On the Finsler variational nature of autoparallels in metric-affine geometry

This paper resolves the longstanding problem of Finsler metrizability for torsion-free affine connections with vectorial nonmetricity by deriving the necessary and sufficient conditions for their autoparallels to be variational and explicitly constructing the corresponding Finsler Lagrangians.

The Gist

Imagine you are driving a car through a strange, shifting landscape. In our everyday understanding of physics (General Relativity), the road is paved with a perfect, unchanging grid. If you let go of the steering wheel, your car follows a straight line (a "geodesic") that minimizes the distance traveled. This path is also the one that nature "prefers" because it minimizes the energy you spend. In physics terms, the path is both a geometric straight line and a variational solution (it comes from a rule of least effort).

However, the paper you provided explores a much weirder universe: Metric-Affine Geometry.

The Problem: Two Different Roads

In this weird universe, the "road" (the metric) and the "rules of the road" (the connection) are two different things.

• The Metric tells you how to measure distance.
• The Connection tells you how to steer (how to keep your direction "parallel" as you move).

Usually, in this weird universe, if you let go of the steering wheel, the car follows the autoparallel (the path dictated by the connection). But here's the catch: This path doesn't minimize distance, and it doesn't come from a "least effort" rule. It's like driving a car that follows a set of instructions but somehow refuses to obey the laws of physics that say "nature loves efficiency."

Physicists have been stuck on a question for a long time: Is there any way to describe these "autoparallel" paths as if they were following a "least effort" rule? Can we find a hidden "map" that makes these weird paths look normal?

The Solution: The Finsler Map

The authors of this paper say: Yes, but you need a new kind of map.

They introduce a concept called Finsler Geometry.

• Riemannian Geometry (The old map): Imagine a map where the cost of driving depends only on where you are. The road is the same whether you drive north or east.
• Finsler Geometry (The new map): Imagine a map where the cost of driving depends on where you are AND which direction you are facing. It's like driving in a strong wind or on a slippery slope; going uphill costs more than going downhill, even if the distance is the same.

The paper asks: Can we find a specific "Finsler map" (a specific rule for cost) that makes the weird autoparallel paths look like normal, efficient paths?

The Specific Case: The "Vectorial" Twist

The authors focus on a specific type of weird universe where the "rules of the road" are distorted by a single, invisible arrow field (a "one-form"). Think of this as a universal wind blowing in a specific direction that changes how distances are measured.

They looked at three famous types of these "windy" universes:

1. Weyl Geometry: The wind changes the size of things (lengths) but keeps angles the same.
2. Schrödinger Geometry: The wind is tricky; it keeps the length of your path constant even though the rules are weird.
3. Completely Symmetric Geometry: A balanced, symmetrical version of the wind.

The Big Discovery

The team did the math to see if they could find a "Finsler map" for these three types of universes.

• The Bad News: For the simplest type of map (called an $(\alpha, \beta)$-metric), the Schrödinger universe was a dead end. You couldn't make its paths look efficient with this simple map.
• The Good News: They invented a more complex, generalized map (Generalized $(\alpha, \beta)$-metrics). With this fancy new map, they found that ALL THREE types of universes (Weyl, Schrödinger, and Completely Symmetric) CAN be made to look like they are following a "least effort" rule!

The Analogy: The Invisible Wind

Think of the "connection" as a strong, invisible wind blowing through the universe.

• In the old view, a car driving with the wind follows a path that feels unnatural and doesn't follow the standard laws of physics.
• The authors say: "Wait! If you change your perspective and realize that the wind itself changes the cost of fuel depending on your speed and direction (Finsler geometry), then that car is actually following the most efficient path possible!"

Why Does This Matter?

1. It Saves the Principle of Least Action: It proves that even in these complex, non-standard theories of gravity, nature might still be following a "least effort" rule, we just need the right mathematical lens (Finsler geometry) to see it.
2. New Physics: This opens the door to using these "windy" geometries to explain real-world mysteries like Dark Energy (why the universe is expanding faster) or Dark Matter (why galaxies spin too fast). If the universe is actually a Finsler space, we might not need invisible particles; the "wind" of geometry itself could be doing the work.

Summary

The paper solves a decades-old puzzle by showing that "weird" paths in complex gravity theories aren't actually weird at all. They are just normal, efficient paths viewed through a more sophisticated, direction-sensitive lens called Finsler Geometry. They successfully built the mathematical "glasses" needed to see this, proving that a wide class of these theories can be described by a single, elegant rule of nature.

Technical Summary

1. Problem Statement

In General Relativity (based on pseudo-Riemannian geometry), the autoparallels of the Levi-Civita connection coincide with the geodesics of the metric length functional. Consequently, free-fall trajectories are both autoparallels and extremals of a variational action principle.

However, in metric-affine gravity, where the affine connection $\nabla$ and the metric $a$ are independent, this equivalence breaks down. In generic metric-affine geometries, autoparallels (curves whose tangent vectors are parallel transported by $\nabla$) are not variational; they do not arise as extremals of any action integral. This creates a fundamental ambiguity in the physical interpretation of free-fall trajectories: should they be metric geodesics, autoparallels, or something else?

The central question addressed in this paper is: Under what conditions does a symmetric affine connection with vectorial nonmetricity admit a variational description? Specifically, the authors seek to determine if such connections are Finsler metrizable, meaning their autoparallels can be interpreted as the geodesics of a Finsler Lagrangian (a parametrization-invariant action).

2. Methodology

The authors approach the problem by analyzing the Finsler metrizability of a specific class of connections: torsion-free affine connections with vectorial nonmetricity.

• Geometric Setup: The nonmetricity tensor $Q_{\mu\nu\rho} = -\nabla_\mu a_{\nu\rho}$ is assumed to be algebraically determined by the metric $a_{\mu\nu}$ and a single one-form $b_\mu$. This class includes notable subcases: Weyl, Schrödinger, and completely symmetric connections.
• Variational Condition: The problem is reformulated as solving a system of first-order partial differential equations (PDEs). A connection is Finsler metrizable if there exists a non-degenerate, 2-homogeneous Lagrangian $L(x, \dot{x})$ such that its Euler-Lagrange equations (geodesics) match the autoparallel equations of the connection.
• Ansatz Strategy: The authors employ a two-step classification strategy based on the algebraic dependence of the Finsler Lagrangian on the metric and the one-form:
  1. $(\alpha, \beta)$-metrics: Lagrangians of the form $L = A \Phi(s)$, where $A = a_{\mu\nu}\dot{x}^\mu\dot{x}^\nu$ and $s = B^2/A$ (with $B = b_\mu\dot{x}^\mu$). These do not depend on the norm of the one-form $|b|$.
  2. Generalized $(\alpha, \beta)$-metrics: Lagrangians of the form $L = A \Phi(|b|, p)$, where $p = U^2/A$ and $U$ is the projection of $\dot{x}$ onto the normalized one-form $u = b/|b|$. This allows dependence on the magnitude of the nonmetricity vector.

The authors solve the resulting overdetermined PDE systems (specifically the condition $\delta_\mu L = 0$) to derive necessary and sufficient conditions on the coefficients of the connection and the differential properties of the one-form $b$.

3. Key Contributions and Results

A. Classification of $(\alpha, \beta)$-Metrizability

The authors provide a complete classification for connections metrizable by standard $(\alpha, \beta)$-metrics (which do not depend on $|b|$).

• Necessary Conditions: For a connection with vectorial nonmetricity to be $(\alpha, \beta)$-metrizable, the coefficient $c_2$ (associated with the symmetric part of the nonmetricity) must vanish ($c_2 = 0$).
• Specific Subcases:
  • Weyl Connections ($c_2=c_3=0$): Always $(\alpha, \beta)$-metrizable if the one-form $b$ satisfies specific differential constraints (it must be torse-forming and closed). The resulting Lagrangian is a power law ($L \propto A s^\lambda$).
  • Schrödinger Connections ($c_1 + 2c_2 = 0, c_3=0$): Never $(\alpha, \beta)$-metrizable because the condition $c_2=0$ implies $c_1=0$, which leads to a trivial connection.
  • Completely Symmetric Connections ($c_1=c_2$): Metrizable only if $c_1=c_2=0$ (trivial) or under specific constraints leading to exponential or Kropina-type metrics.

B. Generalized $(\alpha, \beta)$-Metrizability (Main Result)

By extending the search to Lagrangians that depend on the norm $|b|$, the authors significantly broaden the class of metrizable connections.

• General Conditions: A connection is metrizable by a generalized $(\alpha, \beta)$-metric if and only if:
  1. Either $c_3 = 0$ OR $c_1 = c_2 = 0$.
  2. The one-form $b$ satisfies specific differential constraints: it must be closed ($db=0$) and torse-forming (its covariant derivative is proportional to the metric and the one-form itself).
• Explicit Construction: The authors construct the explicit Finsler Lagrangians for all valid cases.
  • Weyl Connections: Metrizable (as before).
  • Schrödinger Connections: Crucially, these are now shown to be Finsler metrizable. The authors derive the specific generalized Lagrangian (involving exponential and polynomial terms of $p$ and $|b|$) that generates Schrödinger autoparallels.
  • Completely Symmetric Connections: Metrizable under the derived constraints.

C. Summary Table of Results

The paper provides a definitive table classifying the metrizability of the main connection types:

Connection Type	$(\alpha, \beta)$-Metrizable?	Generalized $(\alpha, \beta)$-Metrizable?
Weyl	Yes	Yes
Schrödinger	No	Yes
Completely Symmetric	Yes	Yes

4. Significance and Implications

1. Resolution of the Variational Problem: The paper resolves a long-standing open problem by proving that a broad class of metric-affine connections (including the physically relevant Schrödinger connections) do admit a variational principle. This principle is not metric-based but Finsler-based.
2. Physical Interpretation: The existence of a Finsler Lagrangian implies that the autoparallels of these connections can be interpreted as the trajectories of free-falling particles derived from an action principle. This legitimizes the use of these connections in physical theories of gravity where nonmetricity is present.
3. Extension of Gravity Theories: The results support the application of Finsler geometry in gravitational physics. Since the autoparallels of these connections are Finsler geodesics, theories based on vectorial nonmetricity (like Weyl and Schrödinger gravity) can be consistently formulated within the framework of Finsler spacetime.
4. Future Applications: The authors suggest that these newly identified Finsler functions can serve as ansatzes for finding exact solutions in Finsler gravity. This could lead to geometric explanations for cosmological phenomena such as dark energy and dark matter, potentially modeling the gravitational field around compact objects without invoking exotic matter.

In conclusion, the paper demonstrates that the lack of variationality in metric-affine autoparallels is not a fundamental obstruction but rather a limitation of restricting the search to Riemannian metrics. By stepping into the broader realm of Finsler geometry, specifically generalized $(\alpha, \beta)$-metrics, a wide range of metric-affine connections are revealed to be fully variational.