide a coherent geometric framework. It must be positively homogeneous of degree two in the fiber coordinates, meaning that $L(x,\lambda y)=\lambda^{2}L(x,y)$ for any positive scalar $\lambda$, and for all $x\in M$, $y\in T_{x}M$. This property ensures that the notion of length for a causal curve, i.e. a curve $\gamma\colon I\to M$, $\gamma=\gamma(s)$ satisfying $L(\gamma(s),\dot{\gamma}(s))\geq 0$, remains well-defined under orientation preserving reparametrization, where the length of $\gamma$ is defined as $\int_{I}\sqrt{L(\gamma,\dot{\gamma})}ds.$

The fundamental tensor, which plays a role analogous to the metric tensor in Riemannian geometry, is defined through the relation

[TABLE]

where $\mathcal{A}\subset TM\setminus 0$ is an open subset where $L$ is smooth and such that $\pi(\mathcal{A})=M$ ($\pi:TM\to M$ being the canonical projection) and, for each $v\in\mathcal{A}$, $\lambda v\in\mathcal{A}$ for all $\lambda>0$. The crucial difference from Riemannian geometry is that $g$ depends not only on the position $x$ but also on the direction $y$, i.e. $g_{\mu\nu}$ is a section of the pullback bundle $\pi^{*}(TM\otimes TM)\to\mathcal{A}$, not a tensor on $M$.

We require $g(x,y)$ to be of Lorentzian signature type for all $(x,y)\in\mathcal{A}$.111We adopt signature $(+,-,-,-)$; moreover, units with $G=c=1$ are used.

The rich mathematical structure of Finsler geometry for classical Finsler metrics, i.e. when $L$ is positive and $g$ positive definite for any $(x,y)\in TM\setminus 0$, is presented comprehensively in the influential text by Bao, Chern, and Shen [6].

In this work, we consider static Finsler spacetimes where the Finsler structure takes the form

[TABLE]

where $y=(y^{t},y^{r},y^{\theta},y^{\phi})\in T_{x}M$, $x=(t,r,\theta,\phi)$. This ansatz formally resembles a classical static spherically symmetric configuration but includes non-trivial Finslerian effects through the angular sector metric $\overline{F}$ which is a classical Finsler metric on $S^{2}$, i.e. it is non-negative, positively homogeneous of degree $1$ and its fundamental tensor (associated to $L=\overline{F}^{2}$) is positive definite for all $\theta,\phi,y^{\theta},y^{\phi}$ in the slit tangent bundle $TS^{2}\setminus 0$.

Remark 2.1*.*

For the static Lorentz–Finsler metric $L$ in (2), $\overline{F}^{2}$ is $C^{1}$ on $TM$ but it is $C^{2}$ (or more regular) only on $TS^{2}\setminus\{0\}$. Accordingly, the fundamental tensor $g$ in (1) is well defined on the admissible set

[TABLE]

Along directions $(y^{t},y^{r},0,0)$ the $(\theta,\phi)$–block of $g_{\alpha\beta}$ is not defined (the $(t,r)$–block is). Whenever some “tensorial object”, constructed by contraction with $g_{\mu\nu}(x,y)$ or/and its inverse $g^{\mu\nu}(x,y)$, is required at such directions, we agree to define it as a limit along direction in $\mathcal{A}$ (provided that the limit exists).

Metrics of the type (2) were considered in [8]; they belong to the larger class of static Lorentz–Finsler metrics that were first introduced by [15] and subsequently studied in [7] (see also [16]).

2.1. Randers angular sector

As we will see in Section 5, a physically reasonable choice of the angular metric $\overline{F}$ is to take one with constant flag curvature (see [6, §3.9] for the definition of the flag curvature). Unlike the Riemannian case, there exist infinitely many non-isometric Finsler metrics on $S^{2}$ with the same constant positive flag curvature, as shown by Bryant and Shen [9, 10]. In particular, the family of Randers metrics with constant flag curvature $1$ in [10] is constituted by non-locally projectively flat metrics.

A Randers metric is a positive definite Finsler metric of the form $\overline{F}=\alpha+\beta$, where $\alpha$ is the norm of a Riemannian metric $a$ and $\beta$ is a one-form with norm w.r.t. $a$ everywhere strictly less than $1$. More explicitly, we have

[TABLE]

where the indices $A,B$ run over the angular coordinates $\{\theta,\phi\}$. The physical interpretation of a Randers metric with constant positive flag curvature becomes clearer when we consider Zermelo navigation, where we imagine navigating on the round sphere in the presence of a “wind” field $W$ (see [17, 18]). At each point on $S^{2}$, the wind modifies the indicatrix of the round metric by translating it; as a consequence the distance function is modified as well in a direction–dependent way.

Let $K>0$ and let us consider on $S^{2}$:

[TABLE]

Then $\|W\|_{h_{K}}^{2}=\tfrac{\varepsilon^{2}}{K}\sin^{2}\theta$. Let us set

[TABLE]

For a tangent vector $y=y^{\theta}\partial_{\theta}+y^{\phi}\partial_{\phi}$, the Randers metric obtained by Zermelo navigation with data $h_{K}$ and $W$ is (see [17, §1.1.2]):

[TABLE]

Thus, as a particular case of [17, Th. 5.1] (see also [10, Rmk. 3.1]) we have:

Proposition 2.2**.**

Let $\overline{F}$ be a Randers metric on $S^{2}$ of constant positive flag curvature $K$. Then, locally up to isometry, there exist a Killing vector field $X$ of the round metric on $S^{2}$ and a constant $\varepsilon\in[0,\sqrt{K})$, such that in spherical coordinates $(\theta,\phi)$ adapted to the $SO(2)$-action generated by $X$ (i.e. the fixed points lie at $\theta=0,\pi$ and $X=\partial_{\phi}$), $\overline{F}$ can be written as in (5) with the wind $W=\varepsilon X$.

Conversely, for any choice of $K>0$ and $\varepsilon\in\mathbb{R}$ with $|\varepsilon|<\sqrt{K}$, (5) defines a Randers metric on $S^{2}$ of constant flag curvature $K$.

Remark 2.3* (Symmetries of Randers spheres with constant flag curvature).*

According to the classification in [17], any strongly convex Randers metric $\overline{F}$ on $S^{2}$ with constant positive flag curvature $K$ is obtained (up to local isometry) via Zermelo navigation on the round sphere $(S^{2},h_{K})$ under the influence of a Killing wind $W$ satisfying $\|W\|_{h_{K}}<1$.

The symmetries of the Randers metric are precisely the background isometries that leave the wind invariant. Specifically, from [17, Lemma 1.2], the isometry group and its Lie algebra are given by:

[TABLE]

We can, without loss of generality, align the spherical coordinate system so that $W$ coincides with the azimuthal generator (i.e., we rotate the frame such that the axis of rotation generated by $W$ coincides with the standard $z$-axis). In this adapted frame, (6) reduce to $\mathrm{Isom}_{0}(\overline{F})\cong\mathrm{SO}(2)$ and $\mathfrak{kill}(\overline{F})=\operatorname{span}\{W\}$. Besides this, the equatorial reflection $(\theta,\phi)\mapsto(\pi-\theta,\phi)$ also preserves $W$; hence the full isometry group is $\mathrm{Isom}(\overline{F})=\mathrm{SO}(2)\times\mathbb{Z}_{2}$.

Consequently, the Lorentz–Finsler metric in (5) is static and axially symmetric. It recovers the full spherical symmetry only in the limit of vanishing wind, $\varepsilon\to 0$.

3. Circular geodesics, critical impact parameter and a Sagnac–type effect

Any geodesic $\gamma=\gamma(s)$ of $L$, i.e. any critical point of the functional $\gamma:I\subset\mathbb{R}\to M\mapsto\int_{I}L(\gamma,\dot{\gamma}){\rm d}s$, defined on the path space of sufficiently regular curves connecting two fixed points on $M$, satisfies the conservation law:

[TABLE]

where $\kappa$ is a constant which is, by definition, [math] for null geodesics and positive (resp. negative) for timelike (resp. spacelike) ones (see [7, Th. 2.13]).

Let us denote by $E$ the Noether charge associated with the Killing vector field $\partial_{t}$ of the Lagrangian $\frac{1}{2}L$ in (2):

[TABLE]

It is well known that along a geodesic $\gamma=\gamma(s)$, $E\big(\gamma(s),\dot{\gamma}(s)\big)$ is also constant (see, e.g., [19, Rmk. 2.4]); such a constant will be simply denoted with $E$. We note that for causal geodesics (i.e. the ones with $\kappa\geq 0$) $E\neq 0$ and therefore we can divide them in future–pointing and past–pointing ones according to $E>0$ or $E<0$ respectively.

The Euler–Lagrange equation $\frac{d}{ds}(\partial_{y^{r}}L)-\partial_{r}L=0$ and the assumption $\partial_{r}\overline{F}^{2}=0$ yield for a geodesic with $r$–component constant

[TABLE]

3.1. Photon sphere

We now derive the condition that null circular orbits must satisfy without restricting the analysis to equatorial geodesics $\theta=\frac{\pi}{2}$.

Proposition 3.1**.**

Let $(M,L)$ be a Finsler spacetime with $L$ as in (2). If a null geodesic has constant $r$–component $r(s)\equiv r_{C}$ (a circular photon orbit), then $r_{C}$ satisfies

[TABLE]

Moreover, along such an orbit one has the constraint

[TABLE]

Conversely, if $r_{C}$ obeys (9) and there exists a non-constant $\overline{F}$–geodesic on $(S^{2},\overline{F})$ with constant $\overline{F}$–speed such that (10) holds for some $E>0$, then its lifts with $r\equiv r_{C}$ and $\dot{t}=\pm Ee^{-\nu(r_{C})}$ are, respectively, a future–pointing and a past–pointing null circular geodesic.

Proof.

Recalling that the Noether charge (7) is conserved along geodesics, the time component of the velocity satisfies

[TABLE]

For a circular null geodesic (where $L=0$, $\dot{r}=0$, and $r=r_{C}$), substituting this expression for $\dot{t}$ into the null constraint $e^{\nu(r)}\dot{t}^{2}-r^{2}\overline{F}^{2}=0$ yields:

[TABLE]

Substituting this into (8) gives $r_{C}\nu^{\prime}(r_{C})=2$. This shows the radius is purely determined by the Lorentzian sector associated with coordinates $(t,r)$ and not by $\overline{F}$. With $r\equiv r_{C}$, the remaining equations in $\theta$ and $\phi$ in the Euler-Lagrange system reduce to the geodesic equations of the Finsler metric $\overline{F}$ on $S^{2}$ (with constant $\overline{F}$-speed fixed by (10)). The converse follows analogously as $L$ depends on $(\theta,\phi,\dot{\theta},\dot{\phi})$ only through the Finsler metric $\overline{F}$ and (11) implies that $\dot{t}$ is constant if $r$ is constant. ∎

Remark 3.2*.*

Equation (9) is the same as in the spherical symmetric Lorentzian setting. The role of $\overline{F}$ is twofold: (i) it fixes the curves on $S^{2}$ (the $\overline{F}$–geodesics), and (ii) it enters the impact parameter via (10) when an additional axial symmetry is present (see § 3.2).

Let us see some explicit examples:

• •

Schwarzschild-like metric: for the classic case $e^{\nu(r)}=1-\frac{2M}{r}$, we have $\nu^{\prime}(r)=\frac{2M}{r(r-2M)}$. The condition becomes $\frac{2M}{r_{C}-2M}=2$, which yields the unique physically relevant solution $r_{C}=3M$.

• •

Power-law metric: consider the case $e^{\nu(r)}=Cr^{k}$ (so that $\nu(r)=k\ln r+\ln C$). Here, the condition holds identically if $k=2$. This implies that every radius $r$ admits circular null geodesics. If $k\neq 2$, the condition is never satisfied, and no circular null orbits exist.

• •

Linear exponent: if $\nu(r)=kr$ (so the metric function grows exponentially $e^{\nu(r)}=e^{kr}$), the condition becomes $kr_{C}=2$, yielding a unique solution at $r_{C}=2/k$.

• •

Finslerian hairy black hole with flag curvature $K$: Consider the hairy black hole solution discussed in Subsection 5.2 which includes the flag curvature $K$ and a primary hair parameter equal to $\alpha\ell$:

[TABLE]

(see (27) below). The condition $r_{C}\nu^{\prime}(r_{C})=2$ is equivalent to $r_{C}\frac{d}{dr}(e^{\nu(r)})|_{r=r_{C}}=2e^{\nu(r_{C})}$, which in turn gives:

[TABLE]

This generalizes the Schwarzschild condition (recovered when $\alpha=0$ and $K=1$). The left-hand side in (12) is a monotonically decreasing function from $+\infty$ (at $r=0$) to $-2$, while the right-hand side is a bounded function that tends monotonically to zero as $r\to+\infty$. Consequently, the two curves intersect at exactly one point $r_{C}$.

Remark 3.3* (Closed geodesics on $(S^{2},\overline{F})$).*

Proposition 3.1 does not require that the projection of a null geodesic on $(S^{2},\overline{F})$ be closed. When $\overline{F}$ is given by (5), we effectively have an equatorial photon ring $\mathcal{C}:=\{r=r_{C},\ \theta=\frac{\pi}{2}\}$, where $r_{C}$ is a solution of (9). Away from the equator the projection on the photon sphere $r=r_{C}$ depends on the geodesic flow of $(S^{2},\overline{F})$ which can be described in terms of the geodesics of the scaled round metric $h_{K}$ and the Killing vector field $W$, see [12, Th. 2]. Since the reflection around the equator is an isometry for $\overline{F}_{K}$ in (5), $\mathcal{C}$ is a light ring which is associated with two opposite parametrized closed geodesics of $\overline{F}_{K}$. These geodesics have different $\overline{F}_{K}$–length and indeed they give rise to a Sagnac–type effect, see § 3.4.

Proposition 3.4** (Equatorial closed geodesic for constant flag curvature Randers spheres).**

Let $\overline{F}$ be a strongly convex Randers metric on $S^{2}$ with constant positive flag curvature $K$. Up to an $h_{K}$–isometry, choose spherical coordinates $(\theta,\phi)$ adapted to the symmetry axis so that $W=\varepsilon\partial_{\phi}$ (see Remark 2.3). Then the equator $\{\theta=\tfrac{\pi}{2}\}$ is the support of two closed $\overline{F}$–geodesics with opposite parametrizations.

Proof.

Let $\mathcal{R}:(\theta,\phi)\mapsto(\pi-\theta,\phi)$ be the equatorial reflection. Since $\mathcal{R}$ is an $h_{K}$–isometry and $\mathcal{R}_{*}W=W$, $\mathcal{R}$ is a $\overline{F}$–isometry by [17, Lemma 1.2]. Take $p$ on the equator and $v\in T_{p}S^{2}$ tangent to the equator and collinear to $\partial_{\phi}$. Let $\gamma$ be the $\overline{F}$–geodesic with initial data $(p,v)$. Because $\mathcal{R}$ is an isometry, $\mathcal{R}\circ\gamma$ is a $\overline{F}$–geodesic with the same initial data (note that $d\mathcal{R}_{p}v=v$). By uniqueness, $\mathcal{R}\circ\gamma=\gamma$, so $\gamma$ lies in the fixed set of $\mathcal{R}$, i.e. the equator. Since the equator is a circle, $\gamma$ is closed. Replacing $v$ by $-v$ gives the opposite orientation, which is again tangent at $p$ and the same argument applies. ∎

Propositions 3.1 and 3.4 immediately give the following:

Corollary 3.5** (Equatorial light ring).**

For the static Lorentz–Finsler metric $L$ in (2) with $\overline{F}$ as in (5) any solution $r_{C}$ of $r_{C}\nu^{\prime}(r_{C})=2$ yields a photon ring $\mathcal{C}:=\{r=r_{C},\ \theta=\tfrac{\pi}{2}\}$ consisting of the projection of two null circular future–pointing geodesics of $\overline{F}$ distinguished by the sign of $\dot{\phi}$.

3.2. Axial Noether charge and critical impact parameter

For any Finsler metric $\overline{F}$ admitting a Killing vector field $X$, the Noether charge of $\overline{L}:=\frac{1}{2}\overline{F}^{2}$ associated with $X$ is given by

[TABLE]

see, e.g., [19]. For a Randers metric $\overline{F}=\alpha+\beta$ with $\alpha$ and $\beta$ in (3), one has

[TABLE]

hence the constant of motion along a geodesic $\sigma$ is given by

[TABLE]

(as for the Noether charge $E$, we will simply denote such a geodesic–dependent constant with $\overline{\mathcal{J}}$). For the axial symmetry $X=\partial_{\phi}$ of Randers metric $\overline{F}_{K}$ in (5) and for any geodesic $\sigma(s)=(\theta(s),\phi(s))$, we then obtain:

[TABLE]

where

[TABLE]

and $\lambda_{K}$ is in (4). On the photon sphere $r=r_{C}$ the null constraint gives

[TABLE]

By parametrizing $\sigma$ with the arc length $\overline{F}_{K}\big(\theta,\phi,\dot{\theta},\dot{\phi}\big)\equiv 1$ and recalling that $E=\dot{t}e^{\nu(r(s))}$, we then get

[TABLE]

For the full spacetime Lagrangian $L$ and the circular null geodesic $\gamma$ associated with $\sigma$ (recall Proposition 3.1) we have

[TABLE]

Thus the impact parameter $b:=\frac{|\mathcal{J}|}{E}$ at the photon sphere (the so-called critical impact parameter) is equal to

[TABLE]

For the two equatorial geodesics at $\theta=\tfrac{\pi}{2}$, recalling that $0\leq\varepsilon<\sqrt{K}$, one then has

[TABLE]

where $\lambda_{K}^{*}:=1-\varepsilon^{2}/K$, with the minus (resp. plus) sign corresponding to the co-rotating (resp. counter-rotating) geodesic, $\dot{\phi}>0$ (resp. $\dot{\phi}<0$).

Remark 3.6*.*

Lorentz–Finsler metrics of the type (2) with $\overline{F}$ given by (5) have been considered recently in [13]. We clarify two critical points regarding their analysis:

1. (1)

Symmetry assumptions: the authors state that the spacetime is “spherically symmetric” and simplify the equations by focusing on the equatorial plane $\theta=\pi/2$. We emphasize that the Finslerian modification breaks the full $O(3)$ spherical symmetry to the axial group $SO(2)$ (see Remark 2.3). Consequently, generic geodesics are non-planar. The restriction to the equator is a valid particular case only because it is a totally geodesic submanifold invariant under the discrete reflection $\theta\to\pi-\theta$, not because of spherical symmetry (see Proposition 3.4). 2. (2)

Reversibility and impact parameter: crucially, the authors do not account for the fact that $\overline{F}$ is not reversible. Their expression for angular momentum on the plane $\theta=\pi/2$ is $\mathcal{J}=r^{2}(1+\epsilon)^{-2}\dot{\phi}$ (see Eq. (82), where the authors assume $K=1$). This formula is correct only for co-rotating geodesics ($\dot{\phi}>0$). For counter-rotating ones ($\dot{\phi}<0$), recalling that $\dot{\phi}=-(1-\epsilon)$ in the arc-length parametrization of $F_{K}$, the correct relation derived from (13)–(14) is:

[TABLE]

By using a single relation, reference [13] misses in Eq. (88) the splitting of the critical impact parameter into two distinct values $b_{C}^{\pm}$, see (16).

3.3. Orbit equation

While the general Finslerian spacetime lacks full spherical symmetry, the existence of the reflection isometry ensures that the equatorial plane $\theta=\pi/2$ is a totally geodesic submanifold. We restrict our dynamical analysis to this invariant plane to derive an orbit equation.

On the equator $\theta=\pi/2$, (13) and (14) (with $r=r(\tau)$ instead of the constant value $r_{C}$) give:

[TABLE]

where

[TABLE]

Solving for $\dot{\phi}$ and substituting it back into the expression for $\overline{F}_{K}$ allows us to find the squared angular metric in terms of $\mathcal{J}$:

[TABLE]

Using $E=e^{\nu(r)}\dot{t}$, the null condition and (18), we then get

[TABLE]

thus by $\dot{r}=(dr/d\phi)\dot{\phi}$ and (17), we obtain

[TABLE]

The turning points of the orbit must satisfy $dr/d\phi=0$. Setting the term in the square brackets to zero, we find the impact parameter required to reach a turning point at $r_{tp}$ must satisfy

[TABLE]

in accordance to the critical impact parameter derived in (16).

3.4. A Finslerian Sagnac–type effect

The time delay between different images in a lensing system also receives Finslerian corrections. Since co-rotating and counter-rotating paths have different effective metrics due to non-reversibility of $L$, the travel time for light differs depending on the orientation of the path around the lensing mass leading to a gravitational Sagnac–type effect. Let us describe it in detail.

For null future–pointing curves projecting on the photon sphere $r=r_{C}$, we have

[TABLE]

so that along the two equatorial closed geodesics of $\overline{F}_{K}$, $\sigma^{\pm}$, in Proposition 3.4 at the photon sphere $r_{C}$, the lapse of flight for the $t$ coordinate is

[TABLE]

Hence the Sagnac–type time delay between co- and counter-rotating equatorial loops at $r=r_{C}$ as measured by the observer $\partial_{t}/\sqrt{e^{\nu(r_{C})}}$ is

[TABLE]

Note that, as shown in [20], a Sagnac–type delay as above can also arise in Lorentzian stationary spacetimes. This is not surprising as the future (resp. the past) null cones of any stationary spacetime can be locally seen as the future (resp. past) ones of a static Finsler spacetime [16, §7]; moreover these two spacetimes share (up to reparametrization) the same future–pointing (resp. past–pointing) null geodesics [16, §6 and Appendix B]. To assess whether the static Lorentz–Finsler metric (2) produces effects distinguishable from a stationary model, we consider in the next section a purely timelike, clock-based test in two isocausal models.

4. Distinguishing static Finsler from stationary Lorentzian spacetimes under shared optical data

In this section we present a way to discriminate between a stationary Lorentzian metric $g$ and a static Lorentz–Finsler metric $L$ of the form (2). We perform the comparison in the most stringent setting, namely when the angular Randers metric $\overline{F}$ is exactly the one obtained from $g$ by the stationary–to–Randers correspondence [31]. Without loss of generality we choose the time unit so that the lapse equals $1$ on $S_{r_{0}}:=\{r_{0}\}\times S^{2}$ in both models, where $r_{0}$ is a solution of $r\nu^{\prime}(r)=2$. This can be done by a constant rescaling of $t$ and does not affect proper–time measurements along $S_{r_{0}}$. For simplicity, we then take the lapse to be identically $1$ in both models.

Stationary spacetime:

• •

the spacetime is $M=\mathbb{R}_{t}\times\mathbb{R}_{r}\times S^{2}$, and we set $\Sigma:=\mathbb{R}_{r}\times S^{2}$;

• •

let $h$ be a Riemannian metric on $S^{2}$ and $\omega$ a $1$–form on $S^{2}$ with $\|\omega\|_{h}<1$; we extend $\omega$ to $\mathbb{R}_{r}\times S^{2}$ as $\hat{\omega}(r,\cdot):=r\,\omega(\cdot)$, so $\hat{\omega}(\partial_{r})=0$;

• •

each slice $\{t=\bar{t}\}$ is spacelike and the Riemannian metric induced by $g$ is

[TABLE]

• •

the stationary spacetime metric is

[TABLE]

(with the shorthand $(dt-\hat{\omega})^{2}:=(dt-\hat{\omega})\otimes(dt-\hat{\omega})$ and the abuse of notation concerning the pullback of tensors to $M$ that are denoted as the original ones).

Static Lorentz–Finsler spacetime:

• •

the spacetime is $M=\mathbb{R}_{t}\times\mathbb{R}_{r}\times S^{2}$.

• •

the angular Finsler metric on $S^{2}$ is of Randers type

[TABLE]

(here we avoid writing explicitly the base point $x\in S^{2}$ carried by $v$) with the same $h$ and $\omega$ as above.

• •

the Lorentz–Finsler metric is

[TABLE]

for each $(y^{t},y^{r},v)\in\mathbb{R}\times\mathbb{R}\times TS^{2}$.

For the stationary metric (19), the Fermat (optical) metric on the shell $S_{r_{0}}$ is

[TABLE]

for each $v\in TS^{2}$. For $L$, one likewise has $F_{L}|_{TS_{r_{0}}}(v)=r_{0}\,\overline{F}(v)$ (see [31, 16]). Hence both metrics $g$ and $L$ share the same future–pointing null geodesics, with constant radial component $r\equiv r_{0}$, provided they exist, i.e. when $r_{0}\nu^{\prime}(r_{0})=2$ is satisfied (cf. [31, Prop. 4.1] and [16, Prop. B.1] and Proposition 3.1). In particular, the time delay in the Sagnac–type effect described in §3.4 is the same in both geometries. In order to distinguish the two geometries we consider the following test involving some future–pointing timelike curves rather than null geodesics.

Let us fix a smooth closed loop $\sigma:[0,L]\to S_{r_{0}}$, parametrized by the $h$–arclength $s$ on $S^{2}$, so that $T:=\dot{\sigma}$ obeys $h(T,T)\equiv 1$. Let us set $\omega(T)(s):=\omega_{\sigma(s)}\big(T(s)\big)$. A time–stretch profile is a function $\mathfrak{b}:[0,L]\to(1,\infty)$ prescribing the $t$–speed along $\sigma$.

Along $\sigma$ we have $dr\equiv 0$, and so $\frac{dt}{ds}=\mathfrak{b}$ must satisfy the following inequalities if we want that $t=t(s)$ is the component of a future–pointing timelike curve for both orientation of $\sigma$:

[TABLE]

(we stress that the second right-hand sides in both inequalities above are evaluated at $L-s$ and not $s$ when we consider the opposite parametrization of $\sigma$). Hence, recalling that $\|\omega\|_{h}<1$, the condition

[TABLE]

guarantees that the curves $\gamma^{\pm}(s)=\big(\int_{0}^{s}\mathfrak{b}(u)du,r_{0},\sigma^{\pm}(s)\big)$ are future–pointing and timelike in both models and both orientations, $\sigma^{+}$ and $\sigma^{-}$, of $\sigma$. Hence the proper times of these curves, depending on the profile $\mathfrak{b}$, are then

[TABLE]

where we have used the change of variable $s\mapsto L-s$ for $\tau_{g}^{-}$ and $\tau_{L}^{-}$.

Let us write, for brevity, $\omega=\omega(T)(s)$ and $-\omega=\omega(-T)(L-s)$

Stationary:

From (20) and $\mathfrak{b}\gg 1$ we factor the large terms $\mathfrak{b}\mp r_{0}\omega$:

[TABLE]

so that

[TABLE]

Hence

[TABLE]

Static Lorentz–Finsler:

From (21) and $\mathfrak{b}\gg 1$ we have:

[TABLE]

Then

[TABLE]

Hence

[TABLE]

Let us define for $\mathfrak{b}\gg 1$,

[TABLE]

Hence we get the following estimates in both models.

Stationary:

From (22)

[TABLE]

Static Lorentz–Finsler:

As $\mu\leq\min_{s\in[0,L]}\mathfrak{b}(s)$, from (23)

[TABLE]

In particular, in the stationary spacetime $(M,g)$ the leading term is independent of $\mathfrak{b}$, whereas in the static Finsler spacetime $(M,L)$ the leading term depends on the chosen profile.

Thus, on the sphere $S_{r_{0}}$ the two models share the same optical data $F|_{TS_{r_{0}}}=r_{0}\overline{F}$ and the same null geodesics with constant $r$–component, hence the Sagnac–type delays are identical. Nevertheless, the difference of proper time intervals of some future-pointing timelike curves projecting on the same loop $\sigma$ traversed in both orientations, distinguishes the models.

5. Finslerian field equations and their simplification

Black holes in static Finsler spacetimes have recently been studied in [13], where the authors claim to obtain novel solutions via the extended gravitational decoupling (EGD) method [21], which they term Finslerian hairy black holes. In what follows we examine their work and point out an attribution issue.

5.1. The vacuum equations

One of the remarkable features of static Finsler spacetimes with constant flag curvature angular sector is that the field equations simplify dramatically. Li and Chang [8] considered the Finslerian vacuum field equation proposed by Rutz [11], which is a scalar equation on $\mathcal{A}$ given as $\text{Ric}=0$, where Ric is the Ricci scalar of the Finsler metric $L$ (see [6, §7.6]).222Note that the Ricci scalar can be computed in $\mathcal{A}$ starting with the positively homogeneous of degree $2$ function $L$, see e.g. [22]. If $\mathrm{Ric}=0$ on $\mathcal{A}$ then we can say that it is [math] on $TM$, recall Remark 2.1. The constant flag curvature condition implies that the Finslerian Ricci scalar of $\overline{F}$ is also constant, $\overline{\text{Ric}}=K$ (see [6], Exercise 7.6.2). The nice observation in [8] is that when the angular metric $\overline{F}$ has constant Ricci scalar $\overline{\text{Ric}}=K$, the equation $\mathrm{Ric}=0$ reduces to three ordinary differential equations in the radial coordinate alone and the anisotropy, relegated to the angular sector, disappears completely. Actually, the fact that the angular part does not enter the field equations in the spherical symmetric ansatz is well-known in the Lorentzian framework (see [23], p. 266). In conclusion, the equations obtained are the same as in the static, spherical symmetric Lorentzian setting with the only difference given by the curvature constant $K$ that can be different from $1$. Thus the time and radial components of the solution after fixing some constants of integration take the form [8, Eqs. (61)–(62)]:

[TABLE]

where $M$ is a mass parameter.

5.2. The Finslerian hairy black hole solutions

In the presence of matter, the authors in [13] follow [8], where the Akbar–Zadeh Ricci tensor and scalar [25] are considered:

[TABLE]

from which they build the Finslerian Einstein tensor $G_{\mu\nu}=\mathrm{Ric}_{ij}-\frac{1}{2}g_{ij}S$ and then the field equations on $\mathcal{A}$:

[TABLE]

obtained by raising indexes using the inverse $g^{ij}$ of the fundamental tensor $g$ in (1); here $\operatorname{area}_{\overline{F}_{K}}(S^{2})$ is the area of the Randers sphere $(S^{2},\overline{F}_{K})$ with respect to a fixed volume form (e.g. the Busemann–Hausdorff one, see [26, Example 2.2.2]) that the authors of [8, 13] denote with $4\pi_{\overline{F}}$. We emphasize that (25) is introduced by analogy with the Einstein field equations and it is not obtained from a variational principle. It is important to say that action based formulations of Finsler gravity exist in literature at least when $T^{\mu}_{\nu}=0$, see [27, 28, 29, 30].

The authors of [13] start from equation (25) considering a so-called anisotropic fluid: the stress–energy tensor is diagonal with components depending only on $r$ and the tangential and radial pressure are different, i.e. $T_{rr}\neq T_{\theta\theta}=T_{\phi\phi}$.

Again the constant curvature assumption on $\overline{F}$ reduces essentially the field equations to the standard Einstein field equations in the static spherically symmetric setting (see [24, Eq. (82.2)]):

[TABLE]

where

[TABLE]

Equations (25)—(26) reproduce the vacuum solution quoted above when $T^{\mu}_{\nu}\equiv 0$ [8]. To incorporate hair, the authors follow the (extended) gravitational decoupling (EGD) prescription [21]: they deform the vacuum solution (24) as follows:

[TABLE]

so that the empty solution is recovered at the parameter $\alpha=0$. It is then not surprising that the solution in [13] bears a striking resemblance to the hairy black hole solution obtained by Ovalle et al. [14] using the gravitational decoupling method in the same static, spherical framework but with the standard round sphere as the angular sector. In fact, the solution in [13] is (see [13, Eq. (67)]):

[TABLE]

while Ovalle et al.’s solution takes the form (see [14, Eq. (69)]):

[TABLE]

where $\mathcal{M}=M+\alpha\ell/2$. In both cases the constant $\alpha\ell$ is the so-called primary hair. When we account for the different notation and the rescaling by the flag curvature $K$, these solutions are mathematically identical off the angular sector. More specifically, setting $K=1$, (27) reproduces (28) exactly.

In conclusion, the derivation in Nekouee et al. follows precisely the same extended gravitational decoupling method, applying the same techniques to arrive at the same result (comparing pages 8–12 in [13] with pages 3–5 in [14] reveals that the calculations are essentially identical).

Despite this clear correspondence, Nekouee et al. do not cite [14] by Ovalle et al., even though they cite other works by Ovalle and collaborators on the gravitational decoupling method.

6. Conclusions

We have presented an analysis of static Lorentz–Finsler metrics in spherical coordinates where the “Finslerianity” is confined to the sphere $S^{2}$ and we have observed that the condition for circular null geodesics (photon sphere) in these class of static Finsler spacetimes takes the same form as in general relativity:

[TABLE]

This result holds independently of the specific angular Finsler structure, provided this Finsler structure does not depend on the radial coordinate.

When the Finsler metric $\overline{F}$ on $S^{2}$ is a Randers metric with constant flag curvature, we have evaluated the critical impact parameter (defined as the ratio $|\mathcal{J}|/E$ between the constants of motion $\mathcal{J}$ and $E$ at the photon sphere $r=r_{C}$) for any circular null future–pointing geodesic, see (15). We have obtained the orbit equation on the totally geodesic plane $\theta=\pi/2$ and the value of the impact parameter at a turning point. We also have quantified the asymmetry of $\overline{F}$ through a Sagnac–type effect. Since the same effect can be measured in a stationary Lorentzian spacetime, we have showed in Section 4 that this class of spacetimes can be distinguished from static Lorent-Finsler one by a clock-based test.

Finally, we have addressed the relationship with the recent results in [13]. While this reference investigates the thermodynamics of the hairy black hole model discussed therein, its optical analysis is incomplete as it neglects the intrinsic non-reversibility of the Finsler metric. Consequently, it fails to capture the difference between the co-rotating and counter-rotating critical impact parameters. Furthermore, regarding the solution itself, we clarify that the hairy black hole spacetime presented in [13] is not a novel derivation; the metric coefficients in the $(t,r)$–sector are mathematically identical to those obtained by Ovalle et al. [14], differing only by a constant rescaling involving the flag curvature parameter $K$.

Acknowledgments

We sincerely thank the referees for their valuable comments.

This work is partially supported by PRIN 2022 PNRR “P2022YFAJH Linear and Nonlinear PDE’s: New directions and Application”, by MUR under the Programme “Department of Excellence” Legge 232/2016 (Grant No. CUP - D93C23000100001) and by GNAMPA INdAM - Italian National Institute of High Mathematics.