Finsler Gravity Vacuum Equation

• Finsler gravity vacuum equations are source‐free field laws on tangent bundles that generalize Einstein’s vacuum by encoding gravity in a direction-dependent metric.
• These formulations use spray coefficients, non-linear connections, and variational principles to derive curvature quantities which recover Einstein’s equations in the metric limit.
• The equations offer a rich solution space—from Berwald reductions and Ricci-flat sectors to horizon thermodynamic formulations—providing practical insights for advanced gravitational research.

The Finsler gravity vacuum equation denotes the source-free field equation imposed on a Finsler spacetime, where the gravitational field is encoded by a positively homogeneous function on the tangent bundle rather than solely by a pseudo-Riemannian metric on the base manifold. In the literature, the term does not refer to a single universally accepted equation: one finds scalar Ricci-flat proposals, action-based scalar equations of Pfeifer–Wohlfarth type, Einstein-like systems for metric-compatible $d$-connections, Berwald reductions, and more recent formulations based on the vanishing of a Ricci 1-form. What these approaches share is that the geometry is direction dependent, the dynamics lives naturally on $TM$ or a homogeneous quotient of it, and the metric limit is required to reproduce Einstein vacuum gravity (Lämmerzahl et al., 2018, Pfeifer et al., 2018).

1. Geometric setting and curvature objects

A common starting point is a Finsler structure

$F:TM\to [0,\infty),$

or, in Lorentz-Finsler spacetime formulations, a homogeneous function

$$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$

with associated Finsler function $F(x,y)=|L(x,y)|^{1/n}$. The fundamental tensor is defined either as

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$

or, for $L$, as

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$

In the pseudo-Finsler setting this tensor need not be positive definite, and the causal structure is built directly into $L$ through a timelike shell and its null boundary (Fuster et al., 2015, Pfeifer et al., 2018).

The geodesic dynamics is governed by the spray coefficients. One standard presentation writes

$$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$

while the Cartan non-linear connection can be written as

$TM$0

From the spray one defines the Finsler curvature quantity

$TM$1

whose trace

$TM$2

is the Finsler Ricci scalar. A widely used Ricci tensor is Akbar-Zadeh’s

$TM$3

which is symmetric, covariant, and reduces to the usual Ricci tensor in the Riemannian limit (Fuster et al., 2015).

These constructions already indicate why vacuum dynamics is more intricate than in general relativity. Curvature depends on both base position and direction, and field equations can naturally be formulated as scalar equations on homogeneous tangent-bundle spaces, as tensorial equations on $TM$4, or as Einstein-like systems adapted to a horizontal/vertical splitting (Hohmann et al., 2018, Vacaru, 2010).

2. Principal vacuum-equation proposals

One influential proposal identifies vacuum with the vanishing of the Finsler Ricci scalar,

$TM$5

In this line of work, $TM$6 is defined directly from the spray coefficients by

$TM$7

and the motivation is that vacuum should correspond to a trace-free tidal tensor in the Finslerian geodesic-deviation equation. The same survey literature emphasizes that this Rutz-type equation is only one candidate among several, and that no consensus exists on a unique Finsler analogue of Einstein’s vacuum equations (Li et al., 2014, Lämmerzahl et al., 2018).

A second line derives a scalar equation from an action principle. In the formulation on the positive projective tangent bundle $TM$8, the variationally completed vacuum field equation is

$TM$9

where $F:TM\to [0,\infty),$0 is the Finsler Ricci scalar and $F:TM\to [0,\infty),$1 is built from the Landsberg tensor. Hohmann, Pfeifer, and Voicu show that Rutz’s equation is not variational and that its canonical variational completion yields this action-based equation (Hohmann et al., 2018).

Closely related is the Pfeifer–Wohlfarth action on the unit tangent bundle

$F:TM\to [0,\infty),$2

Varying with respect to $F:TM\to [0,\infty),$3 gives

$F:TM\to [0,\infty),$4

and in vacuum

$F:TM\to [0,\infty),$5

The paper states explicitly that in the metric limit this equation is equivalent to Einstein’s equations (Pfeifer et al., 2018).

A distinct school formulates Einstein-Finsler gravity in terms of a metric-compatible distinguished connection $F:TM\to [0,\infty),$6 adapted to an $F:TM\to [0,\infty),$7-connection splitting. The Einstein $F:TM\to [0,\infty),$8-tensor is

$F:TM\to [0,\infty),$9

and the vacuum system is

$$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$0

Here the vacuum equation is $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$1, not a scalar Ricci condition (Vacaru, 2010).

3. Berwald reductions and Ricci-flat sectors

Berwald spacetimes occupy a central place because their spray is quadratic in the fiber variables,

$$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$2

so the canonical non-linear connection induces an affine connection on spacetime. Several Finsler gravity equations simplify sharply in this sector. In the Berwald case considered for Finsler pp-waves, the Pfeifer–Wohlfarth vacuum equation reduces to

$$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$3

with $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$4 built from the fundamental tensor $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$5, the distinguished section $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$6, and the Cartan tensor $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$7. The operational statement used there is that any Berwald metric satisfying

$$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$8

is an exact vacuum solution (Fuster et al., 2015).

A different reduction theorem, proved for the scalar Finsler gravity vacuum equation in both metric and Palatini formulations, shows that the full vacuum equation collapses to Ricci-flatness under explicit regularity assumptions. If there exists a power $$L:TM\to \mathbb R,\qquad L(x,\lambda y)=\lambda^n L(x,y),$$9 that is sufficiently regular on the light cone, if the associated

$F(x,y)=|L(x,y)|^{1/n}$0

is non-degenerate there, and if the Landsberg term vanishes, then the metric vacuum equation implies

$F(x,y)=|L(x,y)|^{1/n}$1

while the Palatini vacuum equation implies

$F(x,y)=|L(x,y)|^{1/n}$2

This generalizes earlier reduction results that required regularity of $F(x,y)=|L(x,y)|^{1/n}$3 itself (Pfeifer et al., 23 Jun 2026).

The Berwald sector also exhibits a more subtle phenomenon: Ricci-flatness need not be necessary for vacuum. For Berwald spacetimes with vanishing Landsberg tensor,

$F(x,y)=|L(x,y)|^{1/n}$4

the action-based vacuum equation reduces to

$F(x,y)=|L(x,y)|^{1/n}$5

Using the Berwald identity

$F(x,y)=|L(x,y)|^{1/n}$6

this becomes

$F(x,y)=|L(x,y)|^{1/n}$7

The recent spherically symmetric analysis emphasizes that $F(x,y)=|L(x,y)|^{1/n}$8 is sufficient but not necessary for this vacuum equation, opening the possibility of non-Ricci-flat exact vacuum solutions (Voicu et al., 3 Jun 2026).

4. Ricci 1-form formulations and horizon thermodynamics

A recent Lorentz-Finsler approach selects the vanishing of the Ricci 1-form as the vacuum gravitational equation: $F(x,y)=|L(x,y)|^{1/n}$9 For a support vector $$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$0,

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$1

and the scalar curvature is

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$2

The paper argues explicitly that the natural vacuum equation is not merely $$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$3, but the stronger vectorial condition $$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$4 (Minguzzi, 13 Mar 2026).

In that framework, vacuum equivalence is expressed by

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$5

where

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$6

The same paper proves the identity

$$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$7

so $$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$8 links directional derivatives of the Ricci scalar to the Ricci 1-form (Minguzzi, 13 Mar 2026).

The physical motivation is horizon rigidity. If $$g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 F^2(x,y)}{\partial y^i\partial y^j}$$9 is a totally geodesic null hypersurface with null generator $L$0, surface gravity

$L$1

is defined from

$L$2

Under the null convergence condition

$L$3

together with $L$4, one obtains

$L$5

equivalently

$L$6

This yields the Finslerian analogue of the zeroth law: for a connected compact totally geodesic null hypersurface there exists a smooth future-directed null tangent field $L$7 such that $L$8 is constant on $L$9 (Minguzzi, 13 Mar 2026).

The same work also proposes a more restrictive “unifying equation”

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$0

which in vacuum, $$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$1, becomes equivalent to $$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$2. In this sense, Ricci-1-form vacuum gravity is presented as a curvature equation tied directly to horizon regularity and thermodynamic interpretation (Minguzzi, 13 Mar 2026).

5. Exact solution sectors

The choice of vacuum equation strongly influences the solution space. In the Berwald vacuum sector of Pfeifer–Wohlfarth gravity, the Finsler pp-wave ansatz

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$3

has spray coefficients quadratic in $$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$4, hence is Berwald. Its only nonzero Ricci-tensor component is

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$5

so the vacuum equation is satisfied iff

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$6

The profile equation is therefore the same harmonicity condition as for ordinary pp-waves in general relativity (Fuster et al., 2015).

For the scalar equation $$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$7, one finds Schwarzschild-type constructions. Under spherical symmetry with a Randers-Finsler angular sector, the exact non-Riemannian vacuum solution takes the form

$$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$8

where $$g^L_{ab}=\frac{1}{2}\bar\partial_a\bar\partial_b L.$$9 is the “Finslerian sphere.” The same line of work states a Birkhoff-type result: a vacuum Finsler spacetime with that symmetry must be static. A later extension replaces $L$0 by

$L$1

and obtains a Finslerian Schwarzschild–de Sitter-type solution with

$L$2

after removing a time-dependent prefactor by redefining time (Li et al., 2014, Nekouee et al., 2023).

Action-based Pfeifer–Wohlfarth vacuum gravity also admits exact non-Berwald Landsberg solutions. The modified Elgendi “unicorn” family is weakly Landsberg, and its Ricci scalar satisfies

$L$3

Because the mean Landsberg curvature vanishes, Ricci-flatness makes this an exact solution of the Pfeifer–Wohlfarth vacuum equation. In cosmological time the conformal factor becomes linear,

$L$4

yielding a linearly expanding or contracting Finsler universe (Heefer et al., 2023).

The Einstein-Kropina sector is markedly rigid. For

$L$5

an Einstein-Kropina metric solves the $L$6-vacuum equation of Pfeifer–Wohlfarth type only if the underlying pseudo-Riemannian metric $L$7 is Ricci-flat, $L$8 is Killing, an additional quadratic condition holds, and

$L$9

In Lorentzian or positive definite signature, dimensions $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$0 and below admit only the trivial local solution, essentially Minkowski or Euclidean space; in dimension $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$1 and higher the local structure is a product of a line with a Ricci-flat factor, and the Finsler metric is Berwald (Heefer et al., 5 Jun 2026).

By contrast, the recent classification of $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$2-symmetric, asymptotically flat Berwald spacetimes yields three families of exact vacuum solutions that are not Ricci flat. These solutions are all of $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$3-type,

$$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$4

with $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$5 absolutely parallel with respect to the pseudo-Riemannian metric $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$6. They provide explicit instances in which the Berwald-reduced Finsler vacuum equation is satisfied even though $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$7 (Voicu et al., 3 Jun 2026).

6. Relation to Einstein gravity and present status

Several formalisms insist on the metric limit as a consistency requirement. In the action-based spacetime formalism, if

$$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$8

then the Finsler gravity equation is equivalent to Einstein’s equations (Pfeifer et al., 2018). In the variational-completion approach, inserting a pseudo-Riemannian $$G^i=\gamma^i_{jk}\,y^j y^k, \qquad \ddot x^i+G^i=0,$$9 into the scalar vacuum equation yields $TM$00 after differentiating with respect to the fiber variables, so the pseudo-Riemannian limit reproduces Einstein vacuum exactly (Hohmann et al., 2018).

Metric-compatible $TM$01-connection formalisms provide another route back to general relativity. Through the distortion relation

$TM$02

Einstein-Finsler equations written for a canonical $TM$03-connection can be recast as Einstein equations for the Levi-Civita connection with an effective source coming from the distortion tensor. Imposing suitable zero-torsion and nonholonomic constraints recovers the Levi-Civita vacuum equations directly (Vacaru, 2010). A different anisotropic theory based on pulling back an Einstein–Cartan-like Lagrangian from the slit tangent bundle to spacetime arrives at vacuum equations

$TM$04

and proves that they force

$TM$05

In that model the vacuum geometry becomes velocity independent, the connection reduces to Levi-Civita, and the dynamics coincides with $TM$06 (Garcia-Parrado et al., 2022).

The survey literature states explicitly that no unique accepted Finsler Einstein equation presently exists (Lämmerzahl et al., 2018). Symmetry classifications can be carried out independently of the final field equation: for cosmological Berwald spacetimes, the most general nontrivial homogeneous and isotropic Lagrangian is

$TM$07

but the determining equation for the free function $TM$08 is left to future Finsler generalizations of Einstein gravity (Hohmann et al., 2020). At the same time, some vacuum sectors retain classical regularity features: for standard static Berwald spacetimes satisfying $TM$09, the associated averaged metric is analytic in harmonic coordinates (Caponio et al., 2020).

The literature represented here therefore suggests that “the Finsler gravity vacuum equation” is best understood not as a single settled law but as a family of closely related source-free curvature equations on $TM$10, distinguished by the choice of geometric variables, connection, homogeneity setting, and physical principle used to generalize Einstein vacuum gravity.