You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We present our Finsler spacetime formalism which extends the standard formulation of Finsler geometry to be applicable in physics. Finsler spacetimes are viable non-metric geometric backgrounds for physics; they guarantee well defined causality, the propagation of light on a non-trivial null structure, a clear notion of physical observers and the existence of physical field theories determining the geometry of space-time dynamically in terms of an extended gravitational field equation. Here we give the precise definition of Finsler spacetimes and the notion of well-defined observers, their measurements, the transformations between them and how to formulate action based field theories.'
address: |
    II Institute for theoretical physics, University of Hamburg,\
    Hamburg, Germany\
    $^*$E-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>
author:
- 'C. Pfeifer$^*$ and M. Wohlfarth'
bibliography:
- 'main.bib'
title: Finlser geometric extension of Einstein gravity
---

Finsler spacetimes
==================

The standard textbook formulation of Finsler geometry [@Chern] is a well known extension of Riemannian metric geometry based on a general length measure $S$ for curves $\gamma$ $$L[\gamma]=\int F(\gamma,\dot\gamma),$$ rather then on a positive definite metric $g$ which induces the length measure via $F=\sqrt{g_{ab}(\gamma)\dot\gamma^a\dot\gamma^b}$. It has the shortcoming that it can not be used as a generalization of Lorentzian metric spacetime geometry, since the geometric objects like connections and curvature are not well-defined in case $F$ has a non-trivial null structure. Our definition of Finsler spacetimes solves this problem and sets the stage for the application of non-metric Finslerian spacetime geometry in physics by introducing a smooth n-homogeneous fundamental geometry function $L(x,y)$ on the tangent bundle. For the tangent bundle TM of the spacetime manifold M we use manifold induced coordinates $(x,y)=Z\in TM, Z=y^a\partial_a{}_{|x}$ and $\{\partial_a=\frac{\partial}{\partial x^a},\bar \partial_a=\frac{\partial}{\partial y^a}\}$ as basis of $T_{(x,y)}TM$, see our articles for all details.

Our definition of Finsler spacetimes guarantees a causal structure in each tangent space: $S_x$ is the shell of unit timelike vectors which defines a cone of timelike directions with null boundary, as displayed in figure \[fig:fst\].

![\[fig:fst\]Causal structure of Finsler spacetime.](fst6.pdf){width="40.00000%"}

The geometry of Finsler spacetimes is solely derived from derivatives of $L$ in terms of the unique Cartan non-linear connection coefficients: $N^a{}_b=\frac{1}{4}\bar\partial_b(g^{Laq}(y^m\partial_m\bar\partial_qL-\partial_qL))$. The connection between our definition of Finsler spacetimes and standard Finsler geometry is given by the following theorem: *Wherever L and F are both differentiable they encode the same geometry, i.e. $N[L]=N[F^2]$*. The fact that Finsler spacetimes are built from a smooth function $L$ enables us to consider for example geometries based on $L=G_{a_1...a_n}y^{a_1}...y^{a_n}$ with non-trivial null-structure. In standard Finsler geometry this would not be possible since the geometric objects based on derivatives of $F=(G_{a_1...a_n}y^{a_1}...y^{a_n})^{1/n}$ would not be well defined on the null-structure of spacetime.

Observers
=========

The connection coefficients $N$ split TTM and T\*TM into horizontal and vertical space by $\{\delta_a=\partial_a-N^b{}_{a}\bar\partial_b,\bar\partial_a\}$ and $\{dx^a,\delta y^a=dy^a+N^a{}_{v}dx^b\}$, as displayed in figure \[fig:hv\]. The horizontal (co-)tangent space is identified with the (co-)tangent space along the manifold directions. Timelike observers move on worldlines $x(\tau)\in M$ with trajectory $(x, \dot x)\in TM$ and $\dot x$ in the cone of timelike vectors. A horizontal orthonormal frame defines their time and space directions along the manifold $\{e_a\}=\{e_0=\dot x^a\delta_a,e_\alpha\}; g^F_{(x,\dot x)}(e_\mu,e_\nu)=-\eta_{\mu\nu}$. Measurable quantities are components of horizontal tensors evaluated in this frame at the observers $TM$ position.

Frames of different observers $e$ and $f$ are located in different tangent spaces to $TM$, even if they are at the same position of spacetime, see fig. \[fig:obstraf\]. Therefore the transformation from one observer frame $e$ to the other $f$ involves first parallel transport of frame $e$, to the tangent bundle position of the other observer frame $f$ along the vertical geodesic $v(t)$. There the transported frame $\hat e$ and $f$ are living in the same tangent space to TM and can therefore be mapped onto each other by a linear transformation, which turns out to be a Lorentz transformation. The combination of parallel transport and Lorentz transformation form a groupoid. All technical details and the proofs of the statements can be found in Ref. .

![\[fig:obstraf\]Right: Transformations between observers.](hv3.pdf "fig:"){width="40.00000%"} ![\[fig:obstraf\]Right: Transformations between observers.](3_frame.pdf "fig:"){width="40.00000%"}

Field theories and gravitational dynamics
=========================================

The geometry of Finsler spacetimes is built from tensors on $TM$; hence physical fields coupling to this geometry will be of the same kind. Lagrange densities on $TM$ require the canonical Sasaki-type $TM$-metric $G=-g^F{}_{ab}(dx^adx^b+F^{-2}\delta y^a\delta y^b)$, which allows us to couple field theories to Finsler spacetime geometry as follows: Pick an action for a $p$-form $\phi(x)$ on $(M,g): S[\phi, g]=\int_M\sqrt{g}\mathcal{L}(g,\phi,d\phi)$, use the Lagrangian for a zero homogeneous $p$-form field $\Phi(x,y)$ on $(TM,G)$, introduce Lagrange multipliers to restrict the $p$-form field to be horizontal, integrate over the unit tangent bundle $\Sigma=\{(x,y)\in TM | F(x,y)=1\}$ to obtain the $p$-form field action $S_m[\Phi, L, \lambda]=\int_\Sigma(\sqrt{g^Fh^F} \mathcal{L}(G,\Phi,d\Phi)+\lambda(1-P^H)\Phi)_{|\Sigma}$. Our coupling principle ensures that in case the Finsler spacetime is metric, field theories and gravitational dynamics equal those of general relativity.

The geodesic deviation on Finsler spacetimes gives rise to a tensor causing relative gravitational acceleration $\nabla_{\dot x}\nabla_{\dot x}V^a=R^a{}_{bc}(x,\dot x)\dot x^bV^c$. This non-linear curvature is built from the non-linear connection coefficients leads to the curvature scalar $\mathcal{R^F}=R^a{}_{ab}y^b$ which we choose as Lagrangian for our Finsler gravity action $S[L,\Phi]=\int_\Sigma(\sqrt{g^Fh^F}\mathcal{R^F})_{|\Sigma}+S_m[L,\Phi]$. Variation with respect to the $L$ yields the Finsler gravity field equation $$g^{Fab}\bar\partial_a\bar\partial_bR^F-\frac{6}{F^2}R^F+2g^F_{ab}\big(\nabla_aS_b+S_aS_b+\bar\partial_a(y^q\nabla_q S_b)\big)=-\kappa T_{|\Sigma}\,.$$ In case the function L is the metric length measure the Finsler gravity equation is equivalent to the Einstein equations. Details and a first order perturbative solution of the Finsler gravity equation can be found in Ref. .