Motivated by the potential connection between metric-affine gravity and

linear Generalized Uncertainty Principle (GUP) in the phase space, we develop a

covariant form of linear GUP and an associated modified Poincar’e algebra,

which exhibits distinctive behavior, nearing nullity at the minimal length

scale proposed by linear GUP. We use 3-torus geometry to visually represent

linear GUP within a covariant framework. The 3-torus area provides an exact

geometric representation of Bekenstein’s universal bound. We depart from

Bousso’s approach, which adapts Bekenstein’s bound by substituting the

Schwarzschild radius ($r_s$) with the radius ($R$) of the smallest sphere

enclosing the physical system, thereby basing the covariant entropy bound on

the sphere’s area. Instead, our revised covariant entropy bound is described by

the area of a 3-torus, determined by both the inner radius $r_s$ and outer

radius $R$ where $r_sleq R $ due to gravitational stability. This approach

results in a more precise geometric representation of Bekenstein’s bound,

notably for larger systems where Bousso’s bound is typically much larger than

Bekensetin’s universal bound. Furthermore, we derive an equation that turns the

standard uncertainty inequality into an equation when considering the

contribution of the 3-torus covariant entropy bound, suggesting a new avenue of

quantum gravity.

## Conclusions

- The development of a covariant form of linear Generalized Uncertainty Principle (GUP) and a modified Poincaré algebra.
- The covariant form of GUP exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP.
- Usage of 3-torus geometry to visually represent linear GUP within a covariant framework.
- The 3-torus area provides a geometric representation of Bekenstein’s universal bound.
- The revised covariant entropy bound is described by the area of a 3-torus, determined by both the inner radius and outer radius, resulting in a more precise geometric representation of Bekenstein’s bound.
- Derivation of an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the 3-torus covariant entropy bound.
- Suggesting a new avenue for investigating quantum gravity.

## Roadmap for Readers

- Introduction to metric-affine gravity and the potential connection with linear GUP in the phase space.
- Explanation of the development of a covariant form of linear GUP and its associated modified Poincaré algebra.
- Exploration of the distinctive behavior exhibited by the covariant form of GUP, focusing on its near-nullity at the minimal length scale proposed by linear GUP.
- Demonstration of the usage of 3-torus geometry to represent linear GUP within a covariant framework.
- Explanation of how the 3-torus area provides an exact geometric representation of Bekenstein’s universal bound.
- Comparison of the revised covariant entropy bound, based on the area of a 3-torus, with Bousso’s approach and its adaptation of Bekenstein’s bound.
- Illustration of the more precise geometric representation achieved by the revised covariant entropy bound, especially for larger systems.
- Derivation and presentation of the equation that turns the standard uncertainty inequality into an equation by considering the 3-torus covariant entropy bound.
- Discussion of the implications of the derived equation and its potential for advancing the study of quantum gravity.

## Potential Challenges and Opportunities

**Challenges:**

- The development of a covariant form of linear GUP and its associated modified Poincaré algebra may require advanced mathematical understanding.
- The visualization and understanding of 3-torus geometry may be challenging for some readers without a strong background in mathematics or physics.
- The comparison between the revised covariant entropy bound and Bousso’s approach may involve complex calculations and concepts.
- The derivation and comprehension of the equation that turns the standard uncertainty inequality into an equation may require a deep understanding of quantum mechanics and gravity.
- The exploration of quantum gravity and its potential new avenues may be subject to ongoing research and scientific debate.

**Opportunities:**

- The development of a covariant form of linear GUP and the modified Poincaré algebra opens up possibilities for further exploration and refinement in the understanding of fundamental physics.
- The usage of 3-torus geometry provides a visual representation that may enhance understanding and aid in future research on linear GUP.
- The revised covariant entropy bound offers a more precise geometric representation of Bekenstein’s bound, allowing for potentially improved calculations and predictions in systems with large entropy and gravitational stability.
- The derived equation relating the standard uncertainty inequality to the 3-torus covariant entropy bound presents a new avenue for investigating the connection between quantum mechanics and gravity.
- The study of quantum gravity continues to be an active field of research, providing opportunities for further discoveries and advancements in our understanding of the universe.

## Sources

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