The early history of symmetric teleparallel gravity: An overlooked period

This paper highlights the overlooked early development of symmetric teleparallel gravity by the authors and their collaborators between 2004 and 2013, while also reviewing their subsequent work and offering perspectives on the field's future.

The Gist

Imagine you are trying to understand how the universe works, specifically how gravity pulls things together. For a long time, scientists had two main "rulebooks" for this:

1. Newton's Rulebook: Gravity is a force that acts instantly across space. It worked great for centuries, but eventually, we realized it couldn't explain everything (like why Mercury orbits the way it does or why the universe is expanding faster).
2. Einstein's Rulebook (General Relativity): Gravity isn't a force; it's the bending of space and time itself, like a heavy bowling ball sitting on a trampoline. This fixed Newton's problems, but now we have new mysteries: Dark Matter (stuff we can't see that holds galaxies together) and Dark Energy (stuff pushing the universe apart).

Einstein's book is the current champion, but scientists are wondering: Is there a different way to write the rules that might solve these new mysteries without inventing invisible "dark" stuff?

The "Forgotten" Chapter

This paper is written by a physicist named Muzaffer Adak. He is essentially saying, "Wait a minute! We wrote the first draft of this new rulebook years ago, but nobody noticed."

Around 2017, a new wave of scientists started getting excited about a theory called Symmetric Teleparallel Gravity (STPG). They thought they were discovering something brand new. Adak and his team, however, had been publishing papers on this exact same topic between 2004 and 2013. They feel their work was overlooked, and this paper is their way of saying, "Here is the history you missed, and here is why it matters."

The Three Ways to Describe Gravity

To understand Adak's point, imagine gravity is like a rubber sheet. There are three ways to describe what happens when you put a heavy object on it:

1. Curvature (Einstein's way): The sheet bends. This is the standard view.
2. Torsion (The "Twist" way): The sheet doesn't bend, but it twists like a corkscrew.
3. Non-Metricity (Adak's focus): The sheet doesn't bend or twist, but the ruler you use to measure it changes size as you move it around.

Adak's team focused on the third option: Non-Metricity. They proposed that gravity comes from the fact that our "rulers" (the geometry of space) change length depending on where you are, even if the space itself isn't bending or twisting.

The "Coincident Gauge" Trick

Here is the clever part of their work. In the past, using this "changing ruler" idea was messy and hard to calculate. Adak and his team discovered a special "gauge" (a mathematical setting, like tuning a radio) that they called the "Natural Gauge" (later renamed the "Coincident Gauge" by others).

Think of it like this:

• Imagine you are trying to navigate a city with a map that keeps changing its scale. It's a nightmare.
• Adak found a way to "lock" the map so that the scale is consistent in a specific coordinate system.
• Suddenly, the messy math became clean. They showed that if you use this specific setting, their theory looks exactly like Einstein's theory in many cases, but it also opens up new possibilities to explain the universe without needing Dark Matter.

Why Was It Overlooked?

Adak admits that his team might have missed the spotlight because of how they wrote their papers.

• The Analogy: Imagine two chefs making the same delicious cake.
  • Chef A writes the recipe using standard kitchen terms (cups, spoons, grams). Everyone understands it.
  • Chef B (Adak's team) writes the recipe using a very complex, high-level mathematical language (exterior algebra) that only a few experts speak.
• Because they used this "secret language," other scientists didn't realize they had already solved the problem. When the "new" wave of scientists arrived in 2017, they reinvented the wheel, not knowing Adak had already built a better one.

What's Next?

The paper isn't just about history; it's about the future. Adak's team is now using these geometric ideas to solve problems in:

• Engineering: Understanding cracks and defects in metal crystals (treating them like "twists" or "bends" in the material's geometry).
• Medicine & Finance: They believe this geometric way of thinking can help model complex systems in biology and economics.

The Bottom Line

This paper is a "correction of the record." It tells us that the exciting new theory of Symmetric Teleparallel Gravity (which is currently a hot topic in physics) actually has roots in work done over a decade ago by Adak and his students. They showed that you can describe gravity not by bending space, but by how the "rulers" of space change, and they did it in a way that is mathematically elegant and potentially more powerful than Einstein's original theory.

They are asking the scientific community to finally read their old papers, recognize their contribution, and join them in using this geometric toolkit to solve the universe's biggest mysteries.

Technical Summary

Based on the provided article, here is a detailed technical summary of the paper "The early history of symmetric teleparallel gravity: An overlooked period" by Muzaffer Adak.

1. Problem Statement

The paper addresses a significant gap in the historical and conceptual understanding of Symmetric Teleparallel Gravity (STPG).

• The Oversight: While STPG and its extension, $f(Q)$ gravity, have seen a massive surge in research interest since 2017–2018 (driven by authors like Koivisto, Golovnev, and Jimenez), the foundational work establishing the geometric framework was published much earlier (2004–2013) by the author and his collaborators.
• The Gap: A recent comprehensive review of STPG [13] completely overlooked these pioneering early works.
• The Motivation: The author aims to clarify the conceptual lineage of STPG, demonstrate the continuity of its development, and highlight that many structures currently discussed in modern formulations were explicitly derived and analyzed by his group a decade prior. The paper also seeks to explain why this early work was overlooked (partially due to the use of exterior algebra/orthonormal coframes vs. the more common tensor component notation).

2. Methodology

The paper is a historical and technical review of the author's specific body of work, rather than a presentation of new experimental data. The methodology involves:

• Chronological Reconstruction: Systematically reviewing a series of papers published between 2004 and 2013, and subsequent work post-2018.
• Mathematical Formalism: The author's group consistently utilized exterior algebra and the orthonormal coframe formalism (using differential forms) rather than standard tensor component notation.
• Gauge Fixing: A core methodological pillar was the exploitation of the gauge freedom in metric-affine geometry. Specifically, the authors set the affine connection to zero in the coordinate basis (later termed the "coincident gauge" by others, or "natural/inertial gauge" by Adak).
• Ansatz Construction: To find solutions, the authors employed spherically symmetric static metrics and independent spherically symmetric static affine connections, solving for unknown functions using zero-curvature and zero-torsion conditions.

3. Key Contributions and Timeline of Results

The paper details the evolution of STPG through specific publications:

A. Foundational Period (2004–2013)

• Nester & Yo (2004): Established the concept of attributing gravity to non-metricity ($Q$) with zero curvature and zero torsion. They formulated an equivalent Lagrangian to General Relativity (GR) but did not express it as a quadratic non-metricity tensor.
• Adak & Sert (2004):
  • First used the term "Symmetric Teleparallel Gravity" (STPG).
  • Explicitly wrote down STPG field equations equivalent to Einstein's equations.
  • Obtained a spherically symmetric static solution and analyzed its singularity structure.
• Adak, Kalay, & Sert (2006):
  • Presented the first quadratic STPG Lagrangian containing five independent coupling constants.
  • Derived variational field equations and found solution families dependent on these coupling constants (not restricted to GR equivalence).
• Adak & Dereli (2007):
  • Demonstrated that while GR in $(1+1)$ dimensions is trivial, STPG defines a genuinely dynamical theory.
  • Constructed a 4-coupling-constant Lagrangian and derived conformal/cosmological solutions.
• Adak (2008):
  • Proved that the 5-parameter STPG Lagrangian reduces to the Riemannian Einstein-Hilbert Lagrangian in $(1+3)$ dimensions for specific coupling constants.
  • Explicitly showed the affine connection can be expressed entirely in terms of metric components via gauge fixing (the "natural gauge").
• Adak et al. (2010):
  • Systematically applied the metric formulation to find a broad class of exact solutions: conformal, spherically symmetric static, cosmological, and pp-waves.
  • First discussion of coupling spin-1/2 fermions to STPG.

B. Post-2018 Resurgence and Extensions

Following the renewed interest in the field, the author's group resumed research:

• Gauge Structure & Particle Motion: Analyzed the gauge structure of STPG and revisited test particle motion (autoparallel curves), linking it to the dark matter problem in galactic dynamics.
• Second Clock Effect: Addressed the "second clock effect" (where proper time depends on path history due to non-metricity). The authors proved that STPG models can be constructed to be free of this pathology, ensuring physical viability.
• Lower Dimensions: Revisited STPG in $(1+2)$ dimensions to reveal dynamical degrees of freedom where GR is trivial.
• General Teleparallel Geometry: Demonstrated that general teleparallel geometry (allowing torsion) admits a metric formulation via gauge fixing, analogous to STPG.
• Couplings: Investigated non-minimal coupling between STPG and Maxwell fields, and derived phenomenological bounds on coupling constants using solar neutrino oscillation data in a rotating STPG background.

C. Interdisciplinary Applications

• Crystal Defects: The group is currently applying STPG geometry to mechanical engineering, specifically modeling crystal defects:
  • Curvature $\to$ Disclinations
  • Torsion $\to$ Dislocations
  • Non-metricity $\to$ Point defects
  • They formulate all defects within a unified general teleparallel framework.

4. Significance

• Historical Correction: The paper corrects the historical record, establishing that the "modern" $f(Q)$ and coincident GR frameworks are built upon geometric structures and solutions explicitly derived by Adak and collaborators a decade earlier.
• Conceptual Clarity: It clarifies that STPG is not just an alternative to GR but a distinct geometric framework where gravity is purely non-metricity, offering advantages in lower dimensions (dynamical degrees of freedom) and in the treatment of defects.
• Physical Viability: By resolving the "second clock effect" issue, the work validates the physical consistency of non-metric theories.
• Methodological Insight: The author suggests that the lack of recognition for early STPG work may be partly due to the preference for tensor component notation in the broader community, whereas the author's group utilized the more abstract but powerful language of exterior algebra.
• Future Outlook: The paper posits that STPG geometry has broad potential applications beyond high-energy physics, extending into engineering (defect theory), health sciences, and finance.

Conclusion

This article serves as both a historical record and a technical manifesto. It asserts that the "surge" in STPG research since 2017 is a continuation of a rigorous program initiated in 2004. The author emphasizes that the geometric foundations, exact solutions, and coupling mechanisms currently being rediscovered were systematically established in the early 2000s, urging the community to recognize this lineage and the unique advantages of the non-metricity-based approach to gravity.