Karmarkar Condition in Compact Stellar Models

• Karmarkar Condition is an algebraic constraint on the Riemann tensor that ensures a spherically symmetric spacetime is of embedding class I.
• Its differential and integrated forms reduce the freedom in metric potentials, simplifying the construction of exact anisotropic compact star and wormhole models.
• The condition is applied across Einstein and modified gravity theories, generating viable solutions that demand careful parameter tuning for physical realism.

Searching arXiv for recent and foundational papers on the Karmarkar condition to ground the article. arxiv_search(query="Karmarkar condition static spherically symmetric embedding class I", max_results=10, sort_by="relevance") The Karmarkar condition is an algebraic constraint on the Riemann tensor that is necessary and sufficient for a four-dimensional spherically symmetric spacetime to be of embedding class I, i.e. locally embeddable in a five-dimensional flat space. In relativistic stellar modeling it functions as a geometric integrability condition: once one metric potential is prescribed, the other is determined by a single differential relation, sharply reducing the freedom in the Einstein or modified field equations. This reduction has made the condition a standard device for constructing exact anisotropic interiors, compact-star models, wormhole geometries, scalar reformulations, and time-dependent radiating solutions (Maurya et al., 2017).

1. Geometric definition and embedding-class interpretation

For a 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 is commonly written as

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

In diagonal spherical metrics one typically has $R_{1224}=R_{1334}=0$, so the condition reduces to a single relation among four curvature components (Pandya et al., 2017).

Its geometric meaning is precise: a four-dimensional spherically symmetric spacetime is, in general, of embedding class two, whereas Karmarkar showed that it is of embedding class one if and only if the above Riemann-tensor identity holds (Maurya et al., 2017). In this sense the condition is not an equation of state, nor a material constitutive law, but a restriction on admissible geometry.

In compact-star literature this embedding interpretation is operational rather than merely formal. The condition is imposed to replace two a priori independent metric functions by one generating function plus integration constants. This is why it appears repeatedly in exact anisotropic interiors in general relativity and in modified theories such as $f(R)$, $f(R,T)$, $f(\mathscr R,\mathscr G)$, and $F(Q)$ gravity (Naz et al., 2021).

2. Differential and integrated forms

Substituting the nonzero Riemann components of the static spherical metric into the algebraic condition yields a differential relation between $\nu(r)$ and $\lambda(r)$. One frequently used form is

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

or equivalently

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

These are the standard “metric” forms of the Karmarkar condition in the static case (Maurya et al., 2017).

A first integration gives the quadrature form

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$1

showing explicitly that once $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$2 is chosen, $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$3 follows by a single integral (Ratanpal et al., 2024). In the literature this same integration is also written in algebraic form as

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$4

or

$$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$5

with the difference residing in the normalization of the integration constant (Sharif et al., 2023).

A distinct reformulation replaces the differential relation between metric coefficients by an algebraic relation among structure scalars obtained from the orthogonal splitting of the Riemann tensor. In that language,

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

and in the static limit $$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}, \qquad R_{2323}\neq 0.$$7 this becomes an algebraic constraint among the physical scalar variables rather than a differential equation for the metric (Ospino et al., 2020). This scalar form is important because it recasts embedding class I directly in terms of density, anisotropy, Weyl curvature, and dissipative flux variables.

3. Role in exact compact-star models

The standard compact-star workflow is to choose one metric potential, derive the other from the Karmarkar condition, solve the field equations 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, $R_{1224}=R_{1334}=0$0, and $R_{1224}=R_{1334}=0$1, and then impose boundary matching to an exterior Schwarzschild geometry. The condition is therefore a solution-generating mechanism rather than a complete model by itself (Singh et al., 2016).

Several exact families illustrate this pattern.

• With the paraboloidal ansatz

$R_{1224}=R_{1334}=0$2

the Karmarkar equation integrates to

$R_{1224}=R_{1334}=0$3

leading to closed-form expressions for density, radial pressure, tangential pressure, and anisotropy, and to models compatible with Her X-1, LMC X-4, EXO 1785-248, PSR J1903+327, Vela X-1, and PSR J1614-2230 (Pandya et al., 2017).

• For

$R_{1224}=R_{1334}=0$4

one obtains

$R_{1224}=R_{1334}=0$5

together with

$R_{1224}=R_{1334}=0$6

and a TOV balance

$R_{1224}=R_{1334}=0$7

used for physically viable pulsar interiors such as 4U 1820-30 (Ratanpal et al., 2020).

• With the Wyman IIa choice

$R_{1224}=R_{1334}=0$8

the Karmarkar constraint yields

$R_{1224}=R_{1334}=0$9

and closed-form expressions for $f(R)$0, $f(R)$1, $f(R)$2, $f(R)$3, $f(R)$4, compactness, and surface redshift (Tello-Ortiz, 2018).

Across these models the same physical tests recur: regularity at the center, positivity of $f(R)$5, $f(R)$6, $f(R)$7, monotonic decrease of density and pressures, causality $f(R)$8, energy conditions, adiabatic index bounds such as $f(R)$9, TOV equilibrium, cracking criteria, Harrison–Zeldovich–Novikov stability, and bounds on compactness and redshift (Maurya et al., 2017). Matching conditions typically require continuity of $f(R,T)$0, $f(R,T)$1, and vanishing radial pressure at the boundary, though some modified-gravity constructions also impose continuity of $f(R,T)$2 (Asghar et al., 2023).

4. Use in modified gravity

The Karmarkar condition has been transplanted essentially unchanged into modified-gravity settings. In these cases it still links the metric potentials, while the matter variables are determined by the modified field equations.

In $f(R,T)$3 gravity, one study employed the class-I ansatz

$f(R,T)$4

and examined both an exponential model,

$f(R,T)$5

and a power-law model,

$f(R,T)$6

with applications to Vela X-1, PSR J1614-2230, 4U 1608-52, Cen X-3, and 4U 1820-30 (Naz et al., 2021).

In $f(R,T)$7 gravity, the exponential model

$f(R,T)$8

has been combined with

$f(R,T)$9

and the ansatz

$f(\mathscr R,\mathscr G)$0

to study LMC X-4, EXO 1785-248, Cen X-3, and 4U 1820-30 (Asghar et al., 2023).

In $f(\mathscr R,\mathscr G)$1 gravity, the Karmarkar relation is combined with Lake’s ansatz

$f(\mathscr R,\mathscr G)$2

for the model

$f(\mathscr R,\mathscr G)$3

and matched to Schwarzschild exteriors for LMC X-4 and EXO 1785-248 (Mustafa et al., 2020).

In linear $f(\mathscr R,\mathscr G)$4 gravity, the condition is used together with

$f(\mathscr R,\mathscr G)$5

to produce

$f(\mathscr R,\mathscr G)$6

with physical analysis for Cen X-3 (Paul et al., 20 May 2025).

These constructions indicate that the Karmarkar condition is theory-agnostic at the geometric level: it constrains the interior metric independently of whether the dynamics are governed by Einstein gravity, curvature-modified gravity, or symmetric teleparallel gravity.

5. Wormholes and shape-function generation

A second major application is static traversable wormholes. In the Morris–Thorne form

$f(\mathscr R,\mathscr G)$7

one has

$f(\mathscr R,\mathscr G)$8

Combining this with the integrated Karmarkar relation gives a direct link between the redshift function and the shape function (Shamir et al., 2020).

A widely used result is

$f(\mathscr R,\mathscr G)$9

which implies

$F(Q)$0

Thus, once $F(Q)$1 is prescribed, the wormhole shape function is no longer arbitrary (Gul et al., 2024).

For the redshift choice

$F(Q)$2

one obtains

$F(Q)$3

and, after introducing a constant to enforce the throat condition,

$F(Q)$4

yielding asymptotically flat traversable wormholes in $F(Q)$5 gravity (Shamir et al., 2020). Closely related constructions appear in $F(Q)$6 and $F(Q)$7 gravity, where the Karmarkar condition is used to generate viable shape functions satisfying throat, flare-out, and asymptotic-flatness conditions (Sharif et al., 2023).

A recurrent question is whether the Karmarkar condition removes the need for exotic matter. The available results are mixed. In some $F(Q)$8 models the resulting shape function can lead to wormholes with less, or possibly negligible, exotic matter for suitable parameter choices (Shamir et al., 2020). By contrast, a later $F(Q)$9 study concluded that applying the Karmarkar condition to $\nu(r)$0-type solutions does not eliminate the necessity of exotic matter, even though stable, static, traversable wormholes can still be obtained (Turkoglu, 2 Jun 2025).

6. Scalar, time-dependent, and uniqueness developments

The scalar reformulation of the Karmarkar condition substantially broadened the subject. In static spherical symmetry it provides a constructive algorithm: once the density profile $\nu(r)$1 is given, one derives

$\nu(r)$2

then

$\nu(r)$3

and finally the temporal potential from an integral involving $\nu(r)$4 (Ospino et al., 2020). This yields all static embedding-class-I solutions compatible with the chosen density profile.

The same paper shows that the adiabatic dynamic case is highly restrictive. Under assumptions such as homogeneous density, vanishing complexity, conformal flatness, isotropic pressures, or shear-free geodesic motion, the Karmarkar condition drives the system back to the static Schwarzschild interior (Ospino et al., 2020). Dissipative dynamics are less restrictive: nontrivial radiating families with heat flow $\nu(r)$5 are obtained in closed form.

A complementary time-dependent development used Lie symmetries to reduce the full Karmarkar partial differential equation for spherically symmetric metrics. For a Weyl-free collapsing ansatz, a radiating solution was constructed that satisfies the Santos heat-flux boundary condition when matched to the Vaidya exterior, thereby yielding a non-adiabatic stellar model with explicit matter variables $\nu(r)$6, $\nu(r)$7, $\nu(r)$8, and $\nu(r)$9 (Paliathanasis et al., 2022).

A separate uniqueness result sharpened the structural role of the condition. For a static anisotropic interior, the simultaneous requirements of vanishing complexity, Karmarkar embedding, and conformal flatness imply

$\lambda(r)$0

so that the unique solution is the Schwarzschild interior metric of constant density (Ratanpal et al., 2024).

7. Recurring interpretive issues

Several points recur across the literature.

First, the Karmarkar condition is a geometric embedding criterion, not a guarantee of physical acceptability. Regularity, positivity, causality, energy conditions, stability inequalities, and appropriate boundary matching must still be checked separately (Pandya et al., 2017). This is why papers using the condition almost invariably supplement it with TOV analysis, sound-speed bounds, redshift estimates, and mass-radius constraints.

Second, embedding class I does not imply isotropic matter. On the contrary, one study states that the resulting model is necessarily anisotropic unless it degenerates into the Schwarzschild interior solution or flat space (Maurya et al., 2017). Much of the practical utility of the condition comes precisely from its compatibility with anisotropic pressures $\lambda(r)$1.

Third, the same condition appears in different algebraic guises because authors use different signatures, index conventions, and choices of dependent variable. The differential relation

$\lambda(r)$2

and the integrated forms involving either $\lambda(r)$3 or $\lambda(r)$4 are all standard manifestations of the same class-I restriction (Singh et al., 2016).

Finally, the method is powerful but not unconstrained. Physical viability can depend sensitively on model parameters. In one anisotropic stellar solution the parameter $\lambda(r)$5 controls the stiffness of the equation of state, with smaller $\lambda(r)$6 leading to stiffer configurations (Singh et al., 2016). In a charged Bardeen-star model, acceptable behavior occurs for

$\lambda(r)$7

while the special values $\lambda(r)$8 lead to pathologies such as superluminal sound speeds or negative pressures (Mustafa et al., 2020). A plausible implication is that the Karmarkar condition narrows geometric freedom, but does not remove the need for delicate parameter selection consistent with observation and stability.

In contemporary relativistic astrophysics and gravitational theory, the Karmarkar condition therefore occupies a dual role: it is both a precise criterion for class-I embeddability and a practical exact-solution framework linking geometry, anisotropy, and boundary data across compact stars, wormholes, and dynamical collapse.