The Road Not Taken: How a Historical Accident Shaped Our Understanding of Gravity

Picture a 33-year-old physics professor in Berlin, 1912, desperately searching for mathematics to express his revolutionary idea. Albert Einstein knew that gravity wasn't a force in the traditional sense—his "happiest thought" had revealed that a freely falling observer experiences no gravitational field. But how could he translate this profound physical insight into precise mathematical language? For five agonizing years, he struggled with tensor calculus, differential geometry, and the mathematical machinery needed to express his vision. The breakthrough would come not from Einstein alone, but from a chance reunion with his former classmate Marcel Grossmann, who introduced him to the obscure mathematics of Riemannian geometry.

This mathematical toolkit—developed by Italian mathematicians Ricci-Curbastro and Levi-Civita just years earlier—would become the foundation of general relativity. Einstein later acknowledged this debt with characteristic wit, remarking that what he liked best about Italy was "spaghetti and Levi-Civita." Yet this choice of mathematical framework wasn't inevitable. It was, in many ways, a historical accident—the only suitable mathematics available in 1912. Torsion-based formulations of gravity, which might have served equally well or better, simply didn't exist in the mathematical landscape at that time.

Today, over a century later, we continue to use this Riemannian formulation almost exclusively. Graduate students learn that gravity means curved spacetime, period. Major textbooks devote perhaps three pages out of thirteen hundred to alternative geometric approaches. Yet these alternatives aren't mere mathematical curiosities—they're completely equivalent descriptions of gravity that solve critical problems plaguing modern physics and offer fresh insights into quantum gravity and cosmology. The marginalization of these approaches reveals a profound truth about scientific progress:

What if some of our most fundamental theories are shaped more by historical accident than physical necessity? And what if those fateful historical decisions ultimately impact our ability to make future progress?

Two Ways to Describe Reality (Both Equally Valid)

The standard picture of gravity that every physics student learns treats spacetime as a curved manifold, where matter and energy bend the fabric of space and time. This curvature, described by Riemannian geometry, tells objects how to move—they follow the straightest possible paths (geodesics) through curved spacetime. It's an elegant picture that has served us well for over a century.

But there's another way to describe gravity that's equally valid yet fundamentally different in conception. In torsion-based approaches, spacetime can remain flat while possessing a property called torsion—a kind of twisting that affects how vectors change as they're transported through space. Where Riemannian geometry uses symmetric connections (mathematical objects that tell us how to compare vectors at different points), torsion theories use antisymmetric connections that introduce this twist.

The most developed of these alternatives is teleparallel gravity, ironically pioneered by Einstein himself in 1928 during his failed attempts at creating a unified field theory. In teleparallel gravity, the curvature of spacetime is exactly zero—space is perfectly flat in the Riemannian sense. Instead, gravity arises from torsion, which can be visualized as a field of "frame dragging" that affects how objects move through flat space. The field equations look different, the conceptual picture is radically altered, yet every prediction matches general relativity exactly.

This isn't an approximation or simplification. Teleparallel Equivalent General Relativity (TEGR) is a completely equivalent description that enables different insights. Where general relativity struggles to define gravitational energy (leading to awkward "pseudo-tensors" that aren't properly covariant), teleparallel gravity provides a clean, well-defined gravitational energy-momentum tensor. Where Einstein's geometric picture eliminates gravity locally through the equivalence principle, the teleparallel picture treats gravity as a true force field that can't be eliminated by coordinate choices.

Modern developments have pushed these ideas even further. For decades, we’ve known that GR cannot be the final word on gravity due to is incompatibility with quantum field theory as well as it’s tendency to result in physically-nonsensical singularities. In attempts to reconcile GR with the quantum world, physicists have explored many modifications to GR. However, most of these are formulated using the same Riemannian geometry. These modifications attempt to do things like: explain cosmic acceleration without invoking dark energy, explain galactic rotation dynamics without dark matter, explain how singularities can be avoided, etc.

It is in these attempts to modify gravity where the story gets interesting—and where historical accidents might have profound consequences.

The Ghost That Haunts Modern Physics

For fifty years, physicists have known that modifying general relativity is a dangerous game. The moment you move beyond Einstein's simple field equations to more complex functions of curvature—say, $f(R) = R + \alpha R^2$ and you unleash mathematical demons. The field equations jump from second-order differential equations (involving second derivatives of the metric) to fourth-order equations. This isn't just mathematical complexity; it's a fundamental pathology.

The problem, known as Ostrogradsky instability, was first identified in classical mechanics but extends to field theory with devastating consequences. Fourth-order equations introduce extra degrees of freedom—ghost modes—that carry negative kinetic energy. In quantum theory, these ghosts destroy unitarity, probability conservation, and the entire framework of quantum field theory. As Richard Woodard emphasized, Ostrogradsky's theorem represents "the strongest restriction on what sorts of interacting local quantum field theories can describe fundamental physics."

The mathematical structure is unforgiving. In f(R) gravity, the field equations take the form:

$$f'(R)R_{\mu\nu} - \frac{1}{2}f(R)g_{\mu\nu} - \nabla_\mu\nabla_\nu f'(R) + g_{\mu\nu}\Box f'(R) = \kappa T_{\mu\nu}$$

That $\Box f'(R)$ term—involving fourth derivatives—is the ghost's calling card. It creates unbounded Hamiltonians where energy can become arbitrarily negative, opening a Pandora's box of instabilities.

Yet here's the remarkable twist in our story: torsion theories avoid this catastrophe entirely. The Weitzenböck connection used in teleparallel gravity has zero curvature by construction. When you build f(T) modifications, the torsion scalar contains only first derivatives of the fundamental fields:

$$T = \frac{1}{4}T^{\rho}_{\mu\nu}T_{\rho}^{\mu\nu} + \frac{1}{2}T^{\rho}_{\mu\nu}T^{\nu\mu}_{\rho} - T_\mu T^\mu$$

No matter how complicated your function f(T), the field equations remain resolutely second-order. No ghosts. No instabilities. No violation of unitarity. As Ferraro and Fiorini noted, "The field equations are always second order, remarkably simpler than f(R) theories."

As I began studying this topic more deeply, I couldn’t help but wonder whether our 'unsolvable' problems are actually artifacts of our chosen frameworks rather than true representations of physical limitations.

For fifty years, the ghost problem has restricted theoretical exploration of modified gravity. Countless papers have been written trying to exorcise these ghosts from f(R) theories through various mechanisms—adding extra fields, invoking quantum corrections, or restricting to special cases. Yet a simple change of geometric framework—using torsion instead of curvature—eliminates the problem at its root.

The Systematic Suppression of Alternatives

Open any standard general relativity textbook and count the pages devoted to torsion. Misner, Thorne, and Wheeler's encyclopedic 1,300-page "Gravitation"—the bible of the field—mentions it briefly in a few paragraphs. Weinberg's field theory approach dismisses it in three pages. Carroll's popular graduate text emphasizes that "the torsion never enters" standard calculations. Wald treats it as an unnecessary mathematical curiosity.

This isn't a conspiracy—it's the natural tendency of established systems to self-perpetuate. Once a particular approach becomes standard, powerful forces maintain its dominance. Professors teach what they learned. Textbooks reflect existing curricula. Research funding flows to established areas. Journal editors and referees, trained in the standard approach, may view alternatives with skepticism. Young researchers, needing jobs and tenure, follow well-worn paths rather than risking their careers on marginalized approaches.

The network effects are profound. A graduate student learning general relativity has access to dozens of excellent textbooks, thousands of worked problems, and a global community of experts—all focused on the curvature-based approach. That same student seeking to learn teleparallel gravity finds a handful of specialized reviews, must piece together knowledge from scattered papers, and joins a much smaller community of researchers.

Yet pockets of resistance persist and grow. The University of Malta has become a center for teleparallel gravity research under Jackson Levi Said's leadership. Brazilian institutions, particularly the Instituto de Física Teórica at UNESP, maintain strong research programs building on decades of foundational work by J.G. Pereira and collaborators. Research groups in Estonia, Germany, and Greece advance both theoretical foundations and observational tests. The annual Teleparallel Gravity Workshop series, now in its fifth iteration, brings together a growing international community.

The research output tells a story of accelerating interest. Papers on torsion-based gravity have increased dramatically over the past two decades. Special journal issues devoted to teleparallel gravity appear regularly. The NSF's Gravitational Physics program explicitly includes "new formulations for theoretical and computational gravitational physics" in its funding priorities. Major conferences now include sessions on alternative geometric formulations.

Again, none of this is a conspiracy. It's simply the natural tendency of established systems to self-perpetuate. Breaking this cycle requires conscious effort, institutional support, and researchers willing to risk working outside the mainstream. The history of science shows that such breaks, while difficult, eventually occur when the advantages become undeniable.

What This Teaches Us About Innovation

The marginalization of torsion gravity fits a troubling pattern in scientific history. Gregor Mendel's laws of inheritance, published in 1866, lay dormant for 35 years before their rediscovery launched modern genetics. Alfred Wegener's continental drift hypothesis, proposed in 1912, was ridiculed for fifty years before plate tectonics vindicated his insights. Barbara McClintock's discovery of genetic transposition in the 1940s was ignored for decades before earning her a Nobel Prize at age 81.

In each case, the pattern is the same: a revolutionary framework exists, often for decades, before the scientific community is ready to embrace it. The barriers aren't always empirical. Mendel had data, Wegener had evidence, McClintock had experiments. Rather, these ideas didn't fit the conceptual frameworks, mathematical tools, or theoretical prejudices of their time.

Torsion gravity's history follows this template with eerie precision. Élie Cartan introduced torsion into general relativity in 1922, just seven years after Einstein's original formulation. Einstein himself spent years exploring teleparallel gravity. Yet these approaches remained curiosities, mathematical exercises disconnected from mainstream physics. Only now, as we grapple with dark energy, quantum gravity, and the limitations of modified theories, do their advantages become apparent.

For modern innovators, this history poses uncomfortable questions: How many 'impossible' problems in your field might be artifacts of unquestioned assumptions? In physics, we've spent decades struggling with ghosts in modified gravity theories, never questioning whether the curvature-based framework itself created these problems. In other fields, similar conceptual lock-ins may be hiding solutions in plain sight.

The courage required to question "solved" territories cannot be understated. It means risking career prospects on unfashionable approaches. It means developing expertise in marginalized areas with limited resources. It means facing skepticism from established experts who've built careers on standard approaches. Yet history shows that precisely these explorations, revisiting foundations with fresh tools and perspectives, often yield the greatest breakthroughs.

The Deeper Principle

The pattern extends far beyond physics. Darwin sat on his theory of evolution for twenty years, knowing it contradicted established religious and scientific doctrine. When he finally published, half his evidence came from pigeon breeding—hardly the grand sweep of nature we now associate with evolutionary theory. He worked with what was acceptable and available, even as he understood the broader implications.

In geology, the idea that continents could move was literally unthinkable without a mechanism. Wegener had compelling evidence—matching fossils across oceans, jigsaw-puzzle coastlines, similar rock formations on different continents. But without understanding seafloor spreading and plate tectonics, the geological establishment dismissed him as a fantasist. The conceptual framework didn't exist to properly evaluate his evidence.

These historical examples reveal a deeper principle about how knowledge advances. Revolutionary frameworks often require not just new ideas but new mathematical tools, new instruments, or new conceptual vocabularies. Einstein couldn't formulate general relativity without Riemannian geometry. Quantum mechanics needed matrix algebra and Hilbert spaces. Modern genetics required understanding DNA's structure.

Perhaps most importantly, competence in the orthodox approach can blind us to revolutionary alternatives. The very expertise that makes someone a leading authority can create conceptual blinders. Those deepest in a paradigm often have the most to lose from its overthrow. This is why revolutionary insights often come from outsiders, young researchers, or those working across disciplinary boundaries.

The polymath advantage is real. Torsion gravity's recent renaissance has been driven partly by researchers comfortable crossing boundaries—between differential geometry and gauge theory, between classical gravity and quantum field theory, between pure mathematics and observational cosmology. These boundary crossers see connections invisible to specialists, apply tools from one field to problems in another, and question assumptions that disciplinary experts take for granted.

A Call to Intellectual Courage

For students entering physics—or any field—the lesson is clear: Don't just learn the standard approach—understand why it became standard.

What historical accidents shaped your field's foundations? What mathematical tools existed when core theories were developed? What alternatives were proposed but abandoned, and why? The answers often reveal as much about sociology and historical contingency as about physical necessity.

For established researchers, the challenge is both harder and more rewarding: What mathematical or theoretical tools exist now that didn't when your field's foundations were laid? Modern mathematics offers structures undreamed of in Einstein's time—category theory, noncommutative geometry, quantum information theory. Which of these might reframe "impossible" problems in your field?

Consider how torsion approaches might have changed physics history. Without the ghost problem, exploration of modified gravity theories could have proceeded decades earlier. Quantum gravity research might have taken different paths with teleparallel formulations available from the start. Cosmologists might have had better tools for understanding dark energy. We can't rerun history, but we can learn from it.

For innovators in any field, I would encourage you to remember that the next breakthrough might not require new discovery, but rediscovery with new tools. Those marginalized approaches, dismissed alternatives, and roads not taken deserve periodic reexamination. What seemed impossible or unworkable with yesterday's tools might be exactly what today's challenges require.

This isn't an argument for abandoning successful theories or chasing every alternative. General relativity remains one of humanity's greatest intellectual achievements, and the curvature-based formulation has proven extraordinarily successful. Rather, it's a call for intellectual humility and openness. When we mistake historical contingency for physical necessity, when we confuse mathematical convenience for fundamental truth, we limit our ability to progress.

In physics, as in all fields, the historical road not taken might not have been rejected…it might simply have not been seen or understood at the time.

Torsion gravity waited a century for its moment, which is just now starting to come into its own. Mendel's genetics waited 35 years. Continental drift waited 50. In each case, the ideas were there, waiting for minds ready to grasp their significance and tools adequate to their development.

What roads not taken exist in your field? What foundations might be reexamined with fresh eyes and modern tools? What assumptions do we mistake for necessities? These questions aren't just academic exercises. The courage to ask them, and the persistence to pursue the answers, is what separates incremental progress from transformative breakthroughs.

Sometimes the solution isn't ahead of us, waiting to be discovered. Sometimes it's behind us, on a road we didn't take, waiting to be rediscovered and understood anew.

If you’d like to learn more about torsion-based approaches to gravity, I put this primer together along with recommended reading materials: https://www.glennclayton.me/torsion