Adler-Finch-Skea Solution in Compact Stars

• The Adler-Finch-Skea solution is a family of exact interior metrics combining a quadratic Adler temporal potential with Finch-Skea radial geometry to model regular compact-star interiors.
• It utilizes an embedding-class-one condition (Karmarkar constraint) to analytically relate the metric potentials, ensuring controlled profiles for density, pressure, and redshift.
• Recent studies extend the model to charged anisotropic fluids, dark-energy stars, and modified gravity, highlighting its adaptability and astrophysical viability.

Searching arXiv for recent and foundational papers on Adler–Finch–Skea solutions and related Finch–Skea compact-star models. The Adler–Finch–Skea solution denotes a class of exact interior metrics for static, spherically symmetric compact stars in which the temporal potential is Adler-type and the radial potential is Finch–Skea-type, either imposed directly or generated through an embedding-class-one constraint such as the Karmarkar condition. In its most recognizable realization, one takes an Adler form for the time metric coefficient,

$e^{\nu(r)}=B(1+Cr^2)^2,$

and the class-one relation then yields a Finch–Skea radial potential linear in $r^2$,

$e^{\lambda(r)}=1+16BC^2Fr^2,$

with constants fixed by boundary matching and matter variables determined from the Einstein or Einstein–Maxwell equations (Bhar et al., 2017). Across the recent literature, the same structural idea has been extended to charged anisotropic fluids, dark-energy stars, $f(Q)$ gravity, $f(\mathcal{R},\mathcal{T})$ gravity, $f(R,T)$ gravity, higher-dimensional spacetimes, and lower-dimensional BTZ-matched interiors, so the term functions both as the name of a specific exact solution and as a label for a broader metric family (Shamir et al., 2021).

1. Definition, scope, and nomenclature

In the narrow sense, the Adler–Finch–Skea configuration is the class-one charged anisotropic solution in which $g_{00}$ is chosen in Adler’s quadratic form and $g_{11}$ becomes Finch–Skea-like after imposing the Karmarkar condition (Bhar et al., 2017). In a broader sense, the surveyed literature uses the label for regular compact-star interiors that combine an Adler-type temporal metric with a Finch–Skea radial structure, or that retain the Finch–Skea radial ansatz while deriving the temporal potential from an embedding relation or other closure condition (Shamir et al., 2021).

This broader usage matters because several later papers work explicitly with Finch–Skea geometry without always naming Adler, yet place their constructions within the same structural lineage. A 2023 $f(Q)$ study describes Finch–Skea as one of the most widely used exact interior solutions and notes that the “Adler–Finch–Skea” label usually denotes a family of regular interiors with a characteristic radial metric structure (Mustafa et al., 2023). A 2023 generalization with

$e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$

treats the $r^2$0 case as the original Finch–Skea geometry and identifies suitable specializations as “Adler–Finch–Skea–type” subcases (Patel et al., 2023). A 2024 decoupling-based extension in $r^2$1 gravity likewise describes the Finch–Skea seed metric

$r^2$2

as belonging to the broader Adler-type family of exact interiors (Naseer et al., 2024).

A common misconception is therefore that “Adler–Finch–Skea solution” refers to a single immutable metric. The literature represented here instead shows a family resemblance: a quadratic Adler-type $r^2$3, a simple Finch–Skea-type $r^2$4, regularity at the center, and analytic control over density, pressure, compactness, and redshift (Bhar et al., 2017).

2. Geometric core and embedding-class-one structure

The common geometric starting point is the static spherical line element

$r^2$5

In class-one constructions, the Karmarkar or Eiesland condition ties the two metric potentials, so specifying one fixes the other up to constants. In one standard form,

$r^2$6

where $r^2$7 is an embedding constant (Shamir et al., 2021).

With Adler’s ansatz

$r^2$8

the class-one condition gives

$r^2$9

which the authors explicitly identify as similar to the Finch–Skea solution (Shamir et al., 2021). The same mechanism appears in later work on charged dark-energy stars, where

$e^{\lambda(r)}=1+16BC^2Fr^2,$0

is presented as the canonical Adler–Finch–Skea interior (Rej, 17 Jun 2026).

The inverse construction also occurs. Instead of beginning from Adler’s $e^{\lambda(r)}=1+16BC^2Fr^2,$1, one may prescribe the Finch–Skea-type $e^{\lambda(r)}=1+16BC^2Fr^2,$2 and derive $e^{\lambda(r)}=1+16BC^2Fr^2,$3 from the class-one relation. In $e^{\lambda(r)}=1+16BC^2Fr^2,$4 gravity this is done with

$e^{\lambda(r)}=1+16BC^2Fr^2,$5

and the Karmarkar condition produces

$e^{\lambda(r)}=1+16BC^2Fr^2,$6

(Mustafa et al., 2023). A generalized class-one Finch–Skea model similarly adopts

$e^{\lambda(r)}=1+16BC^2Fr^2,$7

and determines $e^{\lambda(r)}=1+16BC^2Fr^2,$8 by quadrature, obtaining a family whose stiffness increases with the parameter $e^{\lambda(r)}=1+16BC^2Fr^2,$9 (Singh et al., 2019).

Thus the geometric essence of the Adler–Finch–Skea construction is not merely the pair of functions themselves but the reduction of the stellar interior to a one-function problem through an embedding constraint.

3. Matter models and exact realizations

The most developed Adler–Finch–Skea realizations couple the geometry to anisotropic and often charged matter. In the charged class-one model of 2017, the matter source is an Einstein–Maxwell anisotropic fluid with

$f(Q)$0

and a specific electric profile

$f(Q)$1

The resulting density, radial pressure, tangential pressure, charge density, and anisotropy are all obtained in closed form (Bhar et al., 2017).

A closely related 2021 construction uses the same Adler/Karmarkar mechanism but matches the interior to a Bardeen exterior rather than to Reissner–Nordström. There the matter sector is a charged anisotropic fluid with anisotropy $f(Q)$2, charge function $f(Q)$3, and electric field $f(Q)$4. The authors adopt

$f(Q)$5

and derive explicit analytic expressions for $f(Q)$6, $f(Q)$7, $f(Q)$8, $f(Q)$9, and $f(\mathcal{R},\mathcal{T})$0 (Shamir et al., 2021).

The same metric backbone also supports two-fluid dark-energy interiors. In a 2026 model for Her X-1, the effective source is composed of ordinary matter plus a dark-energy sector obeying

$f(\mathcal{R},\mathcal{T})$1

With the Adler–Finch–Skea metric, the ordinary density, ordinary pressure, electric field, dark density, and dark pressures are again obtained algebraically (Rej, 17 Jun 2026).

By contrast, some Finch–Skea descendants retain the same radial geometry but use isotropic perfect-fluid matter or anisotropy without charge. In $f(\mathcal{R},\mathcal{T})$2 gravity, for example, the Finch–Skea ansatz

$f(\mathcal{R},\mathcal{T})$3

combined with a perfect fluid and $f(\mathcal{R},\mathcal{T})$4 yields the standard Finch–Skea temporal potential in trigonometric form and explicit physical density and pressure profiles for PSR J1614–2230 (Bhar et al., 2021). This usage does not explicitly name Adler, but it remains structurally adjacent to the broader Adler–Finch–Skea family.

4. Junction conditions and exterior completion

The integration constants in Adler–Finch–Skea interiors are not arbitrary. They are fixed by matching the interior solution to an exterior spacetime at the stellar boundary and by imposing $f(\mathcal{R},\mathcal{T})$5.

For the charged anisotropic class-one model, the exterior is Reissner–Nordström, and continuity of $f(\mathcal{R},\mathcal{T})$6, continuity of $f(\mathcal{R},\mathcal{T})$7, and vanishing radial pressure determine the constants $f(\mathcal{R},\mathcal{T})$8, $f(\mathcal{R},\mathcal{T})$9, and $f(R,T)$0 in terms of total mass, radius, and charge (Bhar et al., 2017). In the charged anisotropic Finch–Skea–Bardeen model, the same logic is applied to the Bardeen regular black-hole exterior, so the interior constants $f(R,T)$1 are fixed by $f(R,T)$2, $f(R,T)$3, and the charge parameter through Darmois–Israel matching and $f(R,T)$4 (Shamir et al., 2021).

The dark-energy implementation again matches by Darmois–Israel conditions, this time to Reissner–Nordström, enforcing continuity of $f(R,T)$5, $f(R,T)$6, and $f(R,T)$7, together with $f(R,T)$8, to determine $f(R,T)$9, $g_{00}$0, $g_{00}$1, and $g_{00}$2 (Rej, 17 Jun 2026).

Outside four-dimensional GR, the exterior changes with the theory. In $g_{00}$3 gravity with $g_{00}$4, Finch–Skea interiors are matched to a Schwarzschild–(A)dS-type exterior with $g_{00}$5 (Mustafa et al., 2023). In $g_{00}$6 dimensions, Finch–Skea-type interiors are matched to BTZ exteriors rather than Schwarzschild, showing that the same geometric idea persists even when the ambient theory and dimensionality change (Banerjee et al., 2012).

These matching results make clear that the Adler–Finch–Skea interior is not a complete spacetime by itself; it is an interior patch whose constants acquire physical meaning only after global completion.

5. Physical admissibility and stability

A defining reason for the persistence of the Adler–Finch–Skea family is that it is repeatedly shown to satisfy the standard compact-star admissibility tests. Across the surveyed papers, these include regularity at the center, positivity of density and pressures, monotonic outward decrease of $g_{00}$7, $g_{00}$8, and $g_{00}$9, fulfillment of NEC/WEC/SEC/DEC, causal sound speeds, Tolman–Oppenheimer–Volkoff equilibrium, and sufficiently large adiabatic index (Shamir et al., 2021).

In the 2017 charged anisotropic Adler–Finch–Skea model, electric charge is the decisive modification. Maurya et al. had reported that the neutral $g_{11}$0 counterpart is not well behaved because the radial sound speed is non-decreasing outward, whereas the charged version becomes well behaved with decreasing sound speed outward. For the reported configuration, the central sound speeds are

$g_{11}$1

the compactness is

$g_{11}$2

close to the Buchdahl limit $g_{11}$3, and the model supports a mass $g_{11}$4 with radius $g_{11}$5 km (Bhar et al., 2017).

The Finch–Skea–Bardeen construction verifies a similarly broad set of conditions: central regularity, decreasing density and pressures, all standard energy conditions, $g_{11}$6, the Abreu–Herrera cracking bound

$g_{11}$7

TOV force balance including charge and anisotropy, the Buchdahl compactness bound, and acceptable surface redshift below the Böhmer–Harko and Ivanov limit (Shamir et al., 2021).

Not all anisotropy is stabilizing. A study devoted specifically to anisotropy under Finch–Skea geometry finds that the model is stable for zero anisotropy, while the chosen attractive anisotropy case is less favorable under Herrera’s cracking concept even though it satisfies several other viability conditions (Das et al., 2020). The literature therefore does not treat anisotropy as automatically beneficial; its sign and radial profile matter.

6. Generalizations and extensions

The modern Adler–Finch–Skea literature is best understood as a template that survives changes in gravity theory, matter model, and dimension. Representative extensions are summarized below.

Setting	Retained structure	Notable feature
$g_{11}$8 gravity (Mustafa et al., 2023)	Finch–Skea $g_{11}$9 plus Karmarkar-generated $f(Q)$0	Maximum-mass and $f(Q)$1–$f(Q)$2 analysis
$f(Q)$3 gravity (Naseer et al., 2024)	Finch–Skea seed metric	Anisotropy via gravitational decoupling
$f(Q)$4 gravity (Bhar et al., 2021)	Finch–Skea ansatz $f(Q)$5	Stable isotropic PSR J1614–2230 model
5D Einstein–Gauss–Bonnet (Sardar, 2016)	Finch–Skea $f(Q)$6 in 5D	Matching to EGB Schwarzschild exterior
$f(Q)$7 strange stars (Das et al., 2023)	Finch–Skea geometry in higher dimensions	$f(Q)$8 in four dimensions
$f(Q)$9 dimensions (Banerjee et al., 2012)	Finch–Skea-type $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$0	Matching to BTZ exterior

Two structural developments deserve particular notice. First, the generalized class-one Finch–Skea solution with

$e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$1

shows that the parameter $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$2 stiffens the equation of state and produces a mass at $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$3 lower by about $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$4 from $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$5, which the authors interpret as consistent with an EOS without strong high-density softening from hyperonization or exotic phase transition (Singh et al., 2019). Second, lower- and higher-dimensional analogues demonstrate that the Finch–Skea radial geometry is not tied to four-dimensional GR alone: it remains analytically productive in BTZ-matched $e^{\lambda(r)}=\left(1+\frac{r^2}{R^2}\right)^n$6 interiors and in higher-dimensional strange-star models (Bhar et al., 2014).

Taken together, these developments suggest that the Adler–Finch–Skea solution is best regarded as a robust geometric scheme for exact relativistic stellar interiors rather than as a single closed model. Its defining content is the compatibility of a quadratic Adler-type temporal potential, a Finch–Skea-type radial potential, and a regular, matchable compact-star interior; its continuing relevance lies in how easily that scheme adapts to charge, anisotropy, modified gravity, dark sectors, and altered dimensionality (Rej, 17 Jun 2026).