1 Introduction↩︎

Asymptotically flat, spatially spherically symmetric spacetimes describe the gravitational field of non-rotating compact objects and serve as foundation to find the gravitational field of realistic, axially symmetric, rotating astrophysical systems such as black holes, ordinary and neutron stars, black hole-mergers, dust clouds and many more. As long as the the objects are slowly rotating and can be treated perturbatively, the spherically symmetric spacetime approximation is sufficient to predict observables from these systems with good accuracy.

In general relativity, the Jebsen-Birkhoff Theorem [1], [2] ensures that the only spatially spherically symmetric vacuum solution of the Einstein equations must be the asymptotically flat, static Schwarzschild solution. As the Einstein vacuum equations are identical to the vanishing of the Ricci curvature tensor, one can rephrase the statement of the theorem as: the only spatially spherically symmetric vacuum solutions are given by the family of Ricci-flat, asymptotically flat, static Schwarzschild spacetimes (parametrized by the Schwarzschild radius). Yet, in theories of gravity that go beyond general relativity, this statement is not necessarily true.

In this article, we consider Finsler gravity, based on pseudo-Finsler spacetime geometry, see for example for extensive recent reviews [3], [4]. In the Finslerian setting, the geometry of spacetime is derived from a (pseudo-)norm of tangent vectors – which defines the length of curves in a largely similar way as the geometry of spacetime is derived, in general relativity, from the spacetime metric of pseudo-Riemannian geometry [5]–[7]. Recent results indicate that Finsler geometry is a viable candidate to improve the description of the gravitational field beyond general relativity; and has drawn an increasing attention from both mathematics and theoretical physics communities [8]–[21]. Specific applications of Finsler geometry that are particularly to be highlighted are:

• the description of the gravitational field of kinetic gases through Finsler geometry [22], [23], which has led to the finding of an accelerated expansion of the universe without dark energy [24];

• the interpretation, in metric affine geometry, of autoparallels that arise from parametrization-invariant actions, as geodesics on a Finsler spacetime [25], [26];

• the geometric description of quantum gravity phenomenology effects including deformations of local Lorentz invariance [27]–[33];

• propagation of waves in media and on curved spacetimes [34], [35].

There have been several proposals for suitable field equations that determine the Finslerian geometry of spacetime dynamically, of which the physically best motivated and most promising one is the action-based scalar field equation which naturally couples the kinetic gas to Finsler geometry proposed in [22],[23] – or its Palatini generalization – see also [36]–[38] for a detailed discussion. To describe compact objects, it is important to classify and understand spatially spherically symmetric, asymptotically flat vacuum solutions of the Finsler gravity equations.

Among all Finsler spacetimes, the ones of Berwald type [39]–[43] can be considered closest to pseudo-Riemannian geometry. Their main feature is that the geometry defining length measure for curves is genuinely Finslerian (i.e., not pseudo-Riemannian), however, their geodesics are autoparallels of an affine connection on spacetime; in other words, the so-called canonical Finslerian nonlinear connection is equivalent to an affine connection on spacetime. These classes of Finsler spacetimes are particularly interesting as they are closely related to metric-affine geometry [25], [26], [44]. Moreover, they have the advantage that the usually rather complicated Finsler gravity equation significantly simplifies to \[\begin{align} D^{ab}R_{ab} = 0\,; \end{align}\] here, \(D^{ab}\) is a tensorial object obtained from the Berwald-Finsler metric and the tangent space coordinates, and \(R_{ab}\) are the components of the Ricci tensor of the canonical affine connection, discussed in detail in Section 2.2. In contrast to general relativity, Ricci flatness is not equivalent to solving the Finsler gravity equations – it is only sufficient.

Thus, the following nontrivial question is immediate: What are all spherically symmetric Berwald-Finsler spacetimes vacuum solutions? Can they be classified, what are their properties?

The starting point to answer these questions is the local classification of all spherically symmetric Berwald Finsler spacetimes, which is known from [45], [46]. It has been used to show that the Jebsen-Birkhoff Theorem extends to Berwald Finsler spcetimes in the sense that \(R_{ab}=0\) implies that spatially spherically symmetric Finsler spacetime geometry must be either completely flat, or pseudo-Riemannian Schwarzschild geometry [47].

In this article, we find for the first time spatially spherically symmetric, asymptotically flat Berwald Finsler spacetime vacuum solutions that are neither Ricci flat, nor pseudo-Riemannian; moreover, we find all of them. They are candidates to describe the gravitational field of compact objects, beyond general relativity.

To this aim, we proceed as follows.

• Using the earlier found local classification of all spatially spherically symmetric (technically \(SO\left( 3\right)\)-symmetric), 4-dimensional Berwald-Finsler structures [45], we show that requiring the existence of nondegenerate light cones in each tangent space, together with asymptotical flatness, leaves just one of the five existing classes as candidate.

• For this remaining class (deemed Class 3 in [48]), we completely integrate the Finslerian gravity equation in vacuum. It turns out that there exist three families of solutions that are not Ricci-flat.

• For one of these families, we provide a concrete example which has a well defined causal structure.

The presentation of the results throughout this article is organized as follows: In Section 2 we introduce the necessary language to discuss all classes of spatially spherically symmetric Berwald-Finsler spacetimes, before we continue in Section 3 with defining and characterizing asymptotic flatness in this context. Next, in Section 4 we identify the unique class (Class 3), among all spatially spherically symmetric Berwald Finsler spacetimes, that allows for asymptotically flat geometries with well-defined light cones. Afterwards, we solve the Finsler gravity equation for this class in Section 5 in all generality. In Section 6 we present an explicit example before we come to our outlook and conclusion in Section 7.

2 Spatially spherically symmetric Berwald Finsler spacetimes↩︎

In this section, we briefly introduce the concepts of (pseudo)-Finsler space, Finsler spacetime and Finsler gravity. Moreover, we recall the definition of Berwald-Finsler spacetimes and the classification of all spatially spherically symmetric Berwald-Finsler spacetimes introduced in [45]. All of these concepts lay the foundation for the search of non-Ricci flat, but asymptotically flat (see Sec. 3), causally well-behaved (see Sec. 4), vaccuum solutions in Section 5.

We keep the the presentation here as short as possible and refer to [45], [47], [48] for further details.

2.1 Pseudo-Finsler spaces, Finsler spacetimes and Finsler Gravity↩︎

Consider a smooth \(4\)-dimensional manifold \(M\) (spacetime) equiped with local coordinate charts \(\left( U,\left(x^{a}\right) \right)\), where indices \(a,b,c,....\) run over \(0,1,2,3\). Moreover, let \(TM\equiv \left\{ \left( x,\dot{x}\right) ~|~x\in M,\dot{x}\in T_{x}M\right\}\), be the tangent bundle equipped with naturally induced local charts \(\left( TU,\left( x^{a},\dot{x}^{a}\right) \right)\), that is, \(\dot{x}=\dot{x}^{a}\partial _{a}\) is the decomposition of the tangent vector \(\dot{x}\in T_{x}M\) in the natural local coordinate basis \(\left\{ \partial _{a}\right\}\). If the chart is fixed and there is no risk of confusion, we will typically denote the coordinates \(\left( x^{a},\dot{x}^{a}\right)\) simply as \(\left( x,\dot{x}\right)\). Moreover, denote the natural local coordinate basis of \(T_{(x,\dot{x})}TM\) by \(\{\partial _{a}=\tfrac{\partial }{\partial x^{a}},\dot{\partial}_{a}=\tfrac{\partial }{\partial \dot{x}^{a}}\}\) and its dual, by \(\{dx^a,d\dot{x}^a\}\).

A pseudo-Finsler structure (or pseudo-Finsler metric), [49], on \(M\), is a smooth function \(L:\mathcal{A}\rightarrow \mathbb{R}\) (sometimes also called Finsler Lagrangian), defined on a conic subbundle¹ \(\mathcal{A}\subset TM\backslash \{0\},\) with the following properties:

1. Positive 2-homogeneity:\(L\left( x,\lambda \dot{x}\right) =\lambda ^{2}L\left( x,\dot{x}\right) ,\) \(\forall \lambda >0.\)

2. Nondegeneracy: at any \(\left( x,\dot{x}\right) \in \mathcal{A}\) and in one (then, in any) local chart around \(\left( x,\dot{x}\right)\), the Hessian:\[g_{ab}\left( x,\dot{x}\right) :=\dfrac{1}{2} \dot{\partial}_a\dot{\partial}_b L\] is nonsingular.

We note that any pseudo-Finsler function \(L\) can be continuously prolonged as \(0\) at \(\dot{x}=0\).

In the above, the conic bundle \(\mathcal{A}\) is called the set of admissible vectors and the functions \(g_{ab}=g_{ab}\left( x,\dot{x}\right)\) are the local components of the Finslerian metric tensor attached to \(L,\) which is a mapping \[g:\mathcal{A}\rightarrow T_{2}^{0}M,(x,\dot{x})\mapsto g_{(x,\dot{x})}=g_{ab}dx^{a}\otimes dx^{b}.\] Terminology convention: In the following, we will typically omit the prefix ‘pseudo’ and call for simplicity, Finsler (respectively, Riemannian) spaces, pseudo-Finsler (respectively, pseudo-Riemannian) ones; whenever needed, we will indicate the signature explicitly.

The square root \(F = \sqrt{|L|}\) defines a (parametrization-invariant) length measure for admissible curves \(\gamma\) on \(M\) through \[\begin{align} \label{eq:Flength} \ell[\gamma] = \int F(\gamma(\tau),\dot{\gamma}(\tau))\, d\tau; \end{align}\tag{1}\] the curve \(\gamma\) is called admissible if its canonical lift \((\gamma, \dot{\gamma}:=\frac{d\gamma}{d\tau})\) to \(TM\) has its image completely contained in \(\mathcal{A}\) . Going from Finsler structures to Finsler spacetimes and Finsler gravity can be done in the following steps:

• Finsler spacetimes: In the literature, there exist various definitions of Finsler spacetimes, e.g., [8], [9], [36], [50]–[52], which differ in nuances on the precise differentiability requirements imposed on \(L\). Without entering the precise details of these definitions, they all tend to agree on the following: the manifold \(M\) should be 4-dimensional, connected and orientable and the Finslerian metric tensor \(g\) must be well-defined and have Lorentzian signature \(\left( +,-,-,-\right)\) on an appropriate cone \(\mathcal{T}_{x}\) in each tangent space (on which \(L>0\)); moreover, on the boundary \(\partial \mathcal{T}_{x},\) one must have \(L=0.\).

  The cones \(\mathcal{T}_{x}\) are physically interpreted as future-pointing timelike cones, representing the physical \(4\)-velocities at which massive particles propagate through spacetime. The boundaries \(\partial \mathcal{T}_{x}\), represent the light cones, interpreted as the directions along which massless particles propagate.

• The Cartan tensor: A first measure of the departure of a general Finsler function from a Riemannian one is given by the Cartan tensor \(C=C_{abc}dx^{a}\otimes dx^{b}\otimes dx^{c},\) with local components \(C_{abc}=C_{abc}\left( x,\dot{x}\right)\) given by: \[C_{abc}:=\dfrac{1}{2}\dot{\partial}_c\partial g_{ab} =\dfrac{1}{4}\dot{\partial}_c \dot{\partial}_b \dot{\partial}_a L\,.\label{def:C95a}\tag{2}\] Its trace \(C_{a}:=g^{bc}C_{abc}\) measures the \(\dot{x}\)-dependence of \(\det g:\) \[C_{a}=\dot{\partial}_a \sqrt{\left\vert \det g\right\vert }\,. \label{eq:C95a95det40g41}\tag{3}\]

• Geodesics and canonical nonlinear connection: Curves that extremize the length measure 1 are called Finsler geodesics. In arc-length parametrization, they are given by solutions of the Finsler geodesic equation \[\begin{align} \ddot x^a + G^a(x,\dot{x}) = 0\,. \end{align}\] Here \(G^a\) are the so called geodesic spray components which define the canonical Cartan nonlinear connection coefficients \(G^a{}_b\): \[\begin{align} \label{def:G94a95G94a95b} G^a = \frac{1}{4}g^{ab}(\dot{x}^c \partial_c \dot{\partial}_b L - \partial_b L)\,,\quad G^a{}_b = \dot{\partial}_b G^a\,. \end{align}\tag{4}\] The nonlinear connection coefficiens define the so-called horizontal derivative operators \[\begin{align} \delta_a = \partial_a - G^a{}_b \dot{\partial}_b\,. \end{align}\]

• Dynamical covariant derivative: The canonical Cartan nonlinear connection leads to the definition of the dynamical covariant derivative, that acts on on either horizontal \(X=X^{a}\delta _{a}\) or on vertical \(Y=Y^{a}\dot{\partial}_{a}\) vector fields on \(\mathcal{A}\) as: \[\nabla X=\left( \nabla X^{a}\right) \delta _{a}=\left( \dot{x}^{b}\delta _{b}X^{a}+G_{~b}^{a}X^{b}\right) \delta _{a},~\quad \nabla Y=\left( \nabla Y^a\right) \dot{\partial} _{a}=\left( \dot{x}^{b}\delta _{b}Y^{a}+G_{~b}^{a}Y^{b}\right) \dot{\partial}_{a}.\] Intuitively, the dynamical covariant derivative measures the rate of change of the respective vector fields along (canonically lifted to \(TM\)) geodesics \(\gamma\). It is the canonical generalization of the Levi-Civita covariant derivative to Finsler geometry and also satisfies the properties:\(\nabla L=0\), \(\nabla g=0\).

• Curvature and Finslerian Ricci scalar: The curvature of a nonlinear connection is, by definition the Lie bracket of the horizontal basis elements \(\delta_a\); this is a tensor field \(\mathcal{R}\) on \(TM\): \[\mathcal{R}=\frac{1}{2}R_{~bc}^{a}\dot{\partial}_{a}\otimes dx^{b}\wedge dx^{c},~\;\;\;\left[ \delta _{b},\delta _{c}\right] =:R_{~bc}^{a}\dot{ \partial}_{a}=(\delta _{c}G_{~b}^{a}-\delta _{b}G_{~c}^{a})\dot{\partial} _{a}.\; \label{def:R}\tag{5}\] It measures the non-integrability of the horizontal distribution \(Span\{\delta_a\}\) and serves to characterize Finslerian geodesic deviation (see, e.g., [6]). The trace of the geodesic deviation operator is given by the \(2\)-homogeneous scalar \[R=R_{~ac}^{a}\dot{x}^{c}, \label{def:Finsler95Ricci95scalar}\tag{6}\] called the Finslerian Ricci scalar.

• The Landsberg tensor: The Landsberg tensor \(P = P_{abc}dx^a \otimes dx^b \otimes dx^c\), respectively, its trace \(P_a dx^a\), \[\begin{align} P_{abc} = \nabla C_{abc}\,,\quad P_a = g^{bc}P_{abc}\,. \end{align}\] measure the change of the Cartan tensor (respectively, of its trace) along geodesics on \(M\).

• Finsler gravity: All the structures just introduced allow us to determine Finsler spacetimes dynamically from a physical gravitational field equation, similar to the way Einstein’s equations determine the pseudo-Riemannian geometry of spacetime in general relativity.

  Using the (\(0\)-homogenized) Finslerian Ricci scalar \(L^{-1}R\) as a field-theoretical Langrange function to construct an action on the tangent bundle for Lagrangians \(L\), leads to the most promising Finslerian gravitational field equation, sourced by the 1-particle distribution function \(\varphi\) of a kinetic gas [22]. The resulting field equation can be expressed as \[g^{ab}\dot{\partial}_{a}\dot{\partial}_{b}R-6\dfrac{R}{L}+2 g^{ab}[2 \dot{\partial_a} (\nabla P_b) - \nabla (\dot{\partial_a} P_b)]=\kappa \varphi . \label{eq:field95eqn}\tag{7}\] Here \(\kappa\) is the gravitational coupling constant. Recently, this equation was very successfully applied to determine the evolution of the universe [24]. In homogeneous and isotropic symmetry, it becomes the so-called Finsler-Friedmann equation, leading to an accelerated expansion of the universe without the need of introducing dark energy.

Particular case: Riemannian geometry. For the case of (pseudo-)Riemannian geometry we have that \[L=a_{ab}\left( x\right) \dot{x}^{a}\dot{x}^{b}\,,\] where \(a_{ab}(x)\) are the components of a pseudo-Riemannian metric, and thus \[2G^{a}\left( x,\dot{x}\right) = \Gamma _{~bc}^{a}\left( x\right) \dot{x}^{b} \dot{x}^{c},~\;G_{~b}^{a}\left( x,\dot{x}\right) =\Gamma _{~bc}^{a}\left( x\right) \dot{x}^{c},\]where \(\Gamma _{~bc}^{a}\) are the Christoffel symbols of the metric \(a\) and, accordingly,\[R_{~bc}^{a}(x,\dot{x})=R_{d~bc}^{~a}(x)\dot{x}^{d},~\; R(x,\dot{x})=R_{d~ac}^{~a}(x)\dot{x}^{d}\dot{x}^{c}=-R_{cd}(x)\dot{x}^{c}\dot{x}^{d}\,.\] Here, \(R_{d~bc}^{~a}\) and \(R_{dc}=R_{d~ca}^{~a}\) represent the components of the Riemannian curvature and of the Ricci tensor of \(a.\) We note that the Finsler-Ricci scalar of \(a\) – actually, the Raychaudhuri scalar of \(a\) – is different from the scalar curvature \(a^{cd}R_{cd}\). Also, if one sets \(\varphi =0\) in (7 ) and restricts the search for \(L\) to the class of Riemannian Finsler Lagrangians \(L=a_{ab}\left( x\right) \dot{x}^{a}\dot{x}^{b},\) one obtains that our single equation (7 ) is equivalent to the set of 10 vacuum Einstein equations \(R_{ab}=0.\)
Yet, in general, the functions \(G_{~b}^{a}=G_{~b}^{a}\left( x,\dot{x}\right)\) are nonlinear in \(\dot{x}\); the case when these turn out to be linear but \(L\neq a_{ab}\left( x\right) \dot{x}^{a}\dot{x}^{b}\) deserves a special attention and will be discussed in the next subsection. We will see that, even in this most restrictive Finslerian case, the field equation can have dramatically different solutions from the Einstein equations.

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2.2 Finsler geometry of Berwald type↩︎

Berwald-Finsler structures are defined as (pseudo-)Finsler structure whose arc length parametrized geodesics are autoparallel curves of a symmetric affine connection \(\Gamma\) on the base manifold \(M\); denoting its local coefficients by \(\Gamma _{~ab}^{c}=\Gamma _{~ab}^{c}\left( x\right)\), this is \[\frac{d^{2}x^{a}}{ds^{2}}+\Gamma _{~ab}^{c}(x)\dfrac{dx^{a}}{ds}\dfrac{dx^{b}}{ds}=0. \label{eq:Berwald95geo}\tag{8}\] In other words, a Finsler space \(\left( M,L\right)\) is of Berwald type if in any local chart, the canonical spray coefficients \(G^{c}\) are quadratic, \(G_{~b}^{c}\) are linear and \(\dot{\partial}_{a}G_{~b}^{c}=\Gamma _{~bc}^{a}\left( x\right)\) are independent of \(\dot{x}\): \[2G^{c}=\Gamma _{~ab}^{c}(x)\dot{x}^{a}\dot{x}^{b}~\;\Leftrightarrow ~G_{~b}^{c}=\Gamma _{~ab}^{c}(x)\dot{x}^{a}~\;\Leftrightarrow ~\dot{\partial}_{a}G_{~b}^{c}=\Gamma _{~ab}^{c}\left( x\right) . \label{eq:coeffs95Berwald95spray}\tag{9}\] Thus, the dynamical covariant derivative \(\nabla\) (on \(\mathcal{A} \subset TM\)) induces an affine connection \(\Gamma\) on the base manifold \(M\).

As a consequence, the curvature of the canonical nonlinear connection and the Finslerian Ricci scalar of a Berwald space are given by \[\label{eq:nlcurvberw} R_{~bc}^{a}(x,\dot{x})=R_{d~bc}^{~a}\left( x\right) \dot{x}^{d},\tag{10}\] where \(R_{d~bc}^{~a}\left( x\right)\) are the curvature components of the induced affine connection on \(M\). Moreover, one can show that for Berwald Finsler structures the Landsberg tensor vanishes, see also [5], \[\begin{align} P_{abc} = \nabla C_{abc} = 0\quad \Rightarrow\quad P_a = \nabla C_{a}=0\,. \end{align}\]

The above remarks on the curvature and the Lansberg tensor lead to a considerable simplification of the Finsler gravity equation 7 . In vacuum (\(\varphi =0\)), it reads \[g^{ab}\dot{\partial}_{a}\dot{\partial}_{b}R-6\dfrac{R}{L}=0 \quad \Leftrightarrow \quad \left( g^{ab}(x,\dot{x}) - 3\frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})} \right) R_{ab}(x)= 0. \label{eq:vacuum95Berwald}\tag{11}\]

Remark (Connection Ricci tensor vs. Finsler-Ricci scalar): The \(\dot{x}\)-derivatives of the Finslerian Ricci scalar \(\dot{\partial}_{a}\dot{\partial}_{b} R(x,\dot{x})\) are related to the affine Ricci tensor components \(R_{ab}(x)=R_{a~bc}^{~c}(x)\) as: \[R(x,\dot{x})={\textcolor{blue}{ -}}R_{ab}(x)\dot{x}^{a}\dot{x}^{b}\Rightarrow \dot{\partial}_{a}\dot{\partial}_{b} R=-2R_{ab}. \label{eq:R95to95R95ab}\tag{12}\]

It thus becomes obvious that \(R_{ab} = 0\) is a sufficient condition (but not a necessary one, as we will prove below) for a Berwald-Finsler function \(L\) to represent a vacuum solution of the Finsler gravity equation 11 . As, in spatial spherical symmetry, it constrains a Berwald-Finsler spacetime to be either Riemannian (Schwarzschild), or flat (see [47]), physically interesting vacuum solutions are those that do not satisfy it.

2.3 All spatially spherically symmetric Berwald metrics↩︎

To locally² characterize all spatially spherically symmetric Finsler structures of Berwald type using coordinates, we can assume with no loss of generality that we are working on \(M\) as an open subset of \(\mathbb{R}^{4}\backslash \{0\}\), equipped with spherical coordinates \(\left( t,r,\theta ,\phi \right)\). Moreover, we assume that \(M\) has ends, i.e., it admits points with \(r\rightarrow \infty\).

A pseudo-Finsler function \(L:\mathcal{A\rightarrow }\mathbb{R}\) as above is called spatially spherically symmetric if:

1. \(SO\left( 3\right)\) is a subgroup of the isometry group of \(L\), i.e., \(SO(3)\) acts by diffeomorphisms \(\Psi:M \rightarrow M\) such that \(L(\Psi\left(x\right) ,d\Psi _{x}\left( \dot{x}\right)) = L\left( x,\dot{x}\right),\) for all \(\left( x,\dot{x}\right)~\in~\mathcal{A}\).

2. The elements \(\;\in SO\left( 3\right)\) leave the coordinate \(t\) (usually interpreted as time) invariant.

In naturally induced coordinates \(\left( x,\dot{x}\right) :=\left( t,r,\theta ,\phi ,\dot{t},\dot{r},\dot{\theta},\dot{\phi}\right)\) on \(TM,\) any spatially spherically symmetric Finsler metric is known to be, [53], of the form \[L(x,\dot{x})=L(t,r,\dot{t},\dot{r},w),\quad w^{2}=\dot{\theta}^{2}+\dot{\phi}^{2}\sin ^{2}\theta \,.\]

Nontrivially Finslerian (i.e., non-quadratic)Berwald metrics with spatial spherical symmetry, as well as their canonical affine connections have been locally classified in [45]. It was found that their coefficients \(\Gamma^a{}_{bc}(x)\) of are encoded into 10 functions \(k_{i}=k_{i}(t,r)\) and the curvature tensor \(R^a{}_{bcd}(x)\) of \(\Gamma\) is described by 14 coefficients \(a_{i}=a_{i}(t,r);\) see Appendix 8 for the precise definitions of \(k_{i}\) and \(a_{i}\) and for the relations between them. Depending on which relations hold between \(k_{i}\) and \(a_{i}\), the Berwald condition \[\label{eq:bercond} \delta_a L = \partial_a L - \Gamma^b{}_{ac}(x)\dot{\partial_ b}L = 0\,,\tag{13}\] can be solved and one finds two major cases of spatially spherically symmetric Berwald Finsler structures that contain several subclasses:

• If \(k_{7},k_{8},k_{9},k_{10}\)are not all zero, one may assume with no loss of generality, that \(k_{10}\not=0.\) Denoting\[a:=\dfrac{k_{7}}{k_{10}},~\;\;b=\dfrac{k_{8}}{k_{10}},~\;\;c=\dfrac{k_{9}k_{10}-k_{7}k_{8}}{k_{10}^{2}}, \label{def:abc}\tag{14}\] the connection must satisfy:\[\begin{align} A& :=b\left( aa_{1}+a_{2}\right) +\left( ab+c\right) \left( aa_{3}+a_{4}\right) -a_{5}\left( 2ab+c\right) =0\,, \\ B& :=a\left( aa_{3}+a_{4}\right) -\left( aa_{1}+a_{2}\right) =0, \\ C& :=\left( ab+c\right) a_{3}+b\left( aa_{3}+a_{4}\right) +b(a_{1}-2a_{5})=0, \\ a_{6}& =aa_{7},~a_{8}=ba_{7},~\;a_{9}=\left( ab+c\right) a_{7},~\; a_{10}=aa_{11},~a_{12}=ba_{11},~\;\;a_{13}=\left( ab+c\right) a_{11}. \end{align} \label{eq:ABC}\tag{15}\]

  Further, depending on the quantities\[\begin{align} D &=&aa_{3}-a_{1}+a_{5},~\;\;E=ba_{3},~\;\;F=aa_{3}-a_{1}, \label{eq:DEF} \\ G &=&2\left( k_{1}-k_{4}a\right) ,~\;\;\tilde{G}=G-2k_{8},~\;\;\;H=2\left( k_{2}-k_{6}a\right) ,~\;\;\;\tilde{H}=H-2k_{9}, \end{align}\tag{16}\] one obtains in this case, three classes:

1. \(D\neq 0\). Then, \(\left[ \delta _{t},\delta _{r}\right] \not=0\) and \(L\) has a power law formula:\[L=\vartheta (t,r)u^{2-2\lambda }\left( v+\rho u^{2}\right) ^{\lambda }\,, \label{def:power95law}\tag{17}\] where:\[u=\dot{t}-a\dot{r},\quad v=c\dot{r}^{2}+2b\dot{t}\dot{r}-w^{2}\,. \label{def:uv}\tag{18}\] \(\lambda =\frac{F}{D}\not=1\) is a constant³, \(\rho =\frac{E}{D}\) and \(\vartheta (t,r)=\exp \left( \int \left( G-\lambda \tilde{G}\right)dt+\left( H-\lambda \tilde{H}\right) dr\right)\) (where the integral is taken along an arbitrary path in the \(\left( t,r\right)\) plane, joining some \(\left( t_{0},r_{0}\right)\) to \(\left( t,r\right)\)).

2. \(D=0,\, E,F\neq 0\). Then \(\left[ \delta _{t},\delta_{r}\right] \not=0\) and \(L\) is of exponential type:\[L=\varphi (t,r)u^{2}\exp (\frac{v}{u^{2}}\mu )\,, \label{def95exponential95law}\tag{19}\] where \(\mu =\frac{F}{E}\), \(u,v\) are as above and \(\varphi (t,r)=\exp \left( \int \left( G+2k_{4}b\mu \right) dt+\left( H+2k_{6}b\mu \right) dr\right)\) (where the integral is taken along an arbitrary path in the \(\left( t,r\right)\) plane, joining some \(\left( t_{0},r_{0}\right)\) to \(\left( t,r\right)\)).

3. \(D=E=F=0.\) Then, \(\left[ \delta _{t},\delta _{r}\right] =0\) and:\[L=u^{2}\Xi (z),\quad z=\frac{v(\dot{\tilde{t}},\dot{\tilde{r}},w)}{u(\dot{\tilde{t}},\dot{\tilde{r}},w)^{2}}\,, \label{def95class953}\tag{20}\] where \(\Xi =\Xi (z)\) is an arbitrary function of the single variable \(z=\dfrac{v}{u^{2}}\) and \(u,v\) as defined above are, up to a suitable coordinate change, independent of the new coordinates \(\tilde{t}\) and \(\tilde{r}\). In this case, \(b\) and \(c\) cannot simultaneously vanish.

• If \(k_{7}=k_{8}=k_{9}=k_{10}=0\), we distinguish two more classes:

1. \(\left[ \delta _{t},\delta _{r}\right] =0.\) Then, up to a suitable coordinate change \(\left( t,r\right) \mapsto \left( \tilde{t},\tilde{r}\right)\), \(L\) is given by a free function of two variables: \[L=L(\dot{t},\dot{r},w)=\dot{t}^{2}L(1,p,s)\,,\quad p=\frac{\dot{r}}{\dot{t}},\quad s=\frac{w}{\dot{t}}\,. \label{def95class955}\tag{21}\]

2. \(\left[ \delta _{t},\delta _{r}\right] \not=0\). Then \(a_{1}a_{4}-a_{2}a_{3}\not=0\) and\[L=w^{2}\xi (q)\,,\quad q=\frac{\dot{t}e^{I\left( p\right) -\varphi }}{w},~\; \;\;p=\frac{\dot{r}}{\dot{t}}, \label{def95class954}\tag{22}\] where \(\xi =\xi (q)\) is an arbitrary function of the single variable \(q;\) \(\varphi\) is a function of \(t,r\) only and\[I=\int \dfrac{a_{1}+a_{2}p}{a_{2}p^{2}+\left( a_{1}-a_{4}\right) p-a_{3}}dp. \label{I95p}\tag{23}\]

In the following, we will prove that, in the Finslerian case, there exist vacuum solutions of 11 that are not Ricci flat.

3 Asymptotic flatness↩︎

Finding the most general non-Ricci flat Berwald vacuum solution of the Finsler gravity equation is a very difficult endavour and still work in progress. As a first step towards this goal, we focus in this article on all asymptotically flat solutions. Thus, before proceeding to solve the equation, we clarify the notion of asymptotically flat Berwald-Finsler structures here and identify, among all classes introduced in the previous section 2.3, the only physical viable class, Class 3.

By asymptotic flatness of a spatially spherically symmetric Berwald-Finsler structure, we will understand that, as \(r\rightarrow \infty ,\) all the curvature components of its canonical affine connection tend to zero. In our notations, this is: \[\underset{r\rightarrow \infty }{\lim }a_{i}=0,~\;\;\;i=1,...,14. \label{def:asympt95flatness}\tag{24}\]

Here is an immediate result, yet with quite strong implications.

The statements follow immediately from the identity, see 54 , \[a_{14}=1+k_{7}k_{8}+k_{9}k_{10}=1+k_{10}^{2}\left( 2ab+c\right) , \label{a14}\tag{25}\] which makes it clear that, in any of the two above cases, we get \(a_{14}=1,\) which is incompatible with 24 .

The above remark excludes three of the five classes of Berwald-Finsler \(SO\left( 3\right)\)-symmetric metrics:

• Class 2 (exponential metrics), since, in this case, solving the system \(A=B=C=D,\) we find, [47], that we must necessarily have \(2ab+c=0.\)

• Classes 4 and 5, since for these, \(k_{7}=k_{8}=k_{9}=k_{10}=0\).

One more class, Class 1 (power law metrics), can be excluded as it does not possess well defined light cones. It can in principle be asymptotically flat; yet, see Appendix 10, the asymptotical flatness condition leads in this case to degenerate light cones at all spacetime points, hence to no physically viable solutions.

Conclusion:The only class of \(SO\left( 3\right)\)-symmetric Berwald structures which can contain asymptotically flat spacetime metrics is Class 3 of Section 2.3, given by the condition \[\left[ \delta _{t},\delta _{r}\right] =0,~k_{7},k_{10}~-~not~both~zero.\]In the following, we will therefore focus on this class and assume assume without loss of generality, \[k_{10}\not=0\,.\] The situation when \(k_{7}\not=0\) can be treated in a completely similar manner - just, with the roles of \(k_{8}\) and \(k_{9}\) reversed.

4 The structure of Class 3 metrics↩︎

Having identified Class 3 of Section 2.3 as a viable candidate for non-Ricci flat solutions of the Finsler gravity equation, let us collect and recall several properties of these Berwald-Finsler structures – especially, the fact that Finsler Lagrangians of Class 3 Berwald-Finsler structures as so-called \((\alpha,\beta)\)-metrics \(L=L(\mathbf{A},\mathbf{B})\), that are constructed from a pseudo-Riemannian metric factor \(\mathbf{A} = a_{bc}(x)\dot{x}^b \dot{x}^c\) and a \(1\)-form factor \(\mathbf{B} = b_c(x)\dot{x}^c\). This is of core importance in our study of the field equation, to be done in Section 5.
In the following, assume that our input is an \(SO(3)\) (spatially spherically) symmetric, torsion-free affine connection \(\Gamma\) on \(M\), with connection coefficients \(\Gamma^a{}_{bc}(x)\) characterized by the nonvanishing functions \(k_i(x), i=1,...,10\), which satisfies the Finsler metrizability constraints 13 and the Class 3 conditions, i.e , \[A=B=C=D=E=F=0, \label{eq:class95395AF}\tag{26}\] \(k_{10}\not=0\) and \(b,c\;\)- not both zero. For the covariant derivative of objects on \(M\) with respect to this affine connection we will use the semicolon notation, for example \(b_{a;c}\).
For this class, we already know from [48] the following result:

It was shown in [48] that, if the given connection satisfies the Finsler metrizability conditions, then the functions \(\mathcal{G},\mathcal{K}\) and \(\mathcal{M}\) solving (?? )-(?? ) always exist.
In order to understand the structure of the Class 3 Berwald Finsler geometries, it is very convenient to introduce a \(1\)-form \(\mathbf{b} = b_c(x)dx^c\) and a pseudo-Riemannian metric \(\mathbf{a} = a_{bc}(x)dx^b \otimes dx^c\) on spacetime, as follows:

1. Choosing \(\Theta = 1\) in ?? , one finds \(L=\mathbf{B}^2\), where \[\mathbf{B}=e^{\frac{\mathcal{G}}{2}}u~=~e^{\frac{\mathcal{G}}{2}}\left( \dot{t}-a\dot{r}\right)\,. \label{B}\tag{27}\] As \(\mathbf{B}\) is linear in \(\dot{x}\), it can be understood as \(\mathbf{B}=b_c(x)\dot{x}^c\), induced by the \(1\)-form \[\mathbf{b}=e^{\frac{\mathcal{G}}{2}} dt + -ae^{\frac{\mathcal{G}}{2}} dr\] on \(M\). Moreover, as ?? ?? guarantee that the Berwald condition \(\delta_a L = \delta_a \mathbf{B}^2 = 0\) holds, this \(1\)-form is absolutely parallel with respect to the affine connection on \(M\), i.e., it satisfies \(b_{d;c} = 0\).

2. Alternatively, choosing in ?? , \(\Theta :=id.\), we find that \(L\) is a quadratic function in \(\dot{x}\): \[L=\mathbf{A}:=e^{\mathcal{G}}\left( ve^{-\left( \mathcal{G}-2\mathcal{K}\right)}+\mathcal{M}u^{2}\right) =ve^{2\mathcal{K}}+\left( e^{\mathcal{G}}\mathcal{M}\right) u^{2}\,. \label{A}\tag{28}\] The function \(\mathbf{A}\) defines a pseudo-Riemannian metric \[\mathbf{a} = a_{bc}(x)dx^b \otimes dx^c\] with components \(a_{bc} = \dot{\partial}_b \dot{\partial}_c\mathbf{A}\); this is defined on and nondegenerate at least on a subset \(D\subset M\) (which can be specified by appropriately choosing the range of the \(\left( t,r\right)\) coordinates, see [46]). As ?? ?? guarantee that the Berwald condition \(\delta_a L = \delta_a \mathbf{A} = 0\) holds, the pseudo-Riemannian metric satisfies \(a_{bc;a}=0\) for the covariant derivative defined through the affine connection coefficients \(\Gamma^a{}_{bc}(x)\) and thus \(\Gamma^a{}_{bc}(x)\) must define the Levi-Civita connection of \(\mathbf{a}\).

Using the newly defined quantities \(\mathbf{A}\) and \(\mathbf{B},\) we find that all Berwald Finsler structures of Class 3, ?? , can be expressed as: \[\label{eq:Lpsi} L=\mathbf{B}^{2}\Theta \left( \dfrac{\mathbf{A}}{\mathbf{B}^{2}}\right) = \mathbf{A}\Psi \left(s\right)\,,\tag{29}\] where \[\begin{align} s:=\dfrac{\mathbf{B}^{2}}{\mathbf{A}}, \quad \textrm{and} \Psi \left( s\right) :=s\Theta \left( \dfrac{1}{s}\right)\,. \end{align}\] We have thus proven:

This Lemma means that geodesics of the Finsler metric \(L\) and those of \(\mathbf{A}\) are the same as sets of points - i.e., judging by their shapes only, we could not tell what is the correct model for our universe - the Finslerian or the Riemannian one. Yet, the proper time measurement along these worldlines is different when working with \(L\), compared to \(\mathbf{A}\).
Here are some immediate properties of Class 3 metrics, which we will need over the next sections.

Proof. The first result follows immediately from the Ricci identities \(\mathbf{\tilde{b}}_{~;a;d}^{c}-\mathbf{\tilde{b}}_{~;d;a}^{c}=R_{a~bd}^{~c}\mathbf{\tilde{b}}^{a}\) applied to the vector field \(\mathbf{\tilde{b}}=\mathbf{\tilde{b}}^{a}\partial _{a}\) and to the fact that this vector field is absolutely parallel with respect to \(\Gamma.\) The second one is also obtained as a direct consequence of the fact that the 1-form \(\mathbf{b}\) is covariantly constant and \(\Gamma\) is the Levi-Civita connection of \(a\). Finally, statement 3. follows from \[a_{5}\equiv a_{9}-a_{12}=\left( ab+c\right) a_{7}-ba_{11}=0.\]  

---

Let us quickly summarize what we found so far. Searching for asymptotically flat, spatially spherically symmetric Berwald Finsler spacetime, among all possible classes lead us to the conclusion that the Finsler Lagrangian must be of the Class 3 type ?? . A closer look at these Finsler Lagrangians revealed that they are of \((\alpha,\beta)\)-type. Even more, their canonical nonlinear connection induces an affine connection \(\Gamma\) on spacetime that is the Levi-Civita connection of the metric \(\mathbf{a}\), defined thorugh 28 , and the \(1\)-form \(\mathbf{b}\) employed to define the Finsler Lagrangian 27 is absolutely parallel with respect to this connection. The question we answer next is which kind of Finsler Lagrangians \(L\) of the type ?? are non-Ricci flat and solve the Finsler gravity vacuum equations?

5 Solving the field equation for Class 3 Berwald-Finsler metrics↩︎

Following the preparation in the previous sections, we can now solve the Finsler gravity equation for spherically symmetric Berwald Finsler Lagrangian belonging to Class 3. We find two branches, containing one, respectively two families of solutions that are displayed in ?? and 48 . After having found these three families of general solutions, we will discuss an explicit example in Section 6 below.

Let us rewrite the Finsler gravity vacuum equation 11 for Berwald structures as \[g^{ab}R_{\cdot ab}=6\dfrac{R}{L}, \label{field95eq}\tag{30}\] taking into account the expression 29 of Class 3 metrics.

The fact that our metrics are of \(\left( \alpha ,\beta \right)\)-type allows us to explicitly express the inverse metric \(g^{ab},\) as (see [54]): \[\begin{align} g^{ab} &=&\dfrac{1}{\Psi -s\Psi ^{\prime }}{\Large (}a^{ab}+\dfrac{s^{2}}{\mu \mathbf{A}}\Psi ^{\prime \prime }[\Psi ^{\prime }\left( s-\left\langle \mathbf{b,b}\right\rangle \right) -\Psi ]\dot{x}^{a}\dot{x}^{b} \\ &&+\dfrac{2\mathbf{B}s}{\mu \mathbf{A}}\Psi \Psi ^{\prime \prime }\left( \mathbf{\tilde{b}}^{a}\dot{x}^{b}+\mathbf{\tilde{b}}^{b}\dot{x}^{a}\right) +\dfrac{1}{\mu }\left[ \Psi ^{\prime }\left( \Psi -s\Psi ^{\prime }\right) -2s\Psi \Psi ^{\prime \prime }\right] \mathbf{\tilde{b}}^{a}\mathbf{\tilde{b}}^{b}{\Large )}, \end{align}\]where \(\mu\) is a factor originating from the determinant of the Finsler metric: \[\det g =\left( \Psi -s\Psi ^{\prime }\right) ^{2}\mu \det \left( a\right)\,.\]and given by \[\begin{align} \label{mu} \mu &:=&\Psi \left( \Psi -s\Psi ^{\prime }\right) +\left( \left\langle \mathbf{b,b }\right\rangle -s\right) \left( \Psi \Psi ^{\prime }+2s\Psi \Psi ^{\prime \prime }-s\Psi ^{\prime 2}\right)\,. \end{align}\tag{31}\] A direct calculation, taking into account the identity ?? , shows that the contraction on the left hand side of the Finsler gravity vacuum equation 30 is \[g^{ab}R_{\cdot ab}=\dfrac{1}{\Psi -s\Psi ^{\prime }}\left( \left( a^{ab}R_{\cdot ab}\right) +\dfrac{s^{2}}{\mu \mathbf{A}}\Psi ^{\prime \prime }\left( \Psi ^{\prime }\left( s-\left\langle \mathbf{b,b}\right\rangle \right) -\Psi \right) 2R\right) \,. \label{g95R}\tag{32}\] Remark. In relation (32 ), the term \(a^{ab}R_{\cdot ab}\) is, up to a factor of -2, nothing but the curvature scalar of the pseudo-Riemannian metric \(a;\) therefore, we will call it in the following, the Riemannian Ricci scalar.

Substituting the above identity together with \(L=A\Psi\) into equation (30 ) and taking into account that we have excluded the situation when the Finsler-Ricci scalar \(R\) vanishes, a direct computation leads to:

Expressing the Finsler gravity equation as in ?? has the advantage that the \(\dot{x}\)-dependence of the right hand side is manifestly the ratio between quadratic expressions given by \(\mathbf{A}\), 28 , and \(R\), 10 .

In the following, we will integrate equation (?? ). For a clear discussion, we will split our analysis into two cases, depending on whether the Riemannian Ricci scalar \(a^{ab}R_{\cdot ab}\) vanishes (Section 5.2) or not (Section 5.1) .

5.1 Branch 1: \(a^{ab}R_{\cdot ab}\neq0\)↩︎

In this case we proceed as follows. We note that the left hand side of equation only depends on \(s\) and on the constant \(\left\langle \mathbf{b,b}\right\rangle\), which implies the same for the right hand side. This will reveal two more constraints upon the curvature coefficients \(a_{i}\), which then allow us to rewrite ?? as a second order ODE in \(\Psi =\Psi (s)\) and integrate it.

5.1.1 Computation of \(R\) and of \(a^{ab}R_{\cdot ab}\)↩︎

Throughout this section, we will prove

To justify the above statement, we proceed in several steps:

• Step 1:Show that there exist \(\beta =\beta \left( t,r\right)\) and \(\gamma =\gamma \left( t,r\right)\) such that \[R=\beta \mathbf{A}+\gamma \mathbf{B}^{2}: \label{R95A95B}\tag{33}\] We note that, in equation (?? ), \(a^{ab}R_{\cdot ab}\) only depends on \(t\) and \(r\), whereas \(\Phi\) only depends on the ratio \(s=\dfrac{\mathbf{B}^{2}}{\mathbf{A}}.\) It thus follows that the ratio \(\dfrac{\mathbf{A}}{R} =2 \Phi (a^{ab}R_{\cdot ab})^{-1}\) must be a function of \(t,r\) and \(s\) only. Since, on the other hand, \(R\) is quadratic in \(\dot{x}^{a},\) it must be of the form 33 .

• Step 2:Show that \[\left( 2ab+c\right) a_{11}+\left( ab+c\right) a_{14}=0: \label{rel95a1195a14}\tag{34}\] Let us recall that \(b=\dfrac{k_{8}}{k_{10}}\) and \(ab+c=\dfrac{k_{9}}{k_{10}}\) cannot simultaneously vanish (see the definition of Class 3). To make a choice, let us assume in the following: \[ab+c\not=0 \label{assumption95ab43c}\tag{35}\] (the situation when \(b\not=0\) can be treated completely similarly, by switching the roles of \(a_{7}\) and \(a_{11}\), see relation (?? )).

  Taking into account that \(a_{1}=...=a_{5}=0\) and using (55 ), we first write \(R\) as: \[\begin{align} R &=&-2ba_{7}\dot{t}^{2}-4ba_{11}\dot{t}\dot{r}-2\left( ab+c\right) a_{11} \dot{r}^{2}+\left( aa_{7}+a_{11}-a_{14}\right) w^{2} \\ &&\overset{(\ref{a7a11})}{=}-2a_{11}\dfrac{\left[ b\dot{t}+\left( ab+c\right) \dot{r}\right] ^{2}}{c+ab}+\left( \dfrac{\left( 2ab+c\right) a_{11}}{ab+c}-a_{14}\right) w^{2}. \end{align}\] Further, let us note that (28 ) and (27 ) entail: \(u=e^{-\frac{ \mathcal{G}}{2}}\mathbf{B},~\;v=e^{-2\mathcal{K}}\left( \mathbf{A}-\mathcal{M }\mathbf{B}^{2}\right) .\) This allows us, using the definitions 18 of \(u\) and \(v,\) to express \(\dot{t}\) and \(w\) in terms of \(\mathbf{A}\), \(\mathbf{B}\) and \(\dot{r}\): \[\begin{align} \dot{t} &=&u+a\dot{r}=e^{-\frac{\mathcal{G}}{2}}\mathbf{B}+a\dot{r} \tag{36} \\ w^{2} &=&\left( c+2ab\right) \dot{r}^{2}+2bu\dot{r}-v=\left( c+2ab\right) \dot{r}^{2}+2be^{-\frac{\mathcal{G}}{2}}\mathbf{B}\dot{r}-e^{-2\mathcal{K} }\left( \mathbf{A}-\mathcal{M}\mathbf{B}^{2}\right) . \tag{37} \end{align}\] Direct substitution of these values into the expression of \(R\) then recasts it as: \[\begin{align} R &=&-\dfrac{\left( 2ab+c\right) a_{11}+\left( ab+c\right) a_{14}}{ab+c} \left[ \left( \left( 2ab+c\right) \dot{r}+2be^{-\frac{\mathcal{G}}{2}} \mathbf{B}\right) \dot{r}\right] \tag{38} \\ &&-e^{-2\mathcal{K}}\dfrac{\left( 2ab+c\right) a_{11}-\left( ab+c\right) a_{14}}{ab+c}\mathbf{A} \tag{39} \\ &&+\dfrac{1}{ab+c}\left[ \left( \mathcal{M}e^{-2\mathcal{K}}\left( 2ab+c\right) -2e^{-\mathcal{G}}b^{2}\right) a_{11}-\mathcal{M}e^{-2\mathcal{K }}\left( ab+c\right) a_{14}\right] \mathbf{B}^{2}. \tag{40} \end{align}\]

  Assume, in the following, that \(\left( 2ab+c\right) a_{11}+\left( ab+c\right) a_{14}\not=0.\)

  The last two terms in \(R\) are already a linear combination of \(\mathbf{A}\) and \(\mathbf{B}^{2}\) only (with \(\left( t,r\right)\)-dependent coefficients), whereas the square bracket in the first one is, taking into account 36 , \[\left( \left( 2ab+c\right) \dot{r}+2be^{-\frac{\mathcal{G}}{2}}\mathbf{B} \right) \dot{r}=\left( 2b\dot{t}+c\dot{r}\right) \dot{r}.\] In order to obtain \(R\) as a linear combination of \(\mathbf{A}\) and \(\mathbf{B }^{2}\) as in (33 ), we must then have \[\left( 2b\dot{t}+c\dot{r}\right) \dot{r}=\sigma \mathbf{A}+\tau \mathbf{B} ^{2}\] for some \(\sigma =\sigma \left( t,r\right)\) and \(\tau =\tau \left( t,r\right) .\) In the last relation, the left hand side does not involve \(w,\) whereas, using the expression 56 of \(\mathbf{A,}\) together with \(\mathbf{B}=e^{\frac{\mathcal{G}}{2}}\left( \dot{t}-a\dot{r}\right) ,\) it turns out that the right hand side contains the term \(-\sigma e^{2 \mathcal{K}}w^{2}.\) This implies \(\sigma =0,\) that is, \[\left( 2b\dot{t}+c\dot{r}\right) \dot{r}=\tau \mathbf{B}^{2}.\] But then, \(\mathbf{B}=e^{\frac{\mathcal{G}}{2}}\left( \dot{t}-a\dot{r} \right)\) must be a multiple (with \(\left( t,r\right)\)-dependent coefficient) of \(\dot{r}\) – and hence \(\dot{t}=\rho \dot{r}\) for some \(\rho =\rho \left( t,r\right),\) which is not possible (as \(\dot{t}\) and \(\dot{r}\) are independent coordinates). We thus conclude that our assumption was false, therefore (34 ) must hold true.

• Step 3: Prove that \(\left\langle \mathbf{b,b}\right\rangle \not=0\) and \[R=2a_{14}e^{-2\mathcal{K}}\left( \mathbf{A}\mathcal{-}\dfrac{\mathbf{B}^{2}}{ \left\langle \mathbf{b,b}\right\rangle }\right) . \label{R95A95B95refined}\tag{41}\]

  First, using the expressions of \(\left( a^{ab}\right)\) from 58 and \(\mathbf{b}_{a}\equiv \left( e^{\frac{\mathcal{G}}{2}},-ae^{\frac{\mathcal{G}}{2}},0,0\right) ,\) we find: \[\label{eq:bb} \left\langle \mathbf{b},\mathbf{b}\right\rangle =a^{ab}\mathbf{b}_{a}\mathbf{ b}_{b}=\dfrac{e^{\mathcal{G}}\left( 2ab+c\right) }{\mathcal{M}e^{\mathcal{G} }\left( 2ab+c\right) -b^{2}e^{2\mathcal{K}}}.\tag{42}\] Assuming that \(\left\langle \mathbf{b},\mathbf{b}\right\rangle =0,\) we would obtain \(2ab+c=0;\) but, the latter is impossible, as it implies on the one hand, by (34 ) and (35 ), that \(a_{14}=0\;\), and, on the other hand, \(a_{14}=1+k_{10}^{2}\left( 2ab+c\right) =1,\) which is a contradiction. Therefore, we must have \(\left\langle \mathbf{b}, \mathbf{b}\right\rangle \not=0,\) as stated.

  As a byproduct, we also find that \(2ab+c\not=0,\) which allows us to write

  \[a_{11}=-\dfrac{ab+c}{2ab+c}a_{14}.\] Substituting the latter relation into (38 )-(40 ) we get by direct calculation \[R=2a_{14}e^{-2\mathcal{K}}\mathbf{A}-2a_{14}\dfrac{-b^{2}e^{-\mathcal{G}}+ \mathcal{M}e^{-2\mathcal{K}}\left( 2ab+c\right) }{2ab+c}\mathbf{B}^{2}.\] which, using the expression 42 of \(\left\langle \mathbf{b,b} \right\rangle ,\) is precisely (41 ).

• Step 4: Show that \(a_{14}e^{-2\mathcal{K}}\) is a constant:

  Using 54 and taking into account relations (?? ) and (34 ), we find: \[\left\{ \begin{array}{l} \partial _{r}a_{14}=2k_{10}\left( ab+c\right) a_{14}=2k_{9}a_{14} \\ \partial _{t}a_{14}=2k_{10}ba_{14}=2k_{8}a_{14} \end{array} \right. .\] The statement follows then by taking \(t\)- and \(r\)- derivatives of \(a_{14}e^{-2\mathcal{K}}\) and using \(k_{8}=\mathcal{K}_{,t}\) and \(k_{9}=\mathcal{K}_{,r}\).

• Step 5: Use the above results Step 1-4, to prove (?? ):

  Denoting \(\alpha :=a_{14}e^{-2\mathcal{K}},\) relation (41 ) becomes precisely the first equality (?? ); moreover, since \(R\not=0\) by hypothesis, we must have \(\alpha \not=0.\)

  Then, differentiation with respect to \(\dot{x}^{a}\) and \(\dot{x}^{b}\) of this equation gives \(R_{\cdot ab}=4\alpha \left( a_{ab}-\dfrac{\mathbf{b}_{a} \mathbf{b}_{b}}{\left\langle \mathbf{b,b}\right\rangle }\right)\) and, accordingly (recalling that \(\dim M=4\)), \(a^{ab}R_{\cdot ab}=12\alpha ,\) as stated.

Remark:Lemma 4 tells us, in particular, that any vacuum solution belonging to this class has its underlying Riemannian metric \(\mathbf{A}\) with constant nonzero scalar curvature - and absolutely parallel, non-null 1-form \(\mathbf{B}.\)

5.1.2 Integration of the field equation↩︎

Using relations (?? ), the field equation (?? ) can be rewritten as: \(\Phi =\dfrac{3\mathbf{A}\left\langle \mathbf{b},\mathbf{b}\right\rangle }{\mathbf{A}\left\langle \mathbf{b},\mathbf{b}\right\rangle -\mathbf{B}^{2}},\) which is simply:\[\Phi =\dfrac{3\left\langle \mathbf{b},\mathbf{b}\right\rangle }{\left\langle \mathbf{b},\mathbf{b}\right\rangle -s}. \label{eq:Phi95Case95195simplified}\tag{43}\] Substituting \(\Phi\) from (?? ), a direct computation shows that this is equivalent (after discarding a common factor of \(s\)), to:\[\begin{align} &&s\left[ \Psi +\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle -s\right) \Psi ^{\prime }\right] \times \label{eq:Case95195product} \\ &&\left( 3s^{2}(\Psi ^{\prime })^{2}+3\Psi ^{2}+3\left\langle \mathbf{b},\mathbf{b}\right\rangle \Psi \Psi ^{\prime }-6s\Psi \Psi ^{\prime }-3\left\langle \mathbf{b},\mathbf{b}\right\rangle s(\Psi ^{\prime })^{2}-5s^{2}\Psi \Psi ^{\prime \prime }+5\left\langle \mathbf{b},\mathbf{b}\right\rangle s\Psi \Psi ^{\prime \prime }\right) =0. \notag \end{align}\tag{44}\]

Let us analyze each factor.

• The situation \(\Psi +\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle -s\right) \Psi ^{\prime }=0\) leads to \(\Psi =c_{1}\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle -s\right) ,\) \(c_{1}=const.,\) hence to a quadratic function:\[L=A\Psi =c_{1}\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle A-B^{2}\right) .\]Yet, this is not a valid solution. A quick check shows that its associated Finslerian metric, with components:\[g_{ab}=c_{1}\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle a_{ab}-b_{a}b_{b}\right) ,\]has an identically vanishing determinant.

• It thus remains to integrate the remaining equation:\[\mathbf{\;}3s^{2}(\Psi ^{\prime })^{2}+3\Psi ^{2}+3\left\langle \mathbf{b},\mathbf{b}\right\rangle \Psi \Psi ^{\prime }-6s\Psi \Psi ^{\prime }-3\left\langle \mathbf{b},\mathbf{b}\right\rangle s(\Psi ^{\prime })^{2}-5s^{2}\Psi \Psi ^{\prime \prime }+5\left\langle \mathbf{b},\mathbf{b}\right\rangle s\Psi \Psi ^{\prime \prime }=0 \label{eq:Case95195valid}\tag{45}\] This can be recast in the form: \(\left( \Psi ^{-3}\left( \Psi -s\Psi ^{\prime }\right) ^{5}\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle -s\right) ^{3}\right) ^{\prime }=0,\) which leads to a first integral\[\Psi ^{-3}\left( \Psi -s\Psi ^{\prime }\right) ^{5}\left( \left\langle \mathbf{b},\mathbf{b}\right\rangle -s\right) ^{3}=\tilde{c}_{1}, \label{first95int95Case951}\tag{46}\] where \(\tilde{c}_{1}\in \mathbb{R}\) is a constant. Further, the change of the dependent variable \[f:=\dfrac{\Psi }{s},\]casts equation (46 ) into the form\[f^{\prime }=\tilde{c}_{1}^{\tfrac{1}{5}}s^{-\tfrac{7}{5}}\left( s-\left\langle \mathbf{b},\mathbf{b}\right\rangle \right) ^{-\tfrac{3}{5}}f^{\tfrac{3}{5}}.\]This has the general solution\[f=c_{1}\left[ \left( s-\left\langle \mathbf{b},\mathbf{b}\right\rangle \right) ^{\tfrac{2}{5}}s^{-\tfrac{2}{5}}+c_{2}\right] ^{\tfrac{5}{2}},\]where \(c_{1}=\left( \dfrac{25}{4\left\langle \mathbf{b},\mathbf{b}\right\rangle }\tilde{c}_{1}^{\tfrac{1}{5}}\right) ^{\tfrac{5}{2}}>0\) and \(c_{2}\in \mathbb{R}\) are arbitrary constants.

Reverting to the original variable \(\Psi =sf\) and introducing the factor \(s>0\) into the square bracket, we find:

Remark: In order to ensure the existence of nondegenerate light cones for \(L,\) we must have:\[c_{2}<0.\]

With this we found the first two-parameter family of spatially spherically symmetric asymptotically flat Berwald type solutions of the Finsler gravity vacuum equation, that are not Finsler Ricci flat.

5.2 Branch 2:\(a^{ab}R_{\cdot ab}=0\)↩︎

This case will give rise to two further 2-parameter families of spatially spherically symmetric, asymptotically flat Berwald-type solutions of the Finsler gravity vacuum equations, that are not Ricci flat.

Technically, this case is much simpler, as it allows us to directly rewrite equation (?? ) as\[\Phi =0. \label{eq95Case952}\tag{47}\] and integrate it. Indeed, using (?? ), this is:\[3\left( \Psi -s\Psi ^{\prime }\right) -\dfrac{s^{2}}{\mu }\Psi \Psi ^{\prime \prime }\left( \Psi ^{\prime }\left( s-\left\langle \mathbf{b,b}\right\rangle \right) -\Psi \right) =0,\]where we recall that \(\left\langle \mathbf{b,b}\right\rangle =const.\) and\[\mu =\Psi \left( \Psi -s\Psi ^{\prime }\right) +\left( \left\langle \mathbf{b,b}\right\rangle -s\right) \left( \Psi \Psi ^{\prime }+2s\Psi \Psi ^{\prime \prime }-s\Psi ^{\prime 2}\right) .\]

Integration of the above via Maple leads to a general solution \(L _{\pm } = \mathbf{A} \Psi _{\pm }(s),\, s = \frac{\mathbf{B}^2}{\mathbf{A}},\) with two branches corresponding to the plus or minus sign in front of the square root below:\[\Psi _{\pm }=K_{2}\left( s-\left\langle \mathbf{b},\mathbf{b}\right\rangle \right) \exp \left( -6\left\langle \mathbf{b},\mathbf{b}\right\rangle ^{2}\int \left( \dfrac{1}{\left( s-\left\langle \mathbf{b},\mathbf{b}\right\rangle \right) \left( 5\left\langle \mathbf{b},\mathbf{b}\right\rangle s\pm \sqrt{\left\langle \mathbf{b},\mathbf{b}\right\rangle ^{3}s+K_{1}s\left( s-\left\langle \mathbf{b},\mathbf{b}\right\rangle \right) }\right) }\right) ds\right) , \label{sol:Psi95Case952}\tag{48}\] where \(K_{1},K_{2}\in \mathbb{R}\) are integration constants. The integral above can be explicitly calculated for any value of \(\left\langle \mathbf{b},\mathbf{b}\right\rangle\).
Remark. The choice \(\left\langle \mathbf{b},\mathbf{b}\right\rangle =0\) leads to \(\Psi _{\pm }=K_{2}s\), i.e., to \(L=K_{2}B^{2}\) which is degenerate. Hence, we can assume with no loss of generality that\[\left\langle \mathbf{b},\mathbf{b}\right\rangle \not=0.\]

6 Concrete example↩︎

In the previous section, we have found three 2-parameter families of spatially spherically symmetric, asymptotically flat Berwald Finsler Lagragians which solve the Finsler gravity vacuum equations and are not Ricci flat. We displayed their general form, and in particular their explicit \(\dot{x}\) dependence. Yet, we did not explicitly solve for their dependence on the base manifold coordinates \((x^a)=(t,r,\theta,\phi)\); the latter can be determined by integrating equations ?? ?? with 28 and 27 .
To demonstrate that there exist interesting solutions, we construct a concrete example of Berwald-Finsler spacetime function satisfying the above properties and belonging to Branch 1. We find this example by specifying a torsion-free, \(SO(3)\)-symmetric connection \(\Gamma\) that is the Levi-Civita connection of an \(SO(3)\)-symmetric pseudo-Riemannian metric and at the same time satisfies the Finsler metrizabillity conditions 15 . Its non-vanishing connection coefficients are given by: \[\begin{align} &&k_{5}=\Gamma _{~rr}^{r}=-\dfrac{K-8r^{2}}{r\left( K-4r^{2}\right) };~~k_{9}=\Gamma _{~r\theta }^{\theta }=\Gamma _{~r\varphi }^{\varphi }=-\dfrac{1}{r}\;\;\;\;k_{10}=\Gamma _{~\theta \theta }^{r}=\dfrac{\Gamma _{~\varphi \varphi }^{r}}{\sin ^{2}\theta }=-\dfrac{1}{4}\dfrac{K-4r^{2}}{r}, \\ &&\Gamma _{~\varphi \varphi }^{\theta }=-\sin \theta \cos \theta ,~\;\;\; \Gamma _{~\theta \varphi }^{\varphi }=\cot \theta , \end{align}\]where \(K>0\) is an arbitrary constant.

A direct computation shows that its only nonzero curvature components \(a_{i}\) are \[a_{11}=-\dfrac{1}{4}\dfrac{K}{r^{2}},~\;a_{13}=-\dfrac{K}{r^{2}\left( K-4r^{2}\right) },~\;\;a_{14}=\dfrac{1}{4}\dfrac{K}{r^{2}}\,,\] and that this connection is obviously asymptotically flat, as the curvature components all vanish when \(r\to\infty\). Moreover, \[a=b=0,~\;c=\dfrac{4}{K-4r^{2}}.\] Using these values, we calculate via (?? )-(?? ), the "potentials", \(\mathcal{G},\) \(\mathcal{K}\) and \(\mathcal{M}:\) \[\mathcal{G}=2\ln \alpha _{1},~\;\mathcal{K}=\alpha _{2}-\ln r,~\mathcal{M}=\alpha _{3}.\]where \(\alpha _{1}>0,\alpha _{3}\not=0\) and \(\alpha _{2}\in \mathbb{R}\) are arbitrary constants.

This connection obviously satisfies all the Finsler metrizability conditions 15 , the Class 3 conditions \[A=B=C=D=E=F=0\]and the specific Branch 1 conditions (34 ) and \(a_{14}e^{-2\mathcal{K}}=\dfrac{1}{4}Ke^{-2\alpha _{2}}=const.\)

Using (18 ) and (28 ), we find the pseudo-Riemannian part \(\mathbf{A}\) of \(L\) as\[\begin{align} \mathbf{A}&=\left( \alpha _{3}\alpha _{1}^{2}\right) \dot{t}^{2}-\dfrac{4e^{2\alpha _{2}}}{\left( 4r^{2}-K\right) r^{2}}\dot{r}^{2}-\dfrac{e^{2\alpha _{2}}}{r^{2}}w^{2}\\ \Leftrightarrow \left( a_{ab}\right) &=diag\left( \alpha _{3}\alpha _{1}^{2},-\dfrac{4e^{2\alpha _{2}}}{\left( 4r^{2}-K\right) r^{2}},-\dfrac{e^{2\alpha _{2}}}{r^{2}},-\dfrac{e^{2\alpha _{2}}}{r^{2}}\sin ^{2}\theta \right) . \end{align}\] We note that \(\left( a_{ab}\right)\) is in diagonal form, meaning that its inverse is simply \[\left( a^{ab}\right) =diag\left( \dfrac{1}{\alpha _{3}\alpha _{1}^{2}},-\dfrac{\left( 4r^{2}-K\right) r^{2}e^{-2\alpha _{2}}}{4},-r^{2}e^{-2\alpha _{2}},-r^{2}e^{-2\alpha _{2}}\sin ^{2}\theta \right) .\]

Similarly, we get \[\left( \mathbf{b}_{a}\right) \mathbf{:}\mathbf{=}\left( e^{\frac{\mathcal{G}}{2}},-ae^{\frac{\mathcal{G}}{2}},0,0\right) =\left( \alpha _{1},0,0,0\right) \Leftrightarrow ~\;\mathbf{B}=\alpha _{1}\dot{t}\]and accordingly\[\left\langle \mathbf{b,b}\right\rangle =a^{ab}b_{a}b_{b}=\dfrac{1}{\alpha _{3}}.\]

A direct check shows that \(\Gamma\) is indeed, the Levi-Civita connection of \(\mathbf{A}\) and its curvature scalar is a nonzero constant:\[a^{ab}R_{ab}=-\dfrac{1}{2}a^{ab}R_{\cdot ab}=-\dfrac{3}{2}K\not=0.\]

Finally, we get\[\mathbf{B}^{2}-\left\langle \mathbf{b,b}\right\rangle \mathbf{A}=\dfrac{e^{2\alpha _{2}}}{\alpha _{3}r^{2}\left( K-4r^{2}\right) }\left[ 4\dot{r}^{2}+\left( 4r^{2}-K\right) (\dot{\theta}^{2}+\dot{\varphi}^{2}\sin ^{2}\theta )\right] ; \label{eq:spatial95part95example}\tag{49}\] thus, setting for convenience, \(c_{2}:=-1\) in (?? ), we obtain the pseudo-Finsler function\[\label{eq:ex} L=\mathbf{A}\Psi =c_{1}\epsilon \left\{ -\left( \alpha _{1}\right) ^{\tfrac{4}{5}}\dot{t}^{\tfrac{4}{5}}+p_{2}\left( r\right) \left[ 4\dot{r}^{2}+\left( 4r^{2}-K\right) (\dot{\theta}^{2}+\dot{\varphi}^{2}\sin ^{2}\theta )\right] ^{\tfrac{2}{5}}\right\} ^{\tfrac{5}{2}}\tag{50}\] where\[p_{2}\left( r\right) :=\left( \dfrac{e^{2\alpha _{2}}}{\alpha _{3}r^{2}\left( 4r^{2}-K\right) }\right) ^{\tfrac{2}{5}}>0.\]We note that, for all points \(x\in M\) obeying \[r^{2}>\dfrac{K}{4}.\]the spatial part (49 ) of \(L\) is positive definite and \(L\) has well defined light cones \(L=0\) at all these points. These are actually, quadratic cones: \[L=0\Leftrightarrow ~\;\;\;\alpha _{1}\dot{t}^{2}=\left( p_{2}\right) ^{\tfrac{5}{2}}\left[ 4\dot{r}^{2}+\left( 4r^{2}-K\right) (\dot{\theta}^{2}+\dot{\varphi}^{2}\sin ^{2}\theta )\right] .\] which turn out to coincide with those of \(A\).

7 Conclusion and outlook↩︎

We have found all families of spatially spherically symmetric (\(SO\left(3\right)\)-invariant), asymptotically flat Berwald Finsler structures that are non Ricci flat and solve the Finsler gravity equation 7 in vacuum. There are three types of solutions, called \(L_1\) and \(L_\pm\), as displayed in ?? and 48 . Their derivation was discussed in detail in Section 5.

A remarkable property is that all of these solutions belong to the class of \((\alpha,\beta)\)-Finsler structures, that are built from a pseudo-Riemannian metric \(\mathbf{a}\), 28 , and a \(1\)-form \(\mathbf{b}\), 27 , which is absolutely parallel with respect to the Levi-Civita connection of \(\mathbf{a}\). Moreover, for \(L_1\), the scalar curvature of \(\mathbf{a}\) is a non-zero constant, while for \(L_\pm\), it vanishes.

After having found these general forms, which in particular fix the \(\dot{x}\) dependence of the Finsler Lagrangians, we also derived an explicit example whose complete coordinate dependence has been determined in 50 . This example posseses a physical lightcone structure and represent a viable Finsler spacetime.

With this, we have demonstrated for the first time that, in Finsler gravity, even in the closest case to the Riemannian one – represented by the so-called Berwald-type metrics (whose defining property is that their geodesics are autoparallel curves of an affine connection on the spacetime manifold) – there exist vacuum spacetimes that are not Ricci flat. This is in sharp contrast with general relativity, where any vacuum solution of the Einstein field equations must be Ricci flat.

So far, the collection of explicit solutions to the action-based Finsler gravity equation 7 is not very large. With our findings here, we present new classes of explicit solutions, whose application to describe the gravitational field around physical compact objects will be the topic of a forthcoming paper.

8 Connection coefficients and curvature components↩︎

This section provides the definitions of the coefficients \(k_i\) and \(a_i\) used in Section 2.3.

The most general 4-dimensional, \(SO(3)\)-invariant, torsion-free affine connection \(\Gamma\) on \(M\) has, in spherical coordinates, the coefficients described by 12 smooth functions of \(t\) and \(r\), [45]:\[\begin{align} \Gamma _{tt}^{t}& =k_{1}(t,r), & \Gamma _{tr}^{t}& =k_{2}(t,r), \notag \\ \Gamma _{rr}^{t}& =k_{3}(t,r), & \Gamma _{tt}^{r}& =k_{4}(t,r), \notag \\ \Gamma _{rr}^{r}& =k_{5}(t,r), & \Gamma _{tr}^{r}& =k_{6}(t,r), \notag \\ \Gamma _{\theta \theta }^{t}& =\tfrac{\Gamma _{\phi \phi }^{t}}{\sin ^{2}\theta }=k_{7}(t,r), & \Gamma _{\phi t}^{\phi }& =\Gamma _{\theta t}^{\theta }=k_{8}(t,r), \notag \\ \Gamma _{\phi r}^{\phi }& =\Gamma _{\theta r}^{\theta }=k_{9}(t,r), & \Gamma _{\theta \theta }^{r}& =\tfrac{\Gamma _{\phi \phi }^{r}}{\sin ^{2}\theta }=k_{10}(t,r), \notag \\ \sin \theta \Gamma _{t\theta }^{\phi }& =-\tfrac{\Gamma _{\phi t}^{\theta }}{\sin \theta }=k_{11}(t,r), & \Gamma _{\phi \phi }^{\theta }& =-\sin \theta \cos \theta \notag \\ \sin \theta \Gamma _{r\theta }^{\phi }& =-\tfrac{\Gamma _{r\phi }^{\theta }}{\sin \theta }=k_{12}(t,r), & \Gamma _{\theta \phi }^{\phi }& =\Gamma _{\phi \theta }^{\phi }=\cot {\theta }\,. \label{eq:appcon} \end{align}\tag{51}\] If the connection is Finsler metrizable, then there necessarily holds \[k_{11}=k_{12}=0,\] therefore, the most general Finsler metrizable, \(SO(3)\)-invariant connection is locally characterized, in spherical coordinates, by 10 smooth functions \(k_i,\, i=1..10.\)

The canonical nonlinear connection \(N\) has the coefficients:\[N_{~b}^{a}=\Gamma _{~bc}^{a}x\,^{c}\]and its curvature tensor \(\mathcal{R}\) has the local components \[R^{a}{}_{bc}=\delta _{c}N^{a}{}_{b}-\delta _{b}N^{a}{}_{c}=\partial _{c}N^{a}{}_{b}-N^{d}{}_{c}\dot{\partial}_{d}N^{a}{}_{b}-\partial _{b}N^{a}{}_{c}+N^{d}{}_{b}\dot{\partial}_{d}N^{a}{}_{c};\]in detail:\[\begin{array}{llll} R_{~tr}^{t}=a_{1}\dot{t}+a_{2}\dot{r} & R_{~tr}^{r}=a_{3}\dot{t}+a_{4}\dot{r} & R_{~tr}^{\theta }=a_{5}\dot{\theta} & R_{~tr}^{\phi }=a_{5}\dot{\phi} \\ R_{~t\theta }^{t}=a_{6}\dot{\theta} & R_{~t\theta }^{r}=a_{7}\dot{\theta} & R_{t\theta }^{\theta }=a_{8}\dot{t}+a_{9}\dot{r} & R_{t\theta }^{\phi }=0 \\ R_{~t\phi }^{t}=a_{6}\dot{\phi}\sin ^{2}\theta & R_{~t\phi }^{r}=a_{7}\dot{\phi}\sin ^{2}\theta & R_{~t\phi }^{\theta }=0 & R_{t\phi }^{\phi }=a_{8}\dot{t}+a_{9}\dot{r} \\ R_{~r\theta }^{t}=a_{10}\dot{\theta} & R_{~r\theta }^{r}=a_{11}\dot{\theta} & R_{~r\theta }^{\theta }=a_{12}\dot{t}+a_{13}\dot{r} & R_{~r\theta }^{\phi }=0 \\ R_{~r\phi }^{t}=a_{10}\dot{\phi}\sin ^{2}\theta & R_{~r\phi }^{r}=a_{11}\dot{\phi}\sin ^{2}\theta & R_{~r\phi }^{\theta }=0 & R_{~r\phi }^{\phi }=a_{12}\dot{t}+a_{13}\dot{r} \\ R_{~\theta \phi }^{t}=0 & R_{~\theta \phi }^{r}=0 & R_{~\theta \phi }^{\theta }=-a_{14}\dot{\phi}\sin ^{2}\theta & R_{~\theta \phi }^{\phi }=a_{14}\dot{\theta}\,.\end{array}, \label{eq:appcurv}\tag{52}\] where the quantities \(a_{i}\) are functions of \(t\) and \(r\), expressed as: \[\begin{align} a_{1}& =k_{1,r}-k_{2,t}+k_{3}k_{4}-k_{2}k_{6}\,, \\ a_{2}& =k_{2,r}-k_{3,t}+k_{2}^{2}+k_{3}k_{6}-k_{1}k_{3}-k_{2}k_{5}\,, \\ a_{3}& =k_{4,r}-k_{6,t}+k_{1}k_{6}+k_{4}k_{5}-k_{2}k_{4}-k_{6}^{2}\,, \\ a_{4}& =k_{6,r}-k_{5,t}+k_{2}k_{6}-k_{3}k_{4}\,, \\ a_{5}& =k_{8,r}-k_{9,t}\,, \\ a_{6}& =-k_{7,t}+k_{7}k_{8}-k_{1}k_{7}-k_{2}k_{10}\,, \\ a_{7}& =-k_{10,t}+k_{8}k_{10}-k_{4}k_{7}-k_{6}k_{10}\,, \\ a_{8}& =-k_{8,t}+k_{1}k_{8}+k_{4}k_{9}-k_{8}^{2}\,, \\ a_{9}& =-k_{9,t}+k_{2}k_{8}+k_{6}k_{9}-k_{8}k_{9}\,, \\ a_{10}& =-k_{7,r}+k_{7}k_{9}-k_{2}k_{7}-k_{3}k_{10}\,, \\ a_{11}& =-k_{10,r}+k_{9}k_{10}-k_{6}k_{7}-k_{5}k_{10}\,, \\ a_{12}& =-k_{8,r}+k_{2}k_{8}+k_{6}k_{9}-k_{8}k_{9}\,, \\ a_{13}& =-k_{9,r}+k_{3}k_{8}+k_{5}k_{9}-k_{9}^{2}\,, \\ a_{14}& =1+k_{7}k_{8}+k_{9}k_{10}\,. \end{align} \label{eq:a95i}\tag{53}\] In the above, the subscripts \(_{,t}\) and \(_{,r}\) means partial differentiation with respect to \(t\) and \(r\) respectively. The curvature components \(R_{b~cd}^{~a}\) of the affine connection are obtained from \(R_{~cd}^{a}\) by \(\dot{x}\)-differentiation: \[R_{b~cd}^{~a}=\dot{\partial}_{b}R_{~cd}^{a};\]e.g., \(a_{1}=R_{t~tr}^{~t}\) etc.

Assume \(k_{10}\not=0.\) Then, \(a_{14}\) satisfies ([47], [48] for more details): \[a_{14}=1+k_{10}^{2}\left( 2ab+c\right) ,~\partial _{t}a_{14}=-2k_{10}a_{7}\left( c+2ab\right) ,\;\;\partial _{r}a_{14}=-2k_{10}a_{11}\left( c+2ab\right) . \label{eqs:a14}\tag{54}\] where \(a=\dfrac{k_{7}}{k_{10}},\) \(b=\dfrac{k_{8}}{k_{10}},\) \(c=\dfrac{k_{9}k_{10}-k_{7}k_{8}}{k_{10}^{2}}.\)

Moreover, if the connection \(\Gamma\) is metrizable by a non-Riemannian Finsler function \(L\), then it must obey:\[a_{6}=aa_{7},~~a_{8}=ba_{7},~\;a_{9}=\left( ab+c\right) a_{7},~\;\; a_{10}=aa_{11},~~a_{12}=ba_{11},~\;a_{13}=\left( ab+c\right) a_{11}. \label{eq:metrizabillity95conds95a7}\tag{55}\]

9 Pseudo-Riemannian metric components↩︎

In this Appendix we display the quantities derived from the pseudo-Riemannian part \(\mathbf{a}\) (introduced in 28 ) of a general Class 3 metric, that are needed for the calculations in Section 5.1.1.

Using \(\mathbf{A}=ve^{2\mathcal{K}}+\left( e^{\mathcal{G}}\mathcal{M}\right) u^{2}\) and \(u=\dot{t}-a\dot{r},\) \(v=c\dot{r}^{2}+2b\dot{t}\dot{r}-w^{2},\) we find the explicit expression of \(\mathbf{A}\) in terms of \(\dot{t},\dot{r}\) and \(w^{2}=\dot{\theta}^{2}+\dot{\varphi}^{2}\sin ^{2}\theta :\)\[\mathbf{A}=\left( \mathcal{M}e^{\mathcal{G}}\right) \dot{t}^{2}+2\left( be^{2\mathcal{K}}-\mathcal{M}e^{\mathcal{G}}a\right) \dot{t}\dot{r}+\left( \mathcal{M}e^{\mathcal{G}}a^{2}+ce^{2\mathcal{K}}\right) \dot{r}^{2}-e^{2\mathcal{K}}w^{2} \label{eq:A95trw}\tag{56}\] Then, the components of the pseudo-Riemannian metric \(a=\dfrac{1}{2}Hess_{\dot{x}}\mathbf{A}\) are\[\left( a_{ab}\right) =\left( \begin{array}{cccc} \mathcal{M}e^{\mathcal{G}} & -\mathcal{M}e^{\mathcal{G}}a+e^{2\mathcal{K}}b & 0 & 0 \\ -\mathcal{M}e^{\mathcal{G}}a+e^{2\mathcal{K}}b & \mathcal{M}e^{\mathcal{G}}a^{2}+e^{2\mathcal{K}}c & 0 & 0 \\ 0 & 0 & -e^{2\mathcal{K}} & 0 \\ 0 & 0 & 0 & -e^{2\mathcal{K}}\sin ^{2}\theta\end{array}\right) .\]

• Determinant: \[\det \left( a_{ab}\right) =e^{6\mathcal{K}}\Delta \sin ^{2}\theta ,~\;\;\; \Delta :=\mathcal{M}e^{\mathcal{G}}\left( 2ab+c\right) -b^{2}e^{2\mathcal{K}}. \label{def95Delta}\tag{57}\]

• Inverse metric components:\[\label{eq:ainv} \left( a^{ab}\right) =\left( \begin{array}{cccc} \dfrac{1}{\Delta }\left( \mathcal{M}e^{\mathcal{G}-2\mathcal{K}}a^{2}+c\right) & \dfrac{1}{\Delta }\left( \mathcal{M}e^{\mathcal{G}-2\mathcal{K}}a-b\right) & 0 & 0 \\ \dfrac{1}{\Delta }\left( \mathcal{M}e^{\mathcal{G}-2\mathcal{K}}a-b\right) & \dfrac{1}{\Delta }\mathcal{M}e^{\mathcal{G}-2\mathcal{K}} & 0 & 0 \\ 0 & 0 & -e^{-2\mathcal{K}} & 0 \\ 0 & 0 & 0 & -\dfrac{e^{-2\mathcal{K}}}{\sin ^{2}\theta }\end{array}\right) .\tag{58}\]

10 Cone structure of power law metrics↩︎

This section explicitly proves the statement in Section 3, that asymptotically flat Class 1 (power law) metrics have degenerate light cones \(L=0\) at each point.
To this aim, we start from the definition of Class 1 metrics \[L=\theta \left( t,r\right) u^{2-2\lambda }\left( v+\rho u^{2}\right) ^{\lambda }, \label{L95power}\tag{59}\] where:\[u=\dot{t}-a\dot{r},~\;\;v=2b\dot{t}\dot{r}+c\dot{r}^{2}-w^{2},~\;\;\rho =\dfrac{E}{D}\](with \(D\not=0\)) and\[\lambda =\dfrac{F}{D}=\dfrac{aa_{3}-a_{1}}{aa_{3}-a_{1}+a_{5}}=const. \label{lambda}\tag{60}\] is a constant. Since the case \(\lambda =1\) corresponds to pseudo-Riemannian metrics, the only interesting case to investigate is \(\lambda \not=1,\) that is:\[a_{5}\not=0.\]

More precisely, we investigate the realization of the future-pointing cones axiom:

Actually, as we have seen in [36], \(\mathcal{T}_{x}\) must be convex.

In order to check the above axiom, let us first fix an arbitrary \(x\in M\) and solve the system:

\[A=B=C=0\]in the unknowns \(a_{1},...,a_{5}.\) Taking into account that \(b\) and \(c\) cannot simultaneously vanish (as this would lead to \(\det (g_{ab})=0,\) see [45]), this leads to three possibilities:\[b\not=0,~\;a_{1}=-\dfrac{aba_{3}-ba_{5}+ca_{3}}{b},a_{2}=\dfrac{a\left( ab+c\right) a_{3}}{b},~\;a_{4}=a_{5}-aa_{3},~\;a_{3},a_{5}\in \mathbb{R},\] or\[b=0\;\Rightarrow ~a_{2}=a\left( a_{5}-a_{1}\right) ,a_{3}=0,a_{4}=a_{5},a_{1},a_{5}\in \mathbb{R}, \]or\[2ab+c=0,~\;a_{1}=-a_{4}+2a_{5},~a_{2}=a^{2}a_{3}+2aa_{4}-2aa_{5},~\;\; a_{3},a_{4},a_{5}\in \mathbb{R}. \label{iii}\tag{61}\]

Case (iii) is excluded immediately, as it leads to \(a_{14}=1+k_{10}^{2}\left( 2ab+c\right) =1,\) which is in contradiction with our asymptotic flatness assumption.

The first two cases need a more detailed analysis, which we will do in the following.

Proof. Using the asymptotic flatness hypothesis, we must have, in particular, \(\underset{r\rightarrow \infty }{\lim }a_{14}=0\), which, taking again into account the identity \(a_{14}=1+k_{10}^{2}\left( 2ab+c\right)\) (see (25 )), implies that, for any \(t,\) there exists a value \(r_{0}=r_{0}\left( t\right)\) such that for all \(r\geq r_{0},\) the desired inequality holds. There remains the question whether \(2ab+c\) can change sign. To see this, let us assume that there exists some value \(\left( t_{0},r_{0}\right)\) where \(2ab+c=0\); but, at these points, using (i), respectively, (ii), we find:

(i) If \(b\left( t,r_{0}\right) \not=0,\) then\[\rho =\dfrac{E}{D}=\dfrac{ba_{3}}{aa_{3}-a_{1}+a_{5}}=\dfrac{b^{2}}{c+2ab}\rightarrow \infty , \label{eq:rho95subcase951}\tag{62}\] which makes the formula for \(L\) ill-defined at \(\left( t,r_{0}\right) .\)

(ii) If \(b\left( t,r_{0}\right) =0,\) then \(2ab+c=0\) implies also \(c\left( t,r_{0}\right) =0,\) which entails that at the respective points, \(L\;\)is degenerate.

We conclude that none of these can happen, hence \(2ab+c\) must keep a negative sign for all \(r.\)  

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We are now able to investigate the cones \(L=0\) at each \(x\in M.\) These are contained in the union of the sets:

1. The hyperplane\[\left( H\right) :u:=\dot{t}-a\dot{r}=0.\]

2. The quadric\[\left( C\right) :v+\rho u^{2}=0.\]

Let us investigate these cones in each of the cases \(b\not=0\) and \(b=0.\)

Case (i): \(b\not=0.\)

In this situation, we obtain, we have seen above that \(\rho =\dfrac{b^{2}}{c+2ab},\) which leads to:\[v+\rho u^{2}=\dfrac{\left( b\dot{t}+\left( ab+c\right) \dot{r}\right) ^{2}}{2ab+c}-\dot{\theta}^{2}-\dot{\varphi}^{2}\sin ^{2}\theta . \label{aux95metric}\tag{63}\] Using the inequality \(2ab+c<0,\) this implies that \(v+\rho u^{2}\leq 0\) holds identically on the domain of definition of \(L.\) In particular, the quadric \(\left( C\right) :v+\rho u^{2}=0\) is the line \[\left( C\right) :b\dot{t}+\left( ab+c\right) \dot{r}=0,~\;\dot{\theta}=0,~\; \dot{\varphi}=0,\]which makes the cones of \(L\) degenerate. Thus, this case does not lead to any asymptotically flat Finsler spacetime functions.

Case (ii): \(b=0\)

In this case, the inequality \(2ab+c<0\) entails \[c<0.\] Moreover, we get \(E=ba_{3}=0,\) hence \(\rho =0\) and\[L=\theta \left( t,r\right) u^{2-2\lambda }v^{\lambda }.\]

Since \(c<0\), the quadric \(\left( C\right) :v+\rho u^{2}\equiv c\dot{r}^{2}-\dot{\theta}^{2}-\dot{\varphi}^{2}\sin ^{2}\theta =0\) is just the time axis \(\dot{t}:\)\[\left( C\right) :\dot{r}=0,~\dot{\theta}=0,~\dot{\varphi}=0,\]i.e., we get again a degenerate cone \(\mathcal{T}_{x}\).

References↩︎

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1. A conic subbundle of \(TM\) is defined as an open subset \(\mathcal{A}\subset TM\setminus \{0\}\) with non-empty fibers \(\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M\) at all \(x\in M,\)and which is stable under positive rescaling of vectors, i.e.: \(\left( x,\dot{x}\right) \in \mathcal{A}\Rightarrow \left( x,\lambda \dot{x}\right) \in \mathcal{A},\forall \lambda >0.\)↩︎

2. It might be possible to extend the locally characterized spacetimes to further coordinate patches by following geodesics – as one does, e.g., for the Schwarzschild solution, by passing to Kruskal-Szekeres coordinates. We will not discuss such extensions here.↩︎

3. The value \(\lambda =1\) corresponds to Riemannian metrics, which thus do not satisfy the hypothesis of being properly Finslerian.↩︎