Einstein-Kropina Metrics and Their Application in Finsler Gravity

Abstract: We generalize the Einstein condition for Kropina metrics obtained in the positive definite setting by Zhang, Shen and others to all signatures. As examples of Einstein-Kropina metrics $L=L_{a,b}$, we construct new explicit positive definite ones and the first ones with Lorentzian signature. Next, we classify all Einstein-Kropina solutions to the vacuum equation of Finsler gravity by Pfeifer and Wohlfarth, in arbitrary dimension and including a possible cosmological constant $Λ$. For the Lorentzian and positive definite cases, the local picture is as follows. In dimension 4 or lower, only the trivial solution exists: a Minkowski or Euclidean space, essentially. In dimension 5 and higher, the solutions are those $L_{a,b}$ for which the metric $a$ is a product of the real line with a Ricci-flat metric (Lorentzian or Riemannian) and the vector field $b$ is the unique unit vector on the first factor. As a very surprising rigidity phenomenon, all Einstein-Kropina solutions to the $Λ$-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes.

• The paper presents a systematic classification of Einstein-Kropina metrics, proving that Ricci-flat and Λ=0 conditions are necessary for vacuum Finsler gravity solutions.
• It employs explicit constructions using unit Killing fields and product manifolds to extend the analysis to all signatures including Lorentzian cases.
• The study highlights stringent rigidity in Finsler gravity, limiting nontrivial vacuum solutions and suggesting new directions for research in pseudo-Finsler frameworks.

Einstein-Kropina Metrics and Their Application in Finsler Gravity

Introduction and Context

This paper presents a systematic study of Einstein-Kropina metrics in pseudo-Finsler geometry and analyzes their implications for vacuum solutions in Finsler extensions of General Relativity, particularly within the framework developed by Pfeifer and Wohlfarth [Pfeifer:2011xi]. The analysis generalizes previously established results in the positive-definite setting [ZHANG201380] to all signatures, including Lorentzian, and provides a comprehensive characterization and explicit construction of nontrivial Einstein-Kropina metrics. The authors further provide a complete classification of Einstein-Kropina solutions to the Finsler gravitational vacuum equations—with and without cosmological constant $\Lambda$—in arbitrary dimension and signature.

Generalization of Einstein-Kropina Metrics to All Signatures

Einstein-Finsler metrics, and specifically the Kropina subclass, are of intrinsic geometric and physical interest, with Kropina metrics characterized by the strong rigidity of their Einstein condition. In the pseudo-Riemannian setting, the Einstein condition for a Kropina metric $L = A^2/\beta^2$, constructed from a pseudo-Riemannian metric $a$ and a vector field $b$, is fully determined by the pair $(a, b)$, where $b$ is a unit Killing vector relative to $a$. The authors rigorously extend previous results to arbitrary signature, outlining that the Kropina metric $L$ is Einstein if and only if $a$ is Einstein and $b$ is a unit Killing field. This is established for all $L = A^2/\beta^2$0, but the cases $L = A^2/\beta^2$1 differ in the existence of nontrivial solutions.

In the positive definite and Lorentzian cases (with $L = A^2/\beta^2$2 of respective signature), explicit construction methods are provided. In dimensions $L = A^2/\beta^2$3, new families of nontrivial Einstein-Kropina metrics are constructed using product manifolds and Einstein-Sasaki spaces, encompassing all possible signatures. For $L = A^2/\beta^2$4 and $L = A^2/\beta^2$5, existence is highly constrained; in $L = A^2/\beta^2$6, only flat and Berwald (hence Ricci-flat) examples exist, while in $L = A^2/\beta^2$7 only the round $L = A^2/\beta^2$8 and $L = A^2/\beta^2$9 metrics with constant curvature admit unit Killing fields.

Explicit Construction and Classification

Explicit examples are given for odd-dimensional spheres $a$0 (Riemannian), $a$1 (Lorentzian), product spaces, and Einstein-Sasaki manifolds. In all cases, construction leverages the existence of unit Killing fields, and in Einstein-Sasaki manifolds the Reeb vector provides such a field systematically, yielding explicit families of metrics. In higher dimensions, the product structure allows the generation of Kropina metrics of desired signature, with the product metric being Einstein if both factors have matching (nonzero) Einstein constants.

A salient aspect is the explicit class of positive-definite Einstein-Kropina metrics on $a$2 (parameterized by coprime integers) and Lorentzian cases such as $a$3. This explicitness is crucial for further applications in mathematical physics.

Finsler Gravity and Vacuum Equation Analysis

Moving to the application in Finsler gravity, the core focus is the analysis of vacuum solutions to the Finslerian field equations (Pfeifer–Wohlfarth gravitational equations), including a cosmological constant $a$4. The general $a$5-vacuum equation for a pseudo-Finsler metric $a$6 and dimension $a$7 is:

$a$8

where $a$9 is the Landsberg scalar.

Contrary to the pseudo-Riemannian case, where the Einstein condition ensures a solution for arbitrary $b$0, the authors prove that in the Einstein-Kropina class, only Ricci-flat solutions with zero cosmological constant are allowed. Specifically, for $b$1, if a Kropina metric is an Einstein solution to the $b$2-vacuum field equation, then $b$3 and $b$4 must be Ricci-flat, $b$5 must be parallel (in Riemannian or Lorentzian signatures), and the cosmological constant is dynamically forced to vanish. This constitutes a strong rigidity property, especially in contrast to General Relativity.

For Riemannian and Lorentzian $b$6, this rigidity is even sharper: any Killing $b$7 with $b$8 must be parallel, so the local structure of $b$9 splits as a direct product $(a, b)$0, and $(a, b)$1 is the tangent along the first factor. Thus, all nontrivial Einstein-Kropina vacuum solutions in these signatures and for $(a, b)$2 are locally just Kropina metrics built from Ricci-flat product spaces with parallel translation. No nontrivial (proper) Ricci-nonflat, or $(a, b)$3, solution exists in any signature.

Low-dimensional Nonexistence

For $(a, b)$4, classification is complete: all Einstein-Kropina vacuum solutions are locally flat (Euclidean or Minkowski space with translation direction), hence trivial from the perspective of field equations.

Implications and Theoretical Outlook

The results deliver several noteworthy implications:

• Einstein-Kropina vacuum solutions in Finsler gravity are necessarily Ricci-flat and weakly weakly Landsberg; in Riemannian and Lorentzian cases, they are Berwald. This strengthens the analog to classical General Relativity by reducing the solution space to particularly rigid, highly symmetric metrics.
• The cosmological constant is constrained to $(a, b)$5. This is in stark contrast to the classical situation, where vacuum Einstein manifolds with nonzero $(a, b)$6 are generic, e.g., de Sitter/Anti-de Sitter spaces. The absence of $(a, b)$7 Einstein-Kropina solutions is a nontrivial structural distinction in Finsler gravity.
• Richness only in higher dimensions and signatures: Nontrivial solutions require $(a, b)$8 and Ricci-flat metrics with parallel fields, correlating the Finslerian theory very closely with structural properties of underlying pseudo-Riemannian geometry.
• No evidence for weak-Landsberg non-Berwald (unicorn-type) Kropina solutions in Riemannian or Lorentzian settings. In other signatures, the possibility is not excluded, making this a natural open problem with implications for the Landsberg conjecture.

Future directions include the study of Einstein-Kropina metrics in dimension 4 (open for nontrivial examples), the extension of vacuum solution classification to less rigid Finslerian settings (beyond Kropina), and the search for nontrivial solutions with nonzero cosmological constant in more general Finsler geometries.

Conclusion

This work establishes a comprehensive framework and explicit classification for Einstein-Kropina metrics in all signatures, especially highlighting structural rigidity in their application to Finsler gravity. The detailed results—involving existence, explicit construction, and precise compatibility with gravitational field equations—constitute a compelling bridge between Finsler geometric structures and physically motivated gravity models. The strong constraints on vacuum solutions and the necessary vanishing of the cosmological constant within the Einstein-Kropina class mark significant theoretical advances and prompt further research into the role of Finsler geometry in gravitational physics.

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Reference:

"Einstein-Kropina Metrics and Their Application in Finsler Gravity" (2606.07121)