Embedding Gravity Reformulation

• Embedding Gravity is a reformulation where 4D spacetime is viewed as a surface embedded in a higher-dimensional flat space, with the metric induced by derivatives of the embedding function.
• The Regge–Teitelboim approach replaces the metric with embedding functions, leading to extra non-Einsteinian solutions that can mimic dark-matter effects.
• Canonical and field-theoretic formulations reveal a consistent constrained system with promising applications in cosmology and astrophysical phenomenology.

Searching arXiv for recent and foundational papers on embedding gravity to ground the article in cited literature. Embedding gravity is a reformulation of gravitation in which four-dimensional spacetime is treated as a surface embedded in a flat ambient space of higher dimension, and the metric is not fundamental but induced by the embedding function $y^{a}(x^\mu)$ through

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$

In the Regge–Teitelboim approach, the dynamical variable is therefore the embedding function rather than $g_{\mu\nu}$, and the Einstein–Hilbert action is varied after substitution of the induced metric. This yields equations that contain all Einstein solutions but are more general than Einstein’s equations, so the theory generically admits non-Einsteinian branches often called extra solutions (Sheykin et al., 2014). Subsequent work has developed canonical formulations, field-theoretic reformulations in flat ambient space, weak-field expansions, cosmological mechanisms that suppress extra solutions, and astrophysical interpretations in which the non-Einsteinian sector behaves as an effective dark component (Paston et al., 2020).

1. Geometric definition and ambient-space structure

The basic object of embedding gravity is an isometric embedding of spacetime into a flat ambient space, usually taken to be $\mathbb{R}^{1,N-1}$ with metric $\eta_{ab}$. In this setup, the map

$$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$

defines a four-dimensional surface whose intrinsic geometry is encoded by the induced metric. The central conceptual move is a change of variables in the Einstein–Hilbert action: the metric is replaced by the derivative-built expression $$g_{\mu\nu}=\partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}$$, so gravity becomes a surface-dynamics problem rather than a metric-field problem (Sheykin et al., 2014).

A recurrent dimensional benchmark is the local isometric embedding of generic four-dimensional curved spacetime into a sufficiently high-dimensional flat space. One formulation cites the Friedman theorem and states that a generic 4-dimensional curved spacetime may require $N\ge 10$ for a local isometric embedding; related work invokes the Janet–Cartan theorem in the form $N=d(d+1)/2$, giving $N=10$ for $$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$0 (Sheykin et al., 2014). This is why ten-dimensional flat ambient space $$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$1 is standard in much of the literature (0711.0576).

The geometry of the embedded surface is organized by the tangent vectors $$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$2, the tangent and normal projectors, and the second fundamental form

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$3

which appears directly in the equations of motion (Sheykin et al., 2014). In canonical and differential-geometric treatments, the Gauss relation expresses the intrinsic curvature tensor of spacetime in terms of $$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$4, so the ambient flatness does not trivialize the induced four-dimensional curvature (0711.0576).

This geometric picture is distinct from braneworld constructions in which the ambient space carries its own gravitational dynamics. In the formulation discussed here, the ambient space is flat and gravity is encoded entirely in the shape of the embedded four-surface (Paston et al., 2011). A plausible implication is that embedding gravity uses higher-dimensional geometry as a kinematical scaffold rather than as an independent gravitational sector.

2. Regge–Teitelboim equations and the problem of extra solutions

Varying the Einstein–Hilbert action with respect to the embedding function yields the Regge–Teitelboim equations,

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$5

or, using the Bianchi identities and matter conservation,

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$6

Because the Einstein tensor itself depends on second derivatives of the embedding, these equations remain second order in time derivatives even though the metric is derivative-built from $$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$7 (Sheykin et al., 2014).

The decisive structural fact is that the Regge–Teitelboim equations are more general than Einstein’s equations. Every Einstein solution satisfies them, but the converse fails unless additional conditions are imposed. These non-Einsteinian branches are the extra solutions. Their origin is attributed to the derivative-dependent change of variables: the embedding function is a restricted dynamical variable entering the metric through derivatives, and this enlarges the variational solution space (Sheykin et al., 2014).

A useful reformulation introduces the conserved current

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$8

or equivalently rewrites the theory as

$$g_{\mu\nu}(x)=\partial_\mu y^{a}(x)\,\partial_\nu y^{b}(x)\,\eta_{ab}.$$9

In that form, the extra sector behaves as an additional effective matter source. Several papers interpret it as embedding matter, fictitious matter, or a dark-matter-like contribution (Paston et al., 2018). This reinterpretation does not remove the extra solutions; it reorganizes them into an Einstein equation plus a constrained source.

A major exact result concerns the vacuum, static, spherically symmetric, asymptotically flat sector in six-dimensional ambient space. For embeddings with $g_{\mu\nu}$0 symmetry, six embedding types are listed—elliptic, hyperbolic, spiral, exponential, parabolic, and cubic—and the analysis shows that static spherically symmetric asymptotically flat vacuum embeddings in 6-dimensional flat ambient space do not admit extra solutions of the vacuum Regge–Teitelboim equations. Consequently, in that sector every such solution is an Einstein vacuum solution and therefore corresponds to the exterior Schwarzschild metric (Sheykin et al., 2014). This result constrains a common concern that the embedding formulation necessarily contaminates basic vacuum sectors with non-Einsteinian branches.

A different route to equivalence with general relativity imposes additional Einsteinian constraints such as $g_{\mu\nu}$1. In canonical formulations built around those constraints, the embedding theory becomes equivalent to Einstein gravity for generic initial data satisfying the stated nondegeneracy conditions (0711.0576). The existence of both strategies—explicit constraint imposition and dynamical suppression, discussed below—marks a central internal division in the subject.

3. Canonical, covariant, and field-theoretic formulations

The canonical structure of embedding gravity is nontrivial because the fundamental variables are embedding functions, the induced metric is composite, and the standard gravitational constraints are intertwined with ambient-space geometry. One canonical formulation for the version made equivalent to general relativity imposes additional Einstein constraints and yields a theory with an eight-parameter gauge symmetry. In that construction, the Hamiltonian is a linear combination of eight first-class constraints, and equivalence to Einstein gravity is achieved either by partial gauge fixing or by rewriting the dynamics in terms of a gauge-invariant effective metric (0711.0576).

The general embedding theory without artificially imposed Einsteinian constraints has also been analyzed canonically. In that case one natural constraint cannot be written explicitly because the relation between momentum and the unit timelike vector $g_{\mu\nu}$2 is only implicitly invertible. Nevertheless, the algebra of the four resulting constraints can be computed, and the full set is first class (Paston et al., 2017). This is significant because it shows that the unrestricted theory is a consistent constrained Hamiltonian system even when one of its basic constraints is only implicitly defined.

A further development studies the use of external time, namely the ambient-space time coordinate $g_{\mu\nu}$3, as the canonical evolution parameter. The partial gauge condition $g_{\mu\nu}$4 can be inserted directly into the action without loss of independent equations of motion, and the corresponding external-time formalism yields a first-class constraint algebra equivalent to the gauge-fixed form of the arbitrary-time canonical theory (Paston et al., 2015). This is chiefly motivated by quantization: the ambient Minkowski time provides a distinguished clock not available in ordinary metric gravity.

Embedding gravity has also been reformulated as a field theory in flat ambient space through splitting gravity. Instead of describing a single embedded surface parametrically by $g_{\mu\nu}$5, one defines a family of four-dimensional surfaces as level sets

$g_{\mu\nu}$6

This replaces surface coordinates by scalar fields in the ambient space and introduces a renumeration symmetry $g_{\mu\nu}$7. An action of the form

$g_{\mu\nu}$8

reproduces the embedding equations in splitting form, and the canonical formalism for this theory is constraint-free, although the Hamiltonian is only implicitly defined as a function of canonical variables (Paston et al., 2020). The constraint-free character sharply distinguishes it from ADM general relativity and from constrained canonical versions of the Regge–Teitelboim theory.

Another action-level reformulation casts embedding gravity as general relativity plus additional matter described by conserved currents. The proposed action

$g_{\mu\nu}$9

is presented as a generalization of the perfect-fluid action and as structurally analogous to square-root constructions in bimetric gravity (Paston et al., 2018). This current-based formulation makes the extra sector explicit and is completely equivalent to the original embedding theory, provided the appropriate matrix-root branch is chosen and singular exceptional cases are excluded.

4. Emergence of Einstein dynamics, weak-field structure, and background embeddings

A central program in embedding gravity asks whether general relativity arises dynamically rather than by imposing Einsteinian constraints by hand. In a Friedmann–Robertson–Walker setting, one analysis studies the embedding theory without extra constraints and argues that inflation can suppress the extra sector. Writing

$\mathbb{R}^{1,N-1}$0

the deviation from Einstein dynamics is treated as an effective conserved source, and under an inflationary stage with $\mathbb{R}^{1,N-1}$1 the extra contribution $\mathbb{R}^{1,N-1}$2 is driven rapidly toward zero (Paston et al., 2011). The paper identifies inflation, compensation, and limited scenarios, and in the inflationary branch the ratio $\mathbb{R}^{1,N-1}$3 becomes extremely small. This supports the claim that general relativity can emerge with very high precision after inflation rather than being imposed kinematically.

The weak-field regime requires particular care because the choice of background embedding controls whether the Regge–Teitelboim equations linearize properly. Trivial plane embeddings of flat spacetime are inadequate for this purpose. Work on nontrivial isometric embeddings of flat spaces therefore constructs symmetric and unfolded embeddings of Euclidean three-space and Minkowski space, emphasizing that unfolded backgrounds maximize the rank of the second fundamental form and improve weak-field behavior (Paston et al., 2021). In that framework, for $\mathbb{R}^{1,N-1}$4 and unfolded background embedding, embedding gravity becomes exactly equivalent to general relativity in the weak-field limit; for $\mathbb{R}^{1,N-1}$5, extra solutions remain, but the weak-field equations are still linearizable (Paston et al., 2021).

A dedicated weak-field analysis expands the embedding function as

$\mathbb{R}^{1,N-1}$6

around a flat but nontrivial unfolded background. Requiring the solution to be exactly static at second order yields a nonlinear differential equation for the gravitational potential in the spherically symmetric case, with coefficients determined by the chosen background embedding (Kuptsov et al., 2022). An explicit unfolded, spherically symmetric embedding of Euclidean three-space into nine dimensions is parameterized by one arbitrary radial function, and that remaining freedom can be chosen so that the resulting potential is in good agreement with the observed distribution of dark matter in a galactic halo (Kuptsov et al., 2022).

This weak-field program suggests that background geometry in the ambient space is not a mere technical convenience. A plausible implication is that the phenomenology of the extra sector is partly encoded in the choice of unfolded background, even when the induced zeroth-order metric is flat.

5. Embedding matter, fictitious matter, and dark-matter phenomenology

The most active contemporary reinterpretation of embedding gravity treats the extra Regge–Teitelboim sector as an effective dark component. In the Einstein-like form

$\mathbb{R}^{1,N-1}$7

the tensor $\mathbb{R}^{1,N-1}$8 is called fictitious matter or embedding matter, and in weak, nonrelativistic regimes it can behave like cold dark matter (Paston et al., 2023). This line of work is motivated by the possibility of generating dark-matter-like gravitational effects without introducing new particles.

A non-relativistic limit has been constructed in which embedding matter is described by first-order dynamical equations and constraints consistent with them. In that limit, the embedding matter behaves as a pressureless component at leading order, contributes to the Newtonian potential through

$\mathbb{R}^{1,N-1}$9

and acquires self-interaction terms in its Euler-type equation (Paston, 2020). Closely related analysis studies a stable sector in which the conserved ambient-space currents are non-relativistic in the bulk and the effective stress-energy behaves like dust on large scales while retaining geometry-induced self-interaction (Paston, 2020). In both cases, the self-interaction is suggested as potentially relevant to the core–cusp problem.

Galactic applications push the dark-matter analogy further. One analytical study of matter distribution in galaxies formulates the central density profile in terms of a distribution function over trajectories and shows that the behavior near zero angular momentum determines whether the density is cuspy or cored: $\eta_{ab}$0 yields $\eta_{ab}$1, whereas $\eta_{ab}$2 yields a finite central density (Paston et al., 2023). This does not by itself prove that embedding matter generically resolves the core–cusp issue, but it shows that the effective matter sector is flexible enough to realize both classes of halo profile.

A more specific condensation picture appears in a later treatment of fictitious matter of embedding gravity, abbreviated there as FMEG. In the linear regime the additional source satisfies

$\eta_{ab}$3

with $\eta_{ab}$4, and there is an upper density bound

$\eta_{ab}$5

This bound is used to explain why dark-matter-like effects are absent on small scales (Paston et al., 31 Aug 2025). The same work identifies three possible static condensations—wall, string, and sphere—depending on the eigenvalue structure of $\eta_{ab}$6. In the spherical case the effective medium obeys the linear equation of state $\eta_{ab}$7, behaves as an isothermal ideal gas, and develops an asymptotic density profile $\eta_{ab}$8, matching the large-radius behavior of the pseudo-isothermal halo and producing flat galactic rotation curves (Paston et al., 31 Aug 2025).

These dark-matter interpretations are not equivalent to a proof that the embedding theory reproduces all $\eta_{ab}$9CDM phenomenology. The literature instead presents a sequence of partial results: stable non-relativistic sectors, analytic halo-like profiles, density bounds, and candidate condensation mechanisms (Paston, 2020). This suggests a programmatic rather than a closed status for the dark-sector interpretation.

6. Variants, stellar applications, and terminological scope

The phrase embedding gravity covers several related but nonidentical constructions. One variant is Faddeev’s formulation, in which the independent variable is a non-square vielbein $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$0 rather than the embedding function itself. In that theory the metric is still induced by $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$1, but the integrability condition $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$2 is dropped, and the non-square vielbein is treated as fundamental (Kuptsov et al., 2021). The analysis shows that the torsionless sector behaves like general relativity, but the full theory admits extra solutions with nonzero torsion and effective energy-momentum even without real matter. The theory can therefore be rewritten as general relativity plus an extra action term encoding the non-square-vielbein sector (Kuptsov et al., 2021).

A different, widely used meaning of embedding in relativistic astrophysics concerns embedding class one or class-1 conditions imposed on static spherical metrics. Here the Karmarkar condition,

$$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$3

links the two metric potentials and thereby constrains interior stellar geometry. In modified-gravity compact-star models, this approach is used to generate anisotropic stellar solutions and then match them to the Schwarzschild exterior (Mustafa et al., 2021). In $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$4, the embedding class-1 construction is applied to LMC X-4, Cen X-3, and EXO 1785-248, and the resulting models are reported to be realistic, stable, and free from singularities (Mustafa et al., 2021). A parallel construction in $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$5 gravity uses the same Karmarkar mechanism, again for LMC X-4, Cen X-3, and EXO 1785-248, and likewise concludes that the models are realistic, stable, and singularity-free (Malik et al., 2024). In linear $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$6 gravity, an embedding class-I Vaidya–Tikekar configuration combined with gravitational decoupling introduces a two-parameter deformation $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$7, where $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$8 deforms the geometry and $$y^{a}(x^\mu), \qquad \mu,\nu=0,\dots,3,\quad a,b=0,\dots,N-1,$$9 rescales the matter sector (Chanda et al., 20 Feb 2026).

These stellar applications are geometrically related to the embedding idea, but they are not identical to the Regge–Teitelboim reformulation in which the embedding function is the fundamental gravitational variable. In the stellar literature, the embedding condition is typically used as an integrability restriction on metric ansätze rather than as a replacement of the metric by $$g_{\mu\nu}=\partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}$$0. This suggests a useful terminological distinction between embedding gravity in the strict Regge–Teitelboim sense and embedding-based model building in compact-star physics.

A separate terminological caution concerns algebraic unification. One paper titled "An Explicit Embedding of Gravity and the Standard Model in E8" uses embedding in the Lie-algebraic sense, constructing $$g_{\mu\nu}=\partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}$$1 together with Standard Model gauge generators as a subalgebra of $$g_{\mu\nu}=\partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}$$2 and then of $$g_{\mu\nu}=\partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}$$3 (Lisi, 2010). That usage is conceptually unrelated to the isometric-embedding theory of spacetime as a surface in flat ambient space.

Across these variants, the core controversy remains the status of extra degrees of freedom. In some formulations they are removed by Einsteinian constraints (0711.0576); in others they are retained and interpreted as effective matter (Paston et al., 2018); in inflationary cosmology they are argued to be dynamically suppressed (Paston et al., 2011); and in Faddeev’s variant they survive as torsion-supported extra branches (Kuptsov et al., 2021). The resulting landscape is best understood not as a single closed formalism but as a family of embedding-based approaches organized around one geometric premise: four-dimensional gravitation may be encoded by the way spacetime is embedded in a higher-dimensional flat structure.