Beyond Einstein: Clocks and Waves Hunt for Warped Spacetime

Author: Denis Avetisyan

A new study explores how atomic clocks and gravitational wave detectors could reveal subtle distortions in the fabric of spacetime beyond the predictions of general relativity.

This review investigates the potential of using atomic clocks and gravitational waves to probe non-metricity within the framework of Weyl geometry and modified gravity theories.

While general relativity has consistently passed observational tests, its complete validity and the nature of gravity at its most fundamental level remain open questions. This motivates exploration of modified gravity theories, and in ‘Atomic clocks and gravitational waves as probes of non-metricity’ we investigate the phenomenological consequences of spacetime non-metricity within the framework of Weyl geometry. We demonstrate that high-precision atomic clocks and gravitational-wave observations provide complementary avenues for detecting and constraining deviations from Riemannian geometry, revealing a backreaction on gravitational-wave propagation due to dynamical non-metric degrees of freedom. Given current gravitational-wave data already places stringent limits on these effects, can future, even more sensitive experiments, definitively reveal the presence of non-metricity in the universe?

Beyond Fixed Spacetime: Questioning the Foundations of Gravity

General Relativity, the prevailing theory of gravity, rests upon a foundational premise: the metric of spacetime remains constant. This metric dictates how distances and time intervals are measured, essentially defining the geometry of the universe. A constant metric implies that when an object is moved from one point to another without rotation – a process called parallel transport – its length remains unchanged. This assumption has enabled remarkably accurate predictions, from the bending of light around massive objects to the existence of black holes. However, the unwavering constancy of the metric is not necessarily a given; it is an assumption built into the mathematical framework. Current research exploring the unification of gravity with quantum mechanics, alongside observations hinting at possible variations in fundamental constants over cosmic time, prompt scientists to question whether this fundamental property truly holds under all conditions, pushing the boundaries of gravitational theory beyond the established limits of Einstein’s model.

The enduring success of General Relativity hinges on the premise of a constant spacetime metric, yet accumulating evidence subtly challenges this cornerstone. Precise cosmological observations, coupled with the persistent difficulty of unifying gravity with the quantum realm, suggest that fundamental constants – values once believed immutable – may, in fact, exhibit slight variations across vast cosmic distances and timescales. This isn’t to invalidate Einstein’s theory, but rather to indicate its potential incompleteness as a description of reality at its most extreme scales. The search for quantum gravity, a theory that would seamlessly blend general relativity with quantum mechanics, consistently points towards the necessity of exploring physics beyond the confines of a fixed spacetime geometry, demanding investigations into scenarios where the very fabric of space and time is dynamic and evolving.

The enduring success of General Relativity hinges on the assumption of a fixed spacetime metric, defining distances and angles consistently throughout the universe. However, growing theoretical challenges and subtle observational anomalies are prompting physicists to consider scenarios where this metric isn’t constant, but can vary-a concept known as non-metricity. In non-metric theories, the length of a vector transported parallel to itself changes depending on the path taken, fundamentally altering the geometric understanding of gravity. This departure from Riemannian geometry, the foundation of General Relativity, necessitates a more general framework-one capable of describing a universe where the very fabric of spacetime is dynamic and responsive to the distribution of matter and energy. Exploring non-metric gravity isn’t simply tweaking Einstein’s equations; it’s venturing into a potentially radical restructuring of how gravity operates at the most fundamental level, potentially offering a path towards reconciling gravity with the perplexing realm of quantum mechanics and explaining observed phenomena beyond the reach of current models.

The pursuit of gravitational theories beyond Einstein’s General Relativity demands a radical shift in how spacetime itself is conceived. While General Relativity elegantly describes gravity as the curvature of spacetime, it fundamentally assumes a fixed metric – a consistent rule for measuring distances. However, if fundamental constants vary or a quantum theory of gravity is to emerge, this metric can no longer be considered absolute. This necessitates a geometric framework where lengths aren’t preserved when an object is moved – a process known as parallel transport. Imagine carrying a precisely measured rod along a path; in non-metric spacetime, the rod’s length would subtly change upon returning to its starting point. Describing this phenomenon requires utilizing mathematical tools like torsion and non-metricity, extending beyond the familiar Riemannian geometry and opening up a vast landscape of potential gravitational interactions – and potentially, a deeper understanding of the universe’s fundamental laws.

Weyl Geometry: A Framework for Dynamic Spacetime

Weyl Geometry utilizes a generalized affine connection, denoted as $\Gamma_{\mu\nu\lambda}$ , to define spacetime, differing from Riemannian geometry where the connection is solely determined by the metric tensor. This allows for spacetime where the metric tensor $g_{\mu\nu}$ is not necessarily constant, and is independent of the affine connection; that is, $\nabla_{\lambda} g_{\mu\nu} \neq 0$ . The connection is thus not restricted by the condition of metric compatibility, and introduces the concept of non-metricity, quantifying the failure of parallel transport to preserve lengths. This geometric framework enables the description of spacetime where distances between points are not fixed, but can vary depending on the path taken, and is mathematically formulated through tensors derived from the generalized affine connection.

The Weyl Connection, fundamental to Weyl Geometry, extends the concept of a standard affine connection by allowing for non-metricity, quantified by the Weyl Gauge Field. This field, denoted $W_{\mu}$ , is a vector field that directly measures the failure of the metric tensor to be covariantly constant with respect to the connection. Specifically, non-metricity is defined through the relation $\nabla_{\mu}g_{\nu\lambda} = W_{\mu}g_{\nu\lambda}$ , where $\nabla_{\mu}$ represents the covariant derivative. A zero Weyl Gauge Field indicates a metric-compatible connection, reverting to Riemannian geometry, while a non-zero field signifies a spacetime where lengths are not preserved under parallel transport, and therefore, scales are relative rather than absolute.

The Weyl Curvature Tensor, denoted as $W_{\mu\nu\rho\sigma}$ , represents the tidal forces within Weyl geometry and fundamentally describes the gravitational field. Derived from the Weyl connection and its associated derivatives, this tensor is constructed to be conformally invariant, meaning it remains unchanged under local scale transformations of the metric. Unlike the Riemann curvature tensor in standard General Relativity, the Weyl tensor specifically captures the trace-free component of curvature, representing the gravitational degrees of freedom associated with distortions of spacetime that do not involve overall volume changes. In Weyl geometry, all gravitational effects are attributed to this tensor, as the Ricci tensor and scalar curvature are identically zero by construction, simplifying the field equations and focusing solely on conformal distortions of spacetime.

Weyl geometry fundamentally alters the traditional understanding of spacetime by allowing for scale variations in measured lengths. Unlike Riemannian geometry where the metric tensor defines a fixed relationship between points, Weyl geometry introduces a scale factor that can change from point to point. This means the distance between two events is not absolute but depends on the spacetime location itself, manifesting as intrinsic length variations. These variations are mathematically described by the Weyl Gauge Field and are not artifacts of coordinate choice; they are inherent properties of the spacetime geometry. Consequently, physical laws within this framework must be formulated to be invariant under local Weyl transformations – rescaling of the metric – ensuring consistency despite these varying length scales. $g_{\mu\nu} \rightarrow \Omega^2(x) g_{\mu\nu}$ , where $\Omega(x)$ is the scale factor.

Probing Non-Metricity: Theoretical Models and Observational Signatures

Investigations into non-metric gravity often involve extensions of established gravitational theories. Specifically, Quadratic Gravity, a modification of General Relativity incorporating higher-order curvature terms, can be augmented by the introduction of the Weyl Gauge Field. This field, associated with the non-metricity tensor, represents a departure from the assumption of metric compatibility in the gravitational interaction. By coupling the Weyl field to the quadratic curvature terms, researchers aim to develop a theoretical framework that predicts observable deviations from standard General Relativity, potentially explaining phenomena not currently accounted for within the standard model. The resulting modified gravity theories then allow for the exploration of non-metricity’s impact on gravitational phenomena.

The Proca equation, a generalization of the Maxwell equations for massive vector bosons, serves as the foundational equation of motion for the Weyl Gauge Field when considering non-metric gravity. This equation, expressed generally as $\partial_\mu F^{\mu\nu} + m^2 A^\nu = 0$ where $F^{\mu\nu}$ is the field strength tensor, $A^\nu$ is the vector potential, and m represents the mass of the gauge field, establishes the relationship between the field’s curvature and its mass. In the context of non-metricity, the Proca equation links the Weyl Gauge Field to potential sources driving the non-metric character of spacetime, effectively providing a dynamical description of how these sources influence the gravitational interaction. Solving the Proca equation under specific boundary conditions allows for the determination of the Weyl Gauge Field’s behavior and its contribution to observable phenomena.

Non-metricity, a deformation of the metric tensor, predicts deviations from the standard $\Lambda CDM$ model’s expectations for Gravitational Wave (GW) propagation. Specifically, it introduces a dependence of GW speed on both frequency and polarization. This manifests as a dispersion in GW signals-different frequencies travel at slightly different speeds-and birefringence, where the speed depends on the polarization state. These effects would alter the waveforms observed by detectors like LIGO and Virgo, causing shifts in arrival times and changes in signal amplitude. Analysis of GW events allows for stringent tests of these predictions, constraining the magnitude of non-metricity parameters and providing a potential avenue for detecting new physics beyond General Relativity.

Current gravitational wave detectors, such as LIGO and Virgo, constrain the parameter space of non-metric gravity theories by limiting the strength of the Weyl coupling and the associated background field. Analysis of detector data places an upper bound of ≤ 10^-69 GeV on the product of the square of the Weyl coupling constant α and the background field $ω̄₀$ . This constraint arises from the predicted variations in gravitational wave propagation caused by non-metricity. Planned upgrades to existing detectors, alongside the development of new facilities like the Einstein Telescope and Cosmic Explorer, are projected to improve sensitivity by several orders of magnitude, potentially allowing for significantly tighter constraints or even detection of non-metricity signatures in future observations.

Hunting for the Weyl Field: Observational Prospects

The Weyl field, a geometric entity describing spacetime distortion beyond standard gravity, manifests through a measurable quantity known as its flux. This flux isn’t merely a theoretical construct; it directly correlates with the degree of non-metricity – a deviation from the fundamental assumption that spacetime distances remain constant under parallel transport. Crucially, this provides a concrete observational target for physicists seeking to test the limits of Einstein’s general relativity. Detecting variations in this flux would indicate the presence of non-metricity, offering potential insights into phenomena where general relativity falls short, such as the behavior of dark energy or the reconciliation of gravity with quantum mechanics. The magnitude of this flux, though anticipated to be incredibly small, represents a tangible signature of a fundamentally different aspect of spacetime geometry, awaiting detection by increasingly sensitive instruments.

The search for the Weyl field benefits significantly from advances in atomic clock technology, as these devices possess an extraordinary capacity for discerning minute fluctuations in spacetime. Unlike traditional gravitational wave detectors which measure changes in distance, atomic clocks directly measure the flow of time itself; even the slightest variation in the local gravitational potential, induced by the Weyl field, manifests as a detectable shift in the clock’s frequency. Current atomic clocks achieve fractional frequency instabilities on the order of $10^{-{18}}$ , providing the sensitivity needed to probe the subtle spacetime distortions predicted by non-metric gravity. Future generations of optical lattice clocks promise even greater precision, potentially unveiling the Weyl field’s influence and offering a novel pathway to test fundamental aspects of gravity beyond Einstein’s general relativity.

Future gravitational wave detectors, leveraging advancements in laser interferometry and seismic isolation, are poised to probe spacetime with unprecedented sensitivity. These next-generation ground-based observatories aim to achieve a maximum detectable strain of $h_{min} \sim 10^{-{25}}$ , a threshold that would enable rigorous examination of non-metricity in the gravitational field. This level of precision surpasses current capabilities by several orders of magnitude, opening a new window for testing theories beyond general relativity. By meticulously analyzing subtle fluctuations in spacetime, researchers hope to either confirm the existence of non-metricity – a key prediction of certain modified gravity models – or place extremely stringent limits on its magnitude, thereby refining our understanding of the fundamental nature of gravity and potentially revealing pathways towards a unified theory.

The confirmation of non-metricity in the fabric of spacetime would represent a paradigm shift in physics, fundamentally altering established gravitational theories. Current understanding, largely based on Einstein’s general relativity, assumes a metric field describing the geometry of spacetime, but non-metricity proposes a connection that doesn’t rely solely on this metric. Detecting this deviation could resolve long-standing inconsistencies between general relativity – which excels at describing gravity on large scales – and quantum mechanics, governing the universe at the subatomic level. A theory incorporating non-metricity offers a potential pathway toward a unified framework, suggesting gravity isn’t simply a curvature of spacetime but arises from a more fundamental property linked to how distances are measured, potentially manifesting as additional degrees of freedom and influencing the behavior of gravity at extremely high energies or densities – conditions prevalent in the very early universe and within black holes.

The study meticulously dissects spacetime, seeking deviations from established Riemannian geometry through the lens of Weyl geometry and non-metricity. It posits that gravitational waves and atomic clocks offer distinct yet complementary avenues for detecting these subtle distortions. This pursuit of simplification-reducing complex gravitational phenomena to measurable effects-resonates with a timeless observation: “Knowledge is power.”, as stated by Francis Bacon. The research embodies this principle by attempting to distill the intricacies of modified gravity into observable quantities, effectively empowering future investigations with the ability to probe the fundamental nature of spacetime. The core concept of utilizing both gravitational waves and atomic clocks as independent probes highlights a commitment to rigorous verification, mirroring Bacon’s emphasis on empirical evidence.

Where to Next?

The pursuit of modified gravity, predictably, encounters a wall. Current frameworks, including explorations of non-metricity within Weyl geometry, largely attempt to graft novelty onto established structures. The value lies not in creating ever-more-complex models, but in identifying the simplest possible deviation from General Relativity that aligns with observation. Atomic clocks and gravitational waves, as this work demonstrates, offer complementary avenues for such scrutiny, yet their combined potential remains largely untapped. A truly decisive test requires precision not merely in measurement, but in the theoretical formulation of what constitutes a detectable signal.

The reliance on Weyl geometry, while mathematically elegant, presents a philosophical question. Is this a fundamental property of spacetime, or simply a convenient mathematical tool? The answer, of course, is irrelevant to the mathematics itself. However, a continued focus on a priori geometrical structures risks obscuring potentially more fundamental, physically motivated deviations. The field must resist the temptation to prioritize mathematical beauty over empirical necessity.

Future progress necessitates a shift in perspective. Instead of asking how non-metricity might manifest, the question becomes: what observations would definitively disprove it? A negative result, rigorously obtained, is often more illuminating than a positive one, especially when dealing with increasingly speculative theories. The goal is not to confirm a hypothesis, but to eliminate possibilities, leaving only the essential.

Original article: https://arxiv.org/pdf/2601.19407.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Beyond Fixed Spacetime: Questioning the Foundations of Gravity

Weyl Geometry: A Framework for Dynamic Spacetime

Probing Non-Metricity: Theoretical Models and Observational Signatures

Hunting for the Weyl Field: Observational Prospects

Where to Next?

See also: