---
title: Fictitious Matter of Embedding Gravity
url: https://www.emergentmind.com/topics/fictitious-matter-of-embedding-gravity-fmeg
type: topic
---

# Fictitious Matter of Embedding Gravity

Fictitious Matter of Embedding Gravity (FMEG) is the effective matter sector that appears when Regge–Teitelboim embedding gravity is rewritten in Einstein form as ordinary gravity plus an additional stress-energy contribution. In this framework, four-dimensional spacetime is treated as an embedded surface in a higher-dimensional flat ambient space, the fundamental field is the embedding function \(y^a(x^\mu)\), and the usual spacetime metric is induced rather than fundamental. The resulting equations are more general than Einstein’s equations and admit non-Einsteinian branches; from the metric viewpoint, those branches can be encoded by an effective tensor \(\tau_{\mu\nu}\), which may mimic dark matter- or dark energy-like behavior without introducing new microscopic matter fields [1806.10902]. At the same time, FMEG is not generic to every solution sector: in some physically important classes, such as static spherically symmetric asymptotically flat vacuum exteriors with symmetric six-dimensional embeddings, the extra sector disappears and the theory reduces to the Schwarzschild solution [1402.1121].

## 1. Geometric formulation and origin of the extra sector

Embedding gravity starts from the replacement of the metric \(g_{\mu\nu}(x)\) by embedding functions \(y^a(x^\mu)\) into a flat ambient space with metric \(\eta_{ab}\). The induced metric is
\[
g_{\mu\nu} = \partial_\mu y^a\,\partial_\nu y^b\,\eta_{ab}.
\]
For generic four-dimensional metrics, the natural ambient dimension is \(N=10\), in accord with the Janet–Cartan–Friedman bound \(n(n+1)/2\) for \(n=4\); highly symmetric sectors can require fewer dimensions [1402.1121].

Varying the Einstein–Hilbert action after substituting the induced metric yields the Regge–Teitelboim equations rather than Einstein’s equations. In divergence form they are
\[
D_\mu\!\left((G^{\mu\nu}-\varkappa T^{\mu\nu})\,\partial_\nu y^a\right)=0,
\]
and in geometric form
\[
(G^{\mu\nu}-\varkappa T^{\mu\nu})\,b_{\mu\nu}^a=0,
\qquad
b_{\mu\nu}^a=D_\mu\partial_\nu y^a.
\]
These equations are weaker than \(G^{\mu\nu}=\varkappa T^{\mu\nu}\) because the change of variables from \(g_{\mu\nu}\) to \(y^a\) contains derivatives and restricts the admissible variations. The extra solution branches are therefore structural, not accidental. This suggests that FMEG should be understood as the Einstein-space interpretation of those non-Einsteinian embedding configurations rather than as an independently postulated matter component [1402.1121].

The second fundamental form \(b_{\mu\nu}^a\) is central to this weakening. The equations constrain only the projection of the Einstein-equation residual along extrinsic curvature directions. Unless the embedding is sufficiently free for \(b_{\mu\nu}^a\) to span the full symmetric-tensor space, the Regge–Teitelboim equations need not force the Einstein residual to vanish. That geometric fact is the basic origin of FMEG.

## 2. Einstein-form rewriting and the definition of embedding matter

The standard Einstein-form rewriting is
\[
G^{\mu\nu}=\varkappa\left(T^{\mu\nu}+\tau^{\mu\nu}\right),
\qquad
D_\mu\!\left(\tau^{\mu\nu}\partial_\nu y^a\right)=0.
\]
Here \(\tau^{\mu\nu}\) is the effective stress-energy tensor of embedding matter. It is not fundamental matter added by hand; it is a repackaging of the extra Regge–Teitelboim solutions that arise from the embedding description [1806.10902].

A convenient current representation is
\[
j_a^\mu=\tau^{\mu\nu}\partial_\nu y_a,
\qquad
D_\mu j_a^\mu=0.
\]
In this form, FMEG is a constrained multi-current medium rather than an ordinary fluid. The same paper gives an action for the additional sector in terms of conserved currents,
\[
S_{\text{add}}=\int d^4x\,\sqrt{-g}\,\left(j^\mu_a\partial_\mu y^a-\operatorname{tr}\sqrt{g_{\mu\nu}j^\nu_a j^{\alpha a}}\right),
\]
and shows that it is completely equivalent to the original embedding theory [1806.10902].

This rewriting motivated the description of embedding theory as a “new geometrical mimetic gravity.” The analogy is structural: both mimetic gravity and embedding gravity arise from derivative-dependent variable changes in the gravitational action, both generate extra solutions beyond ordinary GR, and both admit a GR-plus-effective-matter representation. The analogy is not an identity. In the embedding case, the effective sector is generally more complicated than pressureless dust and is encoded by the currents \(j^\mu_a\) together with the embedding fields \(y^a\).

Within this terminology, “embedding matter,” “additional matter,” and “fictional embedding matter” are the most faithful labels used in the literature summarized here. FMEG is therefore a compact label for the non-Einsteinian stress-energy sector \(\tau^{\mu\nu}\) generated by the embedding formulation itself.

## 3. Canonical structure, constraints, and degrees of freedom

The unconstrained embedding theory is larger than GR, and its canonical structure reflects that enlargement. In the general Hamiltonian formulation without artificially imposed Einsteinian constraints, there are four natural constraints, one of which is defined only implicitly through the nonlinear relation between canonical momentum and the normal vector \(n^a\). Even so, the full constraint algebra closes, and all four constraints are first class [1705.07361].

This result is significant for FMEG because it shows that the non-Einsteinian sector is not removed by canonical inconsistency. The unconstrained theory that supports extra solutions has a well-defined first-class gauge structure. A naive phase-space counting is suggestive of more degrees of freedom than in GR, consistent with the existence of an additional embedding-matter-like sector, although the paper does not itself present the extra sector as a phenomenological fluid [1705.07361].

A complementary canonical treatment rewrites embedding gravity directly as GR plus dark matter. After solving simple constraints, the Hamiltonian becomes a linear combination of four first-class constraints with Lagrange multipliers, but six pairs of second-class constraints remain. In that formulation the theory has \(6\) degrees of freedom, interpreted as the usual \(2\) of GR plus \(4\) additional embedding-matter degrees of freedom [2207.13654].

The role of gauge fixing is clarified further by the external-time formalism. Imposing the ambient-time condition
\[
y^0(x^\mu)=x^0
\]
directly in the action does not lose equations of motion: the missing Regge–Teitelboim equation follows from the others. In the GR-equivalent constrained sector, the external-time canonical formalism reproduces the same result as gauge fixing in the constraint algebra. This means that the external-time choice reorganizes the theory but does not itself create or eliminate the extra embedding sector; only the additional Einsteinian constraints remove it and restore strict equivalence to GR [1509.01529].

## 4. Symmetric solution sectors: elimination and persistence of FMEG

The most explicit no-go result for FMEG is the analysis by Sheykin and Paston of static, spherically symmetric, asymptotically flat vacuum metrics embedded symmetrically in six-dimensional flat space. For metrics of the form
\[
ds^2=(1-P(r))dt^2-(1-Q(r))dr^2-r^2d\Omega^2,
\]
with \(P(r),Q(r)\to 0\) as \(r\to\infty\), the authors classify all smooth \(SO(3)\times T^1\)-symmetric embeddings in \(6\) dimensions into six types: elliptic, hyperbolic, spiral, exponential, parabolic, and cubic. They then show that the vacuum Regge–Teitelboim equations admit no extra solutions in this class, so the only solution is the exterior Schwarzschild metric [1402.1121].

This result sharply limits the scope of FMEG. It shows that embedding-induced fictitious matter is not an unavoidable consequence of the formalism. In that vacuum sector, symmetry plus asymptotic flatness force the corresponding conserved quantities to vanish, and the effective embedding current disappears. A plausible implication is that FMEG is highly sector-dependent and controlled by symmetry, boundary conditions, and embedding class rather than by the mere existence of an embedding description.

Cosmological sectors behave differently. For the spatially flat Friedmann model in \(\mathbb{R}^{1,9}\), a classification of fully symmetric embeddings finds exactly two possibilities: the classical five-dimensional Robertson embedding and a new eight-dimensional embedding [2503.00598]. In that setting the fictitious embedding stress tensor is taken in perfect-fluid form,
\[
\tau^\mu{}_\nu=
\begin{pmatrix}
\rho(t)&0&0&0\\
0&-p(t)&0&0\\
0&0&-p(t)&0\\
0&0&0&-p(t)
\end{pmatrix},
\]
and for the new eight-dimensional embedding the paper derives
\[
\rho=\frac{C}{a^{3}\dot f},
\qquad
p=\frac{C\ddot f}{3a^2\dot a\,\dot f^{\,2}},
\qquad
f(t)=\sqrt{a^2(t)-b^2}.
\]
The same study concludes, however, that this new embedding does not improve the cosmological dark-matter scenario: with realistic initial conditions at the start of inflation, the resulting fictitious matter does not make a significant contribution to the present cosmic matter budget [2503.00598].

## 5. Weak-field, nonrelativistic, and galactic realizations

The nonrelativistic sector is the main regime in which FMEG has been developed as an effective dark-matter analogue. In this limit, embedding gravity can be rewritten as GR with an additional density \(\rho_\tau\) entering the Poisson equation,
\[
\Delta\varphi=4\pi G(\rho+\rho_\tau),
\]
while the embedding matter obeys a continuity equation
\[
\partial_t \rho_\tau = -\partial_i(\rho_\tau v^i)
\]
and an Euler-type equation whose force density contains the ordinary Newtonian term together with additional self-interaction terms depending on geometric variables of the embedding [2006.09026]. The papers emphasize that embedding matter is therefore not merely collisionless dust; it is a self-interacting effective medium of geometric origin.

A technically important condition in weak-field theory is that the background embedding of flat spacetime must be unfolded. The trivial planar embedding has vanishing background second fundamental form and does not yield a useful linearization of the Regge–Teitelboim equations. Around a suitable unfolded background, corrections to the embedding function become linear in corrections to the metric, and the weak-field equations can be used to derive a nonlinear equation for a spherically symmetric gravitational potential [2210.13272].

In that framework, an explicit family of spherically symmetric unfolded background embeddings of flat three-dimensional Euclidean space is proposed, parameterized by one function of radius. The resulting nonlinear equation can be solved so that the gravitational potential matches the one associated with a Burkert halo profile. This shows that the non-Einsteinian branch of the embedding equations can reproduce a cored galactic dark-halo potential without introducing particulate dark matter [2210.13272].

The galactic center problem has also been analyzed through the distribution function over trajectories. In that treatment, the central asymptotics of the density profile is controlled by the small-angular-momentum behavior of the distribution function. If \(f(\varepsilon,0)\neq 0\), then
\[
\rho(r)\sim \frac{1}{r}
\]
and the profile is cuspy; if \(f(\varepsilon,l)\sim f'_l(\varepsilon,0)\,l\), the central density remains finite and the profile is cored [2307.01307]. This suggests that FMEG does not predict a unique central profile independently of the dynamical formation history.

The broader phenomenological claim is therefore limited but concrete. Embedding gravity supplies extra degrees of freedom large enough to motivate a dark-matter interpretation, produces a nonrelativistic branch that behaves as cold matter with self-interaction, and admits weak-field halo realizations in galactic settings. It does not yet provide a full replacement for \(\Lambda\)CDM across all data sets.

## 6. Condensation, related generalizations, and scope

The most explicit use of the term itself appears in the study of “fictitious matter of embedding gravity (FMEG)” as a static weak-field medium that can condense into localized structures [2509.00980]. In that paper, the Einstein-form equations are
\[
G^{\mu\nu}=\varkappa\bigl(T^{\mu\nu}+\tau^{\mu\nu}\bigr),
\qquad
D_\mu\tau^{\mu\nu}=0,
\qquad
\tau^{\mu\nu}b^a_{\mu\nu}=0,
\]
and FMEG is treated as moving independently of ordinary matter.

In the weak-field linear regime, the paper derives an upper density bound
\[
\rho_b=\frac{1}{4\pi G L^2},
\]
interpreted as the maximum total density compatible with that regime [2509.00980]. Static condensed configurations are classified into wall, string, and sphere types. In the isotropic spherical case,
\[
\tau^{ik}=w\,\delta^{ik}\rho_\tau,
\qquad
p_\tau=w\rho_\tau,
\]
so the effective medium behaves as an isothermal ideal gas with a linear equation of state. The regular spherical condensations have an outer density asymptotic
\[
\rho_\tau \approx \frac{w}{2\pi G\,r^2},
\]
which yields flat rotation curves with
\[
v(r)\to \sqrt{2w}.
\]
The same paper notes that this large-\(r\) behavior matches the falloff used in pseudo-isothermal halo fitting [2509.00980].

Related generalizations extend the embedding-matter idea beyond four-dimensional Regge–Teitelboim gravity. In a codimension-one extended-objects framework, Lovelock-type brane gravity admits a mimetic reformulation with a conserved worldvolume current \(T^{a\mu}\) and tangential tensor \(T^{ab}\), which function as a fictional energy-momentum sector [2602.23538]. This suggests that the logic of FMEG is not confined to the original RT theory but can be transplanted to broader embedding-based geometries.

Taken together, the literature defines FMEG as a geometric effective source generated by the extra solution space of embedding gravity. Its strongest support comes from exact Einstein-form rewritings, current-based actions, nonrelativistic and weak-field dark-sector behavior, and the existence of halo-like condensations. Its strongest limitations come from the demonstrated absence of extra solutions in important vacuum sectors, the dependence on symmetry and embedding class, and unresolved issues of uniqueness, cosmological viability, and fully predictive phenomenology.

Source: https://www.emergentmind.com/topics/fictitious-matter-of-embedding-gravity-fmeg