Tantum Gravity Limit: Carrollian Triple Scaling

• Tantum Gravity Limit is a Carrollian triple-scaling framework in quantum gravity where c, Gₙ, and ħ are rescaled to isolate a tractable regime with finite black-hole thermodynamics.
• It maintains fixed quantities like Gₘ (Gₙ c⁻⁴) and κ (ħ c) to ensure that key black-hole parameters such as the Schwarzschild radius, Hawking temperature, and entropy remain finite.
• The term also appears in various gravitational limit models, serving as an editorial umbrella for asymptotic procedures that include maximum-force bounds, weak-field limits, and other decoupling schemes.

Searching arXiv for recent and primary sources on “Tantum Gravity” and related usages of “Tantum Gravity Limit.” “Tantum Gravity Limit” is not used uniformly across the gravitational literature. In its most literal recent usage, it denotes a triple-scaling limit of quantum gravity in which $c \to 0$, $G_N \to 0$, and $\hbar \to \infty$, while $G_M := G_N c^{-4}$ and $\kappa := \hbar c$ remain finite; this construction is called “tantum gravity” and is presented as a Carrollian, semi-classical sector in which black-hole thermodynamics remains finite (Ecker et al., 2024). In the supplied literature, however, the same label is also attached to several non-equivalent constructions, including the maximum force $c^4/(4G)$, decoupling limits, weak-field limits, limiting energy density in Riemann–Cartan cosmology, and observational upper bounds on gravitational anisotropy or graviton mass. This suggests that the expression is presently best treated as a context-dependent designation rather than a single standardized term.

1. Terminological scope and status

The papers that explicitly use the term “tantum gravity” define it through a Carrollian triple scaling. The 2024 Letter formulates the limit as $\hbar \to \infty$, $G_N \to 0$, $c \to 0$, with $G_M \equiv G_N c^{-4}$ and $G_N \to 0$0 fixed, and argues that the laws of black hole thermodynamics survive this limit (Ecker et al., 2024). The 2026 thesis uses the same structure, writing $G_N \to 0$1 and $G_N \to 0$2, and refers to the resulting theory as a specific scaling limit useful for understanding quantum gravity in a Carrollian regime (Ecker, 13 Mar 2026).

The broader supplied corpus uses “Tantum Gravity Limit” more elastically. In some cases it is an imposed name for a previously established principle, such as the maximum force or maximum tension principle of general relativity; in other cases it is mapped onto a test-field limit, a weak-gravity linearization, a $G_N \to 0$3 compactification limit, or a large-$G_N \to 0$4 asymptotic regime. One supplied source states explicitly that the paper “does not use the phrase ‘Tantum Gravity Limit’” and that the corresponding concept is instead called the “test-field limit” or “coupling-free limit” (Magnano et al., 2018). Taken together, these materials indicate that the term functions more as an editorial or interpretive umbrella than as a settled technical label.

A recurring commonality is the use of limiting procedures to isolate a tractable sector of gravity. In the maximum-force program, limits organize the Bronshtein cube of theories; in tantum gravity proper, a distinguished corner of the Bronstein cube is selected by requiring finite black-hole thermodynamics; in other works, the relevant limit may be decoupling, weak-field, large-$G_N \to 0$5, or limiting-density. This suggests a family resemblance grounded in asymptotic control rather than a single invariant definition (Schiller, 2023).

2. Triple-scaling tantum gravity

In the Carrollian formulation, the tantum gravity limit is implemented by

$G_N \to 0$6

with $G_N \to 0$7 and $G_N \to 0$8 fixed (Ecker, 13 Mar 2026). The same scaling is presented in the Letter with $G_N \to 0$9 and $\hbar \to \infty$0 held fixed (Ecker et al., 2024). These choices are singled out because the characteristic black-hole quantities remain finite: $\hbar \to \infty$1, the Schwarzschild radius $\hbar \to \infty$2, the Hawking temperature $\hbar \to \infty$3, and the Bekenstein–Hawking entropy (Ecker, 13 Mar 2026).

Within this scaling, the standard Planck units reorganize in a distinctive way. The supplied summary of the Letter states that the Planck length stays finite, the Planck time diverges, the Planck mass diverges, and the Planck energy stays finite (Ecker et al., 2024). For Schwarzschild black holes at fixed energy $\hbar \to \infty$4, the radius becomes $\hbar \to \infty$5, the temperature remains finite as

$\hbar \to \infty$6

and the entropy remains finite as

$\hbar \to \infty$7

with $\hbar \to \infty$8 (Ecker et al., 2024). The surface gravity itself tends to zero, but the combination entering the Hawking temperature remains finite because $\hbar \to \infty$9 diverges while $G_M := G_N c^{-4}$0 vanishes in a compensating way.

The Letter further argues that imposing finiteness of $G_M := G_N c^{-4}$1, $G_M := G_N c^{-4}$2, and $G_M := G_N c^{-4}$3 removes the faces and edges of the Bronstein cube, leaving precisely two antipodal corners, tantum gravity and its dual $G_M := G_N c^{-4}$4, as the surviving limits preserving all of black-hole thermodynamics (Ecker et al., 2024). The thesis makes the same point in closely related language, stating that TG and its antipodal dual are unique corners preserving thermodynamics, whereas all single-scaling faces and most edges do not (Ecker, 13 Mar 2026). This gives the limit a special role among asymptotic reductions of gravity.

3. Carrollian kinematics and gravitational dynamics

The kinematical backbone of tantum gravity is the Carroll contraction of the Poincaré algebra. Starting from the standard commutators of rotations $G_M := G_N c^{-4}$5, boosts $G_M := G_N c^{-4}$6, spatial translations $G_M := G_N c^{-4}$7, and time translations $G_M := G_N c^{-4}$8, the ultra-relativistic limit $G_M := G_N c^{-4}$9 yields the Carroll algebra, in which $\kappa := \hbar c$0 becomes central, the boosts commute, and boosts no longer generate spatial translations (Ecker, 13 Mar 2026). With a cosmological constant retained, the contraction produces the $\kappa := \hbar c$1-Carroll algebra, which differs from Poincaré despite an isomorphism at the vector-space level.

Curved Carroll manifolds are formulated by gauging a kinematical Klein pair $\kappa := \hbar c$2 with Cartan connection

$\kappa := \hbar c$3

The geometric data are a degenerate spatial metric $\kappa := \hbar c$4 and a Carroll vector $\kappa := \hbar c$5 obeying

$\kappa := \hbar c$6

The Carroll extrinsic curvature is

$\kappa := \hbar c$7

and the Carroll scalar curvature entering magnetic Carroll gravity is

$\kappa := \hbar c$8

These structures encode the collapse of the Lorentzian light cone into Carrollian causal geometry (Ecker, 13 Mar 2026).

In the $\kappa := \hbar c$9 limit of ADM gravity, the Hamiltonian and momentum constraints reduce to a Carroll first-class algebra. Two distinct Carroll theories then emerge. Electric Carroll gravity, obtained at fixed $c^4/(4G)$0, has a purely kinetic Hamiltonian constraint and is ultra-local. Magnetic Carroll gravity, obtained at fixed $c^4/(4G)$1, has a purely potential Hamiltonian constraint and is the classical approximation of tantum gravity. Its Cartan action is

$c^4/(4G)$2

and, after integrating out spin connections with minimal torsion, the second-order form becomes

$c^4/(4G)$3

This magnetic theory is the operative classical sector in which Carroll black holes and their thermodynamics are analyzed (Ecker, 13 Mar 2026).

4. Carroll black holes, thermodynamics, and the quantum sector

Because the Lorentzian light cones collapse in the Carroll limit, notions tied to null propagation, such as standard event horizons, are no longer available in their usual form. The thesis therefore defines Carroll black holes as massive vacuum solutions with well-defined thermodynamic properties and a Carroll extremal surface rather than an event horizon (Ecker, 13 Mar 2026). In two dimensions, the extremal surface is characterized by

$c^4/(4G)$4

where $c^4/(4G)$5 is the dilaton. In the first-order description, this locus is also described as the fixed point of Carroll boosts.

Two-dimensional Carroll dilaton gravity provides the explicit laboratory for these constructions. Its first-order action is

$c^4/(4G)$6

with the standard two-derivative potential

$c^4/(4G)$7

The second-order form is

$c^4/(4G)$8

For linear dilaton vacua, one defines $c^4/(4G)$9 and the Casimir

$\hbar \to \infty$0

The thermodynamic quantities then take compact exact forms: $\hbar \to \infty$1 For Carroll–JT, $\hbar \to \infty$2, $\hbar \to \infty$3, $\hbar \to \infty$4, and the thermodynamics is $\hbar \to \infty$5, $\hbar \to \infty$6, $\hbar \to \infty$7, $\hbar \to \infty$8. For Carroll–Schwarzschild in the spherical reduction family, the thermodynamic scalings are $\hbar \to \infty$9, $G_N \to 0$0, $G_N \to 0$1, and $G_N \to 0$2 (Ecker, 13 Mar 2026).

The quantum and semi-classical formulation is organized through the partition function

$G_N \to 0$3

with $G_N \to 0$4. A boundary counterterm is required for finiteness and for a well-defined variational principle,

$G_N \to 0$5

leading to an on-shell free energy

$G_N \to 0$6

From this one recovers

$G_N \to 0$7

in agreement with the first-law derivation (Ecker, 13 Mar 2026).

The same thesis further reports a Carroll–Hawking effect: for the Carroll–Schwarzschild black hole, quantum matter induces a non-vanishing asymptotic energy density in the vacuum. At the level of symmetry algebra, Carroll quantum field theory on curved Carroll backgrounds also yields universal equal-time commutators for the Carroll stress tensor; in two dimensions, a Carroll boost anomaly contributes Schwinger-like terms that reproduce the familiar central extensions of asymptotic symmetries of three-dimensional asymptotically flat Einstein gravity, including the centrally extended $G_N \to 0$8 algebra (Ecker, 13 Mar 2026). This places tantum gravity in direct contact with flat-space holography.

5. Other meanings attached to “Tantum Gravity Limit”

Outside the Carrollian triple-scaling program, the same label is attached in the supplied literature to several unrelated limit constructions and bounds.

Usage in the supplied literature	Defining statement	Representative result
Maximum force / maximum tension principle	Universal bound on gravitational force or tension	$G_N \to 0$9 (Schiller, 2023)
Gravitational anisotropy bound in PPN	Violation of local position invariance via Whitehead parameter $c \to 0$0	$c \to 0$1, $c \to 0$2 (Shao et al., 2013)
Yukawa graviton-mass bound	Finite-range gravity constrained by dipole convergence	$c \to 0$3, $c \to 0$4 (Loeb, 2024)
$c \to 0$5 decoupling limit of dRGT massive gravity	Nontrivial Stückelberg vacua with raised strong-coupling scale	$c \to 0$6 (Rham et al., 2016)
Test-field limit of metric nonlinear gravity	Coupling-free limit for extra modes in higher-derivative gravity	Exists for $c \to 0$7; generic 8 DOF models fail (Magnano et al., 2018)
Thermodynamic limit of “atoms of space”	Gravity as thermodynamic limit of microscopic spacetime DOF	$c \to 0$8 for all null $c \to 0$9 (Padmanabhan, 2015)
Limiting-density regime in Riemann–Cartan cosmology	Maximal energy density where gravity becomes repulsive	$G_M \equiv G_N c^{-4}$0, ultrarelativistic $G_M \equiv G_N c^{-4}$1 (Minkevich, 2021)
Weak-gravity limit in Newer GR	Linearization around Minkowski in symmetric teleparallel gravity	$G_M \equiv G_N c^{-4}$2, $G_M \equiv G_N c^{-4}$3, “three halves” new dynamical modes (Golovnev et al., 2024)
$G_M \equiv G_N c^{-4}$4 compactified Gauss–Bonnet limit	KK reduction plus $G_M \equiv G_N c^{-4}$5	4D Horndeski-type theory with finite entropy (Lu et al., 2020)
Large-$G_M \equiv G_N c^{-4}$6 dimensionally continued gravity	$G_M \equiv G_N c^{-4}$7 with $G_M \equiv G_N c^{-4}$8 fixed	Interactions persist and $G_M \equiv G_N c^{-4}$9 at leading order (Giribet, 2013)
Quantum–gravity clock-stability bound	Joint quantum and gravitational lower bound on clock instability	$G_N \to 0$00 (Sinha et al., 2014)

Among these alternative usages, the maximum-force formulation is especially close in spirit to the “limit-principle” language. In that program, the gravitational limit on the general-relativity edge of the Bronshtein cube is

$G_N \to 0$01

with the same factor $G_N \to 0$02 propagating to equivalent bounds such as $G_N \to 0$03 and $G_N \to 0$04. Combining $G_N \to 0$05, $G_N \to 0$06, and $G_N \to 0$07 then yields a minimum length

$G_N \to 0$08

a minimum area $G_N \to 0$09, and the entropy relation $G_N \to 0$10 (Schiller, 2023). This is conceptually close to tantum gravity proper only in the broad sense that both reorganize gravity around invariant limits.

Several of the other interpretations are observational or model-selective rather than universal. The PPN anisotropy result constrains a possible violation of local position invariance of gravity; the 2MASS dipole argument constrains Yukawa-screened massive-graviton theories; the clock-stability result gives a quantum–gravity lower bound in terrestrial or astrophysical gravitational fields. None of these is equivalent to the Carrollian triple scaling, even though each is presented as a “limit” on gravity in a precise technical sense (Shao et al., 2013).

6. Conceptual issues, exclusions, and open directions

In tantum gravity proper, the principal conceptual shift is that black holes are no longer defined by causal horizons. The Letter states that usual notions based on causal propagation, such as event horizons, are ill-posed in the Carrollian setting, and that “Carroll black holes” are instead defined thermodynamically and via Carroll extremal surfaces (Ecker et al., 2024). The thesis develops this replacement in detail, using holonomy and Gauss–Bonnet smoothness conditions to fix the inverse temperature and employing Noether charges to define energy and entropy (Ecker, 13 Mar 2026). This is a substantive departure from standard semiclassical gravity, not merely a rescaling of parameters.

The supplied works also emphasize several limitations. In the Letter, the evaporation power scales to zero in the tantum limit, so Hawking temperature remains finite while the radiative power tends to zero; Page time is therefore suggested to tend to infinity in the naive scaling analysis (Ecker et al., 2024). Canonical-ensemble subtleties remain, as in ordinary Schwarzschild thermodynamics with negative specific heat, and the usual remedies—placing the system in a finite cavity or in AdS—are still required. The same source notes that rotating black holes are more subtle and that the Carrollian limit of Kerr remains open.

Other uses of the label carry their own scope conditions. The Yukawa graviton-mass bound applies only to theories in which the static potential is well described by a Yukawa screening with a single range $G_N \to 0$11; it is not a generic bound on all massive-gravity frameworks (Loeb, 2024). The compactified $G_N \to 0$12 Gauss–Bonnet construction is explicitly distinguished from the naive Glavan–Lin prescription, because the latter leads to divergences and lacks a well-defined local four-dimensional action, whereas the compactified construction produces a finite Horndeski-type action and finite Wald entropy (Lu et al., 2020). In the test-field analysis of metric nonlinear gravity, the failure of a proper coupling-free limit in generic 8 DOF models is presented as a serious drawback relative to the restricted 7 DOF or 3 DOF subclasses (Magnano et al., 2018).

A further set of exclusions appears in the maximum-force and minimum-length program. There, the minimum length and limit-cube framework are used to argue against measurable trans-Planckian structures, exact continuum manifolds at arbitrarily small scales, point-based discretizations, additional continuous or discrete spacetime symmetries requiring trans-Planckian resolution, and extra elementary particles beyond the Standard Model (Schiller, 2023). These are claims internal to that program rather than consequences of tantum gravity in the Carrollian sense, but they illustrate how the rhetoric of “limit” often functions as a mechanism for theory selection.

The overall picture is therefore plural. In the strictest current sense, the tantum gravity limit is the Carrollian triple scaling $G_N \to 0$13, $G_N \to 0$14, $G_N \to 0$15 with $G_N \to 0$16 and $G_N \to 0$17 fixed, yielding a nontrivial gravitational theory with finite black-hole thermodynamics, a magnetic Carroll classical sector, Carroll black holes, and a controlled saddle-point partition function (Ecker et al., 2024). More broadly, the supplied literature uses the same phrase for a heterogeneous set of asymptotic regimes, bounds, and decoupling procedures. A plausible implication is that any encyclopedia treatment must distinguish sharply between tantum gravity as a specific Carrollian limit and the wider family of editorially labeled “Tantum Gravity Limit” constructions that share only the abstract strategy of defining gravity through a limiting principle.