Beyond Relativity: Exploring the Limits of Spacetime

by

Author: Denis Avetisyan

A new study delves into Carroll symmetries-a non-Lorentzian framework-to uncover surprising connections between field theory, gravity, and the potential foundations of quantum gravity.

This review examines Carroll symmetries, their implications for defining Carroll black holes, and their relationship to ‘tantum gravity’ as a pathway to understanding asymptotic symmetries and quantum gravity.

The pursuit of a consistent theory of quantum gravity necessitates exploring non-Lorentzian geometries and symmetries beyond those traditionally considered. This thesis, ‘Carroll symmetries in field theory and gravity’, investigates the multifaceted implications of Carroll symmetry-a non-Lorentzian contraction of Poincaré symmetry-across field theory and gravity, revealing a universal sector governing commutators in Carroll quantum field theories and establishing connections to holography through anomalies mirroring asymptotic symmetries in three-dimensional gravity. We define Carroll black holes-characterized by Carroll extremal surfaces rather than event horizons-and demonstrate a non-vanishing asymptotic energy density-the Carroll-Hawking effect-suggesting novel thermodynamic properties. Could these findings, particularly within the ‘tantum gravity’ limit, offer a valuable pathway toward understanding quantum gravity in regimes inaccessible through conventional approaches?

Beyond Relativity: Exploring the Limits of Spacetime

The persistent challenges in unifying quantum mechanics and general relativity necessitate innovative theoretical approaches, as conventional methods often break down when confronted with the extreme conditions predicted by black holes or the very early universe. Existing attempts at quantum gravity frequently encounter intractable infinities or lose predictive power at the Planck scale. Consequently, physicists are increasingly turning to simplified models – those that retain the essential physics while stripping away unnecessary complexities – to gain fresh perspectives. These models aren’t simply mathematical tricks; they represent a deliberate strategy to isolate fundamental principles and explore the boundaries of known physics, potentially revealing hidden connections and paving the way for a more complete theory of quantum gravity. By focusing on these tractable scenarios, researchers hope to circumvent the limitations of traditional methods and unlock new insights into the nature of spacetime and gravity.

Carrollian spacetime presents a fascinating departure from conventional physics by considering the limit where the speed of light approaches zero – a concept that, while seemingly paradoxical, provides a powerful new framework for exploring gravity. This isn’t simply a mathematical trick; reducing c → 0 fundamentally alters the relationships between space and time, effectively decoupling them and nullifying Lorentz invariance – the principle that dictates the laws of physics are the same for all observers in uniform motion. Consequently, gravitational phenomena are viewed through a drastically different lens, emphasizing spatial relationships over temporal dynamics and revealing symmetries otherwise obscured in standard relativistic scenarios. This unique perspective allows physicists to investigate the fundamental nature of spacetime and potentially uncover connections to areas like condensed matter physics and non-relativistic quantum mechanics, offering a novel pathway to address challenges in quantum gravity.

The transition to a Carrollian spacetime, where the speed of light approaches zero, is far from a purely abstract mathematical exercise. This limit exposes a surprising wealth of underlying symmetries previously obscured by the conventional framework of Lorentz invariance. Specifically, it highlights a powerful connection between electromagnetism and gravity – in Carrollian space, the roles of electric and magnetic fields become interchanged, revealing a duality that suggests a deeper, unified description of these fundamental forces. Moreover, this seemingly extreme limit demonstrates a surprising relationship to Galilean spacetime, the foundation of classical mechanics, indicating that the principles governing low-energy physics may be intrinsically linked to the behavior of gravity in highly unusual regimes. This interconnectedness isn’t simply a curiosity; it offers a new lens through which physicists can explore the fundamental nature of spacetime and potentially resolve long-standing problems in quantum gravity by uncovering previously hidden mathematical structures and physical principles.

Investigations into Carrollian spacetime, a theoretical construct where the speed of light approaches zero, are increasingly viewed as a pathway toward a deeper comprehension of spacetime’s fundamental nature. This isn’t simply an exercise in mathematical abstraction; by pushing the limits of established physics, researchers are uncovering hidden symmetries and unexpected connections to fields like condensed matter physics and non-relativistic gravity. The resulting framework offers a novel lens through which to examine the very structure of spacetime, potentially revealing that what appears as a rigid, four-dimensional continuum is, in reality, an emergent property of more fundamental, underlying principles. This approach suggests that exploring extreme limits – even those seemingly divorced from everyday experience – can provide crucial insights into the architecture of the universe and the laws governing its behavior, offering a complementary perspective to traditional approaches to quantum gravity.

Field Theories in a Zero-Light-Speed Universe: A Formal Approach

Constructing field theories on Carrollian manifolds necessitates modifications to standard approaches due to the underlying non-Lorentzian kinematic structure. Unlike Minkowski spacetime, where c represents a finite speed of light, Carrollian spacetimes are defined by the limit c \rightarrow 0. This alteration fundamentally changes the dispersion relation, resulting in spatial and temporal roles being reversed compared to conventional relativistic field theory. Consequently, standard techniques for constructing Lagrangian densities, defining momentum and energy, and analyzing symmetries require careful adaptation to accommodate this altered kinematic regime; specifically, the usual distinction between space and time breaks down, necessitating a re-evaluation of causality and locality principles.

The Stress-Energy Tensor, denoted as T^{\mu\nu}, is a second-order tensor that describes the density and flux of energy and momentum in spacetime. Within Carrollian field theories, which model spacetimes where the speed of light approaches zero, the Stress-Energy Tensor takes on particular importance due to the altered causal structure and kinematic constraints. Its components represent the energy density T^{00}, momentum density T^{0i}, and stresses T^{ij} within the field. Crucially, it serves as the source term in the equations of motion for gravity in these non-Lorentzian settings, dictating how energy and momentum influence the geometry of the Carrollian manifold. Analysis of T^{\mu\nu} is therefore essential for understanding the dynamics and conservation laws governing fields in a zero-light-speed universe, and for determining the appropriate generalizations of General Relativity to this novel spacetime.

Ward identities in Carrollian field theories are derived from the requirement that physical observables remain invariant under continuous symmetries of the system. These identities relate the variation of a field’s correlation function with a variation of the action under an infinitesimal symmetry transformation. Specifically, they express a constraint on the form of vertex functions, ensuring that the theory is consistent and free from anomalies. Crucially, these identities allow for the systematic calculation of conserved quantities, such as charge and momentum, even in the absence of Lorentz invariance. The derivation relies on integrating by parts and utilizing the equations of motion, ultimately connecting the symmetry transformations to the conservation laws within the Carrollian spacetime. A general form for a Ward identity can be expressed as \in t d^d x \langle T(x) \delta \phi(x) \rangle = 0 , where T(x) represents a conserved current and \delta \phi(x) is an infinitesimal field variation.

The CarrollSwifton model represents a scalar field theory constructed as a zero-light-speed limit of a conventional relativistic field theory. This limit, denoted as c \rightarrow 0, results in a non-Lorentzian spacetime known as a Carrollian manifold. In this model, spatial and temporal derivatives decouple, leading to a dispersion relation where all modes propagate at infinite speed. The resulting field equations exhibit first-order time derivatives, contrasting with the second-order equations of relativistic scalar fields, and the model’s dynamics are governed by a Hamiltonian that is non-local in time. Consequently, the CarrollSwifton model serves as a tractable example for investigating the behavior of fields and symmetries in a zero-light-speed universe, providing insights into the challenges and modifications necessary when formulating field theories in non-Lorentzian spacetimes.

Carrollian Black Holes: Redefining the Event Horizon

Standard black hole solutions, described by the Schwarzschild or Kerr metrics, are characterized by event horizons which define a one-way membrane for spacetime. Carrollian black holes, however, lack this traditional event horizon due to the infinite speed of light in the Carrollian limit. Consequently, a new geometric object, the Carroll extremal surface, is required to demarcate the region from which nothing can escape. This surface is not a null surface as in the standard case, but rather a spatial hypersurface with specific properties dictated by the Carrollian geometry. Its location defines the effective boundary of the black hole and is crucial for determining the black hole’s thermodynamic characteristics and observable effects, replacing the event horizon as the key feature defining the black hole’s causal structure.

Carrollian black holes are distinguished by the Carroll extremal surface, which functions as the analogue of the event horizon in standard black hole solutions. This surface demarcates the region of spacetime from which neither matter nor radiation can escape, effectively defining the black hole’s “boundary.” Crucially, the location and properties of this Carroll extremal surface directly determine the black hole’s thermodynamic characteristics, including its temperature and entropy. Variations in the surface’s geometry influence the black hole’s energy and dictate the rate at which it interacts with surrounding spacetime, establishing a direct link between geometric properties and observable thermodynamic behavior. The surface’s area is proportional to the black hole’s entropy, mirroring the Bekenstein-Hawking area law, while its curvature is related to the black hole’s energy content.

The CarrollHawkingEffect predicts the emission of radiation from Carrollian black holes, analogous to Hawking radiation in standard general relativity. This effect arises directly from the black hole’s geometry and results in a non-vanishing asymptotic energy density that decreases proportionally to 1/r^2, where ‘r’ is the radial distance from the black hole. Unlike standard Hawking radiation which is a quantum effect, the CarrollHawkingEffect stems from the altered causal structure of Carrollian spacetime, leading to a unique mechanism for energy emission and a distinct asymptotic energy density profile. The predicted energy density indicates a persistent radiative flux emanating from the black hole’s event horizon.

The prediction of a Carroll-Hawking effect in Carrollian black holes introduces a conceptual conflict with the Schwinger effect, which describes pair production due to strong electric fields. Unlike the Schwinger effect, where particle creation is driven by a field strength, the Carroll-Hawking effect arises from the geometry of the Carrollian black hole itself, resulting in an asymptotic energy density proportional to 1/r^2. This difference highlights a unique mechanism for particle creation in Carrollian spacetime. The Carroll-Hawking temperature, defined as T = 1 / (2πr_s) for a black hole with a Schwarzschild radius r_s, further distinguishes this effect and implies a non-standard thermal spectrum for emitted particles compared to standard Hawking radiation.

Symmetries and the Infinite-Dimensional Universe: A Deeper Connection

The fundamental symmetries governing Carrollian spacetime, a theoretical construct where the speed of light approaches zero, are elegantly described by the Carroll Algebra. This algebra isn’t a completely new mathematical structure, but rather a ‘contraction’ of the more familiar Poincaré algebra – the foundation for understanding special relativity. Imagine taking the Poincaré algebra and systematically reducing the influence of boosts – transformations relating different inertial frames – until only translations and rotations remain dominant; this process yields the Carroll Algebra. Consequently, Carrollian spacetime exhibits symmetries reflecting these remaining transformations, highlighting a fascinating connection to flat spacetime but with a drastically altered causal structure. \text{Carroll Algebra} \approx \text{Poincaré Algebra}_{c \rightarrow 0} This mathematical relationship suggests that exploring Carrollian symmetry may provide unique insights into the nature of spacetime and gravity, particularly in scenarios where traditional relativistic assumptions break down.

The symmetries governing Carrollian spacetime are not isolated; they exhibit a profound connection to the Bondi-Metzner-Sachs (BMS) group, an infinite-dimensional symmetry group fundamental to the study of flat spacetime. The BMS group describes the asymptotic symmetries – those preserved at spatial and temporal infinity – of spacetime, revealing transformations that leave the gravitational field unchanged at great distances. This infinite dimensionality is critical, as it suggests that flat spacetime possesses far more symmetries than previously understood, hinting at a richer structure than implied by the finite-dimensional Poincaré group. Consequently, the relationship between Carrollian symmetries and the BMS group provides a novel framework for investigating the fundamental properties of flat spacetime and exploring potential connections to concepts like gravitational charges and memory effects, potentially reshaping how physicists approach the boundaries of spacetime and the information they contain.

The surprising relationship between Carrollian spacetime-a non-relativistic limit where light speed approaches zero-and flat spacetime suggests a profound interconnectedness at the heart of physics. This link isn’t merely mathematical; it hints at the possibility of novel holographic dualities. Holography, traditionally linking gravity in a volume to a quantum theory on its boundary, may find a new expression through this connection, potentially allowing researchers to map gravitational phenomena in flat spacetime to a dual description in a Carrollian framework, or vice versa. Such a duality would represent a significant departure from conventional holographic approaches, offering a fresh perspective on the fundamental relationship between spacetime geometry and quantum information, and potentially resolving long-standing puzzles in both gravity and quantum field theory. The exploration of these symmetries opens avenues for understanding how information is encoded and retrieved from spacetime itself, challenging conventional notions of locality and dimensionality.

Investigations into the symmetries of spacetime, particularly when approached through the limit of an infinitely large Planck constant, are beginning to suggest a profound interplay between gravity, quantum mechanics, and the fundamental nature of information. This unconventional approach-effectively ‘turning off’ quantum effects-reveals symmetries normally obscured by quantum fluctuations, exposing a structure where gravitational symmetries become infinite-dimensional. Such symmetries aren’t merely mathematical curiosities; they hint at a deeper connection between the geometry of spacetime and the storage and retrieval of information, potentially providing a novel framework for understanding black hole entropy or even the emergence of spacetime itself from quantum entanglement. The resulting mathematical structures, like the Carroll Algebra and the BMS group, may offer crucial insights into holographic duality, suggesting that the description of a gravitational system can be encoded on a distant boundary, much like a hologram-a concept with implications for both cosmology and quantum gravity.

The exploration of Carroll symmetry, as detailed within the article, necessitates a careful consideration of limiting cases and the assumptions embedded within any theoretical framework. It’s a reminder that models, even those elegantly describing phenomena like Carroll black holes and their thermodynamics, are fundamentally compromises between knowledge and convenience. As Stephen Hawking observed, “Intellectuals begin by denouncing what they don’t understand.” This sentiment perfectly encapsulates the need for rigorous scrutiny when venturing into non-Lorentzian geometries and their implications for quantum gravity via ‘tantum gravity’. The pursuit of asymptotic symmetries, therefore, isn’t about finding the ‘optimal’ model, but about systematically dismantling those that fail under increasingly precise examination.

Where Do We Go From Here?

The exploration of Carroll symmetry, as presented, offers a peculiar lens through which to view the established frameworks of field theory and gravity. It’s tempting to speak of ‘insights’ gained, but a more honest assessment reveals a proliferation of questions. The construction of Carroll black holes, while mathematically elegant, demands a physical interpretation that remains frustratingly elusive. Are these merely mathematical curiosities, or do they correspond to actual, albeit exotic, physical regimes? The thermodynamic properties, so carefully delineated, beg for a statistical mechanical foundation – a challenge that seems particularly acute given the non-Lorentzian nature of the underlying geometry.

The connection to ‘tantum gravity’ – a limit, not a full theory – feels less like a resolution and more like a shift in the landscape. It offers a potential avenue for relating ultraviolet and infrared physics, but at the cost of embracing a highly specific, and perhaps unphysical, scaling. The role of swiftons, these peculiar massless modes, requires further investigation. Are they genuine degrees of freedom, or artifacts of the chosen limit?

Ultimately, this work serves as a potent reminder: a model isn’t a mirror of reality, it’s a mirror of its maker. The real progress lies not in confirming existing beliefs, but in rigorously testing the boundaries of this non-Lorentzian framework. The next step isn’t to find what fits, but to discover what breaks. Only then can one begin to approach a more robust, and perhaps less comfortable, understanding.

Original article: https://arxiv.org/pdf/2603.12902.pdf

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

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2026-03-16 18:04