Emergent Modified Gravity

• Emergent modified gravity is a framework that reconstructs the observable metric from the structure functions of constraint algebras rather than assuming it as fundamental.
• It modifies the standard ADM approach by achieving on-shell covariance and preserving classical second-order dynamics without introducing extra propagating gravitational modes.
• Applications in spherical symmetry, cosmology, and Gowdy systems demonstrate its utility in explaining nonsingular black holes, cosmic bounces, and matter couplings.

Emergent modified gravity is a canonical, generally covariant framework in which the space-time metric is not taken as a fundamental field of the action but is reconstructed from the constraint algebra and covariance conditions. Its defining move is to distinguish the fundamental phase-space variables of gravity from the emergent metric that governs observable geometry. In this setting, the inverse spatial metric is identified with the structure function in the Poisson bracket of two Hamiltonian constraints, and the full line element is obtained only after anomaly freedom and on-shell covariance have been imposed. This construction enlarges the class of generally covariant modifications beyond action-based approaches, while preserving the classical derivative order and, in the formulations described so far, without introducing extra propagating gravitational degrees of freedom (Bojowald et al., 2024, Bojowald et al., 2023, Bojowald et al., 18 Jul 2025).

1. Conceptual basis and canonical definition

Emergent modified gravity departs from the standard assumption that the metric, or a tetrad directly equivalent to it, is the primary field in a local covariant action. In the spherically symmetric formulation, for instance, the canonical configuration variables are $e_1(x,t)$ and $e_2(x,t)$ with conjugate momenta $k_1$ and $k_2$, while the metric that determines geometry is emergent rather than fundamental (Bojowald et al., 2024). In cosmological and triad formulations, the phase space is built instead from densitized triads and their conjugates, such as $E^a_i$ and $K_a^i$, again without identifying these directly with the observable space-time metric (Bojowald et al., 18 Jul 2025).

The canonical starting point is the hypersurface-deformation algebra. In the classical ADM form, the Hamiltonian and diffeomorphism constraints satisfy

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$

$$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$

The crucial observation is that the structure function in the $\{H,H\}$ bracket, classically $q^{ab}$, need not coincide with the inverse of any fundamental configuration variable once canonical covariance is analyzed without assuming a prior space-time metric (Bojowald et al., 2023, Bojowald et al., 2024).

This is the mechanism by which the standard uniqueness results are evaded. The framework described in "Emergent modified gravity: Covariance regained" and "Emergent Modified Gravity" states that new generally covariant theories arise because one drops the assumption that the spatial metric equals the canonical configuration variable and instead reconstructs an emergent inverse metric from the modified structure function (Bojowald et al., 2023, Bojowald et al., 2024). A closely related synthesis in "Emergent field theory" extends the same logic to gauge fields, defining emergent fields as space-time tensors reconstructed from first-class constraint algebras and covariance conditions rather than postulated as fundamental fields (Duque, 22 Jul 2025).

A common misconception is to identify emergent modified gravity with Verlinde’s emergent gravity. The two are distinct paradigms. The former is a canonical, constraint-based construction of generally covariant modified gravity; the latter is a dark-matter-alternative weak-field proposal with a MOND-like spherical formula that depends on $e_2(x,t)$0 and $e_2(x,t)$1, and which has been tested separately against galaxy rotation curves and Solar System perihelia (Hees et al., 2017). "Emergent field theory" explicitly lists Verlinde’s emergent gravity as a different paradigm (Duque, 22 Jul 2025).

2. Covariance from the constraint algebra

The core technical claim of emergent modified gravity is that first-classness alone is not sufficient for space-time covariance. The metric reconstructed from the modified structure function must transform on shell as a space-time tensor under the gauge flow generated by the constraints. In the general canonical setting, lapse and shift transform off shell as

$e_2(x,t)$2

and the on-shell covariance condition is

$e_2(x,t)$3

with the gauge functions mapped to coordinate displacements by

$e_2(x,t)$4

in the cosmological perturbative formulation, or the corresponding spherical form in the reduced models (Bojowald et al., 2023, Bojowald et al., 18 Jul 2025).

The nontrivial content lies in the spatial metric sector. "Emergent modified gravity: Covariance regained" derives conditions requiring that the gauge flow of the candidate emergent metric be independent, on shell, of spatial derivatives of the normal gauge generator. In schematic form,

$e_2(x,t)$5

which ensures that the canonical gauge transformation of the emergent metric matches its Lie derivative (Bojowald et al., 2023). This is why several anomaly-free effective Hamiltonians that close under Poisson brackets are nevertheless non-covariant in the stronger emergent-modified-gravity sense (Belfaqih et al., 2024).

In spherically symmetric vacuum gravity, one robust covariant construction uses a phase-space dependent linear combination of the classical Hamiltonian and diffeomorphism constraints. The modified smearing takes the form

$e_2(x,t)$6

and covariance fixes

$e_2(x,t)$7

with $e_2(x,t)$8 (Bojowald et al., 2024). The emergent radial metric component is then determined by the modified structure function rather than by $e_2(x,t)$9.

This reconstruction also clarifies why the resulting theories are not of higher-curvature form in the usual action sense. The modified dynamics are encoded in the structure function and the gauge flow, not in higher time derivatives of a fundamental metric. The framework retains the classical derivative order, and the papers repeatedly emphasize the absence of Ostrogradsky instabilities in the resulting second-order canonical theories (Bojowald et al., 2024, Bojowald et al., 2023, Bojowald et al., 18 Jul 2025).

3. Explicit realizations in spherical symmetry, cosmology, and Gowdy systems

The best-developed realizations are symmetry-reduced, but they already exhibit a wide range of nontrivial covariant effects. In spherical symmetry, the classical line element is written as

$k_1$0

with

$k_1$1

in one triad convention, or $k_1$2, $k_1$3 in another (Bojowald et al., 2023, Bojowald et al., 2024). The emergent metric replaces the classical radial component by $k_1$4, where $k_1$5 comes from the modified $k_1$6 bracket. A representative generalized emergent radial metric is

$k_1$7

showing explicit dependence on extrinsic curvature through $k_1$8 (Bojowald et al., 2024).

These spherical models support covariant bounded-curvature black-hole geometries and, depending on the branch, signature change. "Emergent modified gravity: Covariance regained" gives the emergent line element in the form

$k_1$9

making explicit that a change in sign of the structure function induces a change in effective signature (Bojowald et al., 2023). "Emergent field theory" reports nonsingular black-hole solutions, including $k_2$0-charged cases with a cosmological constant, as well as modified quasinormal-mode and evaporation properties in related work (Duque, 22 Jul 2025).

Cosmological realizations were initially restricted to homogeneous models, but "Perturbative emergent modified gravity on cosmological backgrounds: Kinematics" extends the framework to perturbative inhomogeneity on a spatially flat FLRW background (Bojowald et al., 18 Jul 2025). The background fields are

$k_2$1

and the background line element is

$k_2$2

The modified bracket takes the form

$k_2$3

with

$k_2$4

and the emergent line element to linear order is

$k_2$5

(Bojowald et al., 18 Jul 2025). The allowed perturbative scalar modifications are encoded in functions $k_2$6, $k_2$7, and $k_2$8, subject to explicit closure and covariance relations such as

$k_2$9

A further extension to a model with one local propagating gravitational degree of freedom is given by the polarized Gowdy $E^a_i$0 system (Bojowald et al., 2024). There the canonical variables are $E^a_i$1, $E^a_i$2, and $E^a_i$3, and a preferred anisotropy variable is

$E^a_i$4

Covariance allows periodic, holonomy-like dependence only through this anisotropy-weighted combination, not through separate $E^a_i$5 or $E^a_i$6, and forbids area-dependent periodic arguments of the standard $E^a_i$7-type (Bojowald et al., 2024). The linear-combination class is controlled by

$E^a_i$8

with structure function

$E^a_i$9

(Bojowald et al., 2024). In homogeneous limits, this class yields nonsingular Kasner-like bounces for $K_a^i$0, while $K_a^i$1 can produce signature change.

4. Matter and gauge-field couplings

Matter coupling is a stringent test of the framework because covariance must survive once non-gravitational sectors are included. In spherical symmetry with a scalar field, the canonical pairs are $K_a^i$2, $K_a^i$3, and $K_a^i$4, with

$K_a^i$5

(Bojowald et al., 2023). A minimal coupling proposal replaces the classical radial metric by the emergent one in the matter Hamiltonian,

$K_a^i$6

and the paper proves that, if the vacuum modified constraints are anomaly-free and $K_a^i$7 is scalar-independent, the hypersurface-deformation algebra is preserved (Bojowald et al., 2023).

This scalar-coupled theory yields a broad class of second-order modified gravity–scalar systems. It also shows that matter can affect geometry kinematically, not only dynamically: in general, the emergent metric may depend on scalar variables as well as gravitational ones (Bojowald et al., 2023). Physical requirements such as current conservation, existence of a mass observable, partial Abelianization, and various classical limits further organize the solution space. A notable outcome is that some singularity-removal mechanisms found in vacuum are unstable under minimal scalar coupling, while alternative singularity-free classes persist even with bounded matter (Bojowald et al., 2023).

Perfect fluids have likewise been coupled canonically in spherical symmetry (Duque, 2023). The matter Hamiltonian is

$K_a^i$8

with

$K_a^i$9

and the fluid stress tensor recovered from the canonical system is

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$0

(Duque, 2023). This matter coupling underlies the Oppenheimer–Snyder collapse model generalized to emergent modified gravity.

Electromagnetism introduces an additional distinction between fundamental and emergent fields. "Emergent electromagnetism" defines the emergent electric field $$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$1, which enters the Lorentz force and the strength tensor, as distinct from the canonical momentum $$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$2 (Duque, 2024). In four-dimensional Hamiltonian electromagnetism with a topological $$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$3 term,

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$4

and the covariance analysis of the modified theory yields

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$5

as the emergent relation (Duque, 2024). The same work couples emergent electromagnetism to emergent modified gravity in spherical symmetry and obtains nonsingular charged black-hole solutions in which a bounded charge function is crucial for removing singularities and excluding (super)extremal black holes (Duque, 2024).

A broader synthesis appears in "Emergent field theory," which extends the emergent construction to Einstein–Yang–Mills systems. There, emergent Yang–Mills electric components are determined by the Hamiltonian through relations such as

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$6

and spherical $$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$7-charged black holes with a cosmological constant are explicitly constructed (Duque, 22 Jul 2025).

5. Physical implications and representative phenomena

The framework has been used to analyze strong-field geometry, cosmological dynamics, and the relation to loop-quantum-gravity-inspired effective models. In spherical symmetry, vacuum modified Hamiltonians with holonomy-like dependence on $$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$8 produce bounded-curvature black-hole models. "Emergent modified gravity: The perfect fluid and gravitational collapse" applies this to Oppenheimer–Snyder collapse with dust. In the Gullstrand–Painlevé gauge, the modified star radius reaches a minimum

$$\{D[M^a], D[N^a]\} = D[\mathcal{L}_M N^a],\qquad \{H[N], D[M^a]\} = H[\mathcal{L}_M N],$$9

with

$$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$0

for dust from spatial infinity, so the collapsing configuration bounces and re-expands (Duque, 2023). The vacuum geometry is regular at the minimum-radius surface, whereas with dust the emergent geometry is singular there even though the canonical fields remain finite. This permits a meaningful continuation of the canonical solution and opens multiple non-unique global completions, including wormhole formation and black-to-white-hole transition scenarios (Duque, 2023).

The scalar-coupled spherical theory sharpens this point. It identifies three classes of covariant modified gravity–scalar theories and shows that the classical-matter-limit and classical-constraint-surface-limit classes can reintroduce divergences at the maximum-curvature surface, whereas a distinct singularity-free class keeps the Ricci scalar finite even with bounded scalar matter (Bojowald et al., 2023). This makes matter stability a central discriminator among apparently similar vacuum models.

Cosmological applications proceed in two directions. First, the homogeneous limit of the spherically symmetric theory produces new expansion–shear couplings. For Kantowski–Sachs-type cosmologies with scale factors $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$1 and $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$2, the expansion and shear are

$$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$3

and the modified Hamiltonian constraint acquires $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$4-dependent $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$5–$$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$6 couplings and terms involving $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$7 and $$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$8 (Bojowald et al., 2024). Second, the perturbative FLRW formulation of (Bojowald et al., 18 Jul 2025) provides the kinematical basis for detailed analyses of inflationary or bouncing early-universe models without higher-time-derivative instabilities.

The polarized Gowdy model supplies the first inhomogeneous system with a local gravitational wave degree of freedom (Bojowald et al., 2024). It shows that emergent modified gravity can accommodate covariant anisotropy-dependent holonomy functions, preserve second-order dynamics, and in a large subset of the classical-$$\{H[N], H[M]\} = D\!\left[q^{ab}(N\partial_b M - M\partial_b N)\right].$$9-limit class give equal propagation speeds for the local gravitational wave and a minimally coupled massless scalar on the same emergent background, provided $\{H,H\}$0 (Bojowald et al., 2024). This is directly relevant to multimessenger constraints.

Several papers frame these results as lessons for effective loop quantum gravity. "Lessons for loop quantum gravity from emergent modified gravity" argues that anomaly freedom is necessary but not sufficient, and that covariance of the emergent structure function rules out or unifies previously distinct holonomy schemes (Belfaqih et al., 2024). In particular, strictly periodic dependence on the relevant momentum is compatible with covariance only for constant $\{H,H\}$1, while scale-dependent $\{H,H\}$2 can be absorbed by canonical transformations at the price of additional non-periodic terms. The same paper states that $\{H,H\}$3- and $\{H,H\}$4-type schemes are canonically equivalent within the broader emergent-modified-gravity class (Belfaqih et al., 2024).

A more speculative long-range application is MOND-like behavior without extra fields. "Emergent Modified Gravity" gives a representative choice

$\{H,H\}$5

which produces logarithmic corrections to the emergent radial metric and can yield intermediate-scale MOND-like phenomenology in a covariant theory (Bojowald et al., 2024). A plausible implication is that emergent modified gravity can reproduce aspects of modified Newtonian dynamics through the emergent metric itself rather than through additional scalar or vector sectors, but the paper also notes the possibility of large-scale signature change in such constructions (Bojowald et al., 2024).

6. Relations to other approaches, limitations, and open problems

Emergent modified gravity is consistently contrasted with action-based modified gravities. The papers state that it is not an $\{H,H\}$6, Horndeski/DHOST, EFT-of-dark-energy, TeVeS, or Einstein–Æther theory, because it does not begin with a fundamental space-time metric in a local covariant action (Bojowald et al., 2024, Bojowald et al., 18 Jul 2025). It also differs from higher-curvature models because the canonical theories described here preserve the classical derivative order and do not introduce higher-than-classical time derivatives (Bojowald et al., 2023, Bojowald et al., 18 Jul 2025). This is the main reason they avoid Ostrogradsky instabilities while still allowing nontrivial covariant modifications.

The strongest current limitation is scope. Most explicit results are still obtained in symmetry-reduced settings: spherical symmetry, polarized Gowdy, homogeneous cosmology, and perturbative inhomogeneity on a flat FLRW background (Bojowald et al., 2023, Bojowald et al., 2024, Bojowald et al., 18 Jul 2025). Several papers identify the extension to general $\{H,H\}$7 systems as an open problem, even though the canonical construction and covariance principles are already formulated abstractly (Bojowald et al., 2023, Duque, 22 Jul 2025). Matter generalization beyond scalars, perfect fluids, and symmetry-reduced gauge sectors also remains incomplete, with fermions explicitly listed as an open issue in "Emergent field theory" (Duque, 22 Jul 2025).

Another limitation is phenomenology. The cosmological perturbation paper deliberately restricts itself to kinematics and covariance, leaving full dynamical equations and explicit Mukhanov–Sasaki-type systems for later work (Bojowald et al., 18 Jul 2025). The spherical papers provide exact solutions and structural results, but detailed observational predictions for shadows, ringdowns, primordial spectra, or collapse transients are generally deferred (Duque, 2023, Duque, 22 Jul 2025). "Emergent field theory" mentions modified quasinormal modes, evaporation, and MOND-like long-range effects as applications, but often points to related work for explicit calculations (Duque, 22 Jul 2025).

There are also internal consistency issues that function as research drivers rather than failures. Matter can destabilize vacuum singularity-resolution mechanisms (Bojowald et al., 2023). Some covariant branches permit signature change, so the causal interpretation becomes region-dependent (Bojowald et al., 2023, Bojowald et al., 2024). Partial Abelianization simplifies quantization but removes the structure function and therefore the direct space-time interpretation until one transforms back (Bojowald et al., 2023). These are not accidental side effects; they follow from the canonical logic of the framework.

Finally, the terminology itself creates ambiguity. The phrase “emergent gravity” is used in several unrelated literatures, including Verlinde-type dark-sector phenomenology, thermodynamic derivations of Einstein equations, geometric-flow constructions, and emergent-quantum-mechanics scenarios. Within the canonical program surveyed here, emergent modified gravity specifically denotes the reconstruction of the observable metric from the structure functions of a first-class constraint algebra and on-shell covariance conditions (Bojowald et al., 2024, Bojowald et al., 2023). This narrower definition is what unifies the vacuum, matter-coupled, cosmological, Gowdy, electromagnetic, and Yang–Mills developments reviewed above.