Embedding Class-I Solutions in Gravity

• Embedding class-I solutions are gravitational configurations where 4D metrics are locally embedded in a flat 5D space via the Karmarkar condition, reducing independent metric functions.
• They offer a powerful reduction scheme that simplifies solution generation for anisotropic compact stars, wormholes, and modified-gravity models by fixing one metric potential in terms of the other.
• Recent advances extend class-I methods to charged systems, relativistic polytropes, and signature-changing cosmologies, emphasizing analytic closure and rigorous matching conditions for physical viability.

Embedding class-I solutions are solutions of gravitational field equations whose four-dimensional spacetime metric can be locally isometrically embedded in a flat five-dimensional space. For static, spherically symmetric geometries, this requirement is encoded by the Karmarkar condition, which replaces two a priori independent metric potentials by a single functional degree of freedom and thereby turns the construction of interiors, wormholes, and related geometries into a constrained solution-generating problem. In the recent literature, this framework has been used for anisotropic compact stars, relativistic polytropes, electromagnetic mass models, wormholes, periodic signature-changing spacetimes, and modified-gravity stellar configurations (Baskey et al., 2020, Kuhfittig, 2018, Sharif et al., 2023, Chanda et al., 20 Feb 2026).

1. Geometric definition and the Karmarkar condition

An $n$-dimensional Riemannian manifold is of embedding class $m$ if $m+n$ is the smallest dimension of a flat Euclidean or Minkowskian space into which it can be locally isometrically embedded. Embedding class one therefore means embeddability in a flat $(n+1)$-dimensional space. For a four-dimensional spherically symmetric metric, the necessary and sufficient criterion is the Karmarkar condition, written in the literature as a relation among Riemann tensor components such as

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$

or equivalent index permutations under different sign conventions (Kuhfittig, 2018, Bhar et al., 2017, Kuhfittig, 2022).

For the standard static, spherically symmetric line element

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$

the Karmarkar condition reduces to a first-order differential relation between $\nu$ and $\lambda$,

$$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$

which integrates to

$$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$

Equivalent presentations used across the literature include

$m$0

and

$m$1

the difference being a matter of normalization convention for the embedding constant (Baskey et al., 2020, Maurya et al., 2015, Kuhfittig, 2022).

Several papers also give explicit five-dimensional flat embeddings. One representative construction introduces

$m$2

together with the standard Cartesian coordinates on the two-sphere,

$m$3

so that the induced four-metric satisfies the class-I relation $m$4 or its normalized counterpart (Kuhfittig, 2022, Kuhfittig, 2021).

2. Class-I reduction as a solution-generating scheme

The central utility of embedding class I is that it trades two independent metric functions for a single ordinary integral relation. In anisotropic stellar modeling, one begins with

$m$5

or an equivalent anisotropic-fluid form, writes the Einstein equations for $m$6, $m$7, and $m$8, imposes the Karmarkar condition, chooses one metric potential, and then determines the other analytically. The papers repeatedly emphasize this co-dependence of the metric potentials as the hallmark of class-I interiors (Baskey et al., 2020, Bhar et al., 2017).

Once the metric functions are fixed, one computes matter variables, anisotropy $m$9, the mass function $m+n$0, compactness, and redshift, and then imposes junction conditions at the boundary. For neutral stars, the interior is matched to the exterior Schwarzschild line element by requiring continuity of $m+n$1, $m+n$2, and $m+n$3. For charged models, the matching is instead to the Reissner–Nordström exterior with continuity of $m+n$4, $m+n$5, $m+n$6, and $m+n$7 (Baskey et al., 2020, Maurya et al., 2015).

Physical admissibility is not automatic. The standard checks across the literature include central regularity, positivity and monotonic decrease of $m+n$8, $m+n$9, and $(n+1)$0, energy conditions, causality bounds $(n+1)$1, generalized TOV equilibrium, Herrera–Abreu “cracking,” the adiabatic-index bound $(n+1)$2, and, in some studies, the Harrison–Zeldovich–Novikov criterion $(n+1)$3. This suggests that embedding class I is best understood as a geometric reduction principle rather than a substitute for matter modeling or boundary-value analysis (Baskey et al., 2020, Bhar et al., 2017, Singh et al., 2016).

3. Anisotropic compact stars and relativistic polytropes

A large part of the embedding-class-I literature concerns anisotropic compact stars. In one Buchdahl-inspired construction, the choice

$(n+1)$4

yields

$(n+1)$5

together with closed forms for $(n+1)$6, $(n+1)$7, $(n+1)$8, the anisotropy factor, and the mass function

$(n+1)$9

After matching to Schwarzschild, the parameters become

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$0

For the surface density $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$1, the same model gives a peak mass $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$2 at $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$3, $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$4, and $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$5 (Baskey et al., 2020).

Bhar et al. chose

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$6

which implies

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$7

This leads to explicit closed-form expressions for $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$8, $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0,$$9, $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$0, the mass function

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$1

compactness, and gravitational redshift. The solution is described as a versatile four-parameter family, free of central singularities and satisfying the static stability criterion (Bhar et al., 2017).

Singh–Bhar–Pant instead adopted

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$2

which gives

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$3

and analytic formulas for

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$4

as well as $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$5, $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$6, and $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$7. After Schwarzschild matching, the model was fitted to RX J1856–37, Her X–1, Vela X–12, and Cen X–3; for Her X–1 the paper reports $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$8, $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2),$$9, $\nu$0, $\nu$1, $\nu$2, and $\nu$3 (Singh et al., 2016).

Embedding class I has also been combined with relativistic polytropes. In the anisotropic-polytrope formulation,

$\nu$4

the Karmarkar condition fixes the anisotropy function as

$\nu$5

which, when substituted into the generalized Lane–Emden equation, produces the final class-I Lane–Emden system for $\nu$6 and $\nu$7. The same study computes the Tolman–Whittaker mass and notes that the numerical class-I polytropes are necessarily anisotropic (Ramos et al., 2021).

4. Charged matter, electromagnetic mass, and modified gravity

The class-I method extends naturally to charged configurations. Maurya et al. considered the Einstein–Maxwell system under the relation

$\nu$8

and introduced three ansätze for $\nu$9: $\lambda$0, $\lambda$1, and $\lambda$2. For Type I, the resulting mass, electric field, density, and pressure are all given in closed form, and the solutions are described as electromagnetic mass models in which density, pressure, and related physical parameters vanish for vanishing charge. Of the three types, only the Type I model survives all physical and stability tests and is used to represent charged, ultra-compact stars (Maurya et al., 2015).

In modified gravity, Sharif and Naseer studied $\lambda$3, $\lambda$4, together with the MIT bag-model equation of state

$\lambda$5

With the generating function

$\lambda$6

the Karmarkar condition yields

$\lambda$7

and Schwarzschild matching fixes

$\lambda$8

The vanishing radial pressure condition at $\lambda$9 determines the bag constant $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$0, and the numerical analysis with $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$1 concludes that $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$2 is the more suitable choice for stable compact structures (Sharif et al., 2023).

A further development appears in linear $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$3 gravity with gravitational decoupling. There the seed class-I Vaidya–Tikekar geometry is deformed by two parameters $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$4, where $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$5 controls geometric deformation and effective EOS stiffness, while $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$6 rescales the matter sector without altering the metric structure. The undeformed choice

$$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$7

and the class-I relation lead to deformed metric potentials $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$8 and $$\frac{2\nu''}{\nu'}+\nu'=\frac{\lambda' e^\lambda}{e^\lambda-1},$$9, with an analytic compactness bound

$$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$0

The paper states that pure linear $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$1 does not alter the classical compactness limit, but the combined action of $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$2 and $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$3 enlarges the accessible stellar mass window while preserving physical acceptability (Chanda et al., 20 Feb 2026).

5. Wormholes, bridges, lensing, and other nonstellar solutions

Embedding class I has also been used to constrain Morris–Thorne-type wormholes. For

$$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$4

the class-I relation implies

$$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$5

so that the shape function is fixed once $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$6 is chosen. One explicit family is

$$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$7

with a zero-tidal-force special case $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$8, $$e^{\nu(r)}=\left[A+B\int^r\sqrt{e^{\lambda(s)}-1}\,ds\right]^2.$$9. In this construction, the null energy condition at the throat is necessarily violated (Kuhfittig, 2018).

A different line of argument concludes that class-I wormholes are generically nontraversable if one starts from a prescribed shape function $m$00. In that approach, integrating the class-I condition gives

$m$01

and the lower-limit behavior forces $m$02, producing an event horizon at the throat. The same paper reinterprets $m$03 as a stellar mass profile $m$04, for which the lower limit shifts to $m$05 and the interior becomes regular, and it also develops a microscopic charged Einstein–Rosen bridge on a noncommutative background with

$m$06

and effective shape function $m$07 (Kuhfittig, 2021).

The literature is therefore not uniform. One paper presents a complete wormhole solution in the sense of obtaining both the redshift and shape functions, while another finds that a well-defined shape function drives the geometry to a horizon at the throat (Kuhfittig, 2018, Kuhfittig, 2021). This suggests that the class-I constraint is highly sensitive to what is taken as the generating datum.

Recent work has pushed these models toward observables. Two explicit class-I wormhole solutions, Model-I and Model-II, use

$m$08

and

$m$09

respectively, and obtain analytic shape functions satisfying the throat and asymptotic-flatness conditions. That study treats the throat as a photon sphere, derives null geodesics by the Hamilton–Jacobi separation method, and numerically computes the radius of the wormhole shadow, strong deflection angle, and lensing observables for M87* and Sgr A*. It reports that $m$10 in Model-I and $m$11 in Model-II have significant effects on shadow and strong lensing, while most pointwise energy conditions are violated over most of the domain (Molla et al., 2024).

Class-I geometry has also been invoked outside the wormhole/stellar dichotomy. One paper combines the embedding condition with conformal symmetry for a charged wormhole and, in a separate model, derives an effective mass profile

$m$12

as an explanation of flat galactic rotation curves without dark matter (Kuhfittig, 2018).

6. Signature change, higher-dimensional interpretation, and recurring issues

A distinctive cosmological application promotes the embedding parameter $m$13 to a periodic function of time,

$m$14

so that the same four-dimensional metric can be viewed as embedded alternately in a five-dimensional flat space with an extra spacelike dimension and in one with an extra timelike dimension. For $m$15, the ambient signature is $m$16; for $m$17, one reassigns a second negative sign and obtains $m$18. The four-dimensional metric remains continuous, while the signature of the embedding space flips periodically. The paper interprets this as a periodic change in the signature of the embedding space and as a model for alternating phases of accelerated and decelerated cosmic expansion (Kuhfittig, 2022).

Across these applications, several recurring points emerge. First, embedding class I is a geometric constraint, not a unique physical model: the same reduction principle supports anisotropic stars, electromagnetic mass models, polytropes, wormholes, Einstein–Rosen bridges, signature-changing cosmologies, and modified-gravity interiors (Baskey et al., 2020, Maurya et al., 2015, Kuhfittig, 2022). Second, the reduction is powerful precisely because it is restrictive: once one metric potential is chosen, the other is fixed algebraically or by quadrature, which explains why class-I solutions are often analytically closed-form (Bhar et al., 2017, Singh et al., 2016).

Third, the class-I condition does not by itself guarantee physical viability. Compact-star papers impose regularity, matching, energy, causality, and stability criteria before accepting a model, whereas wormhole papers often find unavoidable NEC violation or even event-horizon formation at the throat (Baskey et al., 2020, Kuhfittig, 2018, Kuhfittig, 2021). A plausible implication is that embedding class I should be viewed less as a standalone physical hypothesis than as a highly structured constraint that organizes admissible geometries and exposes how much of the resulting matter content is fixed by geometry.

Finally, recent generalizations indicate that the framework remains active. In $m$19 gravity it is combined with the MIT bag model; in linear $m$20 gravity it is combined with gravitational decoupling and a controlled two-parameter deformation; in observational wormhole studies it feeds directly into shadow and strong-lensing calculations (Sharif et al., 2023, Chanda et al., 20 Feb 2026, Molla et al., 2024). The continued use of the Karmarkar condition in these disparate contexts reflects a persistent theme of the subject: embedding class-I solutions provide a bridge between four-dimensional field equations and higher-dimensional geometric structure, while leaving the substantive questions of matter modeling, stability, and phenomenology to be settled case by case.