Spatially Covariant Gravity (SCG)

• SCG is a class of gravity theories constructed via the ADM decomposition, characterized by invariance under time-dependent spatial transformations rather than full spacetime diffeomorphisms.
• Its Hamiltonian analysis reveals a modified constraint structure that generically propagates two tensor modes and one scalar graviton, with additional degrees of freedom emerging when lapse derivatives are dynamical.
• SCG serves as a unifying framework linking unitary-gauge scalar-tensor theories, beyond-Horndeski models, and cosmological applications through rigorous constraint management and operator classification.

Searching arXiv for core SCG papers and closely related developments. arxiv_search(query="^^^^^^^^2all:\2 covariant gravity\"^^^^^^^^2 OR ti:\2^^^^^^^^"spatially covariant gravity\"", max_results=^^^^2 OR ti:\2all:\2^^^^) arxiv_search(query="(&&&^^^^2all:\2^^^^&&&) OR (&&&^^^^2 OR ti:\2^^^^&&&) OR (Gao et al., 2018) OR (Fujita et al., 2015) OR (Gao et al., 2020) OR (Hu et al., 2021) OR (Zhu, 2022) OR (Zhu et al., 2022) OR (Yu et al., 2024) OR (Wang et al., 2024) OR (&&&^^^^2 OR ti:\2all:\2^^^^&&&) OR (&&&^^^^2 OR ti:\2 OR ti:\2^^^^&&&) OR (&&&^^^^2 OR ti:\22^^^^&&&)", max_results=2^^^^2all:\2^^^^) Spatially Covariant Gravity (SCG) denotes a broad class of gravity theories formulated on a preferred foliation, built directly from the PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^ variables of the ADM decomposition and invariant only under time-dependent spatial coordinate transformations rather than the full spacetime diffeomorphism group of general relativity (GR). In the canonical formulation developed for a wide class of such theories, the decisive structural difference from GR is that once the lapse PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^ enters the Hamiltonian nonlinearly, the primary and secondary constraints associated with $N$ become second class, and the theory generically propagates three physical modes: two tensor polarizations and one scalar graviton (&&&^^^^2all:\2^^^^&&&). SCG has subsequently become a unifying language for unitary-gauge scalar-tensor theories, for theories with a dynamical lapse, for tensor-only constructions with specially tuned constraint structure, and for extensions into non-Riemannian geometry such as nonmetricity-based formulations (Gao et al., 2020, &&&^^^^2 OR ti:\2^^^^&&&, Yu et al., 2024).

^^^^2 OR ti:\2^^^^. Definition, kinematics, and scope

The starting point of SCG is the ADM decomposition

$ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$

with lapse $N$, shift $N^i$, and spatial metric $h_{ij}$. The extrinsic curvature is

$$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$

In this setting, SCG means gravity theories built directly in terms of objects adapted to this $3+1$ splitting and invariant only under time-dependent spatial coordinate transformations (&&&^^^^2all:\2^^^^&&&).

A very broad SCG Lagrangian is

$$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$

where the coefficients PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^ and the potential PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^ are general functions of

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^

namely of time, the spatial metric, the lapse, the spatial Ricci tensor, and arbitrary spatial covariant derivatives thereof. The theory does not include explicit dependence on the shift PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^3 itself. This exclusion is structurally important because PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^4 is retained as the gauge variable associated with spatial diffeomorphisms, whereas explicit PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^5-dependence would alter the constraint structure drastically (&&&^^^^2all:\2^^^^&&&).

Within this broad class, one allows arbitrarily high polynomial dependence on PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^6, arbitrary dependence on PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^7, arbitrary dependence on spatial curvature PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^8, and arbitrary spatial derivatives of PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^9 and $N$^^^^2all:\2^^^^, but no explicit time derivatives of $N$^^^^2 OR ti:\2^^^^, $N$2, or higher time derivatives of $N$3 in the original 2^^^^2all:\2 OR ti:\2^^^^4 framework (&&&^^^^2all:\2^^^^&&&). This includes Hořava-like constructions, EFTs in unitary gauge, and the GLPV/beyond-Horndeski-type unitary-gauge theories as special cases. In the later XG3 construction, the action is again built from ADM scalars,

$N$4

with a maximal operator basis up to three second-order derivative operators per term, and GLPV appears as a strict subclass obtained by imposing specific relations among coefficients (Fujita et al., 2015).

The conceptual contrast with GR is sharp. In GR, full spacetime diffeomorphism invariance implies that lapse and shift act as Lagrange multipliers enforcing first-class constraints. In SCG, time reparametrization symmetry is broken down to spatial diffeomorphisms. The shift still generates spatial gauge transformations, but the lapse generally ceases to be a pure Lagrange multiplier if the Hamiltonian depends nonlinearly on $N$5 (&&&^^^^2all:\2^^^^&&&).

2. Hamiltonian structure and the generic three-degree-of-freedom theory

For the general SCG class without $N$6, the phase-space variables are

$N$7

so there are $N$8 configuration variables and $N$9 conjugate momenta, hence a $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$^^^^2all:\2^^^^-dimensional phase space. Because the Lagrangian contains no time derivatives of $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$^^^^2 OR ti:\2^^^^ or $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$2, their conjugate momenta vanish identically,

$ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$3

yielding the primary constraints (&&&^^^^2all:\2^^^^&&&).

Preserving the primary constraints in time yields the secondary constraints

$ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$4

with

$ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$5

A central result is the modified Poisson bracket between a scalar density $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$6 of unit weight and the momentum constraint,

$ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$7

which differs from the pure Lie-derivative form because $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$8 may depend on $ds^2=-N^2 dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt),$9 (&&&^^^^2all:\2^^^^&&&).

The decisive theorem-like statement is that if $N$^^^^2all:\2^^^^ depends functionally on $N$^^^^2 OR ti:\2^^^^, namely if at least one of

$N$2

does not vanish identically on the constraint surface, then

$N$3

and consequently

$N$4

Thus $N$5 and $N$6 form a second-class pair (&&&^^^^2all:\2^^^^&&&).

Because the bare momentum constraints are modified by the $N$7-dependence, the proper generator of spatial diffeomorphisms is the shifted combination

$N$8

The reorganized constraint set is then

$N$9

with $N^i$^^^^2all:\2^^^^ forming six first-class constraints and $N^i$^^^^2 OR ti:\2^^^^ forming two second-class constraints. The degree-of-freedom count is therefore

$N^i$2

The physical interpretation given in the foundational Hamiltonian analysis is that the theory propagates two transverse-traceless tensor gravitons, as in GR, plus one extra scalar mode, the scalar graviton (&&&^^^^2all:\2^^^^&&&).

This generic result admits caveats. If the kinetic structure becomes degenerate, specifically if the matrix $N^i$3 is not invertible, extra primary constraints may appear and the counting may change. The 2^^^^2all:\2 OR ti:\2^^^^4 paper excludes that degenerate case from its generic analysis, and it explicitly notes the analogy with the $N^i$4 case in Hořava gravity, where an extra primary constraint can eliminate the scalar and leave only two degrees of freedom (&&&^^^^2all:\2^^^^&&&).

3. Dynamical lapse, degeneracy, and the consistency condition

A major extension of SCG allows the Lagrangian to depend not only on $N^i$5 but also on

$N^i$6

the derivative of the lapse along the normal direction to the spatial slices. The general action is

$N^i$7

Once $N^i$8 is included, $N^i$9 can become dynamical, and a second scalar mode generally appears (Gao et al., 2018, &&&^^^^2 OR ti:\2^^^^&&&).

In the Hamiltonian analysis of this enlarged class, a generic theory has $h_{ij}$^^^^2all:\2^^^^ configuration variables after auxiliary-field linearization, $h_{ij}$^^^^2 OR ti:\2^^^^ phase-space variables, $h_{ij}$2 first-class constraints associated with spatial diffeomorphisms, and $h_{ij}$3 second-class constraints, yielding

$h_{ij}$4

Thus generic SCG with lapse velocity propagates two tensor modes and two scalar-type modes (Gao et al., 2018).

The key result is that degeneracy of the kinetic matrix is necessary but not sufficient to eliminate the unwanted extra scalar. The degeneracy condition is the vanishing of the Schur complement

$h_{ij}$5

equivalently

$h_{ij}$6

in compact notation. However, the paper shows that the primary constraint produced by degeneracy does not necessarily induce the secondary constraint needed to remove the unwanted mode. A second requirement, the consistency condition,

$h_{ij}$7

must also be imposed (Gao et al., 2018).

The perturbative counterpart of this statement, derived around an FRW background, reaches the same conclusion. For the prototype action

$h_{ij}$8

the degeneracy condition reduces to

$h_{ij}$9

At quadratic order around FRW this is sufficient to ensure that only one scalar propagates. Yet the unwanted mode reappears pathologically either at nonlinear order around FRW or at linear order around an inhomogeneous background unless the additional conditions

$$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$^^^^2all:\2^^^^

are also imposed (&&&^^^^2 OR ti:\2^^^^&&&).

A useful reformulation of the healthy prototype theory is

$$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$^^^^2 OR ti:\2^^^^

with $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$2 depending only on $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$3. This rewriting makes explicit that healthy dependence on $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$4 enters through the special combination $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$5, not arbitrarily (&&&^^^^2 OR ti:\2^^^^&&&).

A common misconception is that kinetic degeneracy alone is the decisive criterion once $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$6 is present. The Hamiltonian and perturbative analyses agree that this is not the case in SCG with lapse velocity: constraint closure, encoded in the consistency condition, is equally essential (Gao et al., 2018, &&&^^^^2 OR ti:\2^^^^&&&). In cosmological perturbation theory, the extra scalar associated with the dynamical lapse can also be interpreted as a heavy mode with mass

$$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$7

and the auxiliary-lapse limit corresponds to $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$8, providing a heavy-light interpretation of the decoupling regime (Zhu et al., 2022).

4. Correspondence with scalar-tensor theory and the beyond-Horndeski viewpoint

SCG is not only a noncovariant foliation-based gravity framework; it is also a unitary-gauge language for generally covariant scalar-tensor theories with a single scalar degree of freedom. In the original correspondence, full spacetime covariance is restored by the Stückelberg trick: one introduces a scalar field $$K_{ij}=\frac{1}{2N}\left(\partial_t h_{ij}-\nabla_i N_j-\nabla_j N_i\right).$$9 labeling the preferred foliation, with unit normal

$3+1$^^^^2all:\2^^^^

Then

$3+1$^^^^2 OR ti:\2^^^^

and similarly for $3+1$2 and for derivatives of $3+1$3 (&&&^^^^2all:\2^^^^&&&).

This correspondence was systematized in later work through explicit gauge-fixing and recovering maps between generally covariant higher-derivative scalar-tensor theory and SCG. The building blocks on the covariant side are the scalar field, the spacetime curvature tensor, and their generally covariant derivatives, while the SCG side uses the spatially covariant geometric quantities together with their spatially covariant derivatives. For a single scalar degree of freedom with timelike gradient, the two descriptions can be transformed into each other by gauge fixing and recovering procedures, and the paper gives the explicit expressions (Gao et al., 2020).

The same program was then formulated in linear-algebraic terms. One constructs the linear space of linearly independent generally covariant scalar-tensor monomials and the linear space of linearly independent SCG monomials, and argues that these two spaces are isomorphic in the sense of gauge fixing and recovering procedures. In particular, the subspaces in SCG spanned by linearly independent monomials built of the extrinsic and intrinsic curvature, the lapse function, and their spatial derivatives up to the fourth order in the total number of derivatives automatically propagate at most three degrees of freedom. Their images under gauge recovering are therefore automatically subspaces of scalar-tensor theory that propagate up to three degrees of freedom as long as the scalar field is timelike (Gao, 2020).

The 2^^^^2all:\22all:\2^^^^ correspondence paper also gives a systematic classification of all scalar monomials in SCG according to the total number of derivatives up to $3+1$4, and derives the covariant $3+1$5 decomposition without fixing any specific coordinate (Gao et al., 2020). This is conceptually important because it shows that SCG is not merely a coordinate gauge; it is a structured operator language that can generate higher-derivative but ghost-free scalar-tensor theories.

After Stückelberg restoration, the covariant theory generally has higher-order equations of motion. Nevertheless, the Hamiltonian analysis shows that the number of propagating degrees of freedom remains three. The foundational interpretation is therefore that these theories evade Ostrogradsky problems not by keeping equations second order, as in Horndeski, but by maintaining the correct constraint structure. This is precisely the sense in which the broad SCG class lies beyond Horndeski, with GLPV treated as a special case of the larger framework (&&&^^^^2all:\2^^^^&&&).

5. Tensor-only SCG and auxiliary-scalar extensions

A central branch of the literature asks when SCG can propagate only the two tensorial gravitational degrees of freedom. A perturbative construction around cosmological backgrounds derives the no-scalar condition from the quadratic scalar sector. For the general polynomial SCG action organized by derivative order $3+1$6, the coefficient of the effective kinetic structure for the scalar mode must vanish: $3+1$7 Applying this systematically up to $3+1$8, explicit tensor-only Lagrangians were found in the pure $3+1$9, pure $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$^^^^2all:\2^^^^, and mixed $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$^^^^2 OR ti:\2^^^^ sectors (Hu et al., 2021).

For the $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$2 sector, the surviving linear-order tensor-only action is

$$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$3

with $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$4 and $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$5 general functions of $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$6. GR is recovered as a special case of this class (Hu et al., 2021). Extending the perturbative analysis to cubic order imposes additional conditions,

$$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$7

and leads to a more restricted $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$8 tensor-only action (Hu et al., 2021).

The cubic-order program has recently been extended to polynomial-type SCG Lagrangians up to $$\mathcal{L}=\sum_{n=1}\mathcal{K}_n[K]+\mathcal{V}, \qquad \mathcal{K}_n[K] = \mathcal{G}_{(n)}^{i_1j_1,\cdots,i_nj_n} K_{i_1j_1}\cdots K_{i_nj_n},$$9. By requiring the scalar mode to be eliminated up to cubic order in perturbations around a cosmological background, five explicit Lagrangians were obtained: PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2all:\2^^^^, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2 OR ti:\2^^^^, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\22^^^^, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^3, and PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^4. Among these, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^5 and PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^6 are the most physically relevant because they admit a GR limit and contain known 2-DOF models such as the quadratic SCG theory and subclasses of extended cuscuton. A whole quadratic branch, called Case II, is ruled out by cubic scalar interactions (&&&^^^^2 OR ti:\22^^^^&&&).

A parallel extension introduces an auxiliary scalar field PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^7 with no time derivatives. In the perturbative analysis, the general action

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^8

still generically propagates three degrees of freedom, because the auxiliary scalar itself is not dynamical; the would-be scalar graviton remains the relevant issue. The necessary linear-order condition for eliminating that scalar mode is again that the coefficient of PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2all:\2^^^^9 vanish after solving the auxiliary perturbations. In the explicit PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2all:\2^^^^ model,

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2 OR ti:\2^^^^

the auxiliary scalar plays a nontrivial role and allows new terms in the Lagrangian, notably PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\22^^^^, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^3, and PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^4, in a perturbatively tensor-only branch (Wang et al., 2024).

The nonperturbative Hamiltonian analysis of the same auxiliary-scalar framework reaches a sharper conclusion. Generically the theory has PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^5 configuration-space variables, PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^6 first-class constraints, and PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^7 second-class constraints, so

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^8

To reduce this to two propagating degrees of freedom, two independent conditions are needed, each of which can eliminate half a degree of freedom. The second condition splits into two cases according to whether it creates another secondary constraint or promotes a new combination to first-class structure. In the explicit polynomial PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2 OR ti:\2^^^^9 model, the physical branch gives

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22all:\2^^^^

and yields a concrete 2-DOF SCG subclass with an auxiliary scalar field (&&&^^^^2 OR ti:\2 OR ti:\2^^^^&&&).

6. Cosmology, frame transformations, mimetic and nonmetric extensions

SCG has also been developed as a framework for cosmological perturbations and frame transformations. The XG3 theory, defined as a concrete subclass of spatially covariant theories of gravity with up to three second-order derivative operators, is closed under disformal transformation. In unitary gauge the covariant disformal map becomes

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22 OR ti:\2^^^^

and the transformed action can again be written in the same XG3 basis, only with reshuffled coefficients. The perturbations themselves obey

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\222^^^^

and both the scalar and tensor quadratic actions, as well as the resulting primordial spectra, are invariant under the disformal transformation (Fujita et al., 2015).

Because higher spatial derivatives are allowed, the tensor quadratic action generally contains a PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^3 correction to the dispersion relation. Consequently the tensor action itself cannot in general be transformed to the standard Einstein-frame action of GR. Nevertheless, there exists a one-parameter family of frames in which the tensor power spectrum takes the standard GR expression. This distinguishes the strict Einstein frame at the action level from an Einstein-frame-like spectrum frame at the level of observables (Fujita et al., 2015).

With a dynamical lapse, the perturbative SCG literature emphasizes again that two scalar degrees of freedom are generic. For the model

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^4

the linear degeneracy condition is

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^5

After the field redefinition

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^6

the kinetic sector becomes degenerate and PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^7 becomes auxiliary at linear order, leaving one scalar mode. The same paper then imposes the mimetic constraint PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^8, which in unitary gauge fixes PRESERVED_PLACEHOLDER_^^^^2 OR ti:\22^^^^9 and removes the lapse scalar from the start, so the mimetic extension propagates only one scalar degree of freedom (Zhu, 2022).

The cosmological consequences of a genuinely dynamical lapse have also been studied in a heavy-light language. For the scalar action

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2(Gao, 2014) OR (Gao et al., 2019) OR (Gao et al., 2018) OR (Fujita et al., 2015) OR (Gao et al., 2020) OR (Hu et al., 2021) OR (Zhu, 2022) OR (Zhu et al., 2022) OR (Yu et al., 2024) OR (Wang et al., 2024) OR (Chakraborty et al., 2023) OR (Zhu et al., 4 Aug 2025) OR (Yu et al., 16 Apr 2026)2all:\2^^^^

the extra lapse mode has mass PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2(Gao, 2014) OR (Gao et al., 2019) OR (Gao et al., 2018) OR (Fujita et al., 2015) OR (Gao et al., 2020) OR (Hu et al., 2021) OR (Zhu, 2022) OR (Zhu et al., 2022) OR (Yu et al., 2024) OR (Wang et al., 2024) OR (Chakraborty et al., 2023) OR (Zhu et al., 4 Aug 2025) OR (Yu et al., 16 Apr 2026)2 OR ti:\2^^^^, becomes infinitely heavy when the lapse reverts to an auxiliary variable, and can be integrated out to yield an effective single-field theory with modified kinetic normalization and sound speed (Zhu et al., 2022).

SCG has further been applied to inflationary model building in a 2-tensor framework. In a simple SCG model with

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^32

supplemented by a canonical scalar field and the gauge-fixing term

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^33

the condition for two tensorial gravitational degrees of freedom in the presence of matter is derived perturbatively as

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^34

The resulting inflationary phenomenology gives

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^35

and, for PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^36,

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^37

Thus the tensor-to-scalar ratio can either be of order unity or be small depending on the SCG parameter PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^38 (&&&^^^^2 OR ti:\2all:\2^^^^&&&).

A more recent non-Riemannian extension constructs SCG with nonmetricity. In a torsionless but non-metric-compatible geometry, the affine connection is decomposed into the Levi-Civita connection plus a disformation tensor, whose six projected components

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^39

enlarge the usual SCG basis. After imposing unitary gauge, a general scalar-nonmetricity theory becomes a spatially covariant theory whose building blocks are all spatial tensors, and the scalar monomials are classified up to total derivative order PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^4^^^^2all:\2^^^^. In the PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^4^^^^2 OR ti:\2^^^^ sector, the disformation is treated as an auxiliary variable and can be integrated out algebraically, producing an effective metric-only SCG

PRESERVED_PLACEHOLDER_^^^^2 OR ti:\2^^^^42

with modified coefficients. The paper is explicit that this restricted auxiliary sector is designed to stay within a ghostfree one-scalar regime, whereas allowing Lie derivatives of the disformation would change the degree-of-freedom count drastically (Yu et al., 2024).

Taken together, these developments place SCG at the intersection of Hamiltonian constraint theory, unitary-gauge scalar-tensor theory, cosmological perturbation theory, disformal transformations, and non-Riemannian extensions. The core structural lesson remains the one already visible in the earliest Hamiltonian analysis: once full spacetime covariance is weakened to spatial covariance, the fate of the lapse and of the scalar graviton is controlled not by symmetry alone but by the detailed constraint structure of the theory (&&&^^^^2all:\2^^^^&&&).